multiple light scattering. tables, formulas, and applications. volume 1

MULTIPLE LIGHT SCATTERING Tables, Formulas, and Applications

Volume 1

H. C. VAN DE HULST

Astronomical Observatory University of Leiden Leiden, The Netherlands

1980

ACADEMIC PRESS A Subsidiary of Harcourt Brace Jovanovich, Publishers

New York London Toronto Sydney San Francisco

C O P Y R I G H T © 1 9 8 0 , BY A C A D E M I C P R E S S , I N C . ALL R I G H T S R E S E R V E D . N O P A R T O F T H I S P U B L I C A T I O N M A Y B E R E P R O D U C E D O R T R A N S M I T T E D I N A N Y F O R M O R B Y A N Y M E A N S , E L E C T R O N I C O R M E C H A N I C A L , I N C L U D I N G P H O T O C O P Y , R E C O R D I N G , OR ANY I N F O R M A T I O N STORAGE AND R E T R I E V A L S Y S T E M , W I T H O U T P E R M I S S I O N I N W R I T I N G F R O M T H E P U B L I S H E R .

A C A D E M I C P R E S S , I N C . I l l Fifth Avenue, New York, New York 10003

United Kingdom Edition published by A C A D E M I C P R E S S , I N C . ( L O N D O N ) L T D . 24/28 Oval Road, London NW1 7DX

L i b r a r y o f C o n g r e s s C a t a l o g i n g i n P u b l i c a t i o n D a t a

H u l s t , H e n d r i k C h r i s t o f f e l v a n d e .

M u l t i p l e l i g h t s c a t t e r i n g

I n c l u d e s b i b l i o g r a p h i e s a n d i n d e x .

1 . L i g h t — S c a t t e r i n g — H a n d b o o k s , m a n u a l s , e t c .

2 . R a d i a t i v e t r a n s f e r — H a n d b o o k s , m a n u a l s , e t c .

I . T i t l e .

Q C 4 2 7 . 6 . H 8 4 5 3 5 \ 4 7 9 - 5 1 6 8 7

I S B N 0 - 1 2 - 7 1 0 7 0 1 - 0 ( v . 1 )

P R I N T E D I N T H E U N I T E D S T A T E S O F A M E R I C A

80 81 82 83 9 8 7 6 5 4 3 2 1

• Preface

The play of radiation by repeated scattering in a cloud layer or any other slab of particles poses a problem that is common to atmospheric physics, astronomy, ocean optics, and branches of industrial research. Methods for solving this problem in diverse situations have been known for decades but their complexity has given the subject the reputation of being accessible only to specialists.

This book is aimed at the nonspecialist, e.g., an expert in an applied field, who needs a result from multiple scattering theory but does not wish to spend excessive time in solving it himself or searching the very extensive literature.

Numerical results form the core of these two volumes. Since users from diverse fields should be served, the tabulated quantities are named by their physical meaning, e.g., reflection function, gain, diffusion pattern, net flux, but are presented in the form of functions of a few dimensionless parameters. Most tables have five-figure accuracy in order to enable readers to use them for checking their own computer programs. The graphical illustrations have been chosen to serve as a quick orientation and also to highlight key phenomena such as asymptotic behavior.

Special cases such as the limits adopted for each quantity for conservative scattering (a = 1), or in a semi-infinite atmosphere (b = oo), or at large depth (τ ρ 1) have been included in each tabulation. The same is true for moments and bimoments of the functions of the angles of incidence and emergence.

The formulas expressing these results show a similar ramification of special cases and asymptotic forms. For clarity and ease of access, they have been arranged, where possible, in a "Display," which is a collection of formulas in tabular form. Derivations have been kept to a minimum. They are presented in

i x

χ Preface

a form emphasizing the physical content and the use of certain intermediate results. Only rarely does an intricate derivation require the use of numbered equations.

Although the author 's prime intention is to present known results, new discoveries or new light shed on the meaning and use of known forms was unavoidable. The major findings have been published in scientific journals and several have come into general use. Subjects like doubling, similarity relations, reduction to Η functions, and, generally, the interpretation of mathematical results in physical terms, are presented here in their proper context.

The volumes have a strict organization: Par t I on general relations and Part II on isotropic scattering (Volume 1), Par t III on anisotropic scattering and Part IV on applications to selected fields (Volume 2). The division of parts into chapters again follows a strictly logical scheme as the table of contents for each volume shows.

What I started as a sideline has become a major project. This would not have been possible without the help and encouragement of a great many people. Among this long list I wish to record my special gratitude to K. G. Grossman and J. W. Hovenier for their support throughout the work and to W. M. Irvine and V. V. Ivanov, whose enthusiasm helped the project gain momentum in the early years.

• Contents of Volume 2

Part III ANISOTROPIC SCATTERING

10 Phase Functions

11 Results for the Henyey-Greenstein Phase Function, Unbounded and Semi-Infinite Medium

12 Other Phase Functions, Semi-Infinite Atmospheres

13 Henyey-Greenstein Functions, Results for Finite Layers

14 Results for Other Phase Functions, Finite Layers

15 Polarization and Azimuth-Dependent Terms

16 Rayleigh Scattering

Part IV SAMPLE APPLICATIONS

17 Photon Optical Paths and Absorption Lines

18 Planets

19 Scattered Light in the Earth's Atmosphere

20 Miscellaneous Applications

INDEX

xi

1 • Concepts, Terms, Notation

1.1 DIRECTIONS FOR USE

Those who wish numerical results for homogeneous slabs with anisotropic scattering should turn to Chapters 11 and 13 (Volume 2).

Those who wish numerical results for homogeneous slabs with isotropic scattering should turn to Chapters 8 and 9.

Those who wish to avoid certain common traps should glance over this chapter.

Those who have a particular application in mind should see if Chapters 18 to 20 contain anything of interest to them (Volume 2).

All others should consult the Table of Contents and/or the Index.

1.2 SOME HARD CHOICES

It is unavoidable that in a subject so extensively studied different conventions on terms and notat ion have come into use. The following comments are intended to warn the reader of when different conventions might lead to confusion and to explain why I have made a particular choice.

Albedo ( = single scattering albedo). Confusion may arise from this notation. The symbol w (curly pi) used by Chandrasekhar is too exotic. Most authors have, perhaps unintentionally, changed it to ω or ω0. Sobolev and his school use λ. The neutron transport people use c. In marine optics it is written as the ratio b/c. I have chosen a, the first letter of albedo. See also phase function.

3

4 1 Concepts, Terms, Notation

Another source of confusion may be the use of the word albedo in different meanings. In reactor research it is used for what we call diffuse reflection. Throughout astrophysics and geophysics it is also used for the fraction of the incident energy returned from a diffusely reflecting surface. We concur with this use and call it plane albedo, symbol UR. Similarly, the fraction of energy returned for uniform radiance from the entire hemisphere is called spherical albedo, symbol URU or A*.

Angles with the normal. The problem is from which side to measure them, i.e., which sign to give the cosine. The three systems in use are shown side by side in Display 1.1. Most physicists prefer system lb. The astrophysical tradition la, whereby optical depth is measured into a stellar atmosphere but angle 0 is used for the normal direction out of the same atmosphere, is still strong enough to dominate a good part of the literature but it is awkward. In the recent astro-physical literature system lb is fairly common.

The use of system II is mostly limited to the radiation field outside slabs. There it has the advantages of never needing minus signs, having equal symbols for reciprocal quantities, and having very simple matrix formulations. I have chosen to use system II with symbol μ wherever possible, and system lb with symbol u where necessary. The same convention has been adopted by Hansen and Travis (1974).

Gain. This concept, in particular the point-direction gain, has two advantages. It is applicable without change of terminology or notation in two reciprocal experimental situations and it is virtually foolproof against normaliza-

D I S P L A Y 1.1

S i g n C o n v e n t i o n s for C o s i n e of A n g l e w i t h N o r m a l

R a n g e of a n g l e R a n g e of c o s i n e D e f i n e d a t d e p t h

A n g l e 0, c o s i n e 1 is a l o n g

U s e d in w o r k of

U s e d in t h i s b o o k

S y m b o l of c o s i n e u s e d in t h i s b o o k

R e l a t i o n w h e r e b o t h a p p l y

S y s t e m l a

0 - π 1 t o - 1

a n y w h e r e

d e c r e a s i n g o p t i c a l

d e p t h C h a n d r a s e k h a r ; m o s t

a s t r o p h y s i c s t e x t s n o t a t a l l

S y s t e m l b

0 π 1 t o - 1

a n y w h e r e

i n c r e a s i n g o p t i c a l

d e p t h S o b o l e v ; m o s t

p h y s i c s t e x t s w h e n d i s c u s s i n g

i n t e r n a l r a d i a t i o n

field

- 1 < u < 1

S y s t e m I I

0 - π / 2

1 t o O

t o p o r b o t t o m

s u r f a c e n o r m a l , in o r o u t

C h a n d r a s e k h a r ; m y e a r l i e r p a p e r s

w h e n d i s c u s s i n g s l a b

p r o p e r t i e s , o r

w h e n c o m b i n i n g

s l a b s

0 < μ < 1

μ = \η\

1.2 Some Hard Choices 5

tion errors of a factor 2 or π, which are a frequent source of irritation. A disadvantage is that this concept (van de Hulst, 1964) has not become commonly used.

Phase function. Sometimes it is more convenient to put the albedo inside and sometimes more convenient to leave it outside the function describing the local scattering properties. We chose in agreement with most authors to describe these properties by the product aO(cos a), where a is the albedo and <D(cos a) is the phase function, so that the phase function is normalized to 1. In some theoretical developments (Chapter 6) the product aO(cos a) is expanded in a series of Legendre polynomials with coefficients ωη. This makes a = ω0.

Polarization. The usefulness of the Stokes parameters (/, Q, 17, V) is beyond doubt. They require the choice of a plane of reference and express the intensity and state of polarization of a beam of light of intensity / . Other options are possible (Section 16.1.2), but there is a practical advantage in having intensity as the first element of the set of four. Other quantities ( J , B, 7 0, etc.) each have to be similarly completed with three more elements if we wish to express the state of polarization. I have avoided an avalanche of new symbols by employing the words including polarization as a technical term indicating the presence of these three further elements. It works.

Radiance, intensity. The energy stream per unit time per unit area per unit solid angle is by international convention called radiance. In astrophysics, where radiance is usually measured per unit frequency band, it was formerly called specific intensity (of the radiation) or surface brightness (of the emitting body). For brevity, the terms intensity and brightness are also frequently used. I have chosen to use the word "intensi ty" in this book. The writers on the subject are from such diverse fields that one cannot hope for uniform terminology. See translation rules.

The integral of radiance over a solid angle is the irradiance. This is an awkward term, particularly because different integrals have to be distinguished. The two most important ones are

net flux = vector irradiance

and

(spherical) average intensity = ( 4 π )_1

χ scalar irradiance

Reflection and transmission function. The reflection function R used in this book is related to Chandrasekhar 's function S by

Κ(μ, φ; μ 0, φ0) = Ξ(τί ; μ, φ, μ0Ψο)/4μμο

In order to introduce a short notation (Chapters 5 and 7) it was necessary to write the reflection formula for arbitrary incident radiation in a fully symmetric

/ Concepts, Terms, Notation

form. With limitation to azimuth-independent terms and omission of the arguments τχ,φ,ψ0, this might be done in two equivalent ways:

I have selected the second form [Eq. (2)] because of several advantages. The intensities themselves appear and ϋ(μ, μ0) assumes finite nonzero values for μ = 0 or μ 0 = 0 (it becomes oo for μ = μ 0 = 0). The function β(μ, μ0) thus defined is also the ratio of the actual reflection function to the reflection function of an ideal white (Lambert) surface.

In the transmission function, where both Chandrasekhar and I use the symbol Τ(μ, μ0) , a corresponding difference in definition occurs.

Translation rules. Quoting a result from one field and applying it to a different field generally requires a translation of terms and notations. This job in itself is rarely difficult, but it constitutes a psychological barrier. How hard it is to overcome this barrier, even for one well versed in the mathematics and physics required, is candidly expressed by Preisendorfer (1976, p. xxxii): "These concepts in other branches of radiative transfer, notably astrophysical optics, were either nonexistent or in the form of unrealizable mathematical abstractions of no use to one with direct instrumental access to the interior of the optical medium of interest; in our case, the sea."

There is no unique remedy for this problem. I have tried to cope with it by:

(a) Spelling out translation rules where this seems prudent. See examples in Sections 18.1.1, 20.1.1, 20.2.1, and 20.4.

(b) Dealing in the three last chapters with a representative set of applied fields in order to show by examples from that field where the access to the earlier, more abstract, results lies.

Wavelength dependence. This has been omitted everywhere in order not to overburden the notation. The reader who works with dimensionless quantities will find this an advantage. The reader who requires physical quantities will have to complete each definition with a choice of units and a specification of the wavelength or frequency interval. The same quantities are applicable to a wider band or to the entire spectrum, provided the properties are not dependent on wavelength within the chosen band.

I n t h i s l is t o f g e n e r a l r e f e r e n c e s w e h a v e b e e n v e r y r e s t r i c t i v e . W o r k s t h a t d e a l w i t h a s m a l l

s u b s e t o f t h e p r o b l e m s d i s c u s s e d i n t h i s m o n o g r a p h a r e c i t e d i n t h e c h a p t e r s w h e r e t h e y b e l o n g .

T h e c h o i c e is c l e a r l y s u b j e c t i v e .

(i)

(2)

R E F E R E N C E S

References 7

B o o k s

C h a n d r a s e k h a r , S. ( 1 9 5 0 ) . " R a d i a t i v e T r a n s f e r . " O x f o r d U n i v . P r e s s ( C l a r e n d o n ) , L o n d o n a n d

N e w Y o r k ; a l s o D o v e r , N e w Y o r k , 1960 .

K o u r g a n o f f , V . ( 1 9 5 2 ) . " B a s i c M e t h o d s in T r a n s f e r P r o b l e m s . " O x f o r d U n i v . P r e s s ( C l a r e n d o n ) ,

L o n d o n a n d N e w Y o r k ; a l s o D o v e r , N e w Y o r k , 1 9 6 3 .

S o b o l e v , V . V . ( 1 9 5 6 ) . " T r a n s f e r o f R a d i a n t E n e r g y i n t h e A t m o s p h e r e s o f S t a r s a n d P l a n e t s . "

S t a t e P u b l i c a t i o n o f T e c h n i c a l - T h e o r e t i c a l L i t e r a t u r e , M o s c o w ( in R u s s i a n ) . T r a n s l a t e d : ( 1 9 6 3 )

" A T r e a t i s e o n R a d i a t i v e T r a n s f e r . " V a n N o s t r a n d - R e i n h o l d , P r i n c e t o n , N e w J e r s e y .

D a v i s o n , B . ( 1 9 5 7 ) . " N e u t r o n T r a n s p o r t T h e o r y . " O x f o r d U n i v . P r e s s ( C l a r e n d o n ) , L o n d o n a n d

N e w Y o r k .

B u s b r i d g e , I . W . ( 1 9 6 0 ) . " T h e M a t h e m a t i c s o f R a d i a t i v e T r a n s f e r . " C a m b r i d g e U n i v . P r e s s ,

L o n d o n a n d N e w Y o r k .

P r e i s e n d o r f e r , R . W . ( 1 9 6 5 ) . " R a d i a t i v e T r a n s f e r o n D i s c r e t e S p a c e s . " P e r g a m o n , O x f o r d .

C a s e , Κ . M . , a n d Z w e i f e l , P . F . ( 1 9 6 7 ) . " L i n e a r T r a n s p o r t T h e o r y . " A d d i s o n - W e s l e y , R e a d i n g ,

M a s s a c h u s e t t s .

I v a n o v , V . V . ( 1 9 6 9 ) . " R a d i a t i v e T r a n s f e r a n d t h e S p e c t r a o f C e l e s t i a l B o d i e s . " M o s c o w ( R u s s i a n ) ;

T r a n s l a t e d : ( 1 9 7 3 ) " T r a n s f e r o f R a d i a t i o n in S p e c t r a l L i n e s . " N a t i o n a l B u r e a u o f S t a n d a r d s

S p e c i a l P u b l . 3 8 5 , V . S . 9 0 0 1 U . S . G o v t . P r i n t i n g Off ice , W a s h i n g t o n , D . C .

S o b o l e v , V . V . ( 1 9 7 2 ) . " L i g h t S c a t t e r i n g i n t h e A t m o s p h e r e s o f P l a n e t s . " M o s c o w ( R u s s i a n ) ;

T r a n s l a t e d a s " L i g h t S c a t t e r i n g in P l a n e t a r y A t m o s p h e r e s . " P e r g a m o n , O x f o r d , 1 9 7 5 .

P r e i s e n d o r f e r , R . W . ( 1 9 7 6 ) . " H y d r o l o g i e O p t i c s . " U . S . D e p t . o f C o m m e r c e , N a t i o n a l O c e a n i c

a n d A t m o s p h e r i c A d m i n i s t r a t i o n . U . S . G o v t . P r i n t i n g Off ice , W a s h i n g t o n , D . C . I n 6 v o l u m e s .

C o n f e r e n c e R e p o r t s , C o m p e n d i a , and M a j o r R e v i e w s

K e r k e r , M . ( 1 9 6 3 ) . Conf. Electromagn. Scattering, 1962, Potsdam, New York. P e r g a m o n , O x f o r d .

R o w e l l , R . L . , a n d S t e i n , R . S. ( 1 9 6 7 ) . Conf. Electromagn. Scattering 1965, Amherst, Massachusetts.

G o r d o n a n d B r e a c h , N e w Y o r k .

H u n t , G . E . ( 1 9 7 1 ) . T r a n s p o r t t h e o r y ( c o n f e r e n c e 1 9 7 0 , O x f o r d ) , J. Quant. Spectrosc. Radiât. Transfer 13 ,511.

K u r i y a n , J . G . ( 1 9 7 4 ) . UCLA Int. Conf. Radiât. Remote Probing Atmos., L o s A n g e l e s , C a l i f o r n i a ,

1 9 7 3 . W e s t e r n P e r i o d i c a l s C o m p a n y , N o r t h H o l l y w o o d , C a l i f o r n i a .

H a n s e n , J . E . , a n d T r a v i s , L . D . ( 1 9 7 4 ) Space Sci. Rev. 16, 5 2 7 .

I r v i n e , W . M . ( 1 9 7 5 ) Icarus 25, 175 .

L e n o b l e , J . ( 1 9 7 7 ) . S t a n d a r d P r o c e d u r e s t o C o m p u t e R a d i a t i v e T r a n s f e r in a S c a t t e r i n g A t m o

s p h e r e . R a d i a t i o n C o m m i s s i o n , I n t e r n a t i o n a l A s s o c i a t i o n o f M e t e o r o l o g y a n d A t m o s p h e r i c

P h y s i c s , I . U . G . G . , p u b l i s h e d b y N a t i o n a l C e n t e r f o r A t m o s p h e r i c R e s e a r c h , B o u l d e r , C o l .

2 • Exponential Integrals and

Related Functions

2.1 QUICK SURVEY

The functions reviewed and tabulated in this chapter occur in calculations of low-order scattering by a plane-parallel atmosphere with many (simple) phase functions. The exponential integrals {E functions) have been studied before this century. The F functions were first studied by King (1913), the G functions probably by van de Hulst (1948). Their nature and use may be summarized as shown in the accompanying tabulation.

O r i g i n a t e b y i n t e g r a t i n g O r d e r s a n d F u n c t i o n s a p r o d u c t o f O c c u r in a r g u m e n t s

Ε f u n c t i o n s E x p o n e n t i a l a n d n e g a t i v e p o w e r B i m o m e n t s , z e r o o r d e r η; χ ( = e x p o n e n t i a l ( S e c t i o n 9 .1 .1) i n t e g r a l s )

F f u n c t i o n s Ε f u n c t i o n a n d e x p o n e n t i a l M o m e n t s , first o r d e r n; x, s ( D i s p l a y 9.1)

G f u n c t i o n s Ε f u n c t i o n a n d Ε f u n c t i o n o r B i m o m e n t s , first o r d e r η, m; χ F f u n c t i o n a n d n e g a t i v e p o w e r ( D i s p l a y 9 .1)

In the applications χ is usually the optical depth τ or the optical thickness b; s = Ι/μ or — Ι/μ, where μ is the cosine of the angle with normal ; and η and m are integers.

8

2.2 Exponential Integrals {Ε Functions) 9

The summary given suggests that more and more complicated functions can be introduced by simply naming the moments and bimoments of higher order. Some of them (for χ = oo only) have indeed been named in the appendix of Kourganoff (1952). It would hardly serve a purpose to give them all separate names. Many of these functions can be found in their proper physical context from the lines second order and third order in Table 12 (Section 9.1.1).

Notat ions used in this chapter are the following: η = 1, 2, 3 , . . . , except for the Ε functions, where we permit also η = 0; m = 1, 2, 3, . . . ; χ is real > 0, s is real, γ = 0.5772157 is Euler's constant; / = In χ + γ.

2.2 EXPONENTIAL INTEGRALS (E FUNCTIONS)

Definition: /•OO

£ „ ( * ) = J e-x,rndt

Alternative definition : /•OO

En(x) = xn~

l J e-

%t'

nàt

Differentiation :

dEn(x)/dx = - £ π_ ! ( χ ) (ηΦ 0)

Recurrence:

nEn+l(x) = e~x - xEn(x)

Special values:

E0(x) = e~x/x (χ Φ 0)

EM = l/(n - 1) (ηΦ 0)

£.(oo) = 0

Series expansion (convergent for all x, most useful for small x):

E0(x) = χ"1 - 1 + ±x - ix

2 + ^ x

3

E^x) = - / + χ - \x2 + ^ x

3

Ε2(χ) = 1 + (/ - l)x - \x2 + ^ x

3

E3(x) = i - χ + i ( - / + f)x2 + i x

3 - · · ·

Asymptotic expansion (if η > 1, semiconvergent, useful for sufficiently large x):

x L X X J

10 2 Exponential Integrals and Related Functions

T A B L E 1

E x p o n e n t i a l I n t e g r a l s a n d R e l a t e d F u n c t i o n s

X E 0( x ) E , ( x ) E 2( x ) E 3( x ) L - χ e E i ( x )

0 OO CO 1 0 . 5 0 0 0 0 -00 1 -co OO 0 . 0 1 9 9 . 0 0 5 4 . 0 3 7 9 3 . 9 4 9 6 7 . 4 9 0 2 8 - 4 . 0 2 7 9 5 - 9 9 0 0 5 - 4 . 0 1 7 9 3 8 . 9 2 4 6 9 0 . 0 2 4 9 . 0 1 0 3 . 3 5 4 7 1 . 9 1 3 1 0 . 4 8 0 9 7 - 3 . 3 3 4 8 1 . 9 8 0 2 0 - 3 . 3 1 4 7 1 6 . 3 6 2 9 9 0 . 0 5 1 0 . 0 2 5 2 . 4 6 7 9 0 . 8 2 7 8 3 . 4 5 4 9 2 - 2 . 4 1 8 5 2 . 9 5 1 2 3 - 2 . 3 6 7 8 8 3 . 6 9 7 3 9 0 . 1 9 . 0 4 8 3 7 1.82292 . 7 2 2 5 5 . 4 1 6 2 9 - 1 . 7 2 5 3 7 . 9 0 4 8 4 - 1 . 6 2 2 8 1 2 . 2 1 2 1 5

0 . 15 5 . 7 3 8 0 5 1 .46446 . 6 4 1 0 4 . 3 8 2 2 8 - 1 . 3 1 9 9 0 . 86071 - 1 . 1 6 4 0 9 1 .54629 0 . 2 4 . 0 9 3 6 5 1 .22265 . 5 7 4 2 0 . 3 5 1 9 5 - 1 . 0 3 2 2 2 . 8 1 8 7 3 - 0 . 8 2 1 7 6 1.16006 0 . 2 5 3 . 1 1 5 2 0 1 .04428 . 5 1 7 7 3 . 3 2 4 6 8 - 0 . 8 0 9 0 8 . 7 7 8 8 0 - 0 . 5 4 2 5 4 0 . 9 0 7 3 0 0 . 3 2 . 4 6 9 3 9 . 9 0 5 6 8 . 4 6 9 1 2 . 3 0 0 0 4 - 0 . 6 2 6 7 6 . 7 4 0 8 2 - 0 . 3 0 2 6 7 0 . 7 2 9 6 5 0 . 4 1 .67580 . 7 0 2 3 8 . 3 8 9 3 7 . 2 5 7 2 9 - 0 . 3 3 9 0 8 . 6 7 0 3 2 0 . 1 0 4 7 7 0 . 4 9 8 8 3

0 . 5 1 .21306 . 5 5 9 7 7 . 3 2 6 6 4 . 2 2 1 6 0 - 0 . 1 1 5 9 3 . 6 0 6 5 3 0 . 4 5 4 2 2 . 3 5 8 2 8 0 . 6 . 9 1 4 6 9 . 4 5 4 3 8 . 2 7 6 1 8 . 1 9 1 5 5 0 . 0 6 6 3 9 .54881 0 . 7 6 9 8 8 . 2 6 5 9 8 0 . 7 . 70941 . 3 7 3 7 7 . 2 3 4 9 5 . 1 6 6 0 6 0 . 2 2 0 5 4 . 4 9 6 5 9 1.06491 . 2 0 2 2 6 0 . 8 . 5 6 1 6 6 . 3 1 0 6 0 . 2 0 0 8 5 . 1 4 4 3 2 0 . 3 5 4 0 7 . 4 4 9 3 3 1 .34740 . 1 5 6 6 4 0 . 9 . 4 5 1 7 4 . 2 6 0 1 8 . 1 7 2 4 0 . 1 2 5 7 0 0 . 4 7 1 8 6 . 4 0 6 5 7 1.62281 . 1 2 3 0 7

1 . 3 6 7 8 8 . 2 1 9 3 8 . 1 4 8 5 0 . 1 0 9 6 9 0 . 5 7 7 2 2 . 3 6 7 8 8 1 .89512 . 0 9 7 8 4 1.25 . 2 2 9 2 0 . 1 4 6 4 1 . 1 0 3 4 9 . 0 7 8 5 7 0 . 8 0 0 3 6 . 2 8 6 5 0 2 . 5 8 1 0 5 . 0 5 7 3 7 1.5 . 1 4 8 7 5 . 1 0 0 0 2 . 0 7 3 1 0 . 0 5 6 7 4 0 . 9 8 2 6 8 . 2 2 3 1 3 3 . 3 0 1 2 9 . 1 3 5 0 8 2 . 0 6 7 6 7 . 0 4 8 9 0 . 0 3 7 5 3 . 0 3 0 1 3 1 .27036 . 1 3 5 3 4 4 . 9 5 4 2 3 . 0 1 4 2 6 2 . 5 . 0 3 2 8 3 .02491 . 0 1 9 8 0 . 0 1 6 3 0 1.49351 . 0 8 2 0 9 7 . 0 7 3 7 7 . 0 0 6 2 5

3 . 0 1 6 6 0 . 0 1 3 0 5 . 0 1 0 6 4 . 0 0 8 9 3 1 . 6 7 5 8 3 . 0 4 9 7 9 9 . 9 3 3 8 3 . 0 0 2 8 8 3 . 5 . 0 0 8 6 3 . 0 0 6 9 7 . 0 0 5 8 0 . 0 0 4 9 5 1 .82998 . 0 3 0 2 0 1 3 . 9 2 5 3 5 . 0 0 1 3 7

4 . 0 0 4 5 8 . 0 0 3 7 8 . 0 0 3 2 0 . 0 0 2 7 6 1.96351 . 0 1 8 3 2 1 9 . 6 3 0 8 7 . 0 0 0 6 7 5 . 0 0 1 3 5 . 0 0 1 1 5 . 0 0 1 0 0 . 0 0 0 8 8 2 . 1 8 6 6 5 . 0 0 6 7 4 4 0 . 1 8 5 2 7 .00017

OO 0 0 0 0 ω 0 CO 0

Definition of Ε function for negative argument:

Ei(x) = - e-'dt/t = eatjt, (x > 0) J — χ J — OO

This is to be understood as the Cauchy main value; the integral diverges at either side of t = 0. Other notations for this function are Ei(x) (Jahnke Emde), — Ex{ — x) (Kourganoff). In this book Ei(x) occurs only in one of the expressions for the Fx function.

Numerical values: See Table 1 and many standard texts. Abramowitz and Stegun (1965) give five pages with functions simply related to Ex(x) and Ei(x); 9D, χ = 0(0.01) 2.00 (0.1) 10.0 and x~

l = 0.100(0.005) 0. They further give

En(x); ID, n = 2,3, 4, 10, 20, with a similar scope in x. This work also presents fuller details on series expansions and on rational approximations, which are particularly helpful in computer work.

2.3 GENERALIZED EXPONENTIAL INTEGRAL

Definition : /•OO

£ < P+1 > ( X) = Ef\t)dt/t

J X Ef\x) = e~

x

2.4 F Functions 11

Differentiation:

dE[p+1)

(x)/dx = -E[p\x)/x

Recurrence: The integrals of order ρ + 1 cannot be expressed in terms of those of order ρ and lower.

Series expansion (convergent):

E?\x) = ψ + Α π

2 - * + - + -

Asymptotic forms (semiconvergent):

1 1 2 6 £<

2(x) = - E2(x) - - j E3(x) + - 3 EA(x) - E5(x) + ••·

X>

W 2 w χ e~x f 3 10 50

M2)(*) = — < ι - - + - 2 - - 3 + ·

xz [ χ x

z X"

3

Use: This function occurs in the expression for the G functions. Numerical values: See Table 1.

2.4 F FUNCTIONS

Definition:

Fn(x,s)= \estEn{t)dt

Jo

This can be transformed to : f»oo — η Γ00 u~

n

F„(x, s) = [1 - «?-*"-·>] du J i u — s

Derivatives :

dFH(x, s)/dx = esxEn(x)

ÔFn(x, s)/ôs = 1<P-1)X

- l ] / (s - 1) - n F „ + 1( x , s)

Recurrence:

s F M + 1( x , s) = esxEn+1(x) - (1/n) + F„(x, s)

Reduction to known functions :

F t( x , 5) = (l/sWE^x) - E^x - sx) - ln(l - s)~] s < 1 but Φ 0

F j (x ,0 ) = 1 - E2( x ) s = 0

F,(x, 1) = βχΕ,{χ) + / 5 = 1

Fi (x , s) = ( l / s ^ E ^ x ) + Ei(sx - x) - ln(s - 1)] s > 1

T A B L E 2 A

F F u n c t i o n s o f O r d e r 1

F , ( x , -s) e

x sF , ( x , .s)

χ s ~

1 = 0.1 0.3 0.5 1.0 s = 0 s

-1 - 1.0 0.5 0.3 0.1

0 . 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 1 0 . 0 4 8 1 0 . 0 4 9 6 0 . 0 4 9 9 0 . 0 5 0 1 0 . 0 5 0 3 0 . 0 5 0 1 0 . 0 4 9 8 0 . 0 4 9 4 0 . 0 4 7 7 0 . 0 2 0 . 0 7 9 7 0 . 0 8 4 4 0 . 0 8 5 4 0 . 0 8 6 1 0 . 0 8 6 9 0 . 0 8 5 9 0 . 0 8 5 0 0 . 0 8 3 7 0 . 0 7 7 9 0 . 0 5 0 . 1 4 0 4 0 . 1 6 0 5 0 . 1 6 5 0 0 . 1 6 8 5 0 . 1 7 2 2 0 . 1 6 7 3 0 . 1 6 2 7 0 . 1 5 6 7 0 . 1 3 0 8 0 . 1 0 0 . 1 9 1 3 0 . 2 4 2 9 0 . 2 5 5 9 0 . 2 6 6 3 0 . 2 7 7 5 0 . 2 6 1 7 0 . 2 4 7 1 0 . 2 2 9 2 0 . 1 6 1 2

0 . 1 5 0 . 2 1 5 1 0 . 2 9 6 9 0 . 3 1 9 5 0 . 3 3 8 4 0 . 3 5 9 0 0 . 3 2 8 4 0 , . 3 0 1 0 0 . 2 6 8 9 0 . 1 6 1 3 0 . 2 0 0 . 2 2 7 0 0 . 3 3 4 3 0 . 3 6 6 7 0 . 3 9 4 5 0 . 4 2 5 8 0 . 3 7 7 5 0 , . 3 3 5 9 0 . 2 8 9 0 0 . 1 5 0 0 0 . 2 5 0 . 2 3 3 0 0 . 3 6 1 1 0 . 4 0 2 8 0 . 4 3 9 6 0 . 4 8 2 3 0 . 4 1 4 2 0 , . 3 5 7 6 0 . 2 9 6 6 0 . 1 3 5 2 0 . 3 0 0 . 2 3 6 2 0 . 3 8 0 6 0 . 4 3 0 9 0 . 4 7 6 6 0 . 5 3 0 9 0 . 4 4 1 4 0 , . 3 6 9 8 0 . 2 9 5 7 0 . 1 2 0 0 0 . 4 0 0 . 2 3 8 7 0 . 4 0 5 7 0 . 4 7 0 7 0 . 5 3 2 9 0 . 6 1 0 6 0 . 4 7 5 1 0 , . 3 7 4 7 0 . 2 7 9 2 0 . 0 9 3 5

0 . 5 0 0 . 2 3 9 5 0 . 4 1 9 9 0 . 4 9 6 4 0 . 5 7 3 0 0 . 6 7 3 4 0 . 4 8 9 5 0 . . 3 6 3 4 0 . 2 5 3 1 0 . 0 7 3 3 0 . 6 0 0 . 2 3 9 7 0 . 4 2 8 0 0 . 5 1 3 2 0 . 6 0 2 2 0 . 7 2 3 8 0 . 4 9 0 8 0 , . 3 4 3 1 0 . 2 2 4 0 0 . 0 5 8 3 0 . 7 0 0 . 2 3 9 8 0 . 4 3 2 8 0 . 5 2 4 5 0 . 6 2 3 8 0 . 7 6 5 1 0 . 4 8 3 3 0 . . 3 1 8 2 0 . 1 9 5 4 0 . 0 4 7 1 0 . 8 0 0 . 2 3 9 8 0 . 4 3 5 6 0 . 5 3 2 2 0 . 6 3 9 9 0 . 7 9 9 1 0 . 4 6 9 7 0 . . 2 9 1 3 0 . 1 6 8 8 0 . 0 3 8 6 0 . 9 0 0 . 2 3 9 8 0 . 4 3 7 3 0 . 5 3 7 4 0 . 6 5 2 1 0 . 8 2 7 6 0 . 4 5 2 0 0 . . 2 6 4 2 0 . 1 4 5 1 0 . 0 3 1 9

1 . 0 0 0 . 2 3 9 8 0 . 4 3 8 3 0 . 5 4 1 0 0 . 6 6 1 3 0 . 8 5 1 5 0 . 4 3 1 7 0 . . 2 3 7 9 0 . 1 2 4 2 0 . 0 2 6 6 1 . 2 5 0 . 2 3 9 8 0 . 4 3 9 4 0 . 5 4 5 9 0 . 6 7 6 1 0 . 8 9 6 5 0 . 3 7 5 7 0 . , 1 7 9 1 0 . 0 8 3 7 0 . 0 1 7 5 1 . 5 0 0 . 2 3 9 8 0 . 4 3 9 8 0 . 5 4 7 9 0 . 6 8 3 9 0 . 9 2 6 9 0 . 3 1 9 3 0 . . 1 3 2 2 0 . 0 5 6 4 0 . 0 1 1 8 2 . 0 0 0 . 2 3 9 8 0 . 4 3 9 9 0 . 5 4 9 0 0 . 6 9 0 3 0 . 9 6 2 5 0 . 2 2 0 8 0 . , 0 6 9 8 0 . 0 2 6 4 0 . 0 0 5 7 2 . 5 0 0 . 2 3 9 8 0 . 4 3 9 9 0 . 5 4 9 3 0 . 6 9 2 3 0 . 9 8 0 2 0 . 1 4 7 5 0 . . 0 3 6 3 0 . 0 1 2 9 0 . 0 0 2 9

3 . 0 0 0 . 2 3 9 8 0 . 4 3 9 9 0 . 5 4 9 3 0 . 6 9 2 9 0 . 9 8 9 4 0 . 0 9 6 5 0 . . 0 1 8 8 0 . 0 0 6 5 0 . 0 0 1 5 3 . 5 0 0 . 2 3 9 8 0 . 4 3 9 9 0 . 5 4 9 3 0 . 6 9 3 1 0 . 9 9 4 2 0 . 0 6 2 2 0 . , 0 0 9 8 0 . 0 0 3 4 0 . 0 0 0 8 4 . 0 0 0 . 2 3 9 8 0 . 4 3 9 9 0 . 5 4 9 3 0 . 6 9 3 1 0 . 9 9 6 8 0 . 0 3 9 7 0 . . 0 0 5 2 0 . 0 0 1 8 0 . 0 0 0 4 5 . 0 0 0 . 2 3 9 8 0 . 4 3 9 9 0 . 5 4 9 3 0 . 6 9 3 1 0 . 9 9 9 0 0 . 0 1 5 9 0 . . 0 0 1 5 0 . 0 0 0 5 0 . 0 0 0 1

00 0 . 2 3 9 8 0 . 4 3 9 9 0 . 5 4 9 3 0 . 6 9 3 1 1 . 0 0 0 0 0 . 0 0 0 0 0 . . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0

T A B L E 2 B

F F u n c t i o n s o f O r d e r 2

F 2( x , S ) F 2( x , 0 ) e~

xsF2(x,

χ s

1 = 0.1 0.3 0.5 1.0 s = 0

1 = 1.0 0.5 0.3 0.1

0 . 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . . 0 0 0 0 0 . 0 1 0 . 0 0 9 3 0 . 0 0 9 6 0 . 0 0 9 6 0 . 0 0 9 7 0 . 0 0 9 7 0 . 0 0 9 7 0 . 0 0 9 6 0 . 0 0 9 6 0 . , 0 0 9 2 0 . 0 2 0 . 0 1 7 3 0 . 0 1 8 4 0 . 0 1 8 7 0 . 0 1 8 8 0 . 0 1 9 0 0 . 0 1 8 8 0 . 0 1 8 7 0 . 0 1 8 4 0 . , 0 1 7 2 0 . 0 5 0 . 0 3 5 7 0 . 0 4 1 6 0 . 0 4 3 0 0 . 0 4 4 0 0 . 0 4 5 1 0 . 0 4 3 9 0 . 0 4 2 8 0 . 0 4 1 4 0 . , 0 3 5 2 0 . 1 0 0 . 0 5 4 3 0 . 0 7 1 8 0 . 0 7 6 3 0 . 0 7 9 9 0 . 0 8 3 7 0 . 0 7 9 5 0 . 0 7 5 5 0 . 0 7 0 6 0 . , 0 5 1 6

0 . 1 5 0 . 0 6 4 2 0 . 0 9 4 3 0 . 1 0 2 8 0 . 1 0 9 9 0 . 1 1 7 7 0 . 1 0 8 7 0 . 1 0 0 6 0 . 0 9 1 0 0 , . 0 5 7 9 0 . 2 0 0 . 0 6 9 5 0 . 1 1 1 3 0 . 1 2 4 2 0 . 1 3 5 4 0 . 1 4 8 1 0 . 1 3 3 0 0 . 1 1 9 9 0 . 1 0 4 9 0 . . 0 5 8 9 0 . 2 5 0 . 0 7 2 4 0 . 1 2 4 2 0 . 1 4 1 6 0 . 1 5 7 2 0 . 1 7 5 3 0 . 1 5 3 1 0 . 1 3 4 4 0 . 1 1 3 9 0 . . 0 5 7 1 0 . 3 0 0 . 0 7 4 0 0 . 1 3 4 1 0 . 1 5 5 8 0 . 1 7 5 9 0 . 2 0 0 0 0 . 1 6 9 7 0 . 1 4 5 0 0 . 1 1 9 1 0 . . 0 5 3 9 0 . 4 0 0 . 0 7 5 4 0 . 1 4 7 5 0 . 1 7 7 2 0 . 2 0 6 1 0 . 2 4 2 7 0 . 1 9 4 1 0 . 1 5 7 4 0 . 1 2 1 5 0 . . 0 4 6 5

0 . 5 0 0 . 0 7 5 8 0 . 1 5 5 5 0 . 1 9 1 7 0 . 2 2 8 9 0 . 2 7 8 4 0 . 2 0 9 6 0 . 1 6 1 1 0 . 1 1 7 3 0 . , 0 3 9 3 0 . 6 0 0 . 0 7 6 0 0 . 1 6 0 4 0 . 2 0 1 8 0 . 2 4 6 2 0 . 3 0 8 4 0 . 2 1 8 2 0 . 1 5 9 1 0 . 1 0 9 5 0 . , 0 3 3 2 0 . 7 0 0 . 0 7 6 0 0 . 1 6 3 3 0 . 2 0 8 8 0 . 2 5 9 6 0 . 3 3 3 9 0 . 2 2 1 6 0 . 1 5 3 3 0 . 1 0 0 0 0 . 0 2 8 1 0 . 8 0 0 . 0 7 6 0 0 . 1 6 5 1 0 . 2 1 3 6 0 . 2 6 9 9 0 . 3 5 5 7 0 . 2 2 1 2 0 . 1 4 5 1 0 . 0 9 0 1 0 . , 0 2 3 9 0 . 9 0 0 . 0 7 6 0 0 . 1 6 6 2 0 . 2 1 7 1 0 . 2 7 7 8 0 . 3 7 4 3 0 . 2 1 7 9 0 . 1 3 5 7 0 . 0 8 0 3 0 . , 0 2 0 4

1 . 0 0 0 . 0 7 6 0 0 . 1 6 6 9 0 . 2 1 9 5 0 . 2 8 4 0 0 . 3 9 0 3 0 . 2 1 2 3 0 . 1 2 5 5 0 . 0 7 1 1 0 . , 0 1 7 5 1 . 2 5 0 . 0 7 6 0 0 . 1 6 7 7 0 . 2 2 2 8 0 . 2 9 4 2 0 . 4 2 1 4 0 . 1 9 2 7 0 . 1 0 0 3 0 . 0 5 1 5 0 . , 0 1 2 1 1 . 5 0 0 . 0 7 6 0 0 . 1 6 7 9 0 . 2 2 4 3 0 . 2 9 9 8 0 . 4 4 3 3 0 . 1 6 9 3 0 . 0 7 7 8 0 . 0 3 6 8 0 . , 0 0 8 5 2 . 0 0 0 . 0 7 6 0 0 . 1 6 8 0 0 . 2 2 5 1 0 . 3 0 4 6 0 . 4 6 9 9 0 . 1 2 3 0 0 . 0 4 4 5 0 . 0 1 8 8 0 . , 0 0 4 3 2 . 5 0 0 . 0 7 6 0 0 . 1 6 8 0 0 . 2 2 5 3 0 . 3 0 6 1 0 . 4 8 3 7 0 . 0 8 5 2 0 . 0 2 4 7 0 . 0 0 9 7 0 . , 0 0 2 3

3 . 0 0 0 . 0 7 6 0 0 . 1 6 8 0 0 . 2 2 5 3 0 . 3 0 6 6 0 . 4 9 1 1 0 . 0 5 7 3 0 . 0 1 3 5 0 . 0 0 5 1 0 , . 0 0 1 2 3 . 5 0 0 . 0 7 6 0 0 . 1 6 8 0 0 . 2 2 5 3 0 . 3 0 6 8 0 . 4 9 5 1 0 . 0 3 7 8 0 . 0 0 7 4 0 . 0 0 2 8 0 , . 0 0 0 7 4 . 0 0 0 . 0 7 6 0 0 . 1 6 8 0 0 . 2 2 5 3 0 . 3 0 6 8 0 . 4 9 7 2 0 . 0 2 4 6 0 . 0 0 4 0 0 . 0 0 1 5 0 , . 0 0 0 4 5 . 0 0 0 . 0 7 6 0 0 . 1 6 8 0 0 . 2 2 5 3 0 . 3 0 6 8 0 . 4 9 9 1 0 . 0 1 0 1 0 . 0 0 1 2 0 . 0 0 0 5 0 , . 0 0 0 1

00 0 . 0 7 6 0 0 . 1 6 8 0 0 . 2 2 5 3 0 . 3 0 6 9 0 . 5 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 , . 0 0 0 0

12

2.5 G Functions 13

Expansions useful for small χ :

F^s) = (-/ + l)x + K-s/ + h + l)x2 + ···

F 2( x , s) = χ + £(/ + s - f )x2 + · · ·

F functions for χ = oo : They exist only for s < 1.

Recurrence:

sFn+1(co,s) = -(1/n) + F„(oo,s)

Special values:

( s # 0 )

F j ioo . s ) = - ( l / s ) I n ( l - s)

F2(oo , s ) = - ( l / s ) - ( l / s2) l n ( l - s)

F3(oo,s) = - ( l / 2 s ) - (1/s2) - ( l / s3) ln( l - s)

( 5 = 0)

F „ ( œ , 0 ) = ( l /n )

( s = - 1 )

F^oo , - 1 ) = I n 2 = 0.69314 7

F2(oo , - 1 ) = 1 - I n 2 = 0.30685 3

F3(oo , - 1 ) = - i + I n 2 = 0.19314 7

F4(oo , — 1 ) = I — I n 2 = 0.14018 6

F5(oo , - 1 ) = — £ + I n 2 = 0.10981 4

Numerical values : Se e Tabl e 2 . Chandrasekha r an d Bree n (1948 ) giv e eight page s o f F function s fo r 1 4 s values , 4 1 χ values < 1 and orders η = 1, 3, and 5;6D.

2.5 G FUNCTIONS

Definitions and symmetry :

Gnm(x) = Gmn(x) = f En(t)Em(t)dt Jo

G'JLx) = G ^ x ) = f EH(t)EJx - t)dt Jo

/•OO

Gn'm(^) = £„(* + t)Em(t) dt (not symmetric)

14 2 Exponential Integrals and Related Functions

Derived relations:

Î00

F„(x, -s)ds/sf

Recurrence and expression in known functions :

(m + n- l)Gm„(x) = xEm(x)En(x) + Fm(x, - 1 ) + Fn(x, -1)

G „ , n + 1( x ) = ( l / 2 n2) - i [ £ n + 1( x ) ]

2

(m + n- l )GL(x) = xEm(x)/(n - 1) - xG^,„_ ,(x) + <>-*[F„(x, 1) + Fm( x , 1)]

G'u(x) = 2 [ £ 1( x ) + ΙΕ2(χ) - xE\2)(x)~]

G;'2(x) = - G ; ' + 1 > 1( x ) + £ n + 1( x )

G„m(œ) = ( m + « - Ι Γ ^ ο ο , - 1 ) + Fm(oo, - 1 ) ]

G „ ( x ) = (/2 - 2/ + 2)x + ( - / + i ) x

2 + · · •

G 1 2( x ) = ( - / + l)x - ( i i2 - Qx

2 + · · ·

G 1 3( x ) = i ( - ' + 1)χ + ^ / χ2+ · · ·

G 2 2( x ) = χ + (/ - f ) x2 + · · ·

6,Ί(χ) = (I2 -21 + 2- \n

2)x + ( - / + f ) x

2 + · · ·

G'1 2(x) = ( - / + l )x - W2 ~ 3/ + f - έπ

2)χ

2 + · · ·

G\3(x) = i(-l+ l )x + i ( / - l ) x2 + · · ·

G'2 2(x) = χ + (/ - f ) x2 + · · ·

GS,(x) = (ψ + W) + (I + 2 In 2 - 3)x + (/ - f )(x2/4) + · · ·

Additional special values and properties of these functions, and further relations with other functions are found in the appendix of Kourganoff (1952), in particular:

M„(x) = G'^{x), J„m = G„m(co), and ξ„(μ) = μ 'Fn(oo , -μ l

).

Expansions useful for small χ :

Special values: G n( c o )

G1 2( c o )

G1 3(oo )

G2 2(oo )

2 In 2

έ ( 4 1 η 2 - 1)

| ( 1 - In 2)

References 15

T A B L E 3

G F u n c t i o n s

X G 1 2( x ) G22{x) G\2(x) G'22(x)

0 . 0 0 0 . 0 1 0 . 0 2 0 . 0 5 0 . 1 0

0 0 . 2 6 3 2 5 0 . 3 9 7 3 4 0 . 6 4 1 5 9 0 . 8 6 5 0 0

0 0 . 0 4 9 0 6 0 . 0 8 3 1 2 0 . 1 5 7 3 5 0 . 2 3 8 9 6

0 0 . 0 2 4 9 4 0 . 0 4 2 7 0 0 . 0 8 2 6 4 0 . 1 2 8 5 6

0 0 . 0 0 9 4 6 0 . 0 1 8 1 2 0 . 0 4 0 7 6 0 . 0 7 0 6 5

0 0 . 2 4 6 9 0 0 . 3 6 4 8 4 0 . 5 6 1 8 0 0 . 7 1 0 1 0

0 0 . 0 4 8 8 2 0 . 0 8 2 2 9 0 . 1 5 3 2 9 0 . 2 2 6 2 4

0 0 . 0 2 4 8 9 0 . 0 4 2 5 1 0 . 0 8 1 5 4 0 . 1 2 4 5 3

0 0 . 0 0 9 4 5 0 . 0 1 8 1 0 0 . 0 4 0 5 4 0 . 0 6 9 5 1

0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0 0 . 4 0

0 . 9 9 8 4 0 1 . 0 8 7 9 9 1 . 1 5 1 9 0 1 . 1 9 9 2 4 1 . 2 6 3 1 8

0 . 2 9 4 5 3 0 . 3 3 5 1 5 0 . 3 6 5 9 8 0 . 3 8 9 9 7 0 . 4 2 4 2 0

0 . 1 6 1 1 3 0 . 1 8 5 6 8 0 . 2 0 4 7 9 0 . 2 1 9 9 8 0 . 2 4 2 2 2

0 . 0 9 3 8 2 0 . 1 1 2 2 3 0 . 1 2 7 1 1 0 . 1 3 9 2 6 0 . 1 5 7 6 0

0 . 7 7 2 8 2 0 . 7 9 5 8 7 0 . 7 9 7 1 4 0 . 7 8 5 5 2 0 . 7 4 1 6 5

0 . 2 7 0 4 5 0 . 2 9 7 9 5 0 . 3 1 4 5 3 0 . 3 2 3 5 4 0 . 3 2 6 7 6

0 . 1 5 2 7 8 0 . 1 7 1 9 3 0 . 1 8 4 8 2 0 . 1 9 3 1 5 0 . 2 0 0 3 8

0 . 0 9 1 0 2 0 . 1 0 7 0 9 0 . 1 1 9 0 0 0 . 1 2 7 6 7 0 . 1 3 7 7 7

0 . 5 0 0 , 6 0 0 . 7 0 0 . 8 0 0 . 9 0

1 . 3 0 2 7 0 1 . 3 2 8 2 5 1 . 3 4 5 3 1 1 . 3 5 6 9 7 1 . 3 6 5 0 8

0 . 4 4 6 6 5 0 . 4 6 1 8 6 0 . 4 7 2 4 0 0 . 4 7 9 8 3 0 . 4 8 5 1 4

0 . 2 5 7 2 5 0 . 2 6 7 6 8 0 . 2 7 5 0 6 0 . 2 8 0 3 5 0 . 2 8 4 1 9

0 . 1 7 0 3 6 0 . 1 7 9 4 2 0 . 1 8 5 9 3 0 . 1 9 0 6 6 0 . 1 9 4 1 4

0 . 6 8 5 5 4 0 . 6 2 6 2 5 0 . 5 6 8 0 1 0 . 5 1 2 8 1 0 . 4 6 1 5 4

0 . 3 1 8 0 7 0 . 3 0 2 9 4 0 . 2 8 4 4 8 0 . 2 6 4 5 7 0 . 2 4 4 3 3

0 . 1 9 9 4 2 0 . 1 9 3 5 7 0 . 1 8 4 7 9 0 . 1 7 4 3 8 0 . 1 6 3 1 6

0 . 1 4 1 1 4 0 . 1 4 0 1 1 0 . 1 3 6 2 1 0 . 1 3 0 4 9 0 . 1 2 3 6 6

1 . 0 0 1 . 2 5 1 . 5 0 2 . 0 0 2 . 5 0

1 . 3 7 0 8 1 1 . 3 7 9 0 2 1 . 3 8 2 7 6 1 . 3 8 5 4 0 1 . 3 8 6 0 5

0 . 4 8 8 9 7 0 . 4 9 4 6 5 0 . 4 9 7 3 3 0 . 4 9 9 3 0 0 . 4 9 9 8 0

0 . 2 8 7 0 1 0 . 2 9 1 2 5 0 . 2 9 3 3 1 0 . 2 9 4 8 6 0 . 2 9 5 2 7

0 . 1 9 6 7 0 0 . 2 0 0 6 2 0 . 2 0 2 5 5 0 . 2 0 4 0 1 0 . 2 0 4 4 1

0 . 4 1 4 5 1 0 . 3 1 5 0 6 0 . 2 3 8 4 7 0 . 1 3 6 1 1 0 . 0 7 7 7 1

0 . 2 2 4 4 8 0 . 1 7 8 8 1 0 . 1 4 0 4 3 0 . 0 8 4 7 2 0 . 0 5 0 3 6

0 . 1 5 1 6 8 0 . 1 2 3 9 1 0 . 0 9 9 3 6 0 . 0 6 1 9 3 0 . 0 3 7 7 4

0 . 1 1 6 2 4 0 . 0 9 7 0 9 0 . 0 7 9 1 7 0 . 0 5 0 5 6 0 . 0 3 1 3 4

3 . 0 0 3 . 5 0 4 . 0 0 5 . 0 0

CO

1 . 3 8 6 2 3 1 . 3 8 6 2 7 1 . 3 8 6 2 9 1 . 3 8 6 2 9 1 . 3 8 6 2 9

0 . 4 9 9 9 4 0 . 4 9 9 9 8 0 . 4 9 9 9 9 0 . 5 0 0 0 0 0 . 5 0 0 0 0

0 . 2 9 5 3 8 0 . 2 9 5 4 2 0 . 2 9 5 4 3 0 . 2 9 5 4 3 0 . 2 9 5 4 3

0 . 2 0 4 5 2 0 . 2 0 4 5 5 0 . 2 0 4 5 6 0 . 2 0 4 5 7 0 . 2 0 4 5 7

0 . 0 4 4 4 9 0 . 0 2 5 5 5 0 . 0 1 4 7 2 0 . 0 0 4 9 3

0

0 . 0 2 9 7 5 0 . 0 1 7 5 1 0 . 0 1 0 3 0 0 . 0 0 3 5 6

0

0 . 0 2 2 7 3 0 . 0 1 3 6 0 0 . 0 0 8 1 1 0 . 0 0 2 8 7

0

0 . 0 1 9 1 2 0 . 0 1 1 5 6 0 . 0 0 6 9 5 0 . 0 0 2 4 9

0

Numerical values: See Table 3. Chandrasekhar and Breen (1948) give two pages of G functions and G' functions for all combinations of orders η and m < 5; 6Z). Chandrasekhar (1949) extends this work with 1 page of G functions, 6D, to order 6.

R E F E R E N C E S

A b r a m o w i t z , M . , a n d S t e g u n , I . ( 1 9 6 5 ) . " H a n d b o o k o f M a t h e m a t i c a l F u n c t i o n s . " D o v e r , N e w

Y o r k .

C h a n d r a s e k h a r , S. ( 1 9 4 9 ) . Astrophys. J. 1 0 9 , 5 5 5 .

C h a n d r a s e k h a r , S . , a n d B r e e n , F . H . ( 1 9 4 8 ) . Astrophys. J. 1 0 8 , 9 2 ( a p p e n d i x ) .

K i n g , L . V . ( 1 9 1 3 ) . Philos. Trans. Soc. London Ser. A 2 1 2 , 3 7 5 .

K o u r g a n o f f , V . ( 1 9 5 2 ) . " B a s i c M e t h o d s in T r a n s f e r P r o b l e m s . " O x f o r d U n i v . P r e s s ( C l a r e n d o n ) .


v a n d e H u l s t , H . C . ( 1 9 4 8 ) . Astrophys. J. 1 0 7 , 2 2 0 .

3 • Reciprocity

3.1 RECIPROCITY AND DETAILED BALANCE

The reciprocity principle in its widest and most general form states that in any linear physical system, the channels which lead from a cause (or action) at one point to an effect (or response) at another point can be equally well traversed in the opposite direction. Let the cause first be placed at Ρ and the effect measured at Q; and in a second experiment, carried out in the same physical system, let the cause be at Q and the effect at P. The reciprocity principle is then expressed by the proportionali ty:

effect at g/cause at Ρ = effect at P/cause at Q

If η points Ρί··Ρη are considered, and Rki is the ratio of effect at Pk to cause at Pi9 the reciprocity principle states that the matrix Rik is symmetric.

A general and strict formulation would require a careful definition of the concepts linear physical system, cause, and effect. We shall not attempt to give such a formulation. In specific physical systems cause and effect are replaced by specific physical quantities. Hence the reciprocity principle appears in a large number of forms in mechanics, acoustics, electromagnetism, atomic collisions, nuclear physics, and radiative transfer. We shall limit the discussion to the emission, absorption, and scattering of light.

Even so, the reciprocity principle can take at least a dozen forms, depending on the system considered and the quantities taken as cause and effect. The purpose of this chapter is (a) to give a number of formulations for reference in later

16

3.1 Reciprocity and Detailed Balance 17

chapters and (b) to indicate, without rigorous proof, how these formulations can be derived from each other and from basic principles. These problems seem trivial, but experience has shown that it is easy to draw erroneous conclusions.

While reciprocity is basically derived from time-reversal invariance, the physical system under consideration may well possess additional symmetries. These lead separately, or combined with reciprocity, to further symmetry relations for phase matrix, reflection function, etc. Such symmetry relations are of great practical help in checking the consistency of analytical formulas or of computational results. They can even be used directly on observations (Minnaert , 1941; see also Section 18.2). The inclusion of polarization in these relations requires particular care (Hovenier, 1969).

This chapter contains a collection of formulations of reciprocity relations in various physical situations. We introduce a quantity as often as possible by its physical meaning, i.e., as the outcome of a well defined thought experiment.

As a consequence of the reciprocity principle, many mathematical functions which are presented in formulas, graphs, or tables in later chapters have in fact two physical meanings, related by time reversal. A simple example is the point-direction gain (Section 3.3.3). Any relation between such functions can similarly be interpreted in two different ways referring to quite different physical experiments.

The reciprocity principle is closely connected to the principle of detailed balance, which states that in thermodynamic equilibrium any detailed process which we choose to consider has a reverse process, and the rates of these processes are in exact balance. This principle was widely applied in astrophysics, long before its basis was well understood (Eddington, 1926, p. 45 ; Rosseland, 1936, p. 356). Later developments in nonequilibrium thermodynamics have provided a firm basis. The principle of detailed balance has been formulated and proven for any system governed by a Hamiltonian which possesses time-reversal invariance, both in classical systems and in quantum mechanics (Wigner, 1954; De Groot and Mazur, 1962; De Groot , 1963). In the usual context one proceeds from this principle to the Onsager relations, which refer to the rate of change of macroscopic variables in nonequilibrium situations; however, this next step is of no interest in the present applications.

It is necessary to note a peculiar inversion of the logical order in some derivations in the following sections. Detailed balance is a statistical concept, because it refers to a situation described statistically ( thermodynamic equilibrium). Reciprocity may or may not refer to a situation described statistically. The logical order would be always to start from the reciprocity in the individual, nonstatistical situation and to proceed towards reciprocity in statistical situations and towards detailed balance. This choice is always open. However, it is often more convenient to start from the other end, reasoning like this: if the principle of detailed balance is to be true, whatever the temperature, the coefficients describing the individual processes must obey certain reciprocity relations.

18 3 Reciprocity

This inverted reasoning is more than a century old. The classical derivation of Kirchhoff's law relating the absorptive and emissive properties of a body, using the second law of thermodynamics, is a good illustration.

Another classical example of an inverse derivation is the determination of the ratio between the cross section of an a tom for photoionization and the corresponding cross section of an ion for electron capture. The direct method for deriving this ratio starts from basic quantum mechanics and does not involve statistics (Landau and Lifshitz, 1957, p. 344; the terminology differs from ours). The inverse method is commonly followed in astrophysics texts, e.g., Rosseland (1936, p. 316). The main advantage of the inverse method is that it is simpler to get all the normalizations correct.

In deriving the light-scattering forms of the reciprocity principle, we shall use both the direct and the inverse method. The inverse method offers advantages if one or several assumptions are involved which make the system resemble a system in thermodynamic equilibrium. Examples of such assumptions are random orientation of particles, incident light with uniform intensity in all directions, and unpolarized light sources.

In all formulations we shall omit the dependence of the quantities on frequency v. The terminology will be correspondingly imprecise: if we mention energy flow, or energy per unit time, this may be read as energy flow in a small interval Δν, or as energy flow in a wide frequency band, or even as light flux measured as a luminous quantity. The distinction must be noted in practical applications but is irrelevant for the purposes of this chapter.

3.2 FAR-FIELD SCATTERING BY A SINGLE PARTICLE

The prototype is scattering by a single particle. Such a particle may be a droplet or crystal in the atmosphere of a planet. It may at times be taken also to mean a total (finite) cloud. It is even possible to consider the entire planet as a particle (Section 18.1.1).

We assume that incident light comes from a distant point (i.e., it is virtually a parallel wave) and that the scattered light is measured at another distant point (i.e., it is virtually an outgoing spherical wave). First, the reciprocity principle is stated for coherent scattering on the basis of electromagnetic theory. Then, by simple statistics, we proceed to formulate the corresponding principle for natural light and for partially polarized light, using various assumptions about the statistics of the orientations.

3.2.1 Coherent Scattering (Amplitude Transformation)

Let the object be fixed and contained in a finite region about the origin (Fig. 3.1). Let k be the propagation constant (27r/wavelength) in the external

3.2 Far-Field Scattering by a Single Particle 19

F i g . 3.1. U n i t v e c t o r s u s e d t o d e s c r i b e fa r - f i e ld s c a t t e r i n g a m p l i t u d e s o f a r b i t r a r y finite b o d y .

medium. Let the radius vector OP be rn and let OQ be r'n', where η and n' are unit vectors, and both kr and kr' are very large. Let m be one of the two unit vectors perpendicular to η and η' (arbitrary choice); let

1 = η x m and F = n' x m

The two experiments, carried out with arbitrary amplitude, phase, and polarization, now give electric fields which may be written as the real parts of the following expressions.

Direct experiment:

( - 4 , 1 + Arm)e+ikr + im

Reverse experiment:

( - Q 1 ' + Crm)eikr' + iaH

(Dtl + Drm)e-ikr+Uot

/ikr

plane incident wave at Ρ

spherical scattered wave at Q

plane incident wave at Q

spherical scattered wave at Ρ

If the direct experiment obeys the matrix relation

(Bl\(R2 R3\(At\ \Br) \R4 R1)\Ar)

20 3 Reciprocity

then the reciprocity principle states that the reverse experiment must obey the relation

DA = I R2 -RAfCA Drj \-R3 Rx)\Cr)

The matrix elements R1-R4. are functions of the frequency ν and of the two directions η, n', but not of r.

This, with minor changes in notation, is the formulation employed by van de Hulst (1957, p. 34). The proof can be given directly on the basis of Maxwell's equations together with the assumption that the dielectric, permeability, and conductive tensors are symmetric. Such a proof in a slightly more general problem (incoming spherical waves from all directions) was presented by Saxon (1955). The reciprocity theorem remains valid in this form if the planes of reference (the planes through η and I) for the incident beam and for the scattered beam are rotated over arbitrary angles.

3.2.2 Incoherent Scattering (Intensity Transformation)

The phases of the scattered waves usually are irrelevant in practical problems. It takes a special effort to secure enough stability in a laboratory experiment to demonstrate optical interference. In turbid media, with many slowly and randomly moving particles, interference effects simply are not seen, because the phases are washed out. What is left is incoherent scattering.

The statistical properties of incoherent light at any wavelength and in any direction can be specified by the Stokes parameters, which add to the intensity / three other quantities Q, U, V, of the same physical dimension, defined with respect to a plane of reference chosen through the direction of propagation. These added parameters are zero for natural (i.e., unpolarized) light. They obey the relation

Q2 + U

2 + V

2 = I

2

for any coherent wave, which is a solution of Maxwell's equations and which, by definition, is fully polarized. The parameters Q and U change if the plane of reference is rotated.

In what follows we shall note the four Stokes parameters as ( 1 = 1, 2, 3,4) and call them the intensity, complete with polarization. The same usage may be applied to quantities with a different physical dimension, e.g., L (energy per unit time per unit solid angle) and S (energy per unit time per unit area) in order to use the letters of the alphabet sparingly.

Let 1 and m be two unit vectors perpendicular to the direction into which a wave travels, such that m χ 1 = n. The plane through η and 1 is called the plane of reference.


F = r

2iM2 + M3 + M4. + Mi) HM2 - M 3

+ M4-Ml) ^ 2 3 + s 4l - 0 2 3 - 0 4 1 ^M2 + M3-M^-M1) - M 3 - M 4 + M , ) ^ 2 3 — S4l - 0 2 3 + 041 S 24 + S 31 - s 3, ^ 2 1 + S 3 4 - 0 2 1 + 0 3 4 " D2* + D31 D 2 4 - 0 3 1 0 2 1 + 034 S 2 1 — ^ 3 4 ,

where Mk = RkR{, Skj = Sjh = ^RjRf + RhRf\ and -Dhj = Djk =

The reciprocity relation between the R matrices transforming the complex amplitudes in the direct and reverse experiments leads to a similar reciprocity relation between the phase matrices F transforming the Stokes parameters. With some care this relation can be read from the explicit form for F given above. It is presented in Display 3.1. The matrix for the reverse experiment is obtained from the matrix for the direct experiment by transposing it and adding minus signs to the nondiagonal elements of the third row and column.

During short intervals of time the field may be written

(£,1 + £Γπ ι ) ^ -ι 7 ί Γη + Ι

'ωί

where £ z and Er are complex amplitudes. The Stokes parameters are defined by average products of these amplitudes, as follows,

1 = 1, = < £ , £ • + ErE*y

Q = I2 = <£,£,* - ErE?}

17 = / 3 = < £ , £ * + £,£,*>

V = I4 = i(EtE? - ErEf}

where the asterisk denotes the complex conjugate value and the angle brackets denote the statistical average. A common constant depending on the choice of units has been omitted.

If this definition is applied both to the incident and to the scattered light, the 2 x 2 matrix of complex numbers,

« 4 Rj

which transforms the complex amplitudes of the incident light into those of the scattered light, gives rise to the phase matrix F , a 4 χ 4 matrix of real numbers, which transforms the corresponding Stokes parameters. The explicit form of this matrix, derived by van de Hulst (1957, p. 44) is

22 3 Reciprocity

D I S P L A Y 3.1

R e c i p r o c i t y in I n c o h e r e n t F a r - F i e l d S c a t t e r i n g b y a F i n i t e O b j e c t

I n c i d e n t q u a s i -

p a r a l l e l b e a m

S c a t t e r e d q u a s i -

s p h e r i c a l b e a m

E n e r g y f low p e r u n i t

a r e a c o m p l e t e w i t h

p o l a r i z a t i o n

E n e r g y f low p e r u n i t

s o l i d a n g l e c o m p l e t e

w i t h p o l a r i z a t i o n E q u a t i o n

D i r e c t e x p e r i m e n t : S,

s c a t t e r i n g f r o m

Ρ t o ρ

R e v e r s e e x p e r i m e n t : S\

s c a t t e r i n g f r o m

Q t o P

L t

L't

Li = Σ Fik Sk k

L'i = Σ

Gik

S'k

k

R e c i p r o c i t y r e l a t i o n : Gik = PiPkFki w h e r e px = p2 = p4 = 1, p3 = - 1 .

3.2.3 Some Consequences

The reciprocity relation in Display 3.1 remains valid whatever the choice of planes of reference in Ρ and Q. This can be most readily seen by formal matrix algebra. Define the matrices

Ύ(φ) =

0 0 cos 2φ 0 — sin2c> 0 0

0 sin 2φ cos 2φ

0

0 \ 0 0 1 /

and

( l 0 0 o\ 0 1 0 0 0 0 - 1 0

l o 0 0 i l

then

Ύ(φ)Ύ( - φ) = 1, P P = 1, and ΡΤ(φ)Ρ = T( - φ)

The equations in Display 3.1 may now be written:

L = FS, L = G S , and G = P F P

where an overbar indicates the transposed matrix. Turning the plane of reference at Q by an angle φ means measuring the scattered light in the first experiment as (n = new)

T(<p)L = L„

and the incident light in the second experiment as

Τ ( - φ ) 8 ' = S'n


We now obtain relations of the form

L n = F nS , L' = G nS n

and ask whether the equation

G n = P F nP

follows from there. This is indeed the case, for we first obtain

F n = TfoOF, F n = FT( - φ), and G n = GT(<p)

and the stated relation easily follows. Contained in the general reciprocity relation of Display 3.1 is the statement

G n = Flt This has the following physical meaning: If unpolarized radiation with energy L per unit area coming from Ρ gives scattered radiation with energy S per unit solid angle into the direction of Q, and if in the reverse experiment energy L per unit area from Q gives scattered energy S' per unit solid angle towards P, then

L'/S' = L/S

The correctness of this interpretation is seen by putting L{ = (L, 0, 0, 0). The scattered radiation will generally be polarized, but the scattered intensity obeys the reciprocity relation.

This form can also be readily derived by the principle of detailed balance. Place absorbing particles (grains of charcoal) of projected areas A and A' at Ρ and Q and enclose everything in a thermostat. Writing Β for the blackbody specific intensity, and equating the energy transferred from Ρ to β to that transferred from Q to P , we find

B(ALA'/r2Sr'

2) = B(A

fL'A/r

,2Sfr2)

from which the reciprocity relation follows. In this proof further refinements may be made. Radiation scattered via the

particle can be separated from direct radiation between Ρ and Q, provided the scattering angle differs from zero. A restriction to any given frequency band (wide or narrow) may be made, if desired, by placing a suitable filter in the beam. Detailed balance is a simple approach in this example, because natural light is a statistical concept.

Further assumptions often simplify the phase matrices considerably. If an average is taken over randomly oriented particles or over particles and mirror particles, some matrix elements become equal, opposite, or zero. The relations for a large variety of such symmetry assumptions have been stated explicitly by van de Hulst (1957, pp. 49-55).

If in any application the need exists to describe not only the state of polarization but also the wavelength dependence, the coherency matrix (Born and Wolf, 1959, p. 542) must be used instead of the Stokes parameters.

24 3 Reciprocity

3.3 ARBITRARY CONFIGURATIONS

Manageable numerical results can be obtained only for schematic problems. For that reason, many simplifying assumptions are made in this book, as in most published papers. In most chapters the scattering particles have convenient phase functions and the clouds containing these particles are plane slabs.

In the present section only we shall consider bodies of arbitrary shape and arbitrarily shaped configurations of scattering particles. F rom the great diversity of theorems that may be formulated for such configurations (all in the general framework of reciprocity and symmetry), we have selected a few whose formulation appears useful for later application to plane cloud layers or to an entire planet.

3.3.1 Kirchhoff's Law for a Finite Body

In Figure 3.2 let Ο be a scattering particle, Ρ a blackbody of surface area A' perpendicular to OP, and Q an arbitrary body capable of absorbing some of the radiation arriving from the direction OQ. The same body Q will then be capable of thermal emission of radiation into the direction Q. The relation between the absorbing and emitting properties of the body at Q, including their dependence on polarization, will be called Kirchhoff"s law.

Q

Ρ

F i g . 3.2. C o n c e p t u a l e x p e r i m e n t t o d e r i v e K i r c h h o f f ' s l a w f o r a f in i te b o d y .

3.3 Arbitrary Configurations 25

Let Ρ and Q be at the same temperature, at which the specific intensity of blackbody radiation is B. Choose a plane of reference through OP and one (possibly another) through OQ. Define the absorption cross section, including polarization, of the body Q for the direction OQ by writing the energy absorbed per unit time in the body Q from incident radiation arriving from OQ as

£ a b s = Σ a

iLi

i Here L t is the energy per unit time per unit area, including polarization, arriving from this direction. Define the emission cross section of the body by writing the energy per unit time per unit solid angle, including polarization, emitted in the direction QO, as

Sem,i = eiB

The relation between at and et is found by applying the principle of detailed balance to the energy transferred per unit time from Ρ to β and conversely, via the scattering particle. We have, through application of the relations in Display 3.1,

EP ^ Q = Σ

α/Γ' ~

2Fnr~

2AB

i

k

Since this is to be equal for any particle Ο obeying the reciprocity relations Fn = PiGu, the absorption and emission cross section of the body Q must obey the relation

et = ρ

where p{ is defined in Section 3.2.2. This is the general formulation of Kirchhoff's law for arbitrarily polarized light (van de Hulst, 1965). It is an extension of the relation e1 = αγ first stated by Kirchhoff roughly a century earlier.

Since this result is so simple, compared to, e.g., the formulation given (without polarization) by Sobolev (1973), a few examples may be helpful.

First, for a blackbody

ei = ai = (G, 0 ,0 ,0)

where G is the geometric cross section in the direction considered. Another very simple example is a thin disk of geometrical area G and optical

thickness b = Δτ < 1. Then, if a is the single-scattering albedo (same letter, now without index) the absorbing power of the disk is described by

a. = (au 0, 0, 0), ax = (1 - α) Δτ G

The cosine of the angle of incidence μ 0 drops out because the depth is lengthened to Δτ /μ0 in the same proport ion as the area is foreshortened to G μ 0. Kirchhoff's

26 3 Reciprocity

law as defined above now states that the energy thermally emitted per unit solid angle by this disk in any direction is

SEM, t = ( S e m, l5 0, 0, 0) S e m, t = (1 - a) AT GB

The three zeros simply mean that the emission is unpolarized. Sobolev writes λ for a and B* for B. He further introduces the symbol B0 for the product (1 — a)B, but I question the usefulness of the introduction of this quantity.

If the optical thickness b of the disk is not infinitesimal, we have

a x = G μ0 [absorbed fraction of incident flux] = Gμ011 - R(a,b,p0) - Γ(αΛμ0) ]

Values of the quantity in brackets are found in Tables 15 (Section 9.2) for isotropic scattering and 39 (Section 13.2) for anisotropic scattering. In a semi-infinite atmosphere, the term Τ drops out.

The emission in any direction by a nontransparent sphere of radius r is

Se m, i = ( S e m, i, 0 ,0 ,0) , S e m, ! = (1 - A*)nr2B

where A* is the spherical albedo (Sections 12.2.3 and 18.1.2).

3.3.2 Gain

Many problems encountered in practical situations are of the form: How much radiation from an isotropic source at Ρ reaches a point If the points Ρ and Q are in free space, the answer is simple. If an arbitrary configuration of scattering and/or absorbing particles is somewhere in the vicinity, the answer will be different. We write this answer as G times the free-space value and call G the gain (van de Hulst, 1964).

The gain may be smaller or larger than 1. For instance, if Ρ is the sun and Q the head of a skier, the amount of sunburn is proport ional to G. Snow on the ground raises G considerably above 1, but snow flurries in the air may bring it far below 1.

The term gain was inspired by a similar concept that has proved useful through many decades in antenna theory. The name is neutral in that it does not refer specifically to one or the other of the two reciprocal situations. The gain of an antenna used in emitting radiation is the same as the gain of that antenna used with a receiver.

The definition of gain adopted in this book is different from that in antenna theory in that we consider incoherent scattering and take an isotropic source emitting natural (i.e., unpolarized) radiation, rather than a dipole, as reference source. At the receiving point we assume a detector that is isotropic and records intensity regardless of polarization.

The configuration of particles is fully arbitrary. It may be a cloud, or an infinite medium in which the distribution function of particles sizes depends


on position. It may contain solid walls, e.g., a cloud layer into which a snow-covered mountain protrudes. The configuration is assumed to be macroscop-ically stationary. This means that to a normal observer it would exhibit no changes with time. Of course, the changing orientations of the single particles and the continuous variations in their precise relative positions define a new microscopic configuration at any instant. The macroscopic observer measures the average scattering properties of this statistical ensemble of microscopic configurations, which in simple language is the (incoherent) scattering by the configuration.

The points Ρ and Q may be taken in, near, or at some distance away from this configuration. There is no strong restriction on their positions. However, we suppose that both points are in a sufficiently large volume of free space to permit the angular dependence of the specific intensity to have a meaning. This is a mild restriction, which is always implicitly made in radiative transfer theory. We can permit the particles to be closely packed anywhere in the configuration, except in the immediate neighborhood of Ρ or Q.

The experiment by which the gain for the said configuration is measured may now be summarized as follows :

Definition. Place small black spheres in Ρ and Q. These spheres are isotropic radiators of natural radiation. Place a filter around Ρ or Q in order to select a frequency band, if desired. Measure the power transferred from Ρ to Q by emission in Ρ and absorption in Q. Normalize the result by dividing by the power transfer from Ρ to β if all scattering particles are removed. The ratio is called the gain (of this configuration) for the pair of points PQ.

The reciprocity principle states that the gain from Ρ to β equals the gain from Q to P. This follows at once from the principle of detailed balance.

The gain is zero or a positive scalar quantity. It is > 1 if the scattering object does not obstruct the line PQ. In other situations it may be larger or smaller than 1. In circumstances arranged for the purpose, e.g., if the object is a lens or mirror, the gain may be very large.

Extension. To avoid confusion, the simplest and most useful gain definition has been given above. Obviously, it is possible to extend the definition to include arbitrary dependence on polarization and on direction, both for the source and for the detector. We state without proof the reciprocity relation for that situation (see Fig. 3.3 for notation).

Let arbitrary planes of references be chosen for all directions n' through Ρ and for all directions n" through Q. Let Lt(n') be the four-vector specifying the energy, complete with polarization, emitted by a point source at Ρ per unit solid angle into the direction n. This may be defined at frequency v, per unit frequency, but the dependence on ν is not written. Let /,·( — n") complete with polarization, be the specific intensity (energy per unit solid angle per unit area per unit frequency) due to this source, arriving at Q from the direction n", i.e., going into the direction — n". Similarly, in an inverse experiment, let a point source

28 3 Reciprocity

η

F i g . 3.3. C o d e s e m p l o y e d in f o r m u l a t i n g r e c i p r o c i t y p r i n c i p l e f o r t w o - p o i n t t r a n s f e r f u n c

t i o n s , i n c l u d i n g a r b i t r a r y p o l a r i z a t i o n .

Lj(n") at Q cause the specific intensity / / ( — n') at P. The general relations must then be the following:

direct experiment :

/.<-»") = Σ ί^(η",η')^(η')^η' k J

reverse experiment:

/ / ( - η ' ) = Σ ÎGik(n',n")Lt(n")dn" k J

where the integrals are over the entire solid angle An. The functions F and G depend on the properties of the configuration, the chosen points Ρ and Q, and the chosen planes of reference through those points, but not on the properties of the radiators or detectors placed there. We may call these functions the two-point transfer functions. The reciprocity principle is

Gi k(n',n") = PiPkFki{n\n)

Proofs of this theorem may be constructed in various ways : (a) We may start from the far-field reciprocity discussed in Section 3.2.2. (b) We may put at Ρ and Q arbitrary test bodies obeying Kirchhoif's law (Section 3.3.1) and then invoke detailed balance, (c) We may use the reciprocity relation derived from a very general equation of transfer by Case (1957), although the mathematical rigor necessitates slightly more restrictive assumptions.

By taking the element 11 we return to the relation

O n = f n

which expresses the equality of gain for the situation in which the source emits natural light and the detector records intensity regardless of polarization.

Still further generalizations may easily be made. The concept of importance, which played a role in reactor theory (Lewins, 1965), is an example of such a very wide generalization.


3.3.3 Point-Direction Gain

The following application of the .gain concept of Section 3.3.2 deserves separate mention. One of the two points, say β , is placed very far from the scattering object. This means either the body should be finite and the distance to β should be much larger than its dimensions, or the body may be infinite, like a plane-parallel atmosphere, and the distance to β should be much larger than the part of the body which effectively contributes to the scattering process. In either case only the direction in which β is seen is important . The gain then becomes the point-direct ion gain (Fig. 3.4).

Definition. The point-direction gain of a configuration of scatterers for a point Ρ and a direction η equals the two-point gain between the point Ρ and any sufficiently distant point β in the direction η from P. This definition requires little comment. It is easily seen that the distance drops out if r becomes sufficiently large.

It may also be useful to define the point-direct ion gain directly, without reference to the concept of two-point gain. This is done as follows (the words direct and reverse signify no priority):

Direct experiment. An isotropic source of natural light is placed at Ρ near or in a cloud of particles which satisfies the description given earlier. The radiation emitted by the point source and cloud together into the direction η per unit solid angle is G times the radiation of the bare point source per unit solid angle.

Reverse experiment. The radiation density observed at Ρ if a parallel beam of natural light traveling in the direction — η falls on the cloud is G times the radiation density in the original beam.

F i g . 3.4. P i c t o r i a l d e f i n i t i o n o f p o i n t - d i r e c t i o n g a i n .

30 3 Reciprocity

3.4 PLANE SURFACES AND PLANE-PARALLEL SLABS

The most common applications of the reciprocity principle refer to plane surfaces or plane layers which have macroscopic properties independent of the coordinates χ and y parallel to the surface. Light incident on the top surface may emerge from the top (reflection) or from the bot tom (transmission). Most often this is diffuse reflection or diffuse transmission, but specular reflection and direct (unscattered) transmission are contained within the general formulation by the proper use of δ functions.

In the context of this chapter, the light within the layer does not have to obey a simple transfer equation. Examples included are a cloud layer, desert soil, a corrugated surface, a sheet of frosted or opal glass, a photographic emulsion separate from or combined with glass plate and backing layer, an a tmosphere whose composition varies with height and which has a solid surface below, a semi-infinite atmosphere, etc.

The reciprocity formulas follow directly from those discussed for arbitrary configurations. Although the changes in definition are trivial, it is useful to spell them out.

3.4.1 Reflection and Transmission from an Inhomogeneous Slab

The full definition of the reflection and transmission matrices and the reciprocity relations are collected in Fig. 3.5 and Display 3.2. There are four matrices (JR, R*, Τ, T*), each consisting of 16 elements, each of which is a function of 2 directions, i.e., 4 angles. These matrices are implicitly defined by the equations for emerging intensity, where we have tried to gain clarity by labeling u and φ in accordance with Fig. 3.5. The matrices obey three reciprocity relations.

F i g . 3.5. C o d e s u s e d i n d e f i n i n g r e c i p r o c i t y r e l a t i o n s f o r a n i n h o m o g e n e o u s s l a b .

3.4 Plane Surfaces and Plane-Parallel Slabs 31

One way of obtaining the proof of these relations is to apply the finite-particle relations (Display 3.1) to a disk with a large but finite area A cut from this slab. For instance, the substitutions for reflection from the top are

Si = I\1Kul9<p1)du1d<pu I\

4) = 0

Lt = -u2AI\2)(u2,(p2)

Rik = -(n/uiU2A)Fik In later chapters we shall simplify the notat ion by writing μ for | u \. All integrals then cover the positive interval (0,1) in μ, and the arguments in the reciprocity relations change order but not sign. Further, we shall often be content with discussing only radiation fields which are independent of φ (for more detail, see Section 15.1.3).

D I S P L A Y 3 . 2

R e c i p r o c i t y R u l e s i n R e f l e c t i o n a n d T r a n s m i s s i o n b y a n I n h o m o g e n e o u s S l a b

N o t a t i o n :

D i r e c t i o n g i v e n b y <p, Θ{μ = c o s Θ)

S t o k e s p a r a m e t e r s l a b e l e d b y s u b s c r i p t s i, k(= 1, 2 , 3 , 4 )

S i g n f a c t o r s px = p2 = p 4 = 1, p3 = — 1

I n t e n s i t i e s ( i n c l u d i n g p o l a r i z a t i o n ) l a b e l e d b y s u p e r s c r i p t s ( s e e F i g . 3.5)

ί / ί

1 }( ι ι , φ) u > 0

o n t o p < . . . U I

2 )( M , φ) u < 0

ί / < · > , φ) u > 0 a t b o t t o m < W\u, φ) u < 0

E q u a t i o n s for e m e r g i n g i n t e n s i t y :

J '

2n d<px f

1

—- Rik(u2, φ 2; u u (p^I^Xui, (pl)2ui dul „ ο

2π Jo

Γ

2π (Ιφ4 f °

k J ο ζπ J _ !

//3)("3 ,Ψ3) = Σ\ -^Γ \

R*k("3 ><Ρ3 ; " 4 , φΜ*ΧνΑ, φΑ)( - 2 « 4) duA

+ Σ F ~^Γ~ Ί

Tik(

u3> <Ρ3; " ι , <Pi)Ik\uu ^ ι ) 2 « ι dul k ·>ο ^ Jo

R e c i p r o c i t y r u l e s :

Ki f c(u2,<J I>2; M j , ^ ) = PiPkRkii-u^n + φχ \ - u 2, n + <p2) Rfk(u3, <p3; m 4, φΑ) = PiPkRîi{-uA, π + φ 4; - u 3, π + <ρ3) Tfk(u2,(p2;u4,(p4) = PiPkTki(-u4,n + <ρ4; - Μ 2, π + <ρ2)

3 Reciprocity

3.4.2 Kirchhoff's Law for a Surface

If a unit area of a slab exposed to incident specific intensity with arbitrary polarization, / |

υ( ν , u, φ\ where u > 0, absorbs per unit time the energy

then the slab at uniform temperature Τ will emit in the opposite direction ( —w, π + φ) the specific intensity

where B(v, T) is the Planck intensity. The proof of this statement can be derived from Section 3.3.1 by simple substitution, applying the law for a finite body to a large disk cut from the slab.

3.4.3 Point-Direction Gain for an Inhomogeneous Slab

By specifying that the body is a plane-parallel slab (or atmosphere, or cloud layer) we obtain from Section 3.3.3 the following definitions in terms of fictitious experiments :

Direct experiment 1. A source at depth ζ below the top surface isotropically emitting unpolarized radiation with energy flow L per unit solid angle creates a blob of multiply scattered diffuse radiation, such that source plus blob of diffuse light are seen at great distances as one source emitting, per unit solid angle in the direction ( — μ, φ\ the radiation (which may be partially polarized)

Direct experiment 2. Layers of isotropic sources of unpolarized radiation embedded in the slab in such a manner that the energy flow emitted from a unit volume into a unit solid angle is p(z), give rise to (possibly polarized) radiation emerging from the top surface, with the specific intensity ( — μ = u < 0)

where the integral is over the entire depth of the layer.

Reverse experiment. Unpolarized radiation with specific intensity Ι(μ0, φ0) with μ 0 > 0 incident on the top surface creates at depth ζ the radiation density (c = velocity of light)

/ S

2 )( v , - i i , π + φ) = PiAt{v9 u, φ)Β(γ9 Τ)

Lf( - ^ , φ) = LG(z,\i)

References 33

The reciprocity principle states that the function G(z, μ) occurring in each of these definitions is the same. Verification of this statement is left to the reader.

It is interesting to note that the point-direct ion gain discussed here is obtained from the general two-point transfer functions (Section 3.3.2) by narrowing the assumptions down in three ways:

(1) F rom general configuration to plane-parallel slab, (2) F rom two-point transfer function to point-direct ion transfer functions, (3) F rom transfer functions, including anisotropy and arbitrary polariza

tion, to gain.

Each of these specializations can be made independently; the various combinations define the eight corners of a cube. One of these was discussed in Section 3.3.3. There are five more corners to be discussed, but this would be uninteresting.

R E F E R E N C E S

B o r n , M . , a n d W o l f , E . ( 1 9 5 9 ) . " P r i n c i p l e s o f O p t i c s . " P e r g a m o n , O x f o r d .

C a s e , Κ . M . ( 1 9 5 7 ) . Rev. Mod. Phys. 2 9 , 6 5 1 .

D e G r o o t , S . R . ( 1 9 6 3 ) . / . Math. Phys. 4 , 1 4 7 .

D e G r o o t , S . R . , a n d M a z u r , P . ( 1 9 6 2 ) . " N o n - E q u i l i b r i u m T h e r m o d y n a m i c s . " N o r t h - H o l l a n d

P u b l . , A m s t e r d a m .

E d d i n g t o n , A . ( 1 9 2 6 ) . " T h e I n t e r n a l C o n s t i t u t i o n o f t h e S t a r s . " C a m b r i d g e U n i v . P r e s s , L o n d o n

a n d N e w Y o r k .

H o v e n i e r , J . W . ( 1 9 6 9 ) . J. Atmos. Sci. 2 8 , 120 .

L a n d a u , L . D . , a n d L i f s h i t z , Ε . M . ( 1 9 5 7 ) . " Q u a n t u m M e c h a n i c s , N o n - r e l a t i v i s t i c T h e o r y . "

P e r g a m o n , O x f o r d .

L e w i n s , J . ( 1 9 6 5 ) . " I m p o r t a n c e , t h e A d j o i n t F u n c t i o n . " P e r g a m o n , O x f o r d .

M i n n a e r t , M . ( 1 9 4 1 ) . Astrophys. J. 9 3 , 4 0 3 .

R o s s e l a n d , S . ( 1 9 3 6 ) . " T h e o r e t i c a l A s t r o p h y s i c s . " O x f o r d U n i v . P r e s s ( C l a r e n d o n ) , L o n d o n a n d

N e w Y o r k .

S a x o n , D . S . ( 1 9 5 5 ) . Phys. Rev. 1 0 0 , 1 7 7 1 .

S o b o l e v , V . V . ( 1 9 7 3 ) . Astrofizika 9 , 5 1 5 [English transi. : 9 , 3 1 3 ( 1 9 7 5 ) ] .

v a n d e H u l s t , H . C . ( 1 9 5 7 ) . " L i g h t S c a t t e r i n g b y S m a l l P a r t i c l e s . " W i l e y , N e w Y o r k ; a l s o D o v e r ,

N e w Y o r k ( 1 9 8 1 ) .

v a n d e H u l s t , H . C . ( 1 9 6 4 ) . Bull. Astron. Inst. Netherlands 17 , 4 9 9 .

v a n d e H u l s t , H . C . ( 1 9 6 5 ) . Bull. Astron. Inst. Netherlands 1 8 , 1.

W i g n e r , E . P . ( 1 9 5 4 ) . J. Chem. Phys. 2 2 , 1 9 1 2 .

4 • Methods

4.1 POSING THE PROBLEM

4.1.1 Planetary Observations; Direct and Inverse Problems

The emphasis in this book is on results applicable in practical situations. In reviewing methods, we therefore begin with a concrete example: A planetary astronomer has obtained certain photometric, polarimetric, or spectroscopic data. The observed radiation may have reached his instruments along many paths, including multiple scattering in the planetary clouds. He seeks a sound interpretation of these data in terms of the physical properties of the constituents and the structure of the planetary atmosphere.

Often the only feasible approach is to make model computations based on assumed properties. Suppose, therefore, that we have, from independent measurements, from theory, or from guesswork, the structure of the atmosphere, i.e., a table showing as a function of height the number densities and composition of all constituents which may scatter or absorb light. Suppose further, that we know (or can estimate) how the bot tom surface below this atmosphere reflects radiation. The steps then necessary to find the reflecting properties of the planet are shown in Display 4.1.

One of these steps, the black arrow labeled "multiple scattering theory," forms the main topic in this book. Various methods for performing this step are compared in this chapter.

Preparatory to this main problem is the choice of the albedo and phase function of a volume of air at each height. Although the two steps leading to this

3 4

4.1 Posing the Problem 35

D I S P L A Y 4 . 1

S t e p s in C o m p u t i n g Di f fuse R e f l e c t i o n b y a P l a n e t w i t h A t m o s p h e r e

S i ze a n d m a t e r i a l

o f a e r o s o l p a r t i c l e s

t h e o r y o r e x p e r i m e n t

E x t i n c t i o n a n d s c a t t e r i n g

f u n c t i o n of p a r t i c l e s

M o l e c u l a r c o n c e n t r a t i o n

R a y l e i g h s c a t t e r i n g

E x t i n c t i o n a n d s c a t t e r i n g

b y m o l e c u l e s

i n t e g r a t i o n o v e r s i zes

E x t i n c t i o n coe f f i c i en t , a l b e d o , a n d p h a s e f u n c t i o n of a v o l u m e of t u r b i d a i r a t e a c h h e i g h t

m u l t i p l e s c a t t e r i n g t h e o r y

Di f fuse r e f l e c t i o n a n d

t r a n s m i s s i o n b y a t m o s p h e r e

φ, μ 0, φ0), Τ(μ, φ, μ0, φ0)

A l b e d o a n d r e f l e c t i o n

l a w of g r o u n d s u r f a c e

s i m p l e i n t e g r a t i o n

L o c a l r e f l e c t i o n b y p l a n e t

R e f l e c t i o n b y e n t i r e p l a n e t

i n t e g r a t i o n o v e r d i s k

point are by no means trivial, we skip a detailed discussion. Adequate reviews are available (van de Hulst, 1957; Deirmendjian, 1968; Kerker, 1969) and a digest of often needed data is given in Chapter 10. It is often prudent to characterize the local properties by a few parameters only. Foremost among these are the albedo a and the asymmetry factor g, which is the average value of cos α over the phase function. In certain asymptotic problems dealing with thick layers, these parameters even suffice to determine the solution. More generally, however, the phase function has to be fully specified before the problem can be solved precisely.

How the bot tom surface can be taken into account is explained in Sections 4.5.4 and 18.4.1. The final integration over the planetary disk is an obvious necessity for the interpretation of any data obtained by means of a telescope in which this disk is observed in toto, or is unresolved. See details in Section 18.1.

36 4 Methods

In principle, the computat ion sketched in Display 4.1 has to be repeated for each separate wavelength interval in which the absorption and or scattering is different. In some situations an earlier integration, leading to the equivalent width of an absorption line, is possible (cf. Section 17.3).

Note that the quantitative interpretation of all photometric, polarimetric, and spectroscopic data on planets with atmospheres depends on calculations of this type. By varying the assumptions and thus performing many model computations, one can assess the uncertainty with which each physical parameter can be inferred from the observations. This is an adequate but often quite lengthy process.

The inverse problems originate from Display 4.1 by inverting the directions of the arrows. They are of great practical importance, and much attention has been paid to these problems in connection with remote sensing. Roughly, one may state that it is advisable to set up formal solutions of these inverse problems only if data of good accuracy are available in such quantity that an automatic reduction is imperative. In all other cases, the best way to get a feel for the latitude of interpretation is to try out several models or assumptions independently. References to some inversion methods are given in Sections 19.2.3 and 19.2.4.

4.1.2 Reflection, Transmission, Gain

The formulation chosen in Display 4.1 obviously contains a number of tacit assumptions. In order to make the problem accessible to mathematical formulation, we shall now make these assumptions explicit:

1. Each particle scatters independently of all others. 2. Properties of a unit volume of air are independent of its orientation

(e.g., there are no floating ice plates). 3. Atmospheres are plane-parallel (horizontal) with z-direction along the

vertical; local properties independent of χ and y. 4. Processes at different wavelengths are independent. 5. There is steady state, i.e., independence of time. 6. There is no reflecting bot tom surface (it will be added in Section 4.5.4). 7. There is no polarization (it will be added in Chapter 15). 8. Radiation is incident from any direction on top and/or bot tom in in

finitely wide homogeneous beams, so that the intensity does not depend on χ and y.

9. Embedded sources of isotropic radiation, providing a zero-order source function S0, are permitted, but must again be independent of χ and y.

With these assumptions we first find by means of the local extinction coefficient fcext(z) the optical depth τ, measured from the top of the a tmosphere:

4.1 Posing the Problem 37

which from now on replaces ζ as the vertical coordinate. The total optical depth of the atmosphere will be written as τ(0) = b.

Radiation emerging from the top of the atmosphere can originate only from radiation originally incident at the top, at the bottom, or from embedded source layers. Since these contributions add linearly, it must be possible to write the combined intensity emerging from the top in the form

Jom,top(M,<P) = - ί μ'άμ' ί Κ(μ, φ; μ', φ ' ) / ί η,1 ο ρ(μ ' , φ') άφ' π Jo Jo

+ - \ μ'άμ' \ Τ*(μ, φ ; μ', φ')Ιίη b o t t o m( ^ , φ') άφ' π Jo Jo

ι rb

+ - G ( T , μ)50(τ ) άτ μ Jo

By this equation the proportionality functions R, T*, and G are implicitly defined. We can easily verify that R is the reflection function of the entire atmosphere normalized in such way that the reflection function is 1 for a standard white surface following Lambert 's law. Τ is the similarly normalized transmission function; it includes the direct (unscattered, zero-order) transmission. Further, G is the point-direct ion gain, which is 1 in the absence of the atmosphere (cf. Section 3.3.3). The intensity emerging at the bot tom of the atmosphere follows from the same equation, with R replaced by Γ , T* by R*, and G by G * . The distinction by means of asterisks is necessary because we have not made the assumption that the vertical structure of the atmosphere is symmetric, i.e., that the atmosphere " looks the same " from above and from below. In most numerical examples in later chapters this extra condition is fulfilled and no asterisks are necessary.

The main purpose of this book is to provide formulas, tables, and graphs for the three functions R, T, and G under a wide variety of specifications.

The point-direct ion gain for a point outside the atmosphere can be expressed in terms of the transmission or reflection function by the general relations

G ( 0 , A I ) = 1 +^ ί άμ' ί R(μ,φ;μ'9φ')άφ κ Jo Jo

G(b, μ) = - Γ άμ' Γ Τ(μ, φ; μ', φ') άφ π Jo Jo

The derivation of the first of these equations proceeds by imagining a transparent plane layer of isotropic sources above the atmosphere. This layer by itself radiates the intensity c/μ up and c/μ down (where c = constant). The downward radiation hits the atmosphere and gives the reflected intensity which should be added to the c/μ going up. Division by c/μ then gives the gain for a point above the atmosphere, as stated. For isotropic scattering this gain is traditionally represented by X. The derivation of G(b, μ), for isotropic scattering

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4.2 Criteria for a Choice 39

called Y, is similar. The term e~bl* arising from the direct (zero-order) trans

mission does not appear separately in this equation because Τ is defined to include the zero-order part (containing a delta function), which upon integration gives this term. These equations contain the assumption that, for symmetry reasons, the dependence on φ' drops out with the averaging over φ.

4.1.3 The Standard Problem and Its Variations

Within the limitations posed by assumptions 1-9 above, a wide variety of problems remains. Display 4.2 gives a sample of the many specifications which can be chosen.

Lines A and Β are the most general. Both allow the possibility of a vertical structure of the atmosphere with arbitrary variations of sources, phase function, etc., with optical depth τ. The case with polarization (line A) is the more general of the two. Whenever phase relationships between the waves scattered by different particles are washed out (this is certainly true in practice) the four Stokes parameters suffice to specify intensity and state of polarization of a beam of light. Thus, every intensity on line A must be read as a four-vector.

Line C briefly indicates six intermediate possibilities. The combined assumptions in line D are said to define the standard problem. The main simplification is that one phase function and albedo characterize all volume elements. The radiation fields obtained from light incident from the top or from embedded internal sources can be simply added. It is often advisable to discuss these assumptions as defining separate problems. In most tables presented in this book, the additional assumption of azimuth independence (line E) is made. For isotropic scattering, this Imposes no restriction.

Finally, line F marks a number of further specifications that can be made in each of the columns, separately or combined. If a problem has been solved for unidirectional incidence, a simple integration over μ 0 suffices to obtain the corresponding result for any distribution of incident radiation Ι(μ0). Two types of intensity distribution occur so often that we have introduced separate symbols Ν and U to distinguish them. The same symbols will be used for the corresponding integrals over μ, i.e., for the moments of the emergent radiation. See Section 5.1 for further explanation. We have decided in this book whenever possible to tabulate with a function of μ and μ 0 also the corresponding moments defined by the operators Ν and U.

4.2 CRITERIA FOR A CHOICE

4.2.1 Advantages of Different Methods

There is a significant difference between method of derivation, whereby an equation, or set of formulas, is derived from some basic equation or from

40 4 Methods

physical principles, and method of computation, whereby this set of formulas is the starting point for constructing tables and graphs.

Published reviews of methods often fail to distinguish between the two. Another reason that no two reviews are alike is the fact that distinctions judged important by one author (e.g., in physical clarity or mathematical rigor) may be considered trivial by another. A third reason is that it is difficult for anyone to be familiar with the very extensive literature.

Besides the subjective review given below, we recommend the following general reviews of methods: Hunt (1971), Hansen and Travis (1974), Irvine (1975), and an extensive review of numerical and analytical methods published under auspicies of the I.A.M.A.P. (International Association of Meteorology and Atmospheric Physics) under the editorship of Lenoble (1977).

I have decided not to include in this book a review that would give proper credit to all publications and to the many parallel lines of development. But some remarks are necessary because the treatment in the next chapters is in various places unusual, the emphasis being placed on the nonmathematical parts of the method. This is illustrated by the schematic diagram (Figure 4.1) and by the following conversation.

" How did you t ravel?" " I took the train." The respondent forgets to mention that he took his car from his house to the train station at A and had to search for a parking place. Nor does he mention that the walk to his friend's house from the train station at Β took longer than expected and that he once lost his way.

The usual description of the method by which a problem in mathematical physics has been solved is a replica of this situation. Mathematics is the train: it has a firmly laid track from a certain mathematical equation (station A)

F i g . 4 . 1 . I n a d e r i v a t i o n in m a t h e m a t i c a l p h y s i c s a c c e s s f r o m p h y s i c a l c o n c e p t s t o i n t e r m e d i a t e s t o p s , if d o n e c a r e f u l l y , is o f t en a d v i s a b l e . L i k e w i s e , i n t e r p r e t a t i o n of i n t e r m e d i a t e n u m e r i c a l r e s u l t s m a y b e a s i m p o r t a n t a s i n t e r p r e t a t i o n of final n u m b e r s .


to its mathematical solution (station B). The added trips to and from the stations are inconvenient and take time and effort but are not judged worth mentioning; yet they are a necessity. It is first necessary to decide that a certain equation fits the physical situation; and the final formula or procedure has to be numerically evaluated before the results can be used in physical interpretation.

Figure 4.1 shows this common route by the heavy arrows. In my discussion I shall try to stay longer with the physics and proceed sooner to the numerics. This corresponds schematically to the thin lines in the diagram. This is not a fundamentally different approach; it is more a difference of emphasis. But there are some practical consequences.

For instance, it simply is not true that one can board the train of radiative transfer only at the station called equation of transfer. There are many other stops. Chapter 5 will illustrate how one can go a long way by using almost exclusively definitions and equations obtained from physical thought experiments. Nor is it correct to describe such methods as "merely heuristic," implying that boarding the train at a different station is a subversive act requiring additional verification. Getting off at intermediate stops, i.e., evaluating intermediate results numerically for physical interpretation, also deserves much more attention than it is given in most texts.

After this digression we return to the methods. A list based on subjective criteria is given in Display 4.3. Since multiple access to intermediate results is possible anyhow, I have not separated the methods by mode of access. Consequently, words like "probabilistic," "physical," or "mathemat ica l" are not judged important in distinguishing a method. The actual classification criterion is what intermediate stations are visited. In the descriptive column I have given priority to the physical description (if any) but also added the more common mathematical approaches leading to the same (or very similar) results.

The sections in which a brief discussion of these methods and some references are found, are shown in the second column of Display 4.3. Here a few words may be said about the last two sets, which will not be discussed separately.

The discretization methods range from very simple to quite powerful and many of these have proved "op t ima l " in accuracy and economy for certain practical problems. An adequate review of their relative merits, in ease and accuracy, would necessarily be very detailed. I have left this task aside, having generally omitted a discussion of approximate methods and results. More about the two-stream approximation is found in Sections 20.2.1 and 20.2.2. A systematic discussion of the opt imum number of streams to be used in dis-cretizing the direction is given by Whitney (1974).

Hybrid methods may also be surprisingly productive sometimes, although they are hard to classify. For instance, applying an approximate method but singling out a delta function as forward scattering peak often works well. In this class are the delta-Eddington method described and tested independently by Joseph et al. (1976), Wiscombe and Joseph (1977), and by Meador and Weaver (1976) and the quasi-single-scattering approximation of Gordon et al. (references in Sections 20.3.2).

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Ad

van

tag

es (

A)

and

dra

wb

ack

s (D

)

1.

Met

ho

d of

4.

3 P

roce

ed

in s

teps

fro

m

ρ —

1 t

imes

sc

atte

red

succ

essi

ve

rad

iati

on

to ρ

tim

es s

catt

ered

ra

dia

tio

n an

d

ord

ers

sum

th

e se

ries

; th

e re

sult

is

a p

ow

er

seri

es

in

the

sing

le s

catt

erin

g al

bed

o a

Mat

hem

atic

al

equ

ival

ent

: so

lve

inte

gral

equ

atio

n by

Ne

um

an

n se

ries

S

ame

resu

lts

ma

y al

so b

e o

bta

ined

th

rou

gh

add

ing

met

ho

d (e

.g.,

Han

sen

and

Tra

vis

, 19

74)

2.

Am

bu

rtsu

mia

n's

4.

4, 6

.3

Bui

ld u

p at

mo

sph

ere

by a

dd

ing

in e

ach

step

a

met

ho

d ve

ry t

hin

laye

r w

ith

give

n sc

atte

rin

g

pro

per

ties

Mat

hem

atic

al

equ

ival

ent

: in

var

ian

t

emb

edd

ing

So

urc

e fu

ncti

on

and

inte

nsit

y

of ρ

tim

es

scat

tere

d

rad

iati

on

at a

ny

dep

th

Ref

lect

ion

and

tran

smis

sio

n

func

tion

s fo

r la

yers

of

any

thic

kn

ess

from

ze

ro t

o fi

nal

thic

kn

ess

b

A

Sim

ple

co

nce

pt

and

easy

pro

gra

mm

ing

Res

ult

is i

n co

nv

enie

nt

form

for

ob

tain

ing

ph

oto

n p

ath

stat

isti

cs (

Sec

tion

17

.1.2

)

D

Pro

hib

itiv

ely

slow

con

ver

gen

ce

for

thic

k la

yers

(b

> 5,

or

so)

if sc

atte

rin

g is

con

serv

ativ

e o

r n

earl

y so

A

Sim

ple

co

nce

pt

and

easy

pro

gra

mm

ing

Rem

ain

s si

mp

le i

f ap

pli

ed

to

inh

om

og

eneo

us

atm

osp

her

e

Can

se

rve,

to

get

her

w

ith

ho

mo

gen

eity

as

sum

pti

on

and

inv

aria

nce

pr

inci

ple,

to

deri

ve

equ

atio

ns

of

met

ho

d

4(c)

D

Slo

w p

roce

ss t

o re

ach

thic

k

laye

rs

42

3.

Ad

din

g or

d

ou

bli

ng

met

ho

d

4.5

Pil

e tw

o la

yers

wit

h k

no

wn

refl

ecti

on

and

tran

smis

sio

n fu

ncti

ons

on

top

of

each

oth

er

and

sum

th

e in

fini

te

seri

es

wh

ich

desc

ribe

s th

e p

lay

bac

k an

d fo

rth

bet

wee

n

the

laye

rs

If t

he

laye

rs a

re h

om

og

eneo

us

and

iden

tica

l

this

is

call

ed t

he d

ou

bli

ng

met

ho

d

Sam

e pr

inci

ple

in s

yst

emat

ic

mat

hem

atic

al

form

: m

atri

x tr

ansf

er

met

ho

d

Red

uct

ion

to X

6.

4,6.

5 S

ame

or

very

sim

ilar

set

s of

eq

uat

ion

s ca

n be

and

Y re

ach

ed

alo

ng

con

cep

tual

ly

very

di

ffer

ent

func

tion

s ro

ute

s.

(a)

Me

tho

d of

sin

gula

r ei

genv

alue

s (V

an

Ka

mp

en

-Ca

se)

: fin

d al

l m

od

es

of

pro

pag

atio

n in

un

bo

un

de

d m

ediu

m,

incl

ud

ing

tho

se w

ith

disc

rete

ei

genv

alue

s

and

tho

se w

ith

a co

nti

nu

um

of

ei

genv

alue

s

(co

rres

po

nd

ing

to s

ingu

lar

eige

nfun

ctio

ns);

mat

ch

bo

un

dar

y co

nd

itio

ns

by

usi

ng

half

-ran

ge

ort

ho

go

nal

ity

theo

rem

s

(b)

Me

tho

d of

dis

cret

e o

rdin

ates

(Ch

and

rase

kh

ar)

: re

plac

e al

l in

tegr

als

ov

er

dir

ecti

on

by

finite

su

ms

wit

h G

auss

ian

wei

gh

ts ;

fol

low

p

lan

of m

eth

od

(a)

;

bec

om

es

met

ho

d (a

) in

th

e li

mit

of

infi

nite

ly

fine

divi

sion

.

In t

he d

ou

bli

ng

met

ho

d

refl

ecti

on

and

tran

smis

sio

n

for

thic

kn

ess

b0,

2b0,

4b0,

etc.

Als

o at

eac

h st

ep:

mid

lay

er

inte

nsit

y d

istr

ibu

tio

n

Dis

cret

e ei

genv

alue

s,

incl

ud

ing

the

on

e co

rres

po

nd

ing

to

leas

t d

am

pe

d m

od

e =

diff

usio

n so

luti

on

Inte

nsit

y p

atte

rn

corr

esp

on

din

g

to t

hese

m

od

es

Ch

arac

teri

stic

fu

ncti

on

X an

d Y

func

tion

s ) fa

S

ob

ole

v p

oly

no

mia

lsj

H

func

tion

s

>fin

ite

b

Bu

sbri

dg

e p

oly

no

mia

ls

b =

oo

A

Co

nv

enie

nt

pro

gra

mm

ing

Fas

t co

nv

erg

ence

in

an

y st

ep

Fas

t w

ay t

o ge

t fr

om

v

ery

thin

to v

ery

thic

k la

yer

s

Pro

vid

es

star

tin

g p

oin

t fo

r

asy

mp

toti

c fit

ting

(Sec

tio

n

5.6)

D

In c

om

pu

ter

pro

gra

m

dire

ct

tran

smis

sio

n h

as t

o b

e

wri

tten

as

sep

arat

e te

rm

No

t su

itab

le f

or

oth

er

geo

met

ries

A

Pri

nci

ple

of

me

tho

d fa

mil

iar

to a

ny

theo

reti

cal

phys

icis

t

Giv

es i

nten

sity

dis

trib

uti

on

at

any

dep

th

Fas

t co

mp

uta

tio

n if

p

has

e

func

tion

is

iso

tro

pic

or

can

be

exp

ress

ed

in v

ery

few

Leg

end

re

po

lyn

om

ials

Det

aile

d so

luti

on

s of

si

mp

le

exam

ple

s in

li

tera

ture

Th

e di

scre

te f

orm

4(

b)

wit

h

thre

e o

r fo

ur

Gau

ssia

n

divi

sion

s su

ffic

es

for

1 %

accu

racy

D

Fin

al

set

of e

qu

atio

ns

for

gen

eral

ph

ase

fun

ctio

ns

is

mes

sy

4.

43

DIS

PL

AY

4.

3 (c

ontin

ued)

So

me

Met

ho

ds

for

Sol

ving

the

Sta

nd

ard

Pro

ble

m

Co

mm

en

ts

Des

crip

tio

n

Bri

ef

nam

e in

S

ecti

on

(wit

h pr

iori

ty g

iven

to

phy

sica

l d

escr

ipti

on

) In

term

edia

te

pro

du

cts

Ad

van

tag

es (

A)

and

dra

wb

ack

s (D

)

4.

Red

uct

ion

to X

an

d Y

func

tion

s)

(c)

Tra

dit

ion

al

met

ho

d (C

han

dra

sek

har

,

Bus

brid

ge,

So

bo

lev

):

intr

od

uce

Am

bar

tsu

my

an

func

tion

s d

escr

ibin

g

com

bin

ed

rad

iati

on

pat

tern

of

po

int

sou

rce

and

slab

; o

bse

rve

that

th

ey c

an

be

fact

oriz

ed

in X

an

d Y

func

tion

s an

d

po

lyn

om

ials

Ma

ny

inte

rmed

iate

re

sult

s

hav

e n

o o

bv

iou

s ph

ysic

al

mea

nin

g

No

t su

itab

le

for

inh

om

og

eneo

us

laye

rs

5.

Var

iou

s 4.

2.1

Un

der

th

is h

ead

ing

we

lum

p to

get

her

a

vari

ety

No

usef

ul

inte

rmed

iate

M

See

cit

ed p

aper

s an

d ge

nera

l

disc

reti

zati

on

of p

ower

ful

met

ho

ds

that

h

ave

in

co

mm

on

pro

du

cts

Di " re

view

s

met

ho

ds

that

dis

cret

izat

ion

in μ

an

d/o

r τ

red

uce

s th

e

inte

gral

eq

uat

ion

s to

mat

rix

equ

atio

ns

wh

ich

can

then

be

sol

ved

by

form

al

inve

rsio

n or

by

iter

atio

n

Ex

amp

les:

He

rma

n an

d B

row

nin

g (1

965)

;

Dav

e an

d G

az

da

g (1

970)

; E

sch

elb

ach

(197

1);

Dlu

gac

h an

d Y

ano

vit

skii

(1

974)

; se

e al

so

gene

ral

revi

ews

cite

d ab

ov

e

A s

yste

mat

ic d

iscu

ssio

n of

the

o

pti

mu

m

nu

mb

er

of "

stre

am

s" t

o be

use

d in

disc

reti

zing

th

e d

irec

tio

n is

giv

en

by

Wh

itn

ey

(197

4)

6.

Mo

nte

C

arlo

4.

2.1

Def

ine

entr

ance

or

pla

ce o

f or

igin

of

a

ligh

t S

tati

stic

al

dat

a o

n an

y A

S

imp

le

con

cep

t

met

ho

d q

ua

ntu

m

by a

ra

nd

om

n

um

be

r g

am

e;

defi

ne

plac

e of

ab

sorp

tio

n o

r pl

ace

and

dire

ctio

n of

scat

teri

ng

sim

ilar

ly

pro

per

ty

of t

he l

ife

his

tory

of

the

qu

anta

(p

rov

ided

com

pu

ter

ou

tpu

t of

th

ese

Ca

n be

pro

gra

mm

ed

for

any

typ

e of

geo

met

ry

of

the

scat

teri

ng

vo

lum

e

Pro

ceed

un

til

qu

an

tum

is

lo

st

dat

a is

p

rog

ram

med

) D

V

ery

exp

ensi

ve

in

com

pu

ter

Rep

eat

~ 10

0,00

0 ti

mes

ti

me

if h

igh

accu

racy

is

de

sire

d

44


Finally, the Monte Carlo method has a conceptual simplicity that speaks for itself. A broader discussion would be warranted only if we should wish to go into details of techniques and cost. Many refinements in this method have been made. A good reference is Carter et al. (1978).

Variational methods are not mentioned at all in this survey. A thorough review of variational methods applied to transfer problems is found in Lewins (1965).

4.2.2 Our Preferred Method

Clearly, the choice of method in any particular situation should be based on expedience, taking into account the required accuracy, repetition rate and, generally, needs and means. Over the last few decades some new elements have influenced our choice of methods :

(a) At present there is less reluctance to use a long numerical computat ion, provided it can be programmed easily on an electronic computer.

(b) Modern users have a stronger wish to have a product ready for use, thus obviating even simple conversions, multiplications, or substitutions to be done by hand.

(c) A growing interest in more complicated problems is seen, including those involving anisotropic scattering and inhomogeneous atmospheres.

After experimenting with several methods, we found the following most convenient for producing the tables in this book. It is composed of various elements:

1. For thin atmospheres, say b < 2 ~9, or even b < 1, it is simple to use

successive-order scattering. The eigenvalues are low, and hence the convergence is fast. The rate of convergence for anisotropic scattering (Fig. 13.10) is not far from the rate for isotropic scattering.

2. For thicker atmospheres, e.g., 1 < b < 32, we proceed by repeated application of the doubling method. In the simplest case of unpolarized radiation, independent of azimuth, the computing scheme does not have to be changed if we take anisotropic instead of isotropic scattering, but a finer integration mesh usually is necessary.

3. For still thicker atmospheres, the asymptotic results for very thick layers are valid, and we use the asymptotic fitting method (Section 5.6). This method is based on the known general forms of the asymptotic equations. The necessary constants and functions are found by an algorithm fitting the results from the doubling method to these forms.

The method just described owes its simplicity to the fact that an expansion in Legendre polynomials is avoided and that only azimuth-independent, unpolarized radiation is considered. These restrictions are not essential. In

46 4 Methods

2- I O 2 "

5 I 2 4 8 1 6 3 2 2

10 B

F i g . 4 . 2 . T h e d o m a i n of v a l i d i t y of d i f f e ren t a p p r o x i m a t i o n s is s h o w n for t h e f u n c t i o n UR \ in

i s o t r o p i c s c a t t e r i n g . Lef t s i d e , l i g h t s h a d i n g : s i n g l e s c a t t e r i n g suf f ices ; h e a v y s h a d i n g : s i n g l e p l u s

d o u b l e s c a t t e r i n g suff ices. R i g h t s i d e , l i g h t s h a d i n g : t h e v a l u e fo r b = oo su f f i ces ; h e a v y s h a d i n g :

t h e d i f f u s i o n a p p r o x i m a t i o n suff ices . D r a w n l i m i t s c o r r e s p o n d t o a 1 % d e v i a t i o n f r o m t h e c o r r e c t

v a l u e . D a s h e d c u r v e s s h o w w h e r e t h e d e v i a t i o n b e c o m e s 5 % . S c a l e o f b ( o p t i c a l t h i c k n e s s ) l o g a r i t h

m i c , of a ( a l b e d o ) l i n e a r in ^ / ( l - a).

principle, the same results (obtained in vector notation) can also be interpreted as accurate formulas referring to polarized light and to azimuth-dependent terms (Chapter 15). This requires a new definition of the physical meaning of a μ vector and a μμ matrix and new rules for their multiplication. This method has the advantage that most of the tricky "adminis t ra t ion" is left to the machine and that the formulas to be written do not become unwieldy.

A quantitative guide to the ranges of a and b where the limiting formulas suffice is given in Figure 4.2, which shows when a slab is thin enough to do only single scattering or thick enough to be regarded as semi-infinite. Precise error limits at the 1 and 5 % levels are given (refer to the legend for detail). Although this illustration refers only to one function (UR1 is the reflected flux for normal incidence) and to isotropic scattering, it will not change much if other assumptions are made. It is apparent that some 10 doubling steps always suffice to bridge the gap between very thin and very thick layers.

4.3 METHOD OF SUCCESSIVE ORDERS

The concept of this method is very simple: If we know where light originates, we can find where and how it is scattered for the first time, then where and how it is scattered for the second time, etc. Finally, we can sum all these terms and thus

4.3 Method of Successive Orders 47

find the total radiation field. The series must converge, for physical reasons, provided the albedo a for single scattering is < 1. The total energy radiated out cannot surpass the total energy given by the incident light or by embedded sources. Furthermore, the scattering process naturally distributes the radiation smoothly (over depth and direction), so that no local accumulation at any depth or in any direction can cause a divergence.

The steps required to compute the successive orders are shown in Display 4.4. They are based on the very general set of assumptions (inhomogeneous atmospheres, anisotropic scattering) given in line Β of Display 4.2. Note that the optical depth τ is measured from the top down, and that the cosine of angle u is measured with such a sign that u = + 1 for radiation traveling with increasing τ, i.e. down (cf. Section 1.2).

Since azimuth-dependent problems may be treated either by writing explicit functions of φ or by Fourier analysis in φ, both forms are spelled out in Sections C and D of Display 4.4. The Fourier components are completely uncoupled and can be computed separately by successive scattering from the proper starting functions. In the half-steps in each Fourier component, P z( M ) and P?(u) are ordinary and associated Legendre functions. The symmetric function h

m(u, u') appearing here is the redistribution function for the mth Fourier

component. Similarly h0(u, u') is the redistribution function for the azimuth-independent terms. Often we shall omit the index 0.

The following changes in Display 4.4 are necessary if we change the assumptions in various ways.

Problems with polarization (line A of Display 4.2): Add terms involving sin m(q> — φ0) and add the proper indices ί and j (Section 15.1.4).

Light incident only from top (line D of Display 4.2): One of the possible terms of the starter function J 0 is missing.

Homogeneous atmosphere (line D of Display 4.2): In half-steps from /„_ 1 to J we may cross out the τ-dependence of α(τ\ Φ(τ, cos α), and /Γ(τ, w, u'\ m > 0.

Azimuth-independent problem (line Ε of Display 4.2): Retain from equations in Section D of Display 4.4 only the terms with m = 0.

Unidirectional incidence, narrow-layer incidence N, or uniform (Lambert) incidence U (line F of Display 4.2): Specify I0 as given in Display 7.3.

Isotropic scattering (line F of Display 4.2): Put 0(cos a) = 1, h0(u, u') = a.

A few further comments should be made. The half-step from Jn to /„ corresponds to a differential equation, valid at any point within the atmosphere:

u dl„(z, Μ, φ)/δτ = -Ιη(τ, u, φ) + J „ ( T , Μ, φ)

as may be verified by differentiation. Summing this equation over all orders we obtain the equation of transfer

u δΙ(τ, u, φ)/οτ = — / ( Τ , W, φ) + . / ( Τ , u, φ)

48 4 Methods

D I S P L A Y 4.4

S t e p s R e q u i r e d t o C o m p u t e S u c c e s s i v e O r d e r s

A . S t a r t i n g f u n c t i o n s

If l i g h t o r i g i n a t e s f r o m e m b e d d e d s o u r c e s :

J 0( x , Μ, φ)

If l i g h t o r i g i n a t e s f r o m e x t e r n a l s o u r c e s :

I0 = 7 (0 , u, (p)e~

T/u, (u > 0 ) ( l i g h t i n c i d e n t f r o m t o p )

I0 = I(b, u, (p)e~

(b~

T)n~

u) (u < 0 ) ( l i g h t i n c i d e n t f r o m b o t t o m )

B . P r o g r a m

o n e s t e p in /

S u m m i n g o v e r a l l o r d e r s g i v e s

i n t e g r a l e q u a t i o n for /

( i n t e g r a t e d f o r m of t r a n s f e r

e q u a t i o n , o r " f o r m a l s o l u t i o n " )

C . F u l l e q u a t i o n s fo r t h e r a d i a t i o n field, d e p e n d e n t o n φ H a l f - s t e p f r o m / „ _ l t o Jn:

+1

i n t e g r a t i o n

o v e r u, φ i n t e g r a t i o n

o v e r τ i n t e g r a t i o n

o v e r u, φ

o n e s t e p in J

S u m m i n g o v e r a l l o r d e r s g i v e s

i n t e g r a l e q u a t i o n for J

( g e n e r a l i z e d M i l n e e q u a t i o n

o r " a u x i l i a r y e q u a t i o n " )

w i t h

c o s α = uu' + (1 - w

2)

1 / 2( l - u'

2)

112 cos(q> - φ')

H a l f - s t e p f r o m J „ t o / „ :

ο (M > 0 )

(u < 0 )

Ιη(τ, 0 , φ) Λ ( τ , 0 , φ), (κ = 0 )

4.3 Method of Successive Orders 49

Readers familiar with the traditional treatment may wonder why the equation of transfer has not been made the starting point. We feel that the integrated form, in one order, which has been written in Display 4.4 as the half-step from /„ to Jn, can as solidly and as easily be based on direct physical insight as can the differentiated form, summed over all orders, which is the traditional equation of transfer. The present way of introducing the equations is no better or worse but simply has the advantage of providing a more convenient starting point for a program of numerical computation.

Combinat ion of two half-steps into one, i.e., elimination of / or J from the equations, does not generally lead to easier programming. However, if the scattering diagram O(cosa) is expressible in a few Legendre polynomials (orders 0 - λ γ ) , the expansion of J in Legendre polynomia is also limited to Ν + 1 terms, whereas / has an infinite expansion in Legendre polynomials. In this case, elimination of / is a more practical proposition than elimination of J. In particular, for isotropic scattering (N = 0), J is a function of τ only, and the integral equation for J ( T ) , i.e., the Milne equation, traditionally called the auxiliary equation, forms the simplest starting point for computing both J ( T ) and /(τ, t/, φ). For detailed formulas see Section 7.2.

D I S P L A Y 4 . 4 (continued)

D . E q u a t i o n s for s e p a r a t e F o u r i e r c o m p o n e n t s

D é f i n i t i o n s :

oo

Ιη(τ, u, φ) = lnc0(z, u) + 2 £ [ / " ( τ , u) c o s νηφ + / " ( τ , w) s in m<p] m= 1

oo

J„(T, Μ, φ) = Jnc0(x, u) + 2 X [J™ (τ , U ) c o s mcp + J™ (τ , W) s in mcp] m = 1

00 00

α(τ)Φ(τ, c o s α) = £ œ /( i ) P i( c o s α) = / ι0( τ , u, u') + 2 ] Γ ^ ( τ , w, u') c o s m(<p — φ')

1 = 0 m=\

w i t h r e d i s t r i b u t i o n f u n c t i o n

h

m(T, u, κ') - Σ ω , ( τ ) τ |—^ w i > ' ) , (m = 0 , 1, 2 , . . . )

i = m (l + m)\

H a l f - s t e p f r o m / „ _ x t o Jn:

• C c f r u) = ^ J Λ " ( τ , w, t / ' ) c - I , C ( T , Μ ' ) ( m = 0, 1, 2 , . . . )

1 f

1

M) = - J Λ " ( τ , M , ι / ) / ™ _ L F β( τ , Μ ' ) < /Μ ' , (m = 1, 2 , . . . )

H a l f - s t e p f r o m J „ t o / „ :

O m i t f r o m e q u a t i o n s in S e c t i o n C t h e a r g u m e n t φ a n d a d d s a m e i n d i c e s m, a n d c o r s, t o

b o t h s i d e s

50 4 Methods

For the low orders it may be helpful, in the interest of increased accuracy, to perform the integrations over τ analytically. Hovenier (1971) gives examples. Uesugi and Irvine (1970), and Hansen and Travis (1974) show in detail how the successive order terms for low order may also be obtained from an invariance principle, i.e., from Ambartsumian's method. Such useful hybrid forms are not uncommon.

4.4 AMBARTSUMIAN'S METHOD

The method first used by Ambartsumian (1943) and worked out in different variations by many authors is to determine how the radiation field inside and outside a scattering atmosphere is changed if a very thin layer is added to the top or bot tom of the atmosphere. Mathematically, this means that partial derivatives of various functions with respect to fc, the total optical depth, are considered. The derivation is simplest by the physical method, i.e., by visualizing what happens, rather than by formal manipulations with equations. We shall follow this method here.

Ambartsumian's method is seen in its most elegant form if we impose two assumptions upon the properties of the added layer.

Assumption 1: The added layer is very thin, dz < 1. Consequence: At most one scattering in the added layer need be con

sidered.

Assumption 2: The added layer has isotropic scattering. Consequence: Its effect is expressible in terms of the point-direction

gain. This function was defined in Section 4.3.3. In the present context we need only the values for a point just outside the original slab, which are traditionally expressed for isotropic scattering as

Ο(0,μ) = Χ(μ\ G(b,p)= Υ(μ)

If assumption 1 is dropped, the necessity arises of expressing the play of reflection back and forth between the original slab and the added layer, in an infinite series. This is done in the adding method (Section 4.5). If only assumption 2 is dropped, it becomes necessary to describe the properties of the original slab in more detail than by its point-direction gain. See full discussion for general phase functions in Sections 5.2.2 and 6.3.1. In case of doubt it is always possible to refer to the formulas for the adding method, and take the added layer b'( = άτ) so small that one term suffices.

Display 4.5 summarizes the results obtained on the basis of assumptions 1 and 2. It shows four basic equations (two for dR and two for dT). They can be written at once from the physical definitions of R, Τ, X, and Y. We shall briefly explain this for the first of these equations. The first term expresses the weakening

4.4 Ambartsumian's Method 51

D I S P L A Y 4.5

A m b a r t s u m i a n ' s M e t h o d for F i n i t e , H o m o g e n e o u s L a y e r s "

L a y e r a d d e d o n t o p L a y e r a d d e d t o b o t t o m

R e f l e c t i o n :

/ 1 \ \ adx

dR = l )àxR+- Χ(μο)Χ(μ) V μ0 μι

4μομ

α dx dR = --Υ(μ0)Υ(μ)

4μ0μ

T h e e q u a l i t y o f t h e s e e x p r e s s i o n s for h o m o g e n e o u s l a y e r s g i v e s

a

4(μ + μ0) T r a n s m i s s i o n :

1 adx dT = - - dx Τ + — - Χ(μ0)Υ(μ)

μ0 4μ0μ

ίΧ(μ)Χ(μ0) ~ Υ(μ)Υ(μο)~]

1 adx dT= - - d x T + - Υ(μ0)Χ(μ)

μ 4 μ 0μ T h e e q u a l i t y o f t h e s e e x p r e s s i o n s for h o m o g e n e o u s l a y e r s g i v e s

U ίΧ(μ)Υ(μ0) - Υ(β)Χ(μ0)1

dΥ(μ)

4 ( μ 0 - μ ) Τ(μ, μ0) = <

δ(μ- - μ ο )

for μ φ μ0

-β-""* + -\Χ(μ) 2 μ 0 4 L

Υ(μ)—ζ— for μ = μ0 ομ ομ

α n o t s c a t t e r e d in a d d e d l a y e r ; o n c e s c a t t e r e d in a d d e d l a y e r ; c u r v e s s ign i fy z e r o o r

m o r e s c a t t e r i n g s .

of the diffuse reflection from the original layer by the additional paths άτ/μ0 on entering and dx/μ on leaving. The second term is the radiation scattered once in the added layer. This layer by itself would give R = α άτ/4μμ0 from the definitions of reflection function, albedo, and optical depth. However, this amount is enhanced by a factor Χ(μ0) by any number of scatterings occurring in the original layer before the light is scattered in the added layer and, subsequently, by a factor Χ(μ) because of scattering in the original layer after this moment.

The other three equations have a similar explanation.

52 4 Methods

(1)

RESULTING EQUATIONS FOR A HOMOGENEOUS ATMOSPHERE WITH ISOTROPIC SCATTERING

The assumption of homogeneity enters when we equate the two expressions for dR and the two for dT. The equation for R thus obtained and given in Display 4.5 requires no comment. The corresponding expression for Τ is undetermined for μ = μ 0. The last line of Display 4.5, therefore, gives separately the expression for Τ in the case μ = μ0. Its first term is the directly transmitted (unscattered) light T0, which will be referred to as zero-order transmission or, in the tables, as P E A K .

We have shown before (Section 4.1.2) that Χ(μ) and Υ (μ) may in turn be expressed in terms of R and Τ by equations which, in the absence of φ dependence, take the form

Χ(μ) = 1 + 2μ f Λ ( μ , μ 0) άμ0 Jo

Υ(μ) = 2μ Τ ( μ , μ 0) άμ0 Jo

Substituting the expressions from Display 4.5 for R and Τ in the integrand we obtain the simultaneous nonlinear integral equations

y / ι \ « , αμ Γ1 b, μ)Χ(α, ft, μ 0) - Y (a, μ) Y (a, b, μ 0)

Λ ( α , 6 , μ ) = 1 + - - - • άμ0 2 J o μ + μο

Υ(α b μ) = e~bl» + — C

Χ^

9 ^

b" ^ ~

Υ^ h

" b

'μ ο)

άμ 2 Jo μο - μ

0

These are usually considered as the defining equations of the X and Y functions for isotropic scattering. The generalization of these equations for arbitrary phase functions is discussed in Sections 6.3 and 6.4.

RESULTING EQUATIONS FOR AN INHOMOGENEOUS ATMOSPHERE WITH ISOTROPIC SCATTERING

The sole assumption underlying the four basic equations in Display 4.5 is that the added layer has isotropic scattering, with albedo a. The original layer may be inhomogeneous and may have anisotropic scattering. Under these more general assumptions, the idea of invariance is lost, and no quadratic expressions for R and Τ in terms of X and Y functions result.

Let us assume for a moment that the layer has isotropic scattering throughout but is inhomogeneous, so that the single scattering albedo is a given function α(τ). The reflection found by adding thin layers until the slab is complete is then expressed by the integral

(2)

Κ * ( τ , μ , μ 0) = Γ exp -{τχ - τ ) (— + - ) J o L \ μ ο μ/_

4 μ 0μ Χ*(τ,μ)Χ*(τ,μ0)</τ (3)

4.4 Ambartsumian's Method 53

where

Χ*(τ, μ) = 1 + 2μ f Κ*(τ, μ, μ 0) άμ0 (4) Jo

This equation can be made the basis of a numerical method to compute the reflection function. It is necessary to integrate by small increments Δτ using at each step the then available X function from Eq. (4).

This method has been used by Bellman et al. (1963). (The tables in their book are for homogeneous atmospheres.) The name invariant embedding used for this method may^be justified in a wider mathematical context but does not seem a happy choice in this application. Embedding means that b is treated as a variable, rather than as a constant. Invariant conveys an association with the fact that in the case of a homogeneous atmosphere, the use of an invariance principle in conjunction with this method permitted Ambartsumian to derive simple closed relations between R, Τ, X and Y (Display 4.5). Under the more general assumptions underlying Eqs. (3) and (4), such an invariance cannot be used.

Equation (3) gives the reflection from the side of the slab labeled τ = xu which we usually call the bot tom. In concurrence with Section 3.4.1, we therefore have distinguished it by an asterisk. Three more equations of the same nature can be written, giving reflection from the side τ = 0 and transmission functions either way. Each of these equations can be written directly from physical definitions. In order to do so, the various multiscattering paths of the light quanta must be considered to be classified according to the depth of the topmost layer in which a scattering occurs. (See Fig. 4.3.) Let this depth fall in the interval dx. The exponential factor signifies the probability that scattering does not occur in higher layers; the factor α(τ)άτ/4μμ0 is simply the reflection or transmission by scattering in the singled-out layer; the X or Y factors are the enhancements by any scatterings in the layer below τ before and after the scattering in dx.

F i g . 4 . 3 . S k e t c h (1) is u s e d in d e r i v i n g t h e i n v a r i a n t e m b e d d i n g e q u a t i o n ( 3 ) . S k e t c h e s (2 ) ,

(3 ) a n d (4) s i m i l a r l y y i e l d t h r e e m o r e e q u a t i o n s , n o t s e p a r a t e l y r e p r o d u c e d .

54 4 Methods

Obviously such a "physical derivat ion" may be replaced by a more formal one. This is necessary if we adopt the traditional atti tude that the "physical pic ture" is suspect and can at most serve heuristically. But this attitude must be paid for by derivations that run to 125 numbered equations (Ueno, 1960) or 76 equations (Busbridge, 1961). I hope to have convinced at least some readers that the same result can be written down in a straightforward and reliable manner from the physical definitions.

For readers wishing to make a detailed comparison it may be best to identify one equation precisely. With the appropriate changes in notation, normalization and sign conventions, Eq. (3) above is equivalent to Sobolev (1956), Eq. 20; Bellman and Kalaba (1956), Eq. 3.8; Ueno (1960), Eqs. 48, A28, A47; Busbridge (1961) Eq. 3.29.

Historical note. The paper by Ambartsumian (1943) derives Eqs. (1) and (2) in a different notation, essentially by the method of Display 4.5.1 have not seen the original paper, since few of these wartime publications reached the West. It is found in Ambartsumian (1960, Vol. I, pp. 232-237). In this volume, 13 papers on radiative transfer are reprinted. I have adopted the notat ion most common now, first introduced by Chandrasekhar in 1947. Ambartsumian did pursue, in other papers, the generalization to anisotropic scattering (see Section 6.3.1). He did not pursue the obvious generalization to inhomogeneous atmospheres expressed in Eqs. (3) and (4) above. Nevertheless, I feel Ambartsumian's pioneering work justifies the title of this entire section.

4.5 THE ADDING OR DOUBLING METHOD

4.5.1 Concept

Imagine a plane-parallel slab with optical thickness V placed on top of one with optical thickness h". By the adding method we mean a method for computing the reflection and transmission functions of the combined layer of optical thickness b = V + b" starting from the known reflection and transmission functions in the separate layers. The point-direction gain can also be obtained by this method.

If b' = b" and if the layers are identical in all other respects, we refer to this method as the doubling method. If, on the other hand, V is infinitesimal, the adding method reduces to Ambartsumian's method (Section 4.4).

Since our first proposal (van de Hulst, 1963) this method has proved quite convenient. With certain variations in form it has been used among others by Twomey et al. (1966, 1967), van de Hulst (1967), Irvine (1968), Hansen (1969), van de Hulst and Grossman (1968). The majority of the tables reproduced in this book have been obtained by the doubling method.

4.5 The Adding or Doubling Method 55

in

S u m - R ( 2 b ) (reflection)

2b Sum- A , D (mid-layer

radiation field)

S u m - T ( 2 b ) (transmission)

F i g . 4 . 4 . C o n c e p t o f t h e d o u b l i n g m e t h o d . E a c h f o r k r e p r e s e n t s a n y n u m b e r o f s u c c e s s i v e

s c a t t e r i n g s t h a t m a y o c c u r w i t h i n t h e h a l f - l a y e r . T h e g a p b e t w e e n t h e h a l f - l a y e r s is a n a r t i f i c e t h a t

c a n b e r e m o v e d .

The concept of the doubling method in its simplest form is illustrated by Fig. 4.4. Here the two sublayers are identical and symmetric, and a gap has been left to show the midlayer radiation field. Each straight line outside and between the sublayers symbolically represents an azimuth-independent intensity distribution /(μ), where μ is the positive cosine of the angle with the vertical, 0 < μ < 1. Each fork represents the sum of all possible scattering events, single and successive, that may take place in that sublayer before the radiation emerges. The ongoing branch of each fork includes the unscattered radiation (i.e. zero-order transmission).

The same concept is easily put into equations. The reflection function #(μ, μ 0) of a separate layer is an operator that transforms any incident intensity distribution Ιιη(μ0) into the reflected intensity distribution 7Γ 6 ί(μ) by an operation which, in full, reads

but which we may abbreviate to the symbolic form

7 r ef = RIin and similarly for transmission. Writing R and Τ for the reflection and transmission of the separate layers we can write, from Fig. 4.4,

^ (double layer) = R + TRT + TRRRT + · · ·

T(double layer) = TT + TRRT + TRRRRT + · · ·

These are the basic equations for the doubling method.

56 4 Methods

The program to go from these very simple forms to a complete treatment now reads as follows:

1. Generalize formulation for layers of unequal thickness (adding). 2. Generalize formulation for inhomogeneous layers. 3. Write the equations for midlayer intensity distribution. 4. Write the equations for point-direction gain. 5. Separate the terms involving zero-order transmission for practical

computation. 6. Extend or translate formulation to include azimuth-dependent terms. 7. Extend or translate formulation to include polarization. 8. Replace infinite sums by matrix inversion.

We shall treat points 1-5 together in Section 4.5.2, where to some extent we also anticipate point 8. The extensions (6 and 7) as well as some more analytic extensions based on point 8 will be discussed in Section 4.5.3.

4.5.2 Adding Inhomogeneous Layers

We follow the same principle as before, of representing in equations how the radiation can get into and out of the combined layer. N o other assumptions are made besides the plane-parallel character of the separate layers. Each, or both, may be inhomogeneous; they may be different ; they may be conservative or not. The scattering pattern may be anisotropic and may vary with height in the layer. None of these data enter explicitly into the equations, because we assume that they have already been rigorously taken into account in deriving reflection, transmission, and point-direction gain for the separate layers.

Inhomogeneity necessitates a notation which distinguishes not only between the upper layer (primes) and the lower layer (double primes), but also for each of these layers between reflection against the top side and against the bot tom side (asterisks). Similar distinctions are necessary for transmission functions and for point-direct ion gain. The complete set of notations for these functions is given in Figure 4.5. It is not necessary to specify from where and in what direction we count the optical depth. The representation is again symbolic: The forks and straight lines stand for any combination of light paths containing zero or more scattering points.

The full adding equations can be read from Fig. 4.6, where the same symbolism of straight lines has been used. The four parts of this figure correspond to four separate physical experiments. The first two experiments serve to find reflection, transmission, and point-direct ion gain of the combined layer. Experiments 3 and 4 serve exclusively to find point-direct ion gain, starting from its reciprocal definition (Section 4.4.3). In each experiment the upper and lower layer have been separated in the drawing in order to show the radiation field at the separating layer, denoted by an intensity distribution D (down) and a distribution A (up, arriba in Spanish).


upper layer

in R" T*" G"

lower layer

Τ in G * "

G G

combined layer

Fig. 4.5. N o t a t i o n s u s e d in f o r m u l a t i n g t h e a d d i n g m e t h o d fo r d i s s i m i l a r , v e r t i c a l l y i n -

h o m o g e n e o u s l a y e r s .

A few further technicalities have to be explained before we can use Fig. 4.6 to write the full set of equations given in Display 4.6.

1. The symbolic notation where the functions ϋ(μ, μ 0) and Τ (μ, μ0) can be written as operators without arguments is explained in Section 7.1.1 (Matrix Notat ion), to which we refer for more detail. Note that the symbol written to the left always operates on the symbol written to the right, so that the chronological order of events corresponds to reading from right to left.

2 . The point-direct ion gain must be divided by 4μ before it expresses a function that can be written as a matrix consistent with the notat ion of Section 7 .1 . We therefore call G ( T , μ)/4μ the matrix Κ when used as a (τ, μ) matrix and Κ when used as a (μ, τ) matrix, and similarly with asterisks and/or primes.

3 . The suffix 1, 2 , 3 , . . . (which we shall call j) is not the number of scatterings, for in each step any number of scatterings in the layer is taken into account. Consequently, the convergence of the series is much faster than in the successive order method. The meaning of the suffix can be read from the figure. For instance, in the first and third experiments J is the number of times the photon has emerged from the top layer, or alternately, j — 1 is the number of times the photon has crossed the separation going up.

58 4 Methods

F i g . 4 . 6 . F o u r t h o u g h t e x p e r i m e n t s w h i c h m a y s e r v e t o find r e f l e c t i o n , t r a n s m i s s i o n , a n d

g a i n of a c o m b i n e d l a y e r t o g e t h e r w i t h full c h e c k s o n t h e r e c i p r o c i t y r e l a t i o n s : ( a ) first e x p e r i m e n t ,

i n c i d e n c e f r o m t o p ; ( b ) s e c o n d e x p e r i m e n t , i n c i d e n c e f r o m b o t t o m ; ( c ) t h i r d e x p e r i m e n t , i s o t r o p i c

s o u r c e l a y e r in u p p e r s l a b ; a n d ( d ) f o u r t h e x p e r i m e n t , i s o t r o p i c s o u r c e l a y e r in l o w e r s l a b .

4. The infinite series can be summed by matrix inversion (denoted by superscript — 1), for they all contain factors of the form

1 + R*'R" + R*R'R*R" + · · = (1 - R+R")1

or

1 + R'R*' + R'R*'R"R*' + · · · = (1 - R'R*')1

Since all of the R matrices (independently of the asterisks or primes) are symmetric, the first factor is the transposed matrix of the other factor. Fur thermore it is easy to verify the relation

R"(l - R+R'Y1 = (1 - / T R * ' )

1/ * "

where the matrix on both sides of the equation is symmetric.

The preparation for the formulas collected in Display 4.6 is now complete. The reader who takes his time to refer back to Figs. 4.5 and 4.6 should find that he can verify these results without pencil and paper. We have not written all the


D I S P L A Y 4 . 6

F o r m u l a s for A d d i n g M e t h o d , I n h o m o g e n e o u s L a y e r s

F i r s t e x p e r i m e n t , c o n i c a l i n c i d e n c e f r o m t o p :

D = (1 — R*'R'Ylr

A = R'D

R = R' + T*'A

Τ = T"D

[K"D,

for p o i n t in u p p e r l a y e r

for p o i n t in l o w e r l a y e r

S e c o n d e x p e r i m e n t , c o n i c a l i n c i d e n c e f r o m b o t t o m :

A* = (1 - R"R*'y

lT*"

D* = R*'A*

R* = R*" + T'D*

Τ* = T*'A*

(K*'A*

IK*" + KD*

for p o i n t in u p p e r l a y e r

fo r p o i n t in l o w e r l a y e r

T h i r d e x p e r i m e n t , l a y e r o f i s o t r o p i c s o u r c e s in u p p e r l a y e r :

D = (1 - R*'R"y

lK*'

( A a n d D d i f f e ren t f r o m first e x p e r i m e n t A = R'D

Κ = K' + T*'A

K* = T'D

F o u r t h e x p e r i m e n t , l a y e r o f i s o t r o p i c s o u r c e s in l o w e r l a y e r :

A = (1 - R"R*')~

lK"

D = R*'A

Κ = T*'A

Κ = Κ*" + T'D

A a n d D d i f fe ren t a g a i n

alternative forms, but we have chosen the ones which seemed simplest and most readily adapted for a computer program.

It is also possible to verify from these equations that the validity of the reciprocity rules (Sections 3.4.1 and 3.4.3) for reflection, transmission, and gain of the separate layers automatically leads to the validity of those rules for the combined layers. We shall illustrate this by one example.

The two reciprocal definitions of gain must give the same answer with the arguments τ, μ in reverse order. Hence the functions found in both ways must

60 4 Methods

be the same but must be represented by a matrix and its transposition. Choose the direction at top and the point in the upper layer. Then the two results, read from Display 4.6, are

Κ = K' + K*'R"(\ - R*'R"ylV (experiment 1)

Κ = K' + T*'R"(\ - R*'R"ylK*' (experiment 3)

The second expression is the transposed matrix of the first, because the factors appear in reversed order and are all t ransposed; the middle combination is symmetric and T*' is the transposed matrix of Τ by the reciprocity principle.

Many further checks may be made. For instance, the point-direction gain for a point at the separation layer and a direction μ at the top can be found in at least five different ways :

G(T9 μ) 4μ

m(K' + K*'A) (first experiment) m{K"D) (first experiment) {N(A + D) (first experiment) (K' + T*'A)m (third experiment) T*'Am (fourth experiment)

Here Ν is the μ vector defined in Section 7.1, and m represents the τ vector, which is equivalent to the instruction to set τ in the preceding or following matrix equal to the depth of the separation layer. Upon elaborating these expressions in terms of the given reflection and transmission functions for the two separate layers, each of the first three gives

£JV(1 + Z O O - R*fR")~

lT'

and the last two give the transposed matrix. Similarly, five different expressions may be written for the gain between a point at the separation layer and a direction at the bottom. These checks in the equation are almost trivial, but they may be very helpful in judging the accuracy and correctness of numerical work.

The values of A and D (first experiment) for μ = 0, i.e., corresponding to a direction parallel to the slab, must be the same if both the phase function Φ(τ, cos a) and the albedo α(τ) are continuous across the separation layer. This is stated without proof. For isotropic scattering this common value must also be am times the average intensity jN(A -f D) found in the previous check. This gives a sixth way to check the same number.

The conclusion is that the adding method permits a large complex of problems to be treated, without excessive administration, in a form which is ready to be programmed for a machine, and which has many internal checks. The only basic machine operations required are integrations over μ in the interval (0, 1).

One more technical point should be made. The equations are correct and complete but in one respect deceivingly simple. Care must be exercised in writing them in the forms of integrals, because the matrix D contains, like T',


T", and T, a singular as well as a diffuse part. Many authors avoid this problem by separating immediately, as follows.

Τ = T0 + Td i ff This separation would have greatly complicated the formulas presented above, as I found out in my first attempts. However, since most machines cannot accept the same instructions in dealing with δ functions as they can in dealing with continuous functions, this separation has to be made when translating the matrix equations into a program (or into machine instructions).

Display 4.7 shows an example. The computat ion of R and Τ by means of the first experiment have been fully written out in the form of integrals. Further we have distinguished in this display positive cosines of angles with the normal as follows:

for incident radiation at top : μ 0 for downward radiation at interface: w, v, w for upward radiation at interface: ζ for emergent radiation at top or bot tom : μ

D I S P L A Y 4 .7

A d d i n g M e t h o d , S a m p l e of T r a n s l a t i o n i n t o F o r m u l a s for C o m p u t e r U s e

M a t r i x f o r m F u n c t i o n a l f o r m o f n o n s i n g u l a r t e r m s

Ql = R*'R" Qx(u, ν) = f R*'(u, z)R"(z, v)2zdz Jo

Qn+1 = QlQn Qn+M ν) = Ί Β ^ Μ , w)Q„(w, v)2w dw

S = Σ Q„ S (u , ο) = Σ QJiu, v) n=1 n= 1

D = (1 + S)(T0 + T d i f f) = T'0 + Ddi{{ Ddm(u, μ0) = Tdi{{(u, μ0) + S(u, μ ο ) * " *

7"

0

fs(u, v)T'difi(v, μ0)2ν dv

A = R'XTO + D d i f f) A(z, μ0) = R"(z, / ί ο )*"*

7"

0 + Ί R"fr u)DdiSt(u, μ0)2η du

•>o

R = R' + (Τξ' + T%n)A RQi, μ0) = Κ ' ( / ι , μ0) + ί Γ * ' Μ ( μ , μ0)

+ Τ%ί{(μ, z)A(z, μ0)2ζ dz Jo

T=(T'0 + TJ i f f)(r0 + D d i f f) Td i f f( / i , μ0) = e -

b l» D d i { {^ , μ0) + T"diffa / ί ο ) * " *

7"

0

— To + 7^,i ff ç ι + Tdif^,u)Ddif{(u^0)2udu

Jo

62 4 Methods

The same equations may be written out to include polarization. See Section 15.2.4, Hansen and Travis (1974), Lacis and Hansen (1974). Similar separations can be made for the other experiments.

In two situations the separation just described is unnecessary.

(a) If we decide to perform the μ integration by discretizing (e.g., in 100 equal intervals, or by Gaussian arguments and weights) and if μ 0 coincides with one of the discrete values, we do not have to separate out the discrete part because everything is discrete.

(b) If the incident radiation is not conical with one μ 0, but has a smooth distribution in μ 0, it is again unnecessary to make the separation, because everything is continuous. For instance, if we seek to compute RU and TU, the reflected and transmitted intensities for incident radiation of uniform intensity, we must postmultiply both sides of each equation in Display 4.6 (first experiment) by U. The products R'U and TU, which are then needed, may be taken directly from the corresponding tables for the separate layers, which are assumed to be given.

4.5.3 Matrix-Transfer Method

The basic equation in Section 1.2 can, in the absence of embedded sources, be written in symbolic form as

This is called the interaction principle (Grant and Hunt, 1969a; Hunt, 1971; Preisendorfer, 1976). The adding method basically consists of stringing two of these equations together and making the appropriate matrix multiplication. This more formal approach is known from many other branches of physics. It has been worked out for radiative transfer under various titles, e.g., matrix transfer theory or matrix operator theory by Redheffer (1962), Aronson and Yarmush (1966), Plass et al. (1973), and probably others. Readers from different fields may prefer one or the other presentation. Many of these developments were quite independent, and none of the papers presents the complete story— nor does this book (see introduction) !

Depending on the author 's preference this method can be pushed by further matrix algebra to a formal analytical solution, or it can be employed in numerical computation. The analytical follow-up is well presented by Aronson and Yarmush (1966). Ready formulas for a finite slab with an arbitrary phase matrix, including polarization, are given by Aronson (1972). The computational use was discussed by Hunt (1971) and Plass et al. (1973). Pahor et al. (1974) discretize the problem by approximately representing each of the intensity distributions by a linear combination of a finite set of assumed functions in the domain (0, 1) and by further employing matrix equations for the sets of coefficients.


In any method a matrix has to be inverted at some point. This can be done by many methods. The series expansion used in the presentation of Section 4.5.2 is but one of those.

The formal approach of the matrix transfer theory not only reveals the full mathematical structure of the solution; it may also help in certain technical details. One of these is the question : at what value of b should we start the doubling method in practical computat ions? Grossman and I initially made all our test runs with a starting b of | , \ , or 1, the reflection and transmission functions for these starting values being found by successive scattering. Later, Hansen did not wish to mix two different methods in one calculation and started doubling from some extremely low value, e.g., 2 "

9, or even 2 "

2 5. The starting

R and Τ may then be taken with single scattering only. The exact propagation of possible errors in the starting functions has been traced by Gran t and Hunt (1969b).

Another practical point is the rapidity of convergence of the series used in the formulation of Section 4.5.2. Experience shows that usually only three or four terms of these series have to be computed, because they very rapidly assume the character of a geometric series. The corresponding ratios have been extracted from the numerical data and are presented for isotropic scattering (Section 9.6.4) and for anisotropic Henyey-Greenstein scattering (Section 13.4). In Section 13.4 we also show that the next eigenvalue of the relevant integral equation is much smaller than the first, which explains the rapid approach to a geometric series.

The extensions to azimuth-dependent terms and to the scattering of polarized light will be discussed in Chapter 15. A new formulation is not even necessary. The symbols of the matrix equation in the first lines of this section can be defined to include these more general assumptions. In the same way, the drawings (Figs. 4.4, etc.) in Section 4.5.1 can be defined to represent this more general situation, and all developments based on these drawings remain valid up to the point where the symbols have to be translated into sets of functions.

4.5.4 Adding a Ground Surface

The influence of a reflecting ground surface of a planet on the radiation field in the atmosphere above it can be found by simple application of the adding method. The ground serves as the lower of the two layers. Hence, after setting R" equal to the reflection function of the ground surface, we can employ all formulas of Section 4.5.2, except those referring specifically to light transmitted by the lower layer.

In the special case of specular reflection with albedo 1, the problem degenerates to simple doubling. This situation never arises in practice, because even a smooth sea is not a perfect mirror. See Section 18.4.1(b) for a discussion of the specular reflection with an albedo different from 1.

The assumption most commonly made is Lambert reflection with ground albedo Ag. The calculation then is greatly simplified, for the radiation returned

64 4 Methods

D I S P L A Y 4 . 8

A t m o s p h e r e a b o v e G r o u n d S u r f a c e R e f l e c t i n g b y L a m b e r t ' s L a w w i t h A l b e d o Ag I n h o m o g e n e o u s a t m o s p h e r e

( m a t r i x n o t a t i o n )

H o m o g e n e o u s a t m o s p h e r e

( f u n c t i o n a l n o t a t i o n )

A t m o s p h e r e spec i f i ed b y r e f l e c t i o n t r a n s m i s s i o n g a i n

D e r i v e d f u n c t i o n s

G r o u n d s u r f a c e spec i f i ed b y

A t m o s p h e r e p l u s g r o u n d s u r f a c e

Τ', T*' >as in S e c t i o n 4 .5 .2

K\ K*j

C* = R*'U = UR*'

r* = UC*

W* = T*'U = uv

L* = K*'U = UK*'

R" = A U U

/ ? ' (μ , μ 0) Τ'(μ, μ0) G'(t, μ 0) / 4 μ 0

ι(μ)= Κ(β,μο)2μ0άμ0 Jo

= Γχ(μ)2μάμ Jo

bo(b ~ τ )

/ Γ ( μ , μ ' ) = Λ

T h e e q u a t i o n s of first experiment in F i g . 4 .6 a n d D i s p l a y 4 .6 b e c o m e t h e f o l l o w i n g :

R e f l e c t e d r a d i a t i o n R = R ' + ) _ \ r* W* • W* R'(V> fo) + yzJTr A A

R a d i a t i o n n e a r g r o u n d , d o w n D = Τ + C* · W* Τ'(μ, μ0) + ?— r^)t Λμ0) 1 - Agr* 1 - Agr

A A R a d i a t i o n n e a r g r o u n d , u p A = - U · W* — £ ι ( μ 0) 1 - Agr* 1 - Agr

( 4 μ 0Γ1 t i m e s p o i n t - d i r e c t i o n ^ Ag + G(x, μ0) 1 Ag

g a i n for o u t w a r d d i r e c t i o n ^ - * + \ - A r* ' 4μ0 2

9 ο{ ~ 1 - A r

l ( M o)

μ0 a n d p o i n t a t o p t i c a l d e p t h τ b e l o w t o p s u r f a c e

from the ground is unpolarized, has a fixed angular distribution (constant brightness), and has an intensity proport ional to the flux which hits the ground. This is expressed by the equation

κ ( μ , μ ο ) = a g which in matrix notat ion (Section 5.1) has to be written in the form R" = AU U. The infinite series in the adding method then becomes a geometric series with ratio A%r*, where

r* = UR*fU

is the fraction of the flux reflected off the bot tom of the atmosphere. This series can now be summed.

References 65

The resulting formulas are collected in Display 4.8. They have been formulated for atmospheres with arbitrary stratification and arbitrary scattering patterns. If the atmosphere is homogeneous, all asterisks may be omitted. The derivation is a matter of simple algebra. Some practical details are discussed in Section 18.4.1.

R E F E R E N C E S

A m b a r t s u m i a n , V . A . ( 1 9 4 3 ) . Dokl. Akad. Nauk SSSR 3 8 , 2 5 7 .

A m b a r t s u m i a n , V . A . ( 1 9 6 0 ) . " N a u c h n i T r u d i " (Sc i en t i f i c w o r k s ) ( V . V . S o b o l e v , e d . ) , 2 v o l u m e s .

I z d . A k a d . N a u k A r m y a n s k o i S S R , Y e r e v a n .

A r o n s o n , R . ( 1 9 7 2 ) . Astrophys. J. 177 , 4 1 1 .

A r o n s o n , R . , a n d Y a r m u s h , D . L . ( 1 9 6 6 ) . J. Math. Phys. 7 , 2 2 1 .

B e l l m a n , R . F . , a n d K a l a b a , R . E . ( 1 9 5 6 ) . Proc. Nat. Acad. Sci. 4 2 , 6 2 9 .

B e l l m a n , R . E . , K a l a b a , R . E . , a n d P r e s t r u d , M . C . ( 1 9 6 3 ) . " I n v a r i a n t E m b e d d i n g a n d R a d i a t i v e

T r a n s f e r in S l a b s o f F i n i t e T h i c k n e s s . " E l s e v i e r , N e w Y o r k .

B u s b r i d g e , I . W . ( 1 9 6 1 ) . Astrophys. J. 1 3 3 , 198 .

C a r t e r , L . L . , H o r a k , H . G . , a n d S a n d f o r d , M . T . ( 1 9 7 8 ) . J. Comput. Phys. 2 6 , 119 .

D a v e , J . V . , a n d G a z d a g , J . ( 1 9 7 0 ) . Appl. Opt. 9 , 1 4 5 7 .

D e i r m e n d j i a n , D . ( 1 9 6 8 ) . " E l e c t r o m a g n e t i c S c a t t e r i n g o n S p h e r i c a l P o l y d i s p e r s i o n s . " E l s e v i e r ,

N e w Y o r k .

D l u g a c h , J . M . , a n d Y a n o v i t s k i i , E . G . ( 1 9 7 4 ) . Icarus 2 2 , 6 6 .

E s c h e l b a c h , G . ( 1 9 7 1 ) . J. Quant. Spectrosc. Radiât. Transfer 1 1 , 7 5 7 .

G r a n t , I . P . , a n d H u n t , G . E . ( 1 9 6 9 a ) . Proc. R. Soc. London Ser. A 3 1 3 , 1 8 3 .

G r a n t , I . P . , a n d H u n t , G . E . ( 1 9 6 9 b ) . Proc. R. Soc. London Ser. A 3 1 3 , 199 .

H a n s e n , J . E . ( 1 9 6 9 ) . Astrophys. J. 1 5 5 , 5 6 5 .

H a n s e n , J . E . , a n d T r a v i s , L . D . ( 1 9 7 4 ) . Space Sci. Rev. 16 , 5 2 7 .

H e r m a n , Β . M . , a n d B r o w n i n g , S. R . ( 1 9 6 5 ) . J. Atmos. Sci. 2 2 , 5 5 9 .

H o v e n i e r , J . W . ( 1 9 7 1 ) . Astron. Astrophys. 1 3 , 7 .

H u n t , G . E . ( 1 9 7 1 ) . J. Quant. Spectrosc. Radiât. Transfer 1 1 , 6 5 5 .

I r v i n e , W . ( 1 9 7 5 ) . Icarus 2 5 , 175 .

I r v i n e , W . M . ( 1 9 6 8 ) . Astrophys. J. 1 5 2 , 8 2 3 .

J o s e p h , J . H . , W i s c o m b e , W . J . , a n d W e i n m a n , J . A . ( 1 9 7 6 ) . J. Atmos. Sci. 3 3 , 2 4 5 2 .

K e r k e r , M . ( 1 9 6 9 ) . " T h e S c a t t e r i n g o f L i g h t a n d O t h e r E l e c t r o m a g n e t i c R a d i a t i o n . " A c a d e m i c

P r e s s , N e w Y o r k .

L a c i s , Α . Α . , a n d H a n s e n , J . E . ( 1 9 7 4 ) . J. Atmos. Sci. 3 1 , 1 1 8 .

L e n o b l e , J . ( 1977) . S t a n d a r d P r o c e d u r e s t o C o m p u t e R a d i a t i v e T r a n s f e r in a S c a t t e r i n g A t m o s p h e r e .

R a d i a t i o n C o m m i s s i o n , I n t e r n a t i o n a l A s s o c i a t i o n o f M e t e o r o l o g y a n d A t m o s p h e r i c P h y s i c s

( I . U . G . G . ) , p u b l i s h e d b y N a t i o n a l C e n t e r f o r A t m o s p h e r i c R e s e a r c h , B o u l d e r , C o l o r a d o .

L e w i n s , J . ( 1 9 6 5 ) . " I m p o r t a n c e , t h e A d j o i n t F u n c t i o n . " P e r g a m o n , O x f o r d .

M e a d o r , W . E . , a n d W e a v e r , W . R . ( 1 9 7 6 ) . Appl. Opt. 15 , 3 1 5 5 .

P a h o r , S . , S u h a d o l c , Α . , a n d Z a k r a j s e k , E . ( 1 9 7 4 ) . Publ. Math. Dept. Ljubljana 6 , 5 1 .

P l a s s , G . N . , K a t t a w a r , G . W . , a n d C a t c h i n g s , F . E . ( 1 9 7 3 ) . Appl. Opt. 1 2 , 3 1 4 .

P r e i s e n d o r f e r , R . W . ( 1 9 7 6 ) . H y d r o l o g i e O p t i c s , V o l . 2 . U . S . D e p t . o f C o m m e r c e ( N O A A E n v i r o n

m e n t a l R e s e a r c h L a b o r a t o r i e s ) , H o n o l u l u .

R e d h e f f e r , R . ( 1 9 6 2 ) . J. Math. Phys. 4 1 , 1.

S h e t t l e , E . P . , a n d G r e e n , A . E . S. ( 1 9 7 4 ) . Appl. Opt. 1 3 , 1 5 6 7 .

S o b o l e v , V . V . ( 1 9 5 6 ) . Dokl. Akad. Nauk 1 1 1 , 1 0 0 0 .

T w o m e y , S . , J a c o b o w i t z , H . , a n d H o w e l l , H . B . ( 1 9 6 6 ) . / . Atmos. Sci. 2 3 , 2 8 9 .

66 4 Methods

T w o m e y , S . , J a c o b o w i t z , H . , a n d H o w e l l , H . B . ( 1 9 6 7 ) . J. Atmos. Sci. 2 4 , 7 0 .

U e n o , S. ( 1 9 6 0 ) . Astrophys. J. 1 3 2 , 7 2 9 .

U e s u g i , Α . , a n d I r v i n e , W . M . ( 1 9 7 0 ) . Astrophys. J. 1 6 1 , 2 4 3 .

v a n d e H u l s t , H . C . ( 1 9 5 7 ) . " L i g h t S c a t t e r i n g b y S m a l l P a r t i c l e s . " W i l e y , N e w Y o r k ; a l s o D o v e r ,

N e w Y o r k , 1 9 8 1 .

v a n d e H u l s t , H . C . ( 1 9 6 3 ) . A N e w L o o k a t M u l t i p l e S c a t t e r i n g . U n n u m b e r e d m i m e o g r a p h e d r e p o r t ,

N A S A I n s t i t u t e f o r S p a c e S c i e n c e , N e w Y o r k ,

v a n d e H u l s t , H . C . ( 1 9 6 7 ) . In " I C E S I I , E l e c t r o m a g n e t i c S c a t t e r i n g " ( R . L . R o w e l l a n d R . S. S t e i n , e d s . ) , p . 7 8 7 .

v a n d e H u l s t , H . C , a n d G r o s s m a n , K . ( 1 9 6 8 ) . In " T h e a t m o s p h e r e s o f V e n u s a n d M a r s " ( J . C . B r a n d t a n d M . B . M c E l r o y , e d s . ) , p . 3 5 . G o r d o n a n d B r e a c h , N e w Y o r k .

W h i t n e y , C . ( 1 9 7 4 ) . J. Quant. Spectrosc. Radiât. Transfer 14 , 5 9 1 . W i s c o m b e , W . J . , a n d J o s e p h , J . H . ( 1 9 7 7 ) . Icarus 3 2 , 3 6 2 .

5 • Very Thick layers with Arbitrary

Anisotropic Scattering

5.1 METHOD AND TERMINOLOGY

This chapter deals with homogeneous atmospheres with arbitrary phase function and arbitrary single-scattering albedo. It presents rigorous results for unbounded media and for semi-infinite atmospheres. It also gives asymptotic results for slabs of large optical thickness. We include in this case a complete derivation. This is not contrary to the plan of this book because the derivation is made by means of thought experiments starting from physical definitions. Many of the intermediate results will be useful in later chapters.

Very thick atmospheres with conservative or nearly conservative scattering were formerly difficult subjects to treat numerically, especially if the phase function was anisotropic. For such atmospheres the method of successive orders has a very poor convergence; the doubling method is better but still slow. Only the method of singular eigenfunctions gives a better convergence for thicker layers.

Logically, the properties of very thin layers follow as a byproduct from the method of successive orders. Likewise, the asymptotic properties of very thick layers follow as a byproduct of the method of singular eigenfunctions. We shall not, however, derive them in this manner, because the plan of this book calls for equations of which the physical contents can be directly visualized. For a full treatment of this method, reference may be made to the literature cited in Section 6.5. Display 4.3 may be consulted for a very brief summary.

67

68 5 Very Thick Layers with Arbitrary Anisotropic Scattering

We shall show in this chapter that the dominant terms in the asymptotic theory can be derived in a simple manner from physical reasoning (van de Hulst, 1968a). We will not obtain the radiation field at arbitrary depth, but only at the boundaries of the atmosphere and in the "diffusion domain" deep inside the atmosphere.

We have limited the derivation in this chapter to scattering without polarization; but it is shown in Section 15.2.1 that most of the results from the present chapter may be translated into a form valid for scattering with polarization.

Since the azimuth-dependent terms of the radiation field are strongly damped in thick layers (Section 15.3.3), we confine the present discussion to fields without azimuth dependence. This means that the intensity, or any other function may depend on the following variables:

Characterizing the phase function : a = ω 0 is the albedo, g — ω1/3α the asymmetry factor, any further parameters, e.g., ω 2, ω 3, . . . ;

Characterizing the cloud layer: b is the total optical thickness;

Characterizing the sources: μ 0 is the cosine of angle of incidence, τ 0 the depth of a source layer;

Specifying the depth and direction in which the intensity is measured: τ is the optical depth from top surface down, u the cosine of angle with downward normal, μ = \u\ for any emerging radiation.

For brevity we shall call a function of one variable μ or μ 0 defined on the interval (0,1) a vector; a function of two such variables is called a matrix. Both vectors and matrices are denoted by capitals. Quantities not dependent on μ are scalars, written in small letters. All of these functions may also depend on a, g, b, and other parameters. The full list of such functions used in this chapter is given in Display 5.1.

The product of two vectors is a scalar defined by

Products of a vector and a matrix, or of two matrices, are similarly defined. The insertion of the factor 2μ (or 2 μ0 if the integration is over μ0) in the definition of all vector and matrix multiplications allows us to arrive at simple formulas.

The physical interpretation of these multiplications is read from right to left. For instance, from the definitions that follow, we interpret

UR = (U operating on R)

5.1 Method and Terminology 69

D I S P L A Y 5 .1

S h o r t h a n d N o t a t i o n U s e d i n T h i s C h a p t e r : L i s t o f V e c t o r s a n d M a t r i c e s

S t a n d s fo r

S h o r t h a n d full f u n c t i o n N a m e a n d c o m m e n t s

M a t r i x R R(a, b, μ, μ 0) R e f l e c t i o n f u n c t i o n , s y m m e t r i c

Rao Roo(a, μ, μ 0) S a m e , fo r s e m i - i n f i n i t e a t m o s p h e r e

τ Γ ( α , b, μ, μ 0) T r a n s m i s s i o n f u n c t i o n , s y m m e t r i c

1 δ(μ ~ μ0)βμ U n i t m a t r i x

ζ δ(μ - μ 0) / 2 μ

2 S i n g u l a r m a t r i x p r o d u c i n g d i v i s i o n b y μ o r μ 0, i n p a r t i c u l a r :

ZR = μ~ lR(a, b, μ, μ 0) , RZ = μό

lR(a, b, μ, μ 0)

/ ι ( μ , μ ' ) / ι ( - μ , - μ ' ) / ι(μ, μ ' ) is f o r w a r d r e d i s t r i b u t i o n f u n c t i o n , s y m m e t r i c , d e Hf 4 μ μ ' 4 μ μ ' f ined i n S e c t i o n 5 .2 .1 , E q . ( 1 )

η(μ, -μ') 7 ι ( - μ , μ ' ) Λ(μ, — μ ' ) is b a c k w a r d r e d i s t r i b u t i o n f u n c t i o n , s y m m e t r i c , Hb 4 μ μ ' 4 μ μ ' d e f i n e d i n S e c t i o n 5 .2 .1 , E q . (1 )

A / ( τ , - μ , μ 0) I n t e n s i t y a t d e p t h τ i n a n u p w a r d d i r e c t i o n , n o t s y m m e t r i c in μ a n d μ 0

D / ( τ , μ, μ 0) I n t e n s i t y a t d e p t h τ i n a d o w n w a r d d i r e c t i o n , n o t s y m m e t r i c i n

μ a n d μ 0 V e c t o r

/ / ( μ ) o r 7 ( μ 0) A n y i n t e n s i t y d i s t r i b u t i o n i n o n e h e m i s p h e r e ; a f u r t h e r s p e c i fication (e .g. , u p , d o w n , i n c i d e n t , e m e r g e n t ) is r e q u i r e d t o

spec i fy w h i c h h e m i s p h e r e

Ρ Ρ(μ) F o r w a r d p a r t o f n o r m a l i z e d d i f f u s i o n p a t t e r n

Q Ρ(-μ) B a c k w a r d p a r t o f n o r m a l i z e d d i f fu s ion p a t t e r n

κ Κ(α, μ ) I n j e c t i o n f u n c t i o n o r e s c a p e f u n c t i o n

Γ^μοΓ1

A s r i g h t f a c t o r : i n c i d e n t r a d i a t i o n a s e m i t t e d f r o m a n a r r o w

Ν l a y e r o f i s o t r o p i c s o u r c e s

1 ( 2 μ ) -

χ A s left f a c t o r : o p e r a t o r d e f i n i n g h e m i s p h e r i c a l a v e r a g e

U 1 A s r i g h t f a c t o r : i n c i d e n t r a d i a t i o n f r o m s t a n d a r d w h i t e ( L a m b e r t ) s u r f a c e ; a s left f a c t o r : o p e r a t o r d e f i n i n g flux/π

t h r o u g h a h o r i z o n t a l u n i t a r e a

W μ 0 o r μ j O p e r a t o r s d e f i n i n g h i g h e r m o m e n t s ; wi l l b e u s e d o n l y in t h e

V μΐ o r μ

2] t r a n s i t i o n t o c o n s e r v a t i v e s c a t t e r i n g ( S e c t i o n 5.4)

This is a function of μ 0 expressing the fraction of the incident flux reflected by the atmosphere, if incidence occurs from direction μ 0. Often this integral is simply called the albedo of the atmosphere.

RU = (R operating on U)

This is a function of μ expressing the intensity reflected in direction μ by an atmosphere exposed to incident radiation with uniform intensity from all directions in the hemisphere.


5.2 BASIC CONCEPTS AND RELATIONS

5.2.1 Physical Definitions

We shall now, within the limitations described in the preceding section, introduce the basic concepts and derive the formulas by which they are related.

The redistribution function h(u, v) is the product of albedo and phase function averaged over azimuth (see Display 4.4)

ii, i?) = ^ - Γ αΦ[ιιι; + (1 - w2)1 / 2

( l - v2)1''

2 cos((? - φ')] άφ (1)

2π J 0 The expression in brackets is the cosine of the angle α between the incoming direction (ν, φ') and the outgoing direction (w, φ) in any individual scattering event, and <I>(cos a) is the phase function. Consequently we have

integrals of phase function

I f1 I f

1

- 0(cos a) d(cos a) = 1 , - <P(cos a)cos ad(cos a) = g 2 J _ ι 2 J _ ι

integrals of redistribution function

I f1 I f

1

- J h(u, v)dv = a, - J h(u, v)v dv = agu same, in shorthand notat ion of Display 5.1

U(H( + Hb) = 2aN, W(H( - Hh) = agU (2)

Box 1 of Fig. 5.1 shows in these notations the effects of a thin scattering and/or absorbing layer, depth άτ, on an arbitrary radiation field incident from below. The unscattered radiation is weakened to (1 — Ζάτ)Ι, but part of the energy reappears as radiation scattered into the forward hemisphere (H{I dx) or into the backward hemisphere (HbI dx). The full intensity emerging in any forward direction thus is / + ( — Ζ + Hf)I dx.

A diffusion domain is physically defined as the range of optical depths deep inside an atmosphere, far enough from the boundaries (and from internal source layers) to let the radiation be transported by the simplest diffusion mode, as if the medium were unbounded. See Section 6.2.1 for more detail.

Let the medium be nonconservative, a < 1. The physically plausible form for the intensity in the diffusion domain is

/(τ, m) = Sle-kTP(u) + s2e

kTP(-u) (3)

where k is the diffusion exponent ( f c-1 the diffusion length), P(w) the diffusion

pattern, s1 the strength of the diffusion stream in the positive τ direction, and 52 the strength of the diffusion stream in the negative τ direction. The values s1 and s2 depend on the zero level from which τ is measured.

Fig

. 5.

1.

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Equation (3) is based on the assumption that any energy pumped into the atmosphere must somehow be transported and that, given enough optical depth, the transport process will settle down to a fixed angular pattern and fixed damping, which is not as strong as in a straight beam (i.e., k < 1). The pattern must have a forward bias, i.e., P ( l ) > P(— 1), and probably should be a positive steadily increasing function of u.

The integral equation from which P(u) and k follow will appear in the course of the derivation [Eq. (12)]. The same integral equation also gives the higher modes, if any exist, characterized by stronger damping, i.e., by a larger value of k and a different function P(u) that changes sign.

We now turn to box 2 of Figure 5.1. It shows in shorthand notation a diffusion stream of strength 1 in the negative τ direction, not accompanied by any stream in the positive τ direction. The intensity at the level τ = 0, which is drawn as a gap for clarity, is by definition Ρ up and Q down.

The reflection function ^^(μ, μ0) and the injection function Κ(μ) are physically defined in box 3 of Figure 5.1. If light, normalized to flux π, is conically incident at an angle with the normal of cosine μ0, then the intensity reflected from a semi-infinite atmosphere in direction μ is Λ «Χμ, /^ο)· The same incident light sets up an internal radiation field. This field has a complicated form in the layers near the surface. But as the radiation trickles down and penetrates deep inside the atmosphere, i.e., in the diffusion domain, it assumes the character of a downward diffusion stream. The yet unknown strength of this stream is defined as Κ(μ0). Applying these same definitions to light incident from all directions with intensity /(μ), we obtain the reflected intensity R^I and the diffusion stream KI, as shown in box 3.

Careful attention must be given to box 4, which describes the effects of a diffusion stream of strength 1 approaching a free surface. Par t of the radiation escapes with an angular intensity pat tern which we call mK. The coefficient m is yet unknown, but the factor Κ = Κ(μ) must be the same as in box 3 because of reciprocity. Hence we also refer to Κ(μ) as the escape function.

In addition, the absence of backscatter from layers outside the atmosphere upsets, in the atmospheric layers near the surface, the standard pattern of the diffusion stream. The intensity is necessarily less than in an unbounded medium. This negative correction is strong near the surface and makes its influence felt further down as an inward diffusion stream of negative strength. See van de Hulst and Terhoeve (1966) for a more complete description. The net effect on the deep layers, shown in box 4, is that the upward diffusion stream suffers an internal reflection at the surface with a negative internal reflection coefficient —/, where Ζ is a positive constant still to be determined. A fully equivalent description is that reflection occurs with a reflection factor — 1 at a virtual level at optical depth q outside the atmosphere. Then

/ = exp( -2 / c<?) (4)

We shall call q the extrapolation length. In nuclear physics q is called the

5.2 Basic Concepts and Relations 73

extrapolated endpoint. If the two terms of Eq. (3) are identified with an outgoing diffusion stream and an ingoing reflected stream of negative strength, then si = —ls2, and — q is the value of τ at which the asymptotic radiation field given by Eq. (3) would have zero radiation density.

5.2.2 Relations

The basic relations between the functions just defined follow from simple combinations of the preceding physical definitions. Boxes 5, 6, and 7 repeat the situations of boxes 2, 3, and 4 with a thin layer dx added. The strengths of the diffusion streams are defined with respect to the τ = 0 level indicated. Their values follow from the simple invariance principle that, by relabeling τ, it must be possible to restore the situation in the box above. For instance, if in box 6 the level τ = 0 is chosen on top of the added layer, the injected stream is KI. The same stream referred to the τ = 0 level shown has the strength KI exp( — kdx) = KI(l - kdx).

Consider box 2. Apply definitions from boxes 3 and 4. We can write at once:

As an immediate consequence we may derive an interesting property of the diffusion pattern. By taking Ρ x Eq. (6) — Q χ Eq. (5) and employing the symmetry of R œ an d Eq . (7 ) w e find th e result , writte n bot h i n vecto r for m an d in ful l notat ion :

P = R xQ + mK

(5)

(6)

(7)

(8)

1 = KP

0 = KQ-l

(9)

Consider box 5. Appl y definition s fro m bo x 1 an d writ e

Ρ = (1 - Ζ dx + Ht dx)P(l + kdx) + Hb dx Q

Q(l + kdx) = (1 - Zdx + H(dx)Q + HbdxP(l + kdx)

Hence, keeping terms proport ional to dx:

(Z - k)P = H(P + HbQ

(Z + k)Q = HfQ + HbP

Together, these two equations represent the integral equation

(10)

(11)


This is the fundamental equation, from which the eigenvalue fe, the eigen-function P(w), and any additional eigenvalues and functions may be solved. It plays a central role in many theories.

Consider box 6. Apply definitions from boxes 1 and 3. We may then write

« J = (1 ~ Zdr + HfdrXR^ + R^^dxRJ x (1 - Zdx + Hfdx)I + Hbdxl

KI{\ -kdx) = K(l + HhdtRJ(i -Zdx + H(dx)I

This must hold for any / . The terms linear in dx give

Consider box 7. Apply definitions from boxes 1, 3, and 4 and write

mK(\ -kdx) = (l-Zdx + H{dxXRnHhdx + l )mX

-1(1 - 2kdx) = -I + KHbdxmK

from which

It would be unfair at this stage to insist on precise answers to the question of whether this set of equations allows a unique solution of R^, P, Q, K, fe, m, and I if the phase function, and therefore H{ and Hb, are given. The flavor of the derivation is in the physics and if we wish to have a more elaborate mathematical foundation, there are very adequate methods, e.g., the matrix transfer method (references in Section 4.5.3) or the method of singular eigenfunction expansions (references in Section 6.5).

The following remarks about the uniqueness of the solution of the system of Eqs. (2), (5)- ( l 1), and (13)—(15) may be made. First, a normalization convention is necessary to define Ρ and Q. This convention, given in Display 5.2, affects Κ and m, but not R^, fe, and Z. Second, the solution for k and P(u) from Eq. (12) is not necessarily unique; some data about additional roots k are found in Section 12.3.2. Third, at least some redundancy exists. For instance, Eq. (15) can be derived by subtracting Eq. (14a) χ Q from Κ x Eq. (11) and simplifying the result by means of Eqs. (6) and (8).

The adopted normalization integral is shown in Display 5.2 with two additional moments. The first and second moments are easily derived by combining Eqs. (10) and (11) with the simple identities UZ = 2N9 and WZ = U and with the integrals [Eq. (2)] of H{ and Hh. Writing the full integrals is just as easy. Higher moments can be obtained by the recurrence relations in Section

ZR^ + R œZ = H b + HfR^ + R^Ht + R„H bR{ K(Z - k) = KH^ + KH (

oo (13)

(14a)

(Z-k)K = R xHbK + H fK

KHbK = Ikl/m

(14b)

(15)

6.2.2.

5.2 Basic Concepts and Relations 75

D I S P L A Y 5 .2

M o m e n t s o f t h e D i f f u s i o n P a t t e r n

F u l l n o t a t i o n S h o r t h a n d R e s u i t S y m b o l N a m e

1 Γ

1

- J P(u)du = έ Ν ( Ρ + ρ ) = 1 N o r m a l i z a t i o n

Ι R

1

P(u)udu 2 J _ j

= i i / ( P - Q ) = (1 -a)/k = y S i m i l a r i t y p a r a m e t e r

1 f

1

P(u)u

2du

2 J - 1

= i ^ ( P + Q ) = (1 - a)(\ - ag)/k

2 = D D i f f u s i o n coef f i c ien t

The normalization integral occurs in the radiation density, the first moment y in the net flux, and the second moment D in what Eddington called the Κ integral, all referring to a simple diffusion stream in an unbounded medium. The use of the similarity parameter is explained in Sections 12.2.4 and 14.1.1.

5.2.3 The Diffusion Approximation

As a side point, let us show the relation of these equations to the diffusion approximation, which is one of the approximation methods for solving transfer problems. In plane geometry the equation of transfer is

u dl(u)/dx = -I(u) + J(u) (16)

with

J(M) = l f h(u,v)I(v)dv (17)

Defining

e = \ Γ / ( m ) du' *f=\ f du> κ = \ f,w2/(w) du

we find, upon integrating Eq. (16) with the factors 1 and u in the integrand:

UF/dx = - ( 1 - a)E, dK/dx = - £ ( 1 - ag)F (18)

These equations are still exact. The factor £ appearing with F in the definition and in the differential equations has been inserted in order to remain consistent with the symbol F used in nF, net flux, throughout this book and in most classical texts.

In the rigorous solution of any problem, the ratio K/E is a function of optical depth, angle of incidence, etc. The diffusion approximation occurs when we assume that this ratio is constant and equal to D, which is the value it must have in a diffusion domain (far from boundaries or sources). If this approximation is made, Κ may be replaced by DE.


The system [Eqs. (18)] is then closed and takes the form

d2Ejdx

2 = (1 - a)(l - ag)E/D (19)

which can be solved by standard methods. The famous Eddington approximation is based on D = which is the correct

value for g = 0, a = 1. In other circumstances it would be more appropriate to choose the value from Display 5.2; see for isotropic scattering also the values of D in Table 5, Section 8.2.1. A multitude of variations is possible. See, e.g., Meador and Weaver (1977) for a practical improved diffusion approximation.

5.3 VERY THICK LAYERS

We maintain the assumption a < 1 and consider a layer of finite total thickness b, large enough to permit a diffusion domain to exist between the boundary zones near top and bottom. N o restriction is placed on the product kb: the losses of the diffusion stream in penetrating the entire layer may be large, intermediate, or almost negligible.

We assume radiation with intensity 7(μ0) incident on the top surface and absence of internal sources. Figure 5.2 defines the two unknown diffusion streams. They have strengths d and — c when τ = 0 is the top surface, but d exp( — kb) and — c exp(kb) when τ = 0 is the bot tom surface. By the definitions from Fig. 5.2, boxes 3 and 4, we may write

RI = R^I - cmK

ΤI = dexp(-kb)mK

d = KI + lc

— c exp(feft) = —Id exp( — kb)

These four equations must be satisfied for arbitrary I. This permits the four unknowns to be expressed in terms of quantities pertaining to the semi-infinite atmosphere.

R = R„- mf(l -f2ylexp(-kb)K-K (20)

Τ = m(l -f2)~

lexp(-kb)K-K (21)

d = (l -f2Y

lKI (22)

c = / ( l -f2Y

lexp{-kb)KI (23)

where Κ · Κ is matrix notation for the product Κ(μ)Κ(μ0), and

/= le~kb = e~

k ( b + 2 q) (24)

Upon substitution of s1 = d, s2 = — c into Eq. (5), we obtain the intensity at any depth and direction in the diffusion domain. Representing this by

5.3 Very Thick Layers 77

F i g . 5 . 2 . T h e r a d i a t i o n field we l l i n s i d e a v e r y t h i c k h o m o g e n e o u s l a y e r is t h e s u m o f a

p o s i t i v e d i f fu s ion s t r e a m d o w n (d) a n d a n e g a t i v e d i f f u s i o n s t r e a m u p ( — c).

DI for u > 0, μ = u (downward intensity), and by AI for u < 0, μ = — u (upward intensity), we obtain the asymmetric matrices

D = (1 -ΡΥι[β'^Ρ -fe

k(T-b)Q] · Κ (25)

A = (1 -f2ylle-

kTQ -fe

k(T~

b)P] · Κ (26)

The average intensity at the midlayer level, τ = b/2, is

±N(A + D)7 = (1 +fYle-

kbl2KI (27)

and the net flux/π flowing down through this level is

U(D - A)I = 4(1 - a)k-\\ -fyl

e-kbl2

KI (28)

Heavy losses through the atmosphere correspond to kb > 1, and hence f < 1. The terms remaining when f = 0 correspond to a semi-infinite a tmosphere. The extra terms due to / φ 0 can be visualized as a consequence of repeated internal reflections of the diffusion stream, which lead to geometric series. That exactly these sums must appear, namely (1 — f

2)~

l in Eqs. (20)

and (21), and (1 + / ) "1 and (1 -f)~

l in Eqs. (27) and (28) can be verified at

once from Figs. 5.1 and 5.2. The derivation just completed follows closely van de Hulst (1968a). Many

of these equations had also been derived by other methods by Germogenova (1961), Mullikin (1968), and Sobolev (1968). The connection with earlier work of Kuscer, Maslennikov, Mullikin, McCormick, and others is explained by van de Hulst (1968a). A concise review is also given by Sobolev (1975, pp. 60-65). Parallel developments, in particular by Germogenova and Konovalov, are referred to by Lenoble (1977).

Some authors prefer to write the expressions 1 — f2, etc. in terms of hyper

bolic functions. The translation formulas are easy.


In consulting Sobolev's or Germogenova's work it should be noted that, unfortunately, the normalizations differ from ours. Sobolev's escape function has an additional factor a (his λ\ compared to our Κ(μ\ his diffusion pattern has a factor a "

1 compared to our P(u\ and his M equals our m/a

2. This is not

apparent in most of the equations. For instance, his Eq. (2.77) shows no extra factors compared with Eq. (6) above, but the actual translation in my shorthand notation is :

(P/a) = (M/a2)(Ka) + (Q/a)R

Finally we may add that an atmosphere consisting of several homogeneous layers, each of them very thick, can be treated in the same way. One option is to write the equations at each interface and to solve for the scalars that define the up and down diffusion streams in each sublayer. Alternatively, we may apply the adding method. Such equations have been worked out by Germogenova and Konovalov (1974) and by Ivanov (1976a,b).

5.4 TRANSITION TO CONSERVATIVE SCATTERING

5.4.1 Expansions for Nearly Conservative Scattering

Conservative scattering, a = 1, k = 0, was excluded from the preceding derivations for obvious mathematical reasons. The physical reason for breakdown is that in this limit a positive diffusion stream down is indistinguishable from a negative diffusion stream up. Rather than treat this case separately (for which, of course, good methods are available), we shall first present some expansions in /c, and in Section 5.4.3 make the transitions k -+ 0. A positive value k <ζ 1 corresponds to an albedo very slightly below 1, i.e., to very weak losses in the atmosphere. This case is not only of interest in making the transition to k 0, a -> 1, but is also of practical importance ; e.g., light scattering in terrestrial clouds falls in this category.

Consider a set of problems in which the phase function keeps the same form but the albedo is varied. The coefficients ωη in the expansion of αΦ in terms of Legendre functions may, for this set, be written as

con = (In + \)abn (29)

where b0 = 1, bn is fixed, and a variable. All functions defined in the preceding sections then depend on a. Among them is the main diffusion exponent k. It is plausible to assume that k increases steadily with decreasing a. This makes it possible to consider k rather than a as the independent variable.

It now turns out that all quantities defined above, which include matrices, vectors, and scalars, can be written in the form of power series in /c, where the coefficients are again matrices, vectors, or scalars. This statement hardly requires proof, for it is made plausible (a) by the solution in the isotropic case,

5.4 Transition to Conservative Scattering 79

(b) by the form of the recurrence relations for gn in Mullikin's (1964b) theory, (c) by the fact that similar expansions have proved useful in neutron scattering (Kuscer, 1967), and (d) by numerical checks presented below. A fuller discussion is given in Section 12.3.2.

We now state without derivation the first few terms of the expansions of each quantity. They can be verified by substitution into Eqs. (5)—(15). The modifications necessary if we wish to read for k another (not the smallest) root of the characteristic equations are also discussed in Section 12.3.2.

The expansions of quantities pertaining to the unbounded medium can be derived with relative ease from the recurrence relations for gn in Mullikin's formulas. The key simplification arises from the fact that the coefficient of k

n

in the pattern P(u) is expressible in a series of (odd or even) Legendre functions up to order η at most. We find

3(1 - g) 45(1 - gf(\ - h)

m - 8

/: , 8 ( 1 6 - 2 1 ^ 4 - 5 ^ ) m- 3 ( l - g )

k+ 45(1 - g)

3(l - h)

+ °

(" } ( 3 1)

P(u) = 1 + - i - ku + — \ - k2P2(u) + 0 ( / c

3) (32)

I - g 3(1 - g)(l - h)

where g = bx is the asymmetry factor, h = b2, and P2(u) = 1 + 3M2).

The expansions of R œ an d Κ are not so obvious. Let quantities for the conservative case (a = 1, k = 0) be represented by the subscript 0. Imagine that the reflection function R(l, oo, μ, μ0) = R0 (we now omit the index oo) has been found from the integral Eq. (13). Then K(l, μ), which is written in vector notat ion as X 0, may be found from the equation mK = (P - Q) + R^P - 6 ) , which is based on Eqs. (5) and (6). Upon substitution of the dominant terms from Eqs. (31) and (32) we find

K0 = frW+R0W) (33)

Further defining

qo = LW - g)lK0W (34)

we find the expansions

R = R0 - [4k/3(l - g)-]K0 · K0 + 0 ( k2) (35)

K = (l - q0k)K0 + 0(k2) (36)

/ = 1 - 2q0k + 2q2

0k2 + 0 ( k

3) (37)

q = q0 + 0(k2) (38)

All of these results can be found from Eqs. (5)-(15) by substituting power series with unknown coefficients. Subsequent coefficients probably involve


solutions of new integral equations, which may not be expressible in terms of the functions or constants already defined. However, some further terms of the moments and bimoments can be expressed without recourse to such new functions. For instance,

5.4.2 Practical Formulas for Combined Losses Due to Finite Depth and Weak Absorption

Both Eqs. (20) and (35) refer to a situation in which weak losses reduce the reflection function slightly below the reflection function # 0( μ , βο) for the conservative case. In Eq. (20) the losses are due to the fact that radiation leaks out of the atmosphere at the bot tom side. In Eq. (35) the losses arise from a small probability of absorption at every single scattering event. At first sight it seems puzzling that the correction terms arising from losses of such a different nature have such a similar form.

The explanation is that both these situations refer to " d e e p " losses. The leak from the bot tom surface occurs by definition deep down in the atmosphere (if b > 1). The influence of this leak propagates through the atmosphere in the form of an upward diffusion stream of negative strength, and this stream "escapes" from the surface with an angular pattern proport ional to Κ(μ). The factor Κ(μ0) enters for symmetry reasons. In the other situation, the small losses at every single scattering reach a large cumulative effect only after many successive scatterings. This means that the energy sink is again effectively located deep down in the atmosphere. Its influence is, therefore, described by the same functions of μ and μ 0.

The next question is, how do these two kinds of losses combine? The answer, like the other equations in this chapter, is valid for any phase function, but it is not simple. It is necessary to start from Eq. (20) and introduce into this equation expressions from Eqs. (24), (31), (35), and (36). The tricky point is that, if both quantities k and (b 4- 2q)~

l are < 1 , the exponent k(b 4- 2q) still assumes all

values from 0 to oo ; thus , /var ies from 0 to 1. Therefore, the factors containing / cannot be simplified.

As a practical expression for the reflection function with combined losses, we propose the equation first given by Danielson et al (1969) for application to planetary cloud layers:

R(a, b, μ, μο) = R(h oo, μ, μ 0) - [ 0 . 5 /1'2 + /

2/ ( l - /

2) ] m X ( l , μ)Χ(1, μ0)

Here / = /(α), m = m(a\ a n d / = f(a,b\ as before, and μ) is the escape function for conservative scattering (abbreviated K0). The corresponding

URU = 1 + [4/3(1 - 0 ) ] ( - / c + q0k2) + 0(fc

3)

KU = 1 - q0k + (K0 V)k2 + 0(k

3)

(39)

(40)

(41)

5.4 Transition to Conservative Scattering 81

equation for the bimoment which expresses the reflected flux in the case of uniform incidence is

URU(a,b) = 1 - [ 0 . 5 /1 /2

+ /2/ ( l - /

2) ] m (42)

The status of these equations should be carefully noted. Equation (41) has a hybrid character. It is rigorous up to first order terms in k. If this is all we want, we may drop the factor /

1 / 2, because m contains a factor k and 1(1) = 1. However,

the corresponding bimoment Eq. (42) is rigorous up to terms proport ional to k2. This may be checked by using Eqs. (39) and (40) together with Eqs. (20),

(35), and (36). The conclusion is that an equation like Eq. (41), but correctly including the second-order terms in fc, would not show the same dependence on direction expressed by the product K(l, μ)Κ(1, μ0). The factor Z

1 /2 in Eq. (41)

assures that errors 0(k2) made by Eq. (41) in any individual set of directions

(μ, μ0) remain small and of varying sign because their bimoment vanishes. The corner of the a,b plane defined by a = 1, b = oo, in which all of the in

cident radiation is reflected without loss, may well be called " the nasty corner." Near this corner the losses due to large but finite depth and those due to slightly nonconservative scattering are intimately connected through the exponent kb contained in the factor / . It would therefore be wrong to assume that we can compute each of these losses independently from the other and then simply add.

I have mapped the losses near this corner in one numerical example in Fig. 19.11. Since this is based on first-order theory, the factor /

1 /2 in Eq. (41) may be

omitted, and the expression within brackets is independent of the assumed phase function. Hence, with a relabeling of the coordinates, the curves of Fig. 19.11 are valid for thick, homogeneous atmospheres with any chosen phase function. For further comments, see Section 19.4.2.

5.4.3 Conservative Scattering

The asymptotic expressions for conservative scattering are simpler and have been derived earlier (Piotrowski, 1955, 1956; Sobolev, 1957) than those for nonconservative scattering. They can be obtained from those in the preceding sections by making the transition k -> 0. Assume an upward diffusion stream c which is reflected at the top boundary into a downward diffusion stream — le. Keeping only terms linear in /c, we find from Eqs. (3), (32), and (37) the intensity in the diffusion domain

/(τ, u) = c(l + kz)[l - ku/(l - g}] - (1 - 2q0k)c(l - *τ)[1 + ku/(l - gj]

By integration over u this gives the net upward flux, independent of τ :

nF = Snkc/3(1 - g)

82 5 Very Thick Layers with Arbitrary Ansiotropic Scattering

The constant c thus has to be large in order to compensate for the near cancellation of the radiation fields of the upward and downward diffusion stream, caused by the smallness of k. Eliminating c and taking the limit k = 0, we find the intensity in the diffusion domain

7( t ,k ) = | F [ ( 1 - </)(τ + 4o) " " ] (43)

This linear form in τ and u is the rigorous asymptotic expression for any phase function. In the same way we find that this flux emerges from the surface with the intensity distribution

7 ( 0 , - μ ) = ΡΚ(1 ,μ) (44)

where the factor F results from the product mc by Eq. (31). The finite-layer Eqs. (25)-(33) are adapted to conservative scattering by

wr i t i ng / = 1 — (b 4- 2q0)k and then letting k 0. In particular:

o o , μ, μ0) - b, μ, μ0) = Γ(1 , b, μ, μ0) = *f ( 1 ' (

^ ° \ (45) 3(1 - g)(b + 2q0)

again a familiar result. The midlayer intensity for the case of normalized incident radiation from direction μ 0 becomes

(D and A) = u) = β + u/(l - <?)(*> + 2^0) ]K(1 , μ0) (46)

This formula is useful in the process of asymptotic fitting described in Section 5.6.

In the conservative case, both Eqs. (5) and (6) reduce to

R0 U = U\ f R(l oo, μ, μ0) 2 μ0 άμ0 = 1 (47) Jo

which means that a conservative, semi-infinite atmosphere reflects the entire incident flux, independently of the direction of incidence or the form of the phase function. For convenience, the shorthand form and the full form of the same equation have been written side by side. Similarly, Eqs. (7), (8), or (40) reduce to

K0U = l, f Κ(1μ)2μάμ= 1 (48) J o

which means that the entire net flux carried towards the surface from large optical depth escapes.

Because of Eq. (48), the quantity K0 W may be interpreted as the average value of μ in the quanta escaping from a conservative atmosphere emitting a constant net flux. The same quantity equals (1 — g)q0 according to Eq. (34), which can be transformed by Eq. (33) into

(1 - g)q0 = K0 W = K i + WR0 W) (49)

5.5 Internal Source Layer 83

As a preview of Section 12.1.3 (Display 12.1) we note that this equation suffices for a fairly accurate evaluation, even if we should have only a wild guess about R0. The "rapid-guess formula" gives

(1 _ g)qo = i j + = 0.708 + 0.005c

The actual values for various phase functions cluster together between 0.709 and 0.715 and are discussed in Section 12.1.2 (Table 30 and Fig. 12.1).

5.5 INTERNAL SOURCE LAYER

The method applied in the preceding sections to the standard problem with radiation incident from outside is equally useful in treating problems with internal source layers.

Figure 5.3 shows a (transparent) layer of isotropic sources, emitting to both sides the intensity | u | "

1 = 2N. This source layer is placed at τ = 0 in an un

bounded medium, creating at its position the actual intensity represented by the vector S which satisfies the equation (see Fig. 5.3)

S-2N = R^S (50)

A possible interpretation of Eq. (50) is that S is made up of 2N and the fields produced by multiple reflection against both sides. At sufficient distance from the source layer, this radiation sets up the diffusion stream KS. Write Eq. (50) in the form

2ΛΓ = (1 — RJS

and multiply both sides by the vector Ρ + Q. Then by Eqs. (5), (6) and Display 5.2 we find

KS = SK = 4m-1 (51)

which is the sought strength of the diffusion stream.

, . , . ί ; . source layer S - 2 N ^

F i g . 5.3. R a d i a t i o n field se t u p b y a l a y e r of i s o t r o p i c s o u r c e s i n s i d e a h o m o g e n e o u s m e d i u m .


Again, a much larger effort is required to find the radiation field at small distances from the source layer. We must then apply Case's method with the higher modes included. An explicit solution based on a similar method was presented by Vanmassenhove and Grosjean (1967).

As an example of the use of Eq. (51) let us compute the asymptotic formula for the point-direction gain. Place the source layer just described at sufficiently large depth τ ΐ5 in a semi-infinite medium. The upgoing diffusion stream gives, by the definition of box 4, the emerging intensity

We show in Display 5.3, in the box a < 1, b = oo the corresponding po in t -direction gain, as defined in Section 3.4.3. This result may be verified in the reciprocal situation by computing from Display 5.2 and box 3 the radiation density at depth τ caused by radiation incident from μ 0 and dividing this by the radiation density in the incident beam.

The corresponding expression for finite b in Display 5.3, valid if both τί and b — τ χ are sufficiently large, may be found from a derivation very similar to that in Section 5.3. To check, we may also derive it from the reciprocal experiment, by dividing the radiation density of Eq. (25) + Eq. (26) by the radiation density in the incident beam. We then obtain the same result with τ χ replaced by τ and μ by μ 0.

The transition to the conservative case by k 0 is obvious. We may check the simple result for b = oo, a = 1 as follows: The source layer emitting the intensity μ

-1 = 2N to either side (up and down) emits the flux 2NU = 2 to

either side. None of this is lost, and because the layer is semi-infinite, the flux 4 will eventually emerge from the top surface, where it has the intensity distribution 4Χ(μ). Division by the original (one-sided) intensity μ "

1 then gives the

result. It is independent of τί9 provided τί > 1. For the midlayer point, τ 0 = \b, of a conservative finite slab, the same

reasoning applies, except for a factor \ because the flux escapes symmetrically to both sides.

D I S P L A Y 5 . 3

A s y m p t o t i c E x p r e s s i o n s for P o i n t - D i r e c t i o n G a i n

0

Am-lmKe-

ktx

F i n i t e b; G(a, b, x u μ) b = oo ; G(a, oo, τ ί, μ)

N o n c o n s e r v a t i v e

a < 1 4μΚ(α, \i)e 4μΚ(α, μ)β

C o n s e r v a t i v e

a = 1 4μΚ(1,μ)

b + q Q~ τχ b + 2q0

4 / Λ φ , μ )

a P o i n t : τί = o p t i c a l d e p t h f r o m t o p s u r f a c e , τχ > \,b - τχ > \. D i r e c t i o n :

μ = c o s i n e of a n g l e f r o m n o r m a l a t t o p s u r f a c e .

5.6 Asymptotic Fitting 85

Some of the restrictive assumptions made above may be lifted without spoiling the simplicity of the result. For instance, Ivanov (1974) considers a layer of sources emitting in one direction u0 and placed inside a semi-infinite medium at a depth τ 0, which may be anywhere from near the surface to far down. The bare source layer emits the intensity à(u — u0). This source layer sets up a radiation field everywhere, which must at large depth (τ > 1 and τ > τ 0) have the character of an inward diffusion stream of s t r e n g t h / ( τ 0, w0). The form of this function is yet unknown, but we can derive it by considering the reciprocal experiment. This is the classical Milne problem, in which sources at finite depth are absent and an outward diffusion stream, say of strength 1, approaches the surface. The intensity emerging at the surface (Section 5.2.1) is then mK(p\ and we may write the corresponding intensity at any depth τ as ηιΙκ(τ, u) so that IK(0, u) = Κ(μ) for u = -μ (μ > 0) and 0 for u > 0.

By the reciprocity principle we see that the two experiments described must lead to the same functional dependence. The full result is

/ ( τ , II) = 2Ικ(τ, -u)

Since this case of reciprocity has not been worked out in detail in Chapter 3, the constant can best be derived by considering the simpler situations in which τ = 0 or τ > 1. Note that not only Ivanov's derivation differs from ours but also his normalization by an extra factor A, the single scattering albedo, in the solution of the Milne problem and A "

1 in the diffusion pattern.

5.6 ASYMPTOTIC FITTING

This section describes a practical method (van de Hulst, 1968b) for computing the functions and constants defined and used in this chapter. The method has been used with considerable success (see numerical data in Chapter 12). As an imput to this method, we obtain by the doubling method the numerical values of R(b, μ, μ 0) and T(b, μ, μ 0) and their moments (if so desired) for three values of b, namely b0, 2 b 0, and 4b0. If b0 is chosen large enough to make the asymptotic expressions [Eqs. (20). and (21)] from Section 5.3 correct, say, to four or six decimals, then these data suffice to find first γ = e x p ( / c è 0) + exp ( — kb 0) and subsequentl y ζ = exp( — kb0) by solving quadratic equations. All other quantities then follow easily. The procedure for the conservative case is even simpler and follows naturally from the asymptotic formulas presented in Section 5.4.3.

The fitting process thus achieved is written out in Display 5.4 in the form of a complete procedure which may be taken as the basis of a computer program. The procedure for the nonconservative case can be verified if we know that ζ = exp( — / c 6 0) , γ = z + ζ "

1, and (5 = lz

2. An enormous amount of redundancy

can be built into the program because many expressions derived as functions of


b0, μ, A*o> or all three should be asymptotically independent of these variables. The footnotes to Display 5.4 specify which results have this internal check. If b0 is not chosen large enough, small variations with the redundant parameter are noticeable. If b0 is chosen too large, cumulative errors or round-off errors may affect the numerical accuracy.

The speed of the method cannot easily be assessed in an objective manner. The asymptotic fitting process adds an insignificant amount of computing time to the time necessary for executing the doubling method, say to b = 32. The conceptual simplicity and the many internal checks are very attractive features of this method.

D I S P L A Y 5 .4

T h e A s y m p t o t i c F i t t i n g M e t h o d

R e c i p e for t h e n o n c o n s e r v a t i v e c a s e . C h o o s e a l a r g e b0 a n d find c o n s e c u t i v e l y :

S t e p 101° α = T(b0, μ, μ 0) / Τ ( 2 / ? 0, μ, μ0) 102" β= Τ(Μ0,μ,μ0)/Τ(4Η0,μ,μ0) 1 0 3 " y = ( 2 α ) -

1 + [ ( 4 α

2) -

1 + β + 1 ]

1 /2

104* ζ = y/2 - (y

2/4 - i)

i/2

105" Κ ^ 0, μ , μ 0) - Κ ^ 0, μ , μ 0)

ο = - -Τ ( 2 6 0, μ, μ0) - z

2T(4b0, μ, μ0)

106 1 = δζ~

2

1 0 7

c k= -(\nz)/b0

108 q = - ( l n / ) / 2 f c

1 0 9

d K oc (μ , Mo) = K ( 2 f c0, μ, μ 0) + ST(2b0, μ, μ 0)

1 1 0

e ηιΚ(μ)Κ(μ0) = (1 - <$

2)ζ~

2Τ ( 2 £ 0, μ, μ 0)

U \

de M0(2b0, μ 0) = J / ( 2 6 0 ,b0, ιι, μ 0) du [= ND + Λ Μ ]

Μ 1( 2 6 0, μ 0) = ί I(2b0,b0,u^0)udu [= {UD - {VA] J -1

113° Κ(μ0) = (1 + < 5 ) Μ 0( 2 6 0, μ 0) / 2 ζ

1 1 4

e k-'K^o) = (1 - 5 ) Λ ί 1( 2 6 0, μ 0) / 2 ( 1 - α)ζ

1 1 5

c/ k = Κ(μ0)/μ-*Κ(μ0Κ

116" m = [ηιΚ(μ)Κ(μ0Κ/Κ(μ)Κ(μ0) 1 1 7 * ' Ρ ( ι ι ) = [ J ( 2 f c0, b 0, u, μ 0) + δΐ (2b0, b0, -u, μ0)1/2Κ(β0)

References

D I S P L A Y 5 .4 {continued)

S t e p

2 0 1 *

R e c i p e for t h e c o n s e r v a t i v e c a s e . C h o o s e a c o n s e c u t i v e l y :

αο(ΑΊ5 μ0) =

R(

b' μο) + T(b, μ, μ 0)

l a r g e b a n d find

2 0 2

e tn(b)= \ T(b, μ, μ 0) μ μ 0 άμ άμ0 [ =

2 0 3 2 g 0 = [ 3 ( 1 - 9) r u] - ' - b

204

β< ' £ , ( 6 , μ 0) = f Τφ,μ,μ0)μάμ [ = i T l / ]

2 0 5

a<

/ Χ ( μ ) = Τφ9μ,μ0)/2ίιφ,μ0)

2 0 6

f l'

9 2q0 = μ, μ 0) / Γ ( 2 6 , μ, μ 0) - Ι ] "

1 - 1} a

T h e s e r a t i o s of t w o f u n c t i o n s of μ a n d μ 0 s h o u l d b e i n d e p e n d e n t o f μ a n d μ 0 ; t h i s p r o v i d e s m a n y c h e c k s . T h e s a m e r a t i o s h o u l d b e o b t a i n e d if n u m e r a t o r a n d d e n o m i n a t o r a r e r e p l a c e d b y i n t e g r a l s o v e r μ o r μ 0, o r b o t h . I n p r a c t i c e w e o f t e n u s e d t h e b i m o m e n t s URU a n d UTU. b

B o t h t h e s e e q u a t i o n s a r e s o l u t i o n s of q u a d r a t i c e q u a t i o n s . E x c e p t in t h e c a s e of v e r y n e a r l y c o n s e r v a t i v e s c a t t e r i n g , w h e r e ζ is c l o s e t o 1, a q u i c k e r s o l u t i o n of t h e s e e q u a t i o n s is o b t a i n e d b y e x p a n s i o n . W e e m p l o y t h e fact t h a t ζ < 1 a n d h e n c e δ < 1. T h e f o l l o w i n g f o r m u l a s a r e a p p r o p r i a t e :

ζ" ' ^

1 ,2 + ( 2

+έ ) ( ά - ^ )

+ °

( ζ 7)

Ζ = β- 1/2 _ ίβ~ 1 ( α- 1 _ β- 1/2) + 0 ( ζ7 ) c T h e s e t w o f o r m u l a s for k p r o v i d e a s e n s i t i v e c h e c k ; f o r m u l a 107 is

t h e m o r e a c c u r a t e o n e . d A n y of t h e s e f u n c t i o n s of μ a n d / o r μ 0 c a n b e i n t e g r a t e d t o find i t s

m o m e n t s . W e c a n a l s o find t h e s e m o m e n t s d i r e c t l y b y s t a r t i n g t o u s e t h e c o r r e s p o n d i n g m o m e n t a t t h e r i g h t - h a n d s i d e . I n o u r d o u b l i n g p r o g r a m , t h e m o m e n t s a n d b i m o m e n t s of R, T, a n d I w e r e r o u t i n e l y c o m p u t e d w i t h e a c h f u n c t i o n . e

I n t h e s e e q u a t i o n s t h e c o r r e s p o n d i n g e x p r e s s i o n in m a t r i x n o t a t i o n is g i v e n in s q u a r e b r a c k e t s . ;

S a m e a s

a, b u t h e r e n u m e r a t o r a n d d e n o m i n a t o r a r e f u n c t i o n s of

o n e v a r i a b l e μ 0, w h i c h d r o p s o u t in t a k i n g t h e r a t i o . 9 T h i s i n d e p e n d e n t w a y t o d e r i v e 2q0 is a d d e d h e r e fo r t h o s e w h o

w i s h t o a v o i d u s i n g t h e b i m o m e n t s .

R E F E R E N C E S

D a n i e l s o n , R . E . , M o o r e , D . R . , a n d v a n d e H u l s t , H . C . ( 1 9 6 9 ) . J. Atmos. Sci. 2 6 , 1 0 7 8 . G e r m o g e n o v a , T . A . ( 1 9 6 1 ) . Zh. Vychisl. Mat. Mat. Fiz. 1, 1 0 0 1 . G e r m o g e n o v a , Τ . Α . , a n d K o n o v a l o v , Ν . V . ( 1 9 7 4 ) . Zh. Vychisl. Mat. Mat. Fiz. 14 , 9 2 8 .


I v a n o v , V . V . ( 1 9 7 4 ) . Astrofizika 10, 193 [English transi: 10, 117] .

I v a n o v , V . V . ( 1 9 7 6 a ) . Tr. Astron. Observ. Leningrad Univ. ( R u s s i a n ) 32, 3 .

I v a n o v , V . V . ( 1 9 7 6 b ) . Tr. Astron. Observ. Leningrad Univ. 32, 2 3 .

K u s c e r , I . ( 1 9 6 7 ) . In " D e v e l o p m e n t s in T r a n s p o r t T h e o r y " ( E . I n ô n u a n d P . F . Z w e i f e l , e d . ) ,

p . 2 4 3 . A c a d e m i c P r e s s , N e w Y o r k .

L e n o b l e , J . ( 1 9 7 7 ) . S t a n d a r d P r o c e d u r e s t o C o m p u t e R a d i a t i v e T r a n s f e r in a S c a t t e r i n g A t m o

s p h e r e . R a d i a t i o n C o m m i s s i o n , I n t e r n a t i o n a l A s s o c i a t i o n o f M e t e o r o l o g y a n d A t m o s p h e r i c

P h y s i c s ( I . U . G . G . ) , p u b l i s h e d b y N a t i o n a l C e n t e r f o r A t m o s p h e r i c R e s e a r c h , B o u l d e r ,

C o l o r a d o .

M e a d o r , W . E . , a n d W e a v e r , W . R . ( 1 9 7 7 ) . J. Opt. Soc. Am. 67, 2 6 1 ( a b s t r a c t ) .

M u l l i k i n , T . W . ( 1 9 6 4 b ) . Astrophys. J. 139, 1267 .

M u l l i k i n , T . W . ( 1 9 6 8 ) . J. Appl. Probability 5, 3 5 7 .

P i o t r o w s k i , S. ( 1 9 5 5 ) . Bull. Acad. Polon. Sci. 3, 3 0 3 .

P i o t r o w s k i , S. ( 1 9 5 6 ) . Acta Astron. 6, 1 6 1 .

S o b o l e v , V . V . ( 1 9 5 7 ) . Astron. Zh. 34, 3 3 6 [English transi.: Sov. Astron.-A.J. 1, 3 3 2 ] .

S o b o l e v , V . V . ( 1 9 6 8 ) . Astron. Zh. 45, 2 5 4 [English transi.: Sov. Astron.-A.J. 12, 2 0 2 ] .

S o b o l o v , V . V . ( 1 9 7 5 ) . " L i g h t S c a t t e r i n g i n P l a n e t a r y A t m o s p h e r e s . " P e r g a m o n , O x f o r d ( o r i g i n a l

R u s s i a n 1972) .

v a n d e H u l s t , H . C . ( 1 9 6 8 a ) . Bull. Astron. Inst. Netherlands 20, 7 7 .

v a n d e H u l s t , H . C . ( 1 9 6 8 b ) . J. Comput. Phys. 3, 2 9 1 .

v a n d e H u l s t , H . C , a n d T e r h o e v e , F . G . ( 1 9 6 6 ) . Bull. Astron. Inst. Netherlands 18, 3 7 7 .

V a n m a s s e n h o v e , F . R . , a n d G r o s j e a n , C . G . ( 1 9 6 7 ) . In " I C E S I I , E l e c t r o m a g n e t i c S c a t t e r i n g "

( R . L . R o w e l l a n d R . S . S t e i n , e d s . ) , p . 7 2 1 . G o r d o n a n d B r e a c h , N e w Y o r k .

6 • Results Obtained by Expanding the

Phase Function in Legendre Polynomials

6.1 INTRODUCTION AND CONCLUSIONS

Almost all transfer problems with anisotropic scattering have first been tackled with the assumption that the phase function was expressible in a finite expansion of Legendre polynomials. The mere fact that much effort has been devoted to these problems and that some published tables are available makes it desirable to summarize the results.

Let the atmosphere be homogeneous and let the product of phase function and albedo be

Ν a<D(cos a) = ] T co„P„(cos a)

The case Ν = 0 (isotropic scattering) is entirely classical; Ν = 1 (linearly isotropic scattering) is usually taken as the next exercise. For this value and also for Ν = 2 a fair amount of tabular material, notably Η functions and their moments, are available. For Ν = 3 and higher, published numerical da ta are confined to a few examples.

The task of summarizing the relevant formulas for arbitrary (finite) Ν is not simple. Many authors prefer to solve a simpler, less general problem in order to make the problem tractable or the explanation readable.

89

90 6 Results Obtained by Phase Function in Legendre Polynomials

The main options are shown in the accompanying tabulation.

G e n e r a l p r o b l e m L i m i t e d p r o b l e m

Ν a r b i t r a r y Ν = 0 ( i s o t r o p i c s c a t t e r i n g )

All m A z i m u t h - i n d e p e n d e n t t e r m s (m = 0 ) o n l y

F i n i t e l a y e r , a n y b, a l s o c a l l e d s l a b S e m i - i n f i n i t e a t m o s p h e r e (b = oo) a l s o c a l l e d

h a l f - s p a c e

A n y v a l u e of s i n g l e s c a t t e r i n g a l b e d o a, C o n s e r v a t i v e s c a t t e r i n g (a = 1) o n l y i n c l u d i n g a = 1

F i n d r a d i a t i o n field a t a n y o p t i c a l d e p t h F i n d e m e r g i n g r a d i a t i o n ( r e f l e c t i o n , t r a n s m i s s i o n ,

e s c a p e ) a n d r a d i a t i o n d e e p i n s i d e (d i f fus ion

d o m a i n )

In this chapter I have decided to treat the general problem, except for the last distinction listed. The emphasis remains on emerging radiation, i.e., on reflection, transmission, and escape functions, as in Chapter 5. Only a brief section (Section 6.5) is devoted to the complete solution for any optical depth τ. In Section 6.4 and Display 6.1 a further simplification is made by dealing only with b = oo, m = 0. This is not an essential limitation but it allows a more concise presentation. Consistent with the plan of this book, I have reduced the derivations to a bare minimum and have tried to emphasize the physical rather than the mathematical explanations.

A block diagram (Display 6.1) may serve to show at what stage in the computation the various functions appear. The tradition in the literature is to treat problems with externally incident radiation (reflection, transmission) separately from the problem of the escape of radiation from a deep atmosphere (Milne problem). Since the solution of the second problem follows easily from the first, they are treated together here.

In isotropic scattering all steps of Display 6.1 are trivial or simple except step 3 from Ψ(μ) to Η(μ). The other steps remain fairly simple for anisotropic phase functions which are expressible in a few Legendre functions (small N). Chandrasekhar, who worked out a number of such cases in his book (1950), took in each case steps 1, 4, 5, and 6 in an ad hoc manner. This explains the strong emphasis in his book on step 3. The other steps are not trivial for general anisotropic phase functions.

The emphasis in this book is on results and on practical recipes. In that sense the block diagram just given represents one method. It should be stated, however, that when one notes the concepts and mathematical techniques used in deriving the equations, one finds that the diagram includes widely different methods. The traditional Chandrasekhar-Busbridge-Mull ikin-Sobolev method starts on a completely different footing from the Case-McCormick-Kuscer method of singular eigenfunction expansions. Yet, when they are both worked out into a procedure for computat ion they prove nearly identical. Some further details are presented in Section 6.5.

6.1 Introduction and Conclusions 91

D I S P L A Y 6 .1

B l o c k D i a g r a m of R e d u c t i o n t o H F u n c t i o n s

S t e p

j P r o p a g a t i o n i n

j u n b o u n d e d m e d i u m

1R e f l e c t i o n o n

o r e s c a p e f r o m

a s e m i - i n f i n i t e

a t m o s p h e r e

T h e s e f u n c t i o n s d o n o t h a v e

a n o b v i o u s p h y s i c a l m e a n i n g

R e s t r i c t i o n s ( n o n e s s e n t i a l ) : N o p o l a r i z a t i o n

O n l y t e r m s i n d e p e n d e n t o f a z i m u t h (m = 0 )

S e m i - i n f i n i t e a t m o s p h e r e (b = oo) O n l y s m a l l e s t k s o u g h t

D i s p l a y

Br i e f d e s c r i p t i o n

G e n e r a l W o r k e d p r o c e d u r e e x a m p l e

1

LA

lb

2 a

2 b

3

F i n d c h a r a c t e r i s t i c f u n c t i o n Ψ ( μ )

F i n d KuSCer p o l y n o m i a l s ρη(μ) a n d c h a r a c t e r i s t i c

b i n o m i a l G(v , μ )

S e t ν = μ a n d a d d f a c t o r \

S o l v e for k f r o m c h a r a c t e r i s t i c e q u a t i o n

F i n d P(u) b y s u b s t i t u t i o n

S o l v e n o n - l i n e a r i n t e g r a l e q u a t i o n o r s i n g u l a r l i n e a r

i n t e g r a l e q u a t i o n w i t h c o n s t r a i n t t o f ind Η f u n c t i o n

Η(μ)

F i n d B u s b r i d g e p o l y n o m i a l s ^„ (μ) f r o m se t o f l i n e a r

e q u a t i o n s

S u m p r o d u c t s t o f ind F ( v , μ)

F i n a l s u b s t i t u t i o n s g i v e r e f l e c t i o n a n d e s c a p e f u n c t i o n s a n d a l l r e q u i r e d c o n s t a n t s

6.2 6 .3 /6 .4

J-(see S e c t i o n 6.2.3)

6.7 -

6.9

6.8

6 .8 /6 .11

6 .10

This leads automatically to the question: How useful is this method in practical computa t ion? Fo r simple phase functions, say with Ν = 0 and 1, the method is fast and elegant. For Ν = 2 and 3 it is still feasible. Regretfully, the conclusion for general phase functions is that it is not useful to make the reduction to Η functions at all. The same assessment holds for the reduction to X and Y functions in the case of a finite atmosphere.

First, the computat ion is not yet complete when we have obtained the H function (or the X and Y functions) from the corresponding integral equations— a step fully clarified by Mullikin's work. Finding and administrating the


Busbridge polynomials (or the corresponding two sets of polynomials for a finite atmosphere) means an additional work load. The equations are available but the solutions for all the moments are cumbersome.

Further support for the same conclusion is that, even if we could find these polynomials with ease, the final form of the reflection function Ν = oo, b = oo is

^{μ -H μ0) n = 0

The simplicity of the first factor is marred by the fact that the second factor, which we have called F(/z, μ0) is, like Κ(μ, μ0) itself, a function of two variables with no attractive properties besides its symmetry. We have therefore gained nothing by separating out the factors Η(μ) and Η(μ0). We believe that for such phase functions the method offers no advantage. A slightly milder conclusion was reached by Kuscer and McCormick (1974), who feel that " the factorization at least retains an aesthetic justification."

6.2 UNBOUNDED MEDIUM

6.2.1 Radiation Field in a Diffusion Domain

A fundamental concept in the theory of radiative transfer is diffusion through an unbounded homogeneous medium. In many rigorous treatments of transfer through bounded or finite layers, this concept has been the starting point. We have derived in Section 5.2.2 the equations satisfied by the constant k and function P(u) describing this diffusion. But we shall first review what the radiation field in this solution looks like.

Let the medium have no boundaries or sources (except at infinity) and let it be characterized at each point by the same single-scattering albedo and phase function. We seek the simplest solution of the equation of transfer in which an actual net flux flows in the positive or negative τ direction. In a conservative medium this solution is linear in τ, the optical depth, and in w, the cosine of the angle with the positive τ axis. In a lossy medium it factorizes into a function of τ and a function of w, which means that the shape of the radiation pattern at each depth is the same.

If ω 0 = 1, the medium is conservative, the flux does not depend on τ, but the intensity depends linearly on both τ and u. The general solution then is

/(τ? u) = A — | F ( 1 — g)x + |Fw intensity

j ( T , u) = A — | F ( 1 — g)x + \Fgu source function

6.2 Unbounded Medium 93

where A and F are arbitrary constants, F is the net flux in the positive τ direction, and g = ωχβ the asymmetry factor of the phase function. A net flux in negative τ direction is obtained by taking F negative. This is the complete and general solution of the radiation field in a diffusion domain with conservative scattering.

If ω 0 < 1, the medium is lossy. The general radiation field in a diffusion domain then is the superposition of a diffusion stream in the positive τ direction and one in the negative τ direction, with arbitrary strengths s t and s 2:

/ (τ , u) = s^i^e'^ + s2P(-u)ekT intensity

J ( T , U) = SI ( l — ku)P(u)e~kx + s 2( l + ku)P( — u)e

kt source function

The attenuation in both streams occurs exponentially and is characterized by the (positive) diffusion exponent k. The reciprocal, written as y = fe

_1, is the

diffusion length, expressed in mean free paths. We define k as the smallest positive eigenvalue of an integral equation [Section 5.2.2, Eq. (12)]

and P(u) as the corresponding eigenfunction. Since the existence of the constant k is physically obvious, we need not worry about the mathematical problem to prove that such a root of the characteristic equation always exists.

Generally, several discrete and a set of continuous eigenvalues and eigenfunctions may exist. In the method of singular eigenfunction expansions (Sections 4.2 and 6.5) they are all used to construct the complete solution. The smallest value k is the only one which persists if we go deep inside the medium and hence is the only one we need in the present section. For the same reason, we can forget about the azimuth-dependent solutions, which are damped strongly (Section 15.3.2).

Physically defined quantities such as k and the associated pattern P(u) can in principle be determined by experiment. The numerical computat ion by means of doubling and asymptotic fitting (Sections 4.5 and 5.6) may be regarded as an experiment yielding these quantities with four- or five-figure accuracy in typical cases. The more direct method is to solve for k and P(u) from the integral Eq. (10) of Section 5.2.2. Several ways of doing so are described in Section 6.2.3.

The physical background of this integral equation can be simply understood. Generally (Display 4.4), J follows from / by a redistribution over the directions involving h(u, v\ and J follows in turn from J by an integration over optical depth. The equations here remain simple because: (1) absence of external or embedded sources makes this computat ion a closed loop, i.e., leads to a homogeneous integral equat ion; and (2) the exponential dependence on ζ makes possible the integration over optical depth, which leads to a factor 1 — ku.

The properties of the redistribution function were derived by Ambartsumian (1941), and the integral equation for the diffusion pattern by Ambartsumian (1942,1944). However, there may well be earlier papers with equivalent results.


6.2.2 Characteristic Function, Kuséer Polynomials

The key function in the traditional method is the characteristic function Ψ(μ), which has no obvious physical explanation. Characteristic functions occur in a wide variety of transfer problems, including those with polarization (Chandrasekhar, 1950) and with redistribution over the frequencies (Ivanov, 1973). We continue the discussion of unpolarized anisotropic scattering in a homogeneous atmosphere.

The most direct procedure for finding the characteristic function for arbitrary m is presented in Display 6.2. Fo r an azimuth-independent radiation field m = 0, omit the upper index m everywhere and use the simplifications shown.

D I S P L A Y 6 . 2

R e c i p e for F i n d i n g K u s c e r P o l y n o m i a l s a n d C h a r a c t e r i s t i c F u n c t i o n for U n p o l a r i z e d P h a s e F u n c t i o n

S t e p 1 G i v e n : coe f f i c i en t s ω„ (η = 0 , 1 , . . . , Ν), t o b e f o u n d if n e c e s s a r y b y e x p a n s i o n o f g i v e n

S t e p 2 F i n d h„ = In + 1 — ωη S t e p 3 C h o o s e m (m = 0, 1 , . . . , N): m = 0 fo r a z i m u t h - i n d e p e n d e n t t e r m

S t e p 4 F i n d c j = ω/J - m) + m ) !, ( ; = m, m + 1 , . . . , Ν)

fo r m = 0 ; c , = ω7·

S t e p 5 F i n d ρ%(μ) = (1 - μ

1)'

τηΙ2Ρ^(μ) b y r e c u r r e n c e :

p h a s e f u n c t i o n (N m a y b e oo ) :

Ν

a<D(cos a ) = £ w„Pn(cos a )

S t e p 6

ρΖ(μ)= 1 · 3 · 5 · · · ( 2 m - 1) for m = 0 : ρ0(μ) = 1

Pm+ ι(μ) = 1 · 3 · 5 · · · ( 2 m + 1)μ ρ^μ) = μ (2j + Ι)μρ^(μ) = (j - m + 1)ρ7+ 100 + 0" + *η)ρ?-ι(μ)

F i n d K u s c e r p o l y n o m i a l s 0™(μ) b y r e c u r r e n c e :

9Ζ(μ)= 1 · 3 · 5 . · · ( 2 ΐ ϋ - 1) fo r m = 0 : βο(μ) = 1

Gm+ ι(μ) = 1 · 3 · 5 · · · ( 2 m - \)Ηημ gx(p) = h0p

hjféjfa) = {j-m + 1 )^+ 1(μ ) + 0 + m)g7-i(ti

S t e p 7 F i n d c h a r a c t e r i s t i c b i n o m i a l :

Ν

S t e p 8 F i n d c h a r a c t e r i s t i c f u n c t i o n :

S t e p 9

Ψ(μ) = Κΐ - μ

2Γ Ο " θ £ , μ )

O n l y fo r m = 0, iV = f in i t e , a l t e r n a t i v e t o s t e p 7 + 8 :

Ψ(μ) = [(Ν + DPMltPs+rQtoM) - ΡΝ(μ)θΝ + 1(μ)1


Display 6.2 shows that Ψ(μ) is obtained by summing the products of two sets of polynomials with the same argument, namely the associated Legendre functions without the factor (1 — μ

2)™

12 and the Kuscer polynomials. An

obvious generalization of this definition is obtained by making the arguments of the two factors unequal. The function of two variables G(v, μ) thus defined (step 7) is called the characteristic binomial. This short, distinctive name has been used before; therefore we have retained it in spite of terminological objections. This asymmetric function occurs without a name in many papers, and probably has a deeper significance than Ψ(μ) itself, which is apparent from the fact (Section 6.2.3) that the source function in the diffusion solution in an unbounded medium is proport ional to G(y, μ).

A few side remarks concerning Display 6.2 may be made.

1. The following definition of #™(Α0 as a

determinant is equivalent to the recurrence relation in step 6:

Κ μ ι

2m + 1 ftm+1/* 2 2m + 2 Κ + 2μ

η — m η + m /ι„μ

0Τ+ι(μ) = 1 · 3 · 5 • · • (2m - 1)

( n - m + 1 ) !

2. If m = 0 and Ν is finite, we may save one summation by choosing step 9 instead of steps 7 and 8 in Display 6.2. This formula can be found by splitting the coefficient in step 7 so that

(Dj = (2j + 1) - (2j + 1 - a>j)

and by applying steps 5 and 6 to the separate terms. 3. The notat ions used in Display 6.2 are mostly Mullikin's (1964a).

Chandrasekhar (1950) gave only an implicit definition of Ψ(μ). The general equation for Ψ(μ) in the rotationally symmetric radiation field, m — 0, was derived by Kuscer (1955, Eq. (11)) and for all m by Mika (1961). Busbridge's book (1960, Section 48) contains a more complicated procedure for finding Ψ(μ) for the azimuth-independent terms (m = 0).

4. The integrals

ί 1^ ι τ / / \ J ι hmhm + i · - - hN

ο2Ψ(μ)άμ = 1 - ( 2 m + 1 ) ( 2w + 3 ). . . ( 2 i V + 1)

may serve as a check. The validity of this formula may be proven from the recurrence relations in Display 6.2 (see the cited work of Kuscer, Mullikin, or Busbridge).


D I S P L A Y 6 . 3

C h a r a c t e r i s t i c F u n c t i o n s for Ν < 3

N o t a t i o n : h0 — 1 — ω0, hi = 3 — ω ΐ5 h2 = 5 — ω2 u p p e r i n d e x m o m i t t e d

m = 0 # o = l Po = 1

# i = Κ μ Ρι= μ 92 = ι ( Μ ι μ

2 - 1) P2 = ΙΜ

2 - i

03 = τίΚΚ^μ3 - ( 4 ^ o + Λ 2) μ ] p 3 = | μ

3 - | μ

2 Ψ ( μ ) = ω0 + £ ω 2 + [ λ 0( ω ι - \ηγω2 + ω 3) + ( - | ω 2 + ^ 2ω 3) ] μ

2

+ [ / ι 0( | Λ ι ω 2 - i h 1h 2œ 3 - | ω 3) - Α & 2ω 3] μ

4 + ι ^ * ο Μ 2ω 3μ

6

C o n s e r v a t i v e c a s e : se t ω 0 = 1, / ι0 = 0

m = 1 0 j = 1 P i = 1

0 2 = ^ μ ρ 2 = 3 μ

0 3 = * Μ 2Μ2- * Ρ 3 = ¥ ^

2- !

2 Ψ ( μ ) = (1 - μ

2) ^ + Α ω 3 + 0 Λ , ω 2 - ^ h xh 2 + 1 5 ) ω3] μ

2 + Α Μ 2ω 3μ

4}

m = 2 g2 = 3 Ρ2 = 3

0 3 = 3 / ζ2μ ρ 3 = 15μ

2 Ψ ( μ ) = (1 - μ

2)

2Κ ω 2 + ω 3Λ 2μ

2)

m = 3 0 3 = 15 ρ 3 = 15

2 Ψ ( μ ) = Α ω3( 1 - / ^

2)

3

For quick reference, we present in Display 6.3 the complete specification of the polynomials g™ and pj and the resulting Ψ functions for Ν = 3. The cases Ν = 0 (isotropic scattering), JV = 1, and Ν = 2 are contained in this result by setting ω Ν +1 and all higher coefficients equal to zero. Conservative scattering has ω 0 = 1, which makes h0 = 0 and gx = 0. As a consequence, the Ψ function for conservative scattering, m = 0, has at most the degree 2N — 2. In particular, conservative scattering with the anisotropic phase function 0 ( c o s a) = 1 + ω1 cos a leads in its azimuth-independent terms to the same characteristic function Ψ(μ) = \ and hence to the same Η function as conservative isotropic scattering.

The resulting characteristic functions for a number of special phase functions are collected in Display 6.4.

6.2.3 Solving the Characteristic Equation

We confine this part of the discussion to intensities independent of azimuth (m = 0) and consequently omit the upper index m everywhere. Many schemes have been devised to obtain the eigenvalue k and eigenfunction P(u) of the


DISPLAY 6.4

C h a r a c t e r i s t i c F u n c t i o n s T h a t A c c o m p a n y S o m e O f t e n U s e d P h a s e F u n c t i o n s

ω0 ωί ω 2 ω 3 m 2 Ψ ( μ )

ω 0 0 0 0 0 ω0 i s o t r o p i c s c a t t e r i n g

ω0 ωι 0 0 0 ω 0 + (1 - ω0)ωιμ2 l i n e a r l y a n i s o t r o p i c s c a t t e r i n g

ω0 ωί 0 0 1 i < ^ i ( l — μ

2)

ω ι ω2 0 0 1 + \ω2 - \ω2μ2 c o n s e r v a t i v e Ν = 2 c a s e

ωχ ω 2 0 1 j(l — μ

2)\_ωί + ω 2( 3 — ω ^ μ

2]

ωι ω2 0 2 | ω 2( 1 — μ

2)

2

0 ^ 0 0 | ( 3 - μ

2) R a y l e i g h p h a s e f u n c t i o n

0 i 0 1 | μ

20 - μ

2)

o i ο 2 A d - μ

2)

2

integral equation cited in Section 6.2.1. The most common procedure is the following:

Define characteristic function Ψ(μ) as in Display 6.2.

Define the dispersion function

Find y = k'1 as the largest positive root of the characteristic equation

T(y) = 0.

Computationally preferable procedures are discussed below. Once γ9 and thus fe, are known, the function P(u) follows from

oo

P(u) = Σ (2" + l)gn{y)Pn{u) (intensity) ( l a ) « = o

oo

(1 - ku)P(u) = Σ œnGn(y)PJM) (sourc e function ) ( l b )

,I = 0 Note th e differen t characte r o f thes e sums . I f Ν is finite, the series in Eq. ( lb ) has Ν + 1 terms, but Eq. ( l a ) is always an infinite series. Hence for small N, Eq. ( lb ) is recommended as the more practical.

The derivation of these equations is quite simple, starting from an expansion with undetermined coefficients as in Eq. ( la) . Upon inserting this in the integral equation, the coefficients of Pn(u) can be equated after eliminating the product uPn(u) by the recurrence relation. The coefficients are found to obey the recurrence relations (Display 6.2, step 6) whereby the Kuscer polynomials are defined, with the special argument y. Equat ion ( lb ) also follows at once. Many authors—preoccupied by the wish to deal only with finite series—have started


with arbitrary coefficients in Eq. ( lb). It is then impossible to express Eq. ( l a ) in terms of these same coefficients, and the relation to the Kuscer polynomials is not so elegantly seen.

Some practical methods for computing the root y (or k) and the coefficients gn(y) were reviewed by van de Hulst (1970). Define the function

I f1 Pn(u)Pi(u)du

Ante) = ~ — \ -, 2 J _ 1 1 — u/z

which can be rigorously expressed, for ζ > 1, by

\zPn(z)Qi(z) for i > η A n i { Z) [zQn(z)Pi(z) for i < η

in which Qn(z) is the Legendre function of the second kind. Substitution of Eq. ( lb ) given above into the integral equation leads to the transcendental equations for y

Ν

gfy) = Σ œ

nAjn(y)gn(y) n = 0

We no w sugges t th e followin g procedure s :

(1) If Ν is small, e.g., 1,2, or 3 substitute j = 0 into the preceding equation, which then reduces to

Ν

Σ <t>nyQn(y)9n(y) =

1

a handy form of the characteristic equation, which can then be solved for y. Expressions appearing in these equations are

AooM = zQo(z) = (*/2)ln[(z + l)/(z - 1)]

A01(z) = zQx(z) = z[A00(z) - 1]

(2) / / Ν is large or infinite, observe that Eq. ( l a ) formally satisfies the integral equation, no matter what value we choose for γ. But a necessary condition for the expansion to converge is

lim gn(y) = 0 n-> 00

With any argument μ different from y, the #„(μ) will "blow u p " with η -> oo. A rapid, accurate numerical procedure is to compute polynomial gnfa) by the recurrence relation in Display 6.2 (m = 0); find its largest root μη; then find

y = lim μη « - • o o

which is the required diffusion length. A detailed discussion with numerical examples is given in Sections 12.3.1 and 12.3.2.

6.3 The Ambartsumian Functions 99

Other methods for solving the characteristic equation were reviewed by Chang and Shultis (1976). These authors recommend a discrete-ordinate solution.

6.3 THE AMBARTSUMIAN FUNCTIONS

6.3.1 Definition; Relations to Reflection and Transmission

The functions φ™(μ) and φ„(μ\ introduced by Ambartsumian (1943) form the basic set of functions from which the reduction to X and Y functions (finite layers) or H functions (semi-infinite atmospheres), explained in Section 6.4, begins. They are defined in set 2, Display 6.5. There are two indices: The upper index m is the order of the Fourier term of the expansion defined in set 1, and the index η is the order of the Legendre polynomial or associated Legendre polynomial adopted in the definition in set 2. Recall (Chapter 1) that μ and ν are defined on the domain (0,1). As earlier, we include in T

m the direct (zero-order,

unscattered) transmission and distinguish the diffuse (nonzero-order) transmission by the subscript diff, so that

Γ τη r

Tim ι

rritn

—

1 0 ~r~

1 diff

In order to avoid misunderstanding, two equivalent ways of defining I /C (aO are shown.

The basic use of these functions is (loosely worded) that any incident radiation field may be converted into incident plus reflected radiation field by simply replacing Legendre functions by Ambartsumian functions; similarly for the transmitted radiation field. This makes it possible to express the source function at the top and bot tom of the slab and, thereby, to add a thin layer to the top or bot tom. As explained by Display 4.5, we may, from that point on, choose either to build up gradually by adding thin layers (invariant embedding) or to apply an invariance relation leading to nonlinear integral equations for reflection and transmission. The set of equations in Display 6.5 replaces those of Display 4.5 in the case of an arbitrary (unpolarized) phase function.

The explanation above. is too loosely worded because incident radiation refers to one hemisphere but an expansion in terms of Legendre functions is uniquely determined only for a function defined on both hemispheres. This flaw is remedied in the following precise, physical definition of the Ambartsumian functions in terms of a thought experiment.

Let μ and ν be in the domain (0,1) as before. Let u be the cosine of the angle with the downward normal, so that u covers the domain ( — 1,1). Place above the atmosphere a plane, fully transparent source layer emitting in any direction (w, φ ) the intensity

I oo η I(u, φ) = — ν £ Ç T f t , , ) COS ηνφ

I U

I n = 0m = 0


DISPLAY 6.5 R e l a t i o n s I n v o l v i n g t h e A m b a r t s u m i a n F u n c t i o n s f o r A r b i t r a r y

P h a s e F u n c t i o n

0'

b

OO

1 ν, φ - φ0) = R°Qi, v) -h 2 ^ / Τ ( μ , ν) c o s ηι(φ - φ0) m = 1

Τ ( μ , ν , φ - φ0) = Τ°(μ, ν) + 2 ^ Γ ^ μ , ν) c o s πι(φ - φ0) m = 1

2 φ:(μ) = Ρ ? ( μ ) + ( - 1)" + ,

" 2 μ ί / Τ ( μ , Ν)/?(Ν) JO

•Ό

« Ο ι ) = ί - " " ^ ) + 2 μ ί 7JS, (μ , ν ) Ρ ; ( ν ) </ ν •Ό

3 « ^ > _ η α ι £ ( _ i r w ( v )^ d& \ μ Ν/ 4 μ ν π^

(ΙΤ

η{μ,ν) 1 1 »

ασ ν 4 μ ν „

Έ/Γ"(μ,Ν) 1 1 " ασ μ 4 μ ν π = „,

5 / Τ ( μ , ν) =

1 Σ ( - 1 ) "

+ π^ [ < ( μ ) < ( ν ) - W 0 # : ( V ) ]

4 ( μ + v)Hfm

Τ ^ μ , Ν) = 1

Σ ^ W W f f ( V ) - <*(Ν)«(Μ)] 4 ( ν - μ ) ~ η α

S e t s 2 - 5 : m = 0, 1, 2 , . . . , oo. h T h e i n d e p e n d e n t v a r i a b l e s t o t a l o p t i c a l d e p t h b, s i n g l e s c a t t e r i n g

a l b e d o a, a s y m m e t r y f a c t o r 0 , a n d a n y f u r t h e r p a r a m e t e r s d e f i n i n g t h e p h a s e f u n c t i o n , r e m a i n u n w r i t t e n .

where /™ is arbitrary. The source layer and atmosphere combined can be observed to emit the intensity

J oo η r(u,q>) = - Σ Σ ( - l)"

+mC<p!TQ*) cos γηφ up from top (μ = -u)

μ„ = o m = 0

j oo π φ) = - Σ Σ C*/C(AO COS mcp down from bot tom (μ = M)

β n = 0m = 0 The reciprocal definition, which is required if we should wish to complete

the derivation in physical terms as in Section 4.4, involves the concept of a


detector of arbitrary angular response over all directions placed above or below a slab illuminated by undirectional radiation. We do not work this out.

A clean derivation of the equations in Display 6.5 may be found in Mullikin (1964a,b) of Sobolev (1969a, 1975). The notat ion of φ™(μ) and ψ™(μ) is identical to that of Sobolev (1975).

As an example, we derive one of the relations in physical terms, completely along the lines of isotropic scattering explained in Display 4.5.

Take incident radiation illuminating the top surface in direction (Μ0, φ) with flux π per unit area. The radiation field above the atmosphere then is

/(w, φ) = Ι0(μ,φ) = (Π/Μ0) δ(μ - Μ0) δ(φ - φ0) for u > 0 (down) = Κ(Μ, φ; Μ 0, φ0) for u < 0, Μ = - w(up)

In the absence of any upward radiation, the expansion of this incident field in spherical harmonics reads:

Incident radiation only:

1 00 00

(n — mV Io(u, Ψ) = -τ- Σ (2 " Km) Σ (2n + 1) ^ ! FMPTiu) cos m(cp - φ0)

The combined up and down field above the atmosphere may now by definition be expanded as :

Incident and reflected radiation combined:

1 00 00

(n — mV /(«, φ) = — Σ (2 - * o , m ) Σ (2» + 1) , j C ^ T O c o s m ^ - φ0)

where the presence of reflection is taken into account by replacing the factor P^iVo) by (PniVoX

as defined in set 2.

Now place in this combined radiation field an extra layer in which albedo and phase function are characterized by the set of coefficients ωη. The source function J in this layer is found from the local intensity by replacing 2n + 1 byœn:

J OO 0 0

J (0 , W , φ ; Μ 0, φ0) = — Σ (2 ~ < V m ) Σ cos τη(φ - φ0)

^ 0 m = 0 η = m

The intensity emitted up and down by this added layer is

(dr/\u\)J(0, Μ , Φ ; Μ 0, ( Ρ 0) The upward emitted intensity is seen directly. The intensity emitted downwards gives, by reflection against the full layer, an added contribution to the upward intensity. Taking the sum of both boils down to replacing the remaining Legendre function P

m(u) (for u = - Μ , Μ > 0) by the corresponding φ

Μ(μ)


as defined in Display 6.5, set 2. We now have the full addition to the upward radiation field above the atmosphere as a consequence of the radiation scattered by the extra layer. Taking into account also the extinction by the added layer (cf. Section 4.4), we obtain the first equation of set 3 of Display 6.5. The other equations of sets 3 and 4 are obtained similarly, or by more formal methods.

The final equations (set 5 of Display 6.5) are obtained by making the assumption at this point that the atmosphere is homogeneous and that the added layer also has the same composition. The two expressions for dR

m/db then must be

equal, and those for dTm/db likewise, from which set 5 emerges.

We insist that only this homogeneity assumption, by which two otherwise unequal expressions may be equated, introduces a notion of invariance. The derivations of the earlier equations (sets 1-4) do not involve such an assumption. It is therefore a bit confusing that the same Chapter VII of Chandrasekhar (1950), entitled "Invariance Principles," contains the equations of set 2 [Eqs. (65)-(66)] and those of sets 3 and 4 [Eqs. (67)-(70)] which do not contain this assumption, as well as the equations for R

m and T

m [set 5 of Display 6.5 = Eqs.

(71)-(72)] which are based on this assumption. Even more confusing is the use of the term "invariant embedding" for a

method in which the equations for dRm/db and dT

m/db are directly integrated

from zero to any desired value of b, using at each step the expression for φ and φ from step 2 (Section 4.4). We do not dispute the fact that for inhomogeneous atmospheres with anisotropic scattering this provides a powerful method of numerical computation. The method is good, but the name is not, since no concept of invariance is used.

6.3.2 Relations for m = 0 in Vector Notation

It is instructive to show briefly that the terms in the preceding equations which are independent of azimuth (m = 0) are already contained in the relations derived in Chapter 5. Introduce the "vec tors"

Nn = Ρη(μ)/2μ

Ln = Nn + (-l)nNnR = <pnM/^

Μη = ΝηΤ=ψη(μ)/2μ

These definitions express in matrix form the equations of Display 6.5, set 2, for m = 0. The generalization of Eq. (13) in Section 5.2.2 to an arbitrary, finite layer, found by a straightforward application of the method employed there, is

RZ + ZR = HfR + RHhR + RHf + Hh - THhT

TZ - ZT= THhR + TH{ - RHbT — H{T


In these equations we may write

Ht = Σω„Ν„-Ν„ ,1 = 0

H b = Σ(-ίΤω„ΝΛ·Ν„ ,1 = 0

By these substitutions the equations just given obtain the form

RZ + ZR = Z ™ n ( - mLn Ln-Mn- Mn) ,I = 0

TZ-ZT= X c o n( M „ L „ - L „ M „ ) ,1=0

which, when written in terms of functions, agrees with Display 6.5, set 5.

6.3.3 Integral Equations

By elimination from sets 2 and 5 of Display 6.5, it is possible to find nonlinear simultaneous integral equations, either for φ™{μ) and ^"(μ) ,

or f °

r R

m(&

v) a n

d T

m(p, v). The latter were just cited in vector form (Section 6.3.2) and are of no

further use to us. The former are presented in Display 6.6, set 1. The same functions also obey a set of linear integral equations. The intro

duction and systematic use of these equations has advanced the theory of radiative transfer substantially. They are cited without derivation from Mullikin (1964b). I have made the notation consistent with earlier sections and, in particular, with Display 6.5. This regrettably makes the translation from Mullikin's equations into ours somewhat of a task [φ and ψ are interchanged and we have an additional factor (1 - μ

2)™

12 in PJ(ji\ φ™(β\ and ψ?(ρ); also his k(a,z)

has the arguments in reverse order compared to our characteristic binomial, and an additional factor (1 — σ

2)™].

Mullikin (1964b) showed that neither the nonlinear integral equations for φ„(μ) and ψ™(μ) (Display 6.6, set 1) nor the linear integral equations (Display 6.6, set 3) generally have unique solutions. A unique solution of the linear integral equations is obtained if they are used together with the linear restraints formulated as set 4 of Display 6.6. The sufficiency of these equations and constraints has been further examined by Leonard and Mullikin (1964).

The formulas for the conservative case, which were not correctly given in Mullikin's original paper, are in set 5 of Display 6.6, as presented by Mullikin himself (1967) and confirmed from a different derivation by Busbridge (1967). A concise presentation of the full equations for reflection and transmission by a slab with conservative anisotropic scattering is given by Busbridge and Orchard (1968). Most of these equations may also be found in Sobolev (1975).


DISPLAY 6.6 I n t e g r a l E q u a t i o n s fo r F i n d i n g φ„{μ) a n d ψ„(μ)

1

00 Γ

1 dv

1 φΚμ) = ΡΧμ) + Σ ( - « ^ Γ — — Ι>Γ(μ)<(ν) - ΨΓ(μ)ΨΓ(ν)]^(ν) 2 i = m J 0 μ + ν

ΨΧμ) = + ^μ Σ <? ί |>Γ(μΧ(ν) - ^Γ(μ)ΆΓ(ν)]Ρ^(ν)

2 D e f i n e ^ (ζ ) a n d G

m( z , s ) ( S e c t i o n 6 .2 .2 ) ; T ( z ) ( S e c t i o n 6.2.3)

3 T h e n

Γ ( ζ )<(ζ ) = (1 - z2)mi2

g:(z) + (1 - ζ2Γ '

2( 1 - σ

2Γ

/2

Γΐ rlG

m(z,a) hl 1 / ^ G ^ z , - σ ) 1

χ - ζ — ^ Κ ( σ ) ^ - ( - 1 ) " β - ^ - ζ V

' ' f l T ( g ) i t o [ 2 J0 σ - ζ 2 J0 σ + ζ J

Τ(ζ)ψΖ(ζ) = (1 - z

2r

/ 2^ ( z ) e ~

b /z + (1 - z

2)

m / 2( l - σ

2Γ '

2

1_2 J 0 σ — z 2 J 0 σ + ζ J

4 C o n s t r a i n t s t o b e u s e d w i t h se t 3 , e x c e p t fo r t h e c a s e d e f i n e d in se t 5 :

F o r ζ e q u a l t o e a c h of t h e r o o t s ζ} o f t h e c h a r a c t e r i s t i c e q u a t i o n T ( z ) = 0, t h e r i g h t - h a n d

m e m b e r s o f t h e e q u a t i o n s of se t 3 m u s t b e z e r o .

5 C o n s t r a i n t s for t h e a z i m u t h - i n d e p e n d e n t t e r m in t h e c o n s e r v a t i v e c a s e , m = 0, ω 0 = 1 :

2δ0η = f

1Ι Ψ » + ( - l ) > n V ) ] da ( a l l n)

2 ( 1 - g)b δ0η = f ίψ°η(σ) - φ„° (σ ) ] [ (1 - g)b + 2 σ ] da ( e v e n η) •Ό

ί = ί ίΨ» + Φ ? ( σ ) ] [ ( 1 - g)b + 2 σ ] <*σ ( o d d η)

A final remark on method: All equations in Display 6.5 and 6.6 occur in pairs in which reflection and transmission play somewhat symmetric roles. Hovenier (1978) has explored the changes arising from a systematic use of the exit function, which is a linear combination of reflection and transmission. He concludes that this leads to savings in both the formulation and the computation.

6.4 REDUCTION TO Η FUNCTIONS

The Ambartsumian functions do not usually appear in schemes of computation, for it is possible and useful to break them down into products of certain polynomials and transcendental functions. This breakdown is discussed in the present section. For clarity we have chosen to present in this section the equations

6.4 Reduction to H Functions 105

for a less general set of assumptions, namely semi-infinite atmospheres (b — oo) and azimuth-independent radiation fields (Fourier order m = 0). The full formulas would contain an upper index m with all quantities, terms proport ional to e~

b/tl in many equations, two functions X

m(b, μ) and Y

m(b, μ) instead of one

function Η (μ), and two sets of polynomials instead of the one set of Busbridge polynomials ί?(μ).

We do not quote the most general formulas since (1) they are cumbersome; (2) they have been published by other authors, e.g., Kuscer and McCormick (1974), Sobolev (1975); and (3) they are rarely used for computat ion (see comments in Section 6.1).

6.4.1 The Η Functions

With any given characteristic function Ψ(μ) a function Η(μ) is uniquely defined as has been thoroughly discussed by Chandrasekhar (1950) and others. The determination of the functions X(b, μ) and Y(b, μ) for layers of finite depth b has been treated by Carlstedt and Mullikin (1966). The transition is Z(oo, μ) = / / (μ) , 7(αο ,μ) = 0.

The key equations, which are of diverse origin, are shown in Display 6.7. A number of comments will be necessary to explain their interrelations. We shall discuss these relations in order of appearance in Display 6.7, although this is neither the historical nor the logical order.

Lines 1-2. Dispersion function. The characteristic function and the corresponding dispersion function T(z), introduced in Section 6.2.3, may be considered as given. The T(z) is well defined for all real or complex z, including the value ζ = y = k'

1 for which it is zero but excluding the real values

— 1 < ζ < 1. For values of ζ on this cut, but not in the endpoints + 1 , it is still possible to use the same integral definition if we use the Cauchy principal value. This value is written as Α0(μ), — 1 < μ < 1. It is possible to remove the singularity from the defining integral, and one of the forms in which the result may then be written is shown on line 2. Since Ψ(μ) is a polynomial, the factor in the integrand may be divided out.

Lines 3-5. Integral equations. The nonlinear integral equation (line 3) is its original and most familiar form. Chandrasekhar found that iteration procedures to solve the nonlinear integral equation, starting with an approximate solution based on Gauss summation, give very satisfactory results, except in the conservative case, when the solution is ambiguous. Mullikin later demonstrated that the solutions are never unique. The full analysis of this difficulty and the way to circumvent it is presented in two papers (Mullikin, 1964c; Carlstedt and Mullikin, 1966).

A byproduct of this analysis is that Χ(μ) and Υ(μ) are shown to satisfy two simultaneous singular linear integral equations. This is an extension of an


D I S P L A Y 6.7

E q u a t i o n s fo r t h e H F u n c t i o n s

G i v e n :

c h a r a c t e r i s t i c f u n c t i o n Ψ ( μ ) ( S e c t i o n 6 .2 .2 )

c h a r a c t e r i s t i c r o o t k = y~

l ( S e c t i o n 6 .2 .3 )

7

d i s p e r s i o n f u n c t i o n T(z) = 1 — ζ *

ν ί"

7 ( S e c t i o n 6 .2 .3 )

J - i

z - μ

Γ

1 ν{μ)άμ

J o A*

2 2 Ψ ( μ ) - Ψ ( ν ) 1 - μ

^ I - ζ— dv + ιιΨ(ιΔ IN 1 +μ

λ0(μ) = 1 + 2 μ

2 I —2 -i-^dv + μ ψ ( μ ) In

I n t e g r a l e q u a t i o n s :

Γ

1 Η ( ν ) Ψ ( ν ) dv

Η(μ) = 1 + μΗ(μ) n o n l i n e a r r Η(ν)ψ(ν)ι

J 0 μ + ν

Γ

1 Η ( ν ) Ψ ( ν ) dv

λ0(μ)Η(μ) = 1 — μ l i n e a r , s i n g u l a r J 0 μ - ν

;

1/ / ( ν )ψ ( ν ) ί / ν

1 = j r e s t r a i n t kv

= r Η ( ν ) ψ (

J 0 1 - ,

E x p l i c i t f o r m u l a ( v a l i d if o n l y o n e p a i r o f r o o t s ±k e x i s t s ) :

1 + ζ Γ f

1 θ(ί) Λ Ι

6 Η ( ζ ) = r - e x p U / A

1 + fcz

FL J o ί(ί + z ) J

w h e r e 0(r ) = ( Ι / π ) a r c t a n ^ i

lF ( i ) / / l 0( f ) ]

R e l a t i o n s :

8 l/H(z)H(-z)= T(z)

9 J Η(μ)Ψ(μ)άμ = 1 - ^ 1 - 2 Ç Ψ(μ) ά μ ^ '

S p e c i a l v a l u e s :

10 H(0) = 1, [ / / ( - y ) ] "

1 = 0, T(y) = 0

for c o n s e r v a t i v e s c a t t e r i n g (ω0 = 1 ) : 2 ί Ψ(μ)άμ = 1, ί Η(μ)Ψ(μ)άμ = 1 Jç> JQ

y = oc , k = 0

earlier result found by Busbridge (1962) for Η (μ). The linear integral equation for Η(μ) is shown in Display 6.7, line 4. The integral is singular because the denominator goes through zero. The integral must be read as the Cauchy principal value. For computational purposes the singularity may be removed as we did in changing from line 1 to line 2.

Mullikin showed that the complete solution of either the nonlinear or the linear integral equations contains certain arbitrary constants. The solution is


made unique by imposing the restraint shown on line 5. We refer to the original papers for details.

Lines 6-7. Explicit expression. Remarkably, it is possible to express the H function explicitly. The full formula (e.g., Kuscer and McCormick, 1974) contains the product ]~[(1 + kjz) over all discrete positive roots, numbered j = 1, . . . , J , of the characteristic equation. The expression presented here is limited to the fairly common case in which only one discrete positive root exists ( J = 1). This expression (or other ones that can be simply reduced to it) have been derived in the literature along two different lines.

Mullikin (1964c) makes use of the fact that the linear, singular integral equation (line 4) can be written in the form of a Fredholm equation, the solution of which can then be written explicitly. An older and more fundamental approach starts from the relation of line 8. This equation, combined with the requirement that H(z) be analytic for Re(z) > 0, may be taken as the definition of H(z) (Kuscer and McCormick 1974). Cauchy's integral theorem permits us to write

By bending the contour and by some further manipulation, the expression of H(z) in lines 6-7 emerges, where we have used the value of T(z) just above and below the cut :

Similar, slightly more complicated expressions exist for X(b, z) and Y(b, z) if b is finite (Mullikin, 1964c).

The properties of a wider class of integral equations and their solutions have been discussed by Leonard and Mullikin (1965).

Lines 8-9. Relations. Among the many relations between Η{μ\ Ψ(μ), and T(z), the factorization shown on line 8 is certainly the most fundamental. The simple requirement that H(z) be analytic for Re(z) > 0 makes this factorization unique and hence makes it possible to take line 8 as the equation defining Η (ζ). Readers not interested in the complex plane but in the numerical behavior of H(z) for real ζ outside the interval (—1, 1) may wish to refer to the illustration for isotropic scattering in Fig. 8.4, which shows that [H(z)] "

1 for real ζ forms a

smooth curve across the boundary ζ — ± oo, if plotted against z ~1. The points

at equal distance from the line z "1 = 0 are related by the equation of line 8.

Line 9 shows the most important of a set of integral relations derived by Chandrasekhar. Corresponding relations between the moments of the Η function exist. Let

lim T(t ± is) = A0(i) ± πΐίΨ(ί) = Α0(ί)[1 ± i t a n ^ ( i ) ]


Then, for a characteristic function of the form

Ψ(μ) = α' + ϋμ2 + c >

4

as occurs if the phase function has Ν = 2 (nonconservative) or Ν = 3 (conservative), the moment relations are

2(α0 — 1) = a'oil + fr'af + c ' (2a1a3 — af)

2 (a2 ~~ i ) =

tf'(2a0a2 — a i ) + b'oi\ -f c'af

This extends an equation given (for d = 0) by Chandrasekhar (1950, p. 109).

Lines 10-11. Special values. These equations are simple reminders of some trivial equalities. The relations for conservative scattering can be extended with many more. See, e.g., the relations for isotropic scattering listed in Section 8.3.3. In nonconservative scattering, the Η function may be expanded in a power series in a. The asymptotic behavior of the terms of this expansion is discussed in a wider context by Ivanov and Sabashvili (1973).

6.4.2 The Busbridge Polynomials

Busbridge (1960) shows that for an arbitrary phase function with finite Ν we have

<Ρη(β) = <?„(μ)# (μ) (2a)

where φη(μ) are the Ambartsumian functions defined in Section 6.3.1, Η(μ) is the Η function defined in Section 6.4.1, and <ζ„(μ) is a set of polynomials of degree N, or, in the conservative case, of degree Ν — 1. We shall call these the Busbridge polynomials. The definition adopted here differs by a factor (—l )

n

from the definition in Busbridge's book (1968) but agrees with the definition in Busbridge's later papers. We may conveniently assume that all results hold also for phase functions that are expandable in an infinite set of Legendre functions (M = oo). In this case the word "polynomia l" is a misnomer, because ^„(μ) are infinite power series.

With these polynomials all quantities sought can be expressed and a number of intriguing relations seen. We follow the presentation by van de Hulst (1970).

The polynomials used are summarized in Display 6.8. The Ρη(μ) and 0„(μ) are even or odd with η even or odd, so that G( — ν, — μ) = G(v, μ), but G(v, μ) is not symmetric. On the other hand <?π(μ) is not odd or even, but F(v, μ) is symmetric. By substituting Eq. (2a) in the reflection function (Display 6.5, set 5 for m = 0) we obtain

Rfo v) = HQi)H(v)FQi, ν)/4(μ + ν) (2b)

6.4 Reduction to Ή Functions 109

D I S P L A Y 6.8

P o l y n o m i a l s O c c u r r i n g i n R e d u c t i o n t o Η F u n c t i o n s

N a m e d a f t e r

L e g e n d r e K u s c e r B u s b r i d g e

P r e s e n t n o t a t i o n

(η = 0 , 1 iV)

D e g r e e for

n o n c o n s e r v a t i v e c a s e

D e g r e e fo r in = 0

c o n s e r v a t i v e <n = 1

c a s e : In > 2

η

0

a η - 2

9η(μ)

Ν

Ν - 1 Α

Ν - 1

B i n o m i a l

O c c u r s i n

B o t h l e a d t o c h a r a c t e r i s t i c

f u n c t i o n

D e g r e e

( n o n c o n s e r v a t i v e )

( c o n s e r v a t i v e )

P r o c e d u r e for c o m p u t a t i o n

G(v, μ) = Σ ωηθη(ν)Ρη(μ) n = 0

d i f fu s ion p a t t e r n ( w i t h ν = γ)

iGOi, μ) = Ψ(μ)

F(v, ^) = Σ ω η( - 1 Μ ν « μ )

r e f l e c t i o n f u n c t i o n

e s c a p e f u n c t i o n ( w i t h ν = — γ)

Μ-μ, μ)

2Ν

2Ν - 2

f r o m r e c u r r e n c e r e l a t i o n s

( D i s p l a y 6.2)

f r o m s y s t e m o f l i n e a r

e q u a t i o n s i n v o l v i n g

m o m e n t s o f Η f u n c t i o n s

( t h i s s e c t i o n ) a E x p r e s s i o n i d e n t i c a l l y z e r o .

Similarly, the nonlinear and linear integral equations for φη(μ) in Display 6.6 lead to nonlinear and linear equations for qn(p) as follows:

Ρη(μ) F(p,v)H(v)Pn(v)dv

and

H{-z) J O V — Ζ

(3)

(4)

where we have used line 8 of Display 6.7. Equations (3) and (4) have a remarkable similarity. By using both we can

reduce the integral

'dX

(5) . (

1H(x)F(ji,x)G(-z,x)x, Ι (

^ > z

> =

? — i — ν — I — ^ — J 0 (μ + x)(z + x)


to a simpler form. In order to do so, we separate

χ 1 ί ζ μ

(μ + χ)(ζ + χ) ζ — μ\ζ + χ μ + χ

and write only in the term with denominator ζ + x: Ν

Fiji, χ) = Y,œ n{--l)nqjji)qn{x)

and only i n th e ter m wit h denominato r μ + χ: Ν

Gi-ζ,χ) = Σ ω „ ( - 1 ) " 3„ ( - ζ ) Ρ π( - χ ) η = 0

Upon using Eqs. (3) and (4) we then find that the complicated integral of Eq. (5) reduces to the simple form

which greatly simplifies the algebra of the final reduction (Section 6.4.3). The rest of this subsection is devoted to the practical computat ion of the

Busbridge polynomials. Either Eq. (3) or (4) may be used and the choice is simple, for Eq. (3) is nonlinear in #„(μ), with all η coupled, and Eq. (4) is linear in 4„(μ), with all η uncoupled.

Use of the nonlinear relations leads to cumbersome algebra (e.g., Horak and Chandrasekhar, 1961). Several authors gave up before reaching the desired result. Busbridge (1968) sketched a general method based on the use of the nonlinear equations, but did not spell out a procedure for actually solving the coefficients. Experience (as described in correspondence by Dr. Busbridge and as confirmed by my own attempts) shows that for Ν = 1 this solution is a chore; for Ν = 2 it becomes unwieldy.

Since then, it has been realized (Kuscer, 1958; Sobolev, 1969a, 1975) that it is much simpler to start from the linear equations. Adding to the members of Eq. (4) the members of the equation

and writing Ψ(ν) = ^G(v, v), we obtain

Ç1G(z,v)qn(v)-G(v9v)qn(z)

Qn(z) = QNIZ) + \Z — H(v) dv (J)

Equation (7) is a Fredholm equation of the type

Qn = Gn + L<ln with the formal solution

qn = (l - L ) "1

^


so that one matrix inversion is required to find the coefficients of all polynomials qn. Since gn(z) is a known polynomial and ν — ζ can be divided out, this leads, for each individual n, to Ν + 1 linear equations from which the coefficients of the polynomial qn(z) can be solved. The coefficients of these equations contain the moments at of Η(μ) up to the order 2N — 1. Since no significant simplification seems possible, it is not attractive to write out the entire procedure.

A snag in this method is that it breaks down in the conservative case, ω 0 = 1, because the determinant of the set of equations becomes zero. This appears from the examples (N = 1, Ν = 2, Ν = 3) which have been worked out in the literature and is probably a general property. The best method seems to be the linear equations with a linear restraint added. The procedure has been worked out and numerically tested by Busbridge and Orchard (1968). It is spelled out in Display 6.9.1 have added to the original work some further symbols in the interest of a condensed notation. Only N(N — 1) coefficients cnj have to be computed, which means six for Ν = 3 and two for Ν = 2. The complete literal solution for JV = 3 is shown in Display 6.10, again based on the work of Busbridge and Orchard. The same Display shows the solutions for Ν = 2 and Ν = 1.

6.4.3 Solution of the Milne Problem

Most authors treat the escape of radiation from a deep atmosphere (the Milne problem) separately from the reflection problem because the equation of radiative transfer had to be solved with a different set of boundary conditions. We find it more helpful to derive the solution of the Milne problem from that of the reflection problem, because the two are simply related.

(a) Nonconservative case. Equation (5), Section 5.2.2, reads in shorthand notation

mK = Ρ — RQ

which is written in full as

τηΚ(μ) = Ρ(μ) - Γ RQJL, v)P(-v)2vdv

By substituting the known diffusion pattern

Ρ(μ) = IvKv - rii](Hy, μ) (8)

and the known reflection function, Eq. (2), we obtain an integral that led to unwieldy algebra in many earlier attempts. Since this integral is of the form of Eq. (5) we can rewrite it in the form of Eq. (6) and obtain

η Λ y

r( Λ fG

( y ^ ) Ffa - y ) 1


A, = f Η (χ )Ψ (χ )χ

η

yta = f #(*)Ο,(* •Ό

:)x" dx

B u s b r i d g e p o l y n o m i a l s

W r i t e

T h e e q u a t i o n s a r e

qM = 9ηΦ) + Ycnj

+1 ( n = 0 , 2 , 3 , . . . , N)

N-2 γ j N - 2 Σ

Cnjfrj-l = -un.L+1 + X

9 n0 7 ,0 + Ô Σ ^ J + L 2 — 2 j =0

N-2

9no<Xo + Σ

C« J

AJ + 1 =

2 [ = 2 for η = 0 a n d 0 for « ^ 0 ]

F o r e a c h η t h e u p p e r se t f o r m s Ν — 2 e q s . (/ = 0 , . . . , Ν — 3) a n d t h e l o w e r se t , o n e e q u a t i o n .

S o l v e Ν — 1 u n k n o w n s cnj f r o m t h e s e Ν — 1 e q u a t i o n s b y s t a n d a r d p r o c e d u r e .

R e p e a t JV t i m e s (n = 0 , 2 , 3 , . . . , N).

DISPLAY 6.9 D e t e r m i n a t i o n o f P o l y n o m i a l s <7„(μ) for C o n s e r v a t i v e S c a t t e r i n g b y t h e B u s b r i d g e - O r c h a r d

M e t h o d

S e e D i s p l a y N o .

P r e p a r a t o r y w o r k

A s s u m e d Ν finite, ω0 = 1, m = 0, b = co

G i v e n ωλ, ω 2, . . . , ω #

F i n d coe f f i c i en t s hn = 2n + 1 — ω„ 6.2

N - 2

p o l y n o m i a l s 0„ (μ ) = Σ gnjrf 6.2 J ' = O

c h a r a c t e r i s t i c f u n c t i o n Ψ ( μ ) 6.2

c h a r a c t e r i s t i c b i n o m i a l ϋ(μ, x ) 6 .2

GOi , x ) - G ( x , x )

N

v

3 ,

p o l y n o m i a l s Q^x) f r o m = > β / (χ )μ

μ - χ | = 0 / / f u n c t i o n Η ( μ ) 6.7

m o m e n t s o f Η f u n c t i o n ct„ = H(x)x

n dx

Jo


D I S P L A Y 6 . 1 0

L i t e r a l S o l u t i o n of C o e f f i c i e n t s for C o n s e r v a t i v e S c a t t e r i n g w i t h Ν < 3

A s s u m e d : Ν = 3 Ν = 2 Ν = 1

ω 0 = 1, m = 0, b = oo

G i v e n : ω , , ω 2, ω 3 P r e p a r a t o r y c o m p u t a t i o n s :

h2, 0oOO t o 0aO4 s e e D i s p l a y 6.3

Q0(x) = h2o)3(3x - 5 j c

3) / 1 2

Η(μ) 1 t o b e n e w l y c o m p u t e d for

a 0, a t, a 2, a 3 j

e a Cn c h a r a c t e r i s t i c f u n c t i o n

7oo = «2^^3(3»! - 5 α 3) / 1 2

7οι = 2 - α 0 - ω 2( α 0 - 3 α 2) / 4 ε0 2 = 7θ2 ~ 2βι = ~ α ι ~ ω 2( α ! - 3 α 3) / 4

s a m e

ω 3 = 0 , ω 2 # 0

s a m e

QoOO = ο s a m e

7οο = 0

7ο ι = 0

s a m e

ω2 = ω 3 = 0

s a m e

0

s a m e

S o u g h t : t h e coe f f i c i en t s in t h e p o l y n o m i a l s

q0Qi) = 1 + ε00μ + οοχμ2

ν2(μ) = - i + ^ 2 0μ + < ?2 1μ

2

S o l u t i o n of l i n e a r e q u a t i o n s :

# = 7 ο ι « 2 - εο2«ι

fc0 = -2s02/D, bx = 2y0JD

-*2/D,

Coo = 7 o o ^ o + (1 - ï « o ) ^ o

c 0i = 7 o o ^ i + ( 1 - jct 0)b1 c20 —

—l 7 0 0

C0 + 4

a0 ^ 0

c2\ = -booC\ + C3 0

— 3 ^ 2

C0 > \ h 2c x 1

N o n z e r o b u t n o t r e q u i r e d .

1 + c 0 0^

- I + c 2 0/ i

/?0 = 2/a l9 b x = 0

c 00 = ( 2 - aoVa j Coi =0

c 20 = a 0/ 2 a !

c 2i = 0

which simplifie s t o

m W= / r<yXy-/ 0

( 9)

which i s th e require d resul t (va n d e Hulst , 1970) . I t i s somewha t puzzlin g t o m e that thi s formul a doe s no t see m t o b e foun d i n th e earlie r literature .

Incidentally, w e ma y not e fo r a chec k tha t upo n computin g Q — RP b y exactly th e sam e method , w e find a n expressio n whic h i s zer o becaus e i t resemble s Eq. (9 ) bu t ha s H( — γ) = oo in the denominator.


The physical significance of the fact that the escape function contains the same expression as the reflection function, with the argument — y instead of μ 0, can be appreciated by the following reasoning. Generally, if the atmosphere is exposed to radiation incident with direction cosine μ 0, the depth τ where the radiation is first scattered is distributed as exp( — τ/μ0)άτ. O n the other hand, radiation fed at a large depth τ into the atmosphere has a chance to escape proport ional to exp( —/CT). Clearly, if it were possible to choose the direction of incidence so that (1/μ0) = — Κ which means μ 0 = — y, the integral over τ would diverge, because any large depth would contribute with equal strength to the emergent radiation. With μ 0 close to — y, the integral would still be convergent, but the contribution of layers near the surface would be negligibly small compared to the contribution of the very deep layers. This means that the reflected radiation must have the same angular distribution as that of radiation emerging from anywhere deep down in the atmosphere. In other words, apart from a proportionality factor, which we know must diverge, Κ(μ) is proportional to Κ(μ, — y). It is easily surmised that the diverging factor is the factor H( — y) contained in R, and that by simply omitting this factor, we have, apart from a constant, the correct solution of the Milne problem.

(b) Conservative case. Earlier derivations for a general anisotropic phase function (Russman, 1965; Busbridge and Orchard, 1968) were based on a fresh solution of the equation of transfer with different boundary conditions. Far more rapidly, we can take Eq. (33) of Section 5.4.1, which reads

Upon writing fv2 = P2(v) + ^P0(v) and referring to the equations defining

φη(μ) (Display 6.5) and ^„(μ) (Section 6.4.2, Eq. 1), we readily see that this may be written in the form

which is the desired result. The polynomials q0 ) and ^2(μ) m a

Y be computed as shown in Displays 6.9 and 6.10.

6.4.4 The Complete Set of Reduced Forms

Having come this far, nothing stands in the way of a complete expression of all functions occurring in Chapter 5 in terms of the Η functions and the polynomials introduced above. The formulas thus found can be real time-savers when it comes to the following applications:

(1) Actual values for Ν = 1, Ν = 2, or Ν = 3 (2) Flux integral, extrapolation length, and other moments

Κ(μ) = (4μ)"1/ / (μ)[^ο(μ) + 2*2(μ)]


We return to these applications in Chapter 12, where numerical results for different phase functions are compared. Without the groundwork of this chapter it would not be possible to bring numerical results from such diverse sources together into the same graphs (e.g., Fig. 12.1).

(a) NONCONSERVATIVE CASE

The results are shown in Display 6.11. The reduced expressions for /, φη(μ), and /„ readily follows from the integrals in Eqs.(3)-(6) defined above. Only the constant m requires further explanation. Starting from the relation m = (mK)P, writing this out as an integral by the recipe of Chapter 5, and substituting for mK from Eq. (9) and for Ρ from Eq. (8), we obtain

where /(μ, ζ) is the integral defined by Eq. (5). In this particular case, because the arguments are equal, we cannot transform the integral to the form of Eq. (6). However, since both arguments are — γ, a simple transition of Eq. (6) to the limit (van de Hulst, 1970) leads to

where T'(y) is the derivative dT(z)/dz at ζ = γ. Substitution into Eq. (10) finally gives the expression for m given in Display 6.11. Incidentally, this result does not contain the H function at all and can be computed directly from the functions defined in Section 6.2. It is obvious that this should be so, for P(u) is expressible in those functions and m is expressible in P(u). It should be possible to derive the final expression for m directly from its integral definition (Section 5.2.2, Eq. 9), but I have found no direct way of doing so.

The two expressions for m in terms of an infinite series in gn(y) follow from the earlier ones by simple manipulation. These series are infinite even if Ν is finite but may nevertheless be useful because of their rapid convergence (see example in section 12.3.1). The series expression for / follows similarly from the reduced form given some lines above; this series is finite for finite Ν because of the coefficient ωη.

(b) CONSERVATIVE CASE

The expressions equivalent to those of Display 6.11 in the case of conservative scattering are shown in Display 6.12. A direct transition to the limit is difficult because of the factors 0 and oo. It is simpler to start from the relations for conservative scattering (Section 5.4.3) and from the escape function already derived in Section 6.4.3. We have included four moments of the escape functions, expressed in terms of the /„ as defined in Display 6.11 and (within parentheses) also in the shorthand notat ion of Chapter 5. The moment of order 2 is the reduced extrapolation length, which will be discussed in detail later.

m = [ 2 y

2/ f f ( y ) ] / ( - y , -y) (10)

/ ( - V , - 7 ) = 4Ψ(?)Η (ν )Τ ' (7) ( H )


K n o w n e n t r i e s R e f e r e n c e s

C o e f f i c i e n t s ω „ , hn K u s c e r p o l y n o m i a l s #„ (μ) C h a r a c t e r i s t i c b i n o m i a l G(v, μ ) C h a r a c t e r i s t i c f u n c t i o n Ψ ( μ ) D i f f u s i o n p a t t e r n P(u) D i s p e r s i o n f u n c t i o n a n d i t s d e r i v a t i v e T ( z ) , T'(z) C h a r a c t e r i s t i c r o o t k I n v e r s e o f c h a r a c t e r i s t i c r o o t y = k ~

1

H f u n c t i o n Η(μ) B u s b r i d g e p o l y n o m i a l s <?„(μ) B i n o m i a l F ( v , μ )

D i s p l a y 6.2 D i s p l a y s 6.2, 6.8 D i s p l a y s 6.2, 6.8 D i s p l a y s 6.2, 6.8 S e c t i o n 6.2.3 S e c t i o n 6.2.3 S e c t i o n 6.2.3 S e c t i o n 6.2.3 D i s p l a y 6.7 D i s p l a y 6.8 D i s p l a y 6.8

B a s i c f u n c t i o n s a n d c o n s t a n t s I n t e g r a l d e f i n i t i o n R e d u c e d f o r m

R e f l e c t i o n f u n c t i o n Η(μ)Η(ν)Ρ(μ, ν)

Κ ( μ , ν) = Κμ }

4 ( μ + ν)

E s c a p e f u n c t i o n γΗ(μ)Ρ(-γ,μ)

m(y - μ)Η(γ)

C o n s t a n t m = J lP(u)Y2udu = 8 7

2Ψ ( 7) Τ ' ( 7)

C o n s t a n t ι 1= f Κ{μ)Ρ(-μ)2μάμ Jo

2 y F ( - y , -γ)

m [ H ( y ) ]

2

M o m e n t s R e f e r e n c e s

φη(μ) = Ρη(μ) + 2 μ f Κ ( μ , ν ) Ρ η( -^ 0

-ν)άν = ηη(μ)Η(μ) D i s p l a y 6.5

Γ1 {-\)

n2qn{

Jo *ηΗ(γ)

-y)

S e r i e s

N e w

m = oc-

8 £ ( n + \)g n{y)gn+ï(y) n = 0

m = 00

4y Σ Λ „ [ 0 „ ( γ ) ]

2

,1=0

I = 00

> 7Χ ω η( - 1 ) " / „

2

n = 0

D I S P L A Y 6 . 1 1

R e f l e c t i o n a n d E s c a p e F u n c t i o n a n d R e l a t e d C o n s t a n t s for N o n c o n s e r v a t i v e S c a t t e r i n g

6.5 The Radiation Field at Arbitrary Depth 117

6.5 THE RADIATION FIELD AT ARBITRARY DEPTH

We briefly review the important question of finding the radiation field at all depths. This adds a new independent variable τ and hence a new dimension to the problem, compared to the preceding sections in which only the emergent radiation (reflection, transmission, escape) and the radiation deep inside the atmosphere (diffusion domain) were sought. In the standard problem of a homogeneous slab of thickness b illuminated by parallel radiation from direction ( μ 0, 0 ) both the source density and the intensity

J(b, μ0 ; Μ, φ, τ) and J(f>, μ0 ; u, φ, τ)

are now functions of five variables besides the set of coefficients ω„, characterizing the single scattering albedo and phase function.

Somewhat surprisingly, this more general problem may be solved using mostly functions that have already been discussed. Without presenting detailed formulas, we shall review two independent solutions of this problem, by what

DISPLAY 6.12 R e d u c t i o n t o H F u n c t i o n s in t h e C a s e o f C o n s e r v a t i v e S c a t t e r i n g

E q u a t i o n s g i v e n in D i s p l a y 6.11 c h a n g e a s f o l l o w s .

B a s i c f u n c t i o n s a n d c o n s t a n t s

k = 0 , 7 = oo

m = 0

/ = 1

Κ ( μ , ν) ( n o c h a n g e ) , φ„(μ) ( n o c h a n g e )

Κ(μ) = (4μ)-^0(μ) + 2 ^ 2( μ ) ] / / ( μ )

M o m e n t s o f e s c a p e f u n c t i o n

ί Κ(μ)άμ = 10 (=NK)

f Κ{μ)μάμ = 1,={ (= iUK) Jo

2 ( " κ ( μ ) / <

2 άμ = q' = (1 - g)q = § / 0 + f h ( = WK)

q is t h e e x t r a p o l a t i o n l e n g t h ; l2 = \q[ — j l 0

2 {

1Κ(μ)μ*άμ = Ι + %

C h e c k : f = 1% + ω2\\ - ω 3/ | + ω411


we call the Sobolev method and the Case method. This serves a dual purpose: (a) It provides a guide to the relevant literature, (b) It illustrates how two conceptually quite different approaches may lead to (virtually) the same set of computational steps once they have been prepared for actual use.

T H E SOBOLEV M E T H O D

Sobolev's work is in the classical tradition of the work of Ambartsumian, Chandrasekhar, and others. Sobolev removes the undue emphasis on functions of angle, which characterizes this tradition, and builds into it functions of optical depth. In his final solution (Sobolev, 1969b) the key function O

m( r , τ 0) obeys

an integral equation with a kernel directly related to the characteristic function. A detailed presentation is available in book form (Sobolev, 1975). Rather than reproducing parts of the derivation, we feel the reader may best be served by a simple bu tcomple te " road m a p " (Display 6.13). This map is in the reverse order of the actual computat ion; it identifies on each line the function to be determined and (by reference to lower lines) the functions that enter into this determination. Sobolev's notat ion is used without change.

Route 1 (lines 1-9), describes Sobolev's method for finding the complete radiation field in a homogeneous slab caused by unidirectional incidence with direction cosine ζ (our μ0) . The slab is characterized by optical thickness τ 0 (our b) and the intensity is found at any depth τ in any direction (η, φ).

D I S P L A Y 6 . 1 3

R o a d m a p t o S o b o l e v ' s G e n e r a l T h e o r y

S o b o l e v

( 1 9 7 5 )

L i n e e q u a t i o n

F u n c t i o n

s o u g h t

F u n c t i o n s e n t e r i n g

f r o m l i n e s O p e r a t i o n

R o u t e 1 : t o f ind i n t e n s i t y a t a r b i t r a r y d e p t h

1 ( 1 . 4 8 M 1 . 4 9 ) Ι(τ9η,ζ9φ9τ0) 2

2 (6 .2 ) Β(τ,η,ζ,φ,τ0) 3

2 I n t e g r a t e o v e r τ '

3 F o u r i e r s u m ( s o u r c e f u n c t i o n )

4 , 8 a , b , 2 2 , 2 3 , 2 7 I n t e g r a t e o v e r η' a n d s u m

5 a , b ( o r 1 5 a , b o r 1 6 a , b ) , 6 I n t e g r a t e o v e r τ ' a n d s u m

6 I n t e g r a t e o v e r τ '

6 I n t e g r a t e o v e r τ'

7 S o l v e i n t e g r a l e q u a t i o n

2 1 I n t e g r a t e o v e r η

3 (6 .9 ) B

m(x9 η9 ζ, τ 0) 4 , 8 a , b , 2 2 , 2 3 , 2 7

4 (6 .19 ) D

m(x9 η9 τ 0) 5 a , b ( o r 1 5 a , b o r

9 a S u m

9 S u m

S o l v e s y s t e m of l i n e a r e q u a t i o n s

R o u t e 1 A : t o f ind r e f l e c t i o n f u n c t i o n v i a r o u t e 1

10 (1 .74 ) ρ ( ! / , £ φ , τ 0) 1 T a k e τ = 0

6.5 The Radiation Field at Arbitrary Depth

D I S P L A Y 6 . 1 3 (continued)

119

S o b o l e v

( 1 9 7 5 ) F u n c t i o n

L i n e e q u a t i o n s o u g h t

R o u t e 2 : t o find r e f l e c t i o n f u n c t i o n

11 (3 .2 ) ρ(η, ζ, φ, τ 0) 12 (3 .16 ) Ρ

η(η, ζ, το)

1 3 a (6 .41 ) <Ρ?(1, τ 0)

1 3 b (6 .42 ) Φ?(η, τ 0)

14a ( 7 . 1 2 3 ) (ff<n, τ 0) 1 4 b ( 7 . 1 2 4 ) *?(η, τ 0) 15a (6 .30 ) X

m( C , T 0)

1 5 b (6 .31 )

16a (6 .32 ) ΧΛΙ τ 0) 1 6 b (6 .33 ) Y

m(C, τ 0)

17a — * : ( τ 0)

1 7 b - 3 ? ( τ 0)

18 ( 7 . 1 2 5 ) C, τ 0) 19a ( 7 . 1 2 7 ) Μ ^ , ζ , ΐ ο ) 1 9 b ( 7 . 1 2 8 )

F u n c t i o n s e n t e r i n g

f r o m l i n e s O p e r a t i o n

12 o r 18 F o u r i e r s u m

1 3 a , b , 2 9 S u m

1 4 a , b , 1 5 a , b , o r 1 6 a , b , 2 7 S u m ( A m b a r t s u m i a n

f u n c t i o n s )

1 4 a , b , 1 5 a , b , o r 1 6 a , b , 2 7 S u m ( A m b a r t s u m i a n

f u n c t i o n s )

1 7 a , b , 2 3 , 2 7 ) c , n.

17a b 2 3 2 7 1 S o l v e s y s t e m of l i n e a r e q u a t i o n s

S o l v e n o n l i n e a r i n t e g r a l e q u a t i o n s

2 0 , 2 1 )

2o 2i ( S o l v e l i n e a r i n t e g r a l e q u a t i o n s

15a o r 16a T a k e m o m e n t s ( i n t e g r a t i o n

o v e r ζ). T h i s s t e p n o t s p e l l e d

o u t b y S o b o l e v

15b o r 1 6 b T a k e m o m e n t s ( i n t e g r a t i o n

o v e r ζ). T h i s s t e p n o t s p e l l e d

o u t b y S o b o l e v

1 5 a , b , o r 1 6 a , b , 1 9 a , b , 2 7 S u m

1 4 a , b , 2 9 S u m

1 4 a , b , 2 9 S u m

A c c e s s t o r o u t e s 1 a n d 2 ( f u n c t i o n s n o t d e p e n d e n t o n τ0 o r τ )

2 0 ( 5 . 6 7 ) T

m(n) 2 1 I n t e g r a t e o v e r ζ

21 (5 .30 ) ψ - f a ) 2 3 , 2 7 P r o d u c t ( c h a r a c t e r i s t i c f u n c t i o n )

2 2 (5 .28 ) GKCiy) 2 3 , 2 5 S u m 2 3 (5 .29 ) 2 6 , 2 7 , 2 9 S u m 2 4 (5 .25 ) 2 5 , 2 6 S u m 2 5 (5 .18 ) Ψΐ(η) 2 6 , 2 7 , 29 S u m

2 6 (5 .15 ) 2 8 R e c u r r e n c e 2 7 (5 .6 ) g i v e n ( a s s o c i a t e d L e g e n d r e

f u n c t i o n s ) 2 8 (5 .9 ) 3 0 a , b E l e m e n t a r y 2 9 (1 .42 ) 3 0 a E l e m e n t a r y 3 0 a - g i v e n ( coe f f i c i en t s o f p h a s e f u n c t i o n ) 3 0 b - λ g i v e n ( s i n g l e s c a t t e r i n g a l b e d o ) a

U s e t h i s e q u a t i o n for τ = 0.


The key lines are 5ab, 6, and 7, in which an integral equation in τ has to be solved, rather than two in ζ (our μ):

Φ"(τ, τ 0) = f V ( | T - ί |)Φ»(ί, τ 0) A + Κ"(τ)

The kernel can be expressed in terms of the characteristic function by (line 7):

KM(T) = f Hr^ye'

xM—

Jo η

Once this integral equation is solved, the traditional X and Y functions follow from (lines 5a,b)

Xm(C,T0)= 1 + f V ( T , T 0> - ^ j T

Jo

Ym(C, τ 0 ) = e~^ + Γ W o " τ , τ ο ) * " * " rfT Jo

In order to obtain the proper coefficients it is further necessary to solve a set of linear equations (line 9ab). The reflection function may be found via this method (Route 1A) by taking τ = 0 and adding the proper normalization factor.

The more traditional method, in which no functions of τ are used, is mapped out as Route 2. It corresponds to formulas spelled out (for τ 0 = oo) in Section 6.4. We easily recognize the X and Y functions at which the two routes touch and the polynomials (line 14ab), with coefficients to be solved from a set of linear equations. If τ 0 -> oo, the polynomials q?ty, τ 0) become qTOl) (Sobolev, Eq. 5.40 or 7.55-56), which in the azimuth-independent case (m = 0) we have called the Busbridge polynomials.

Both Routes 1 and 2 draw upon a certain amount of groundwork which leads to the characteristic function Ψ"

1^ ) and to the dispersion function Τ™(η)

(lines 20,21). The largest zero of Τ™(η) is what we have called γ = k~1. Note that

the functions introduced by Sobolev at this stage are more general than those introduced in Section 6.2, in that some of them have three indices. For instance, the Kuscer polynomials [our g?W]

a re identical to [1 · 3 · 5 · · (2m — l)]R^(yy).

It is evident that the simple integral equation in τ, quoted above as line 6, must have a long history in the literature. We have not tried to trace this history but refer the reader to the clear and systematic treatment given by Ivanov (1973, pp. 198-203). His references go back to 1944 and include work by Fock, Placzek, Sobolev, Case, and Davison.

The resolvent Φ, η(τ, τ 0) may be expressed in terms of a resolvent function of

one variable Φ(τ) by means of / • s m a l l e s t o f τ , τ '

Φ"·(τ, τ') = Φ( | τ - τ ' | ) + Φ(τ - ί)Φ(τ' - t )dt Jo

6.5 The Radiation Field at Arbitrary Depth 121

We refer to Ivanov (1973, pp. 199-200) for further details. For later developments of this approach reference may be made to Ivanov (1975), Yanovitskii (1976), and Fymat and Kalaba (1977).

T H E C A S E M E T H O D

The singular eigenfunction expansions were first used in plasma physics by van Kampen (1955). They were combined with discrete eigenfunctions and thus adapted to radiative transfer by van Kampen (1960) and Case (1960). The method thus originated leads to a complete solution of the radiation field at arbitrary τ. It has the clear advantage of conceptual simplicity. It follows the time-honored concept of any eigenfunction expansion in mathematical physics with the usual steps of (1) finding all eigenvalues and eigenfunctions, (2) proving the completeness theorem that any function must be expandable in these eigenfunctions, (3) deriving the orthogonality relations, (4) setting the boundary conditions, and (5) finding the coefficients necessary to satisfy these boundary conditions by using the orthogonality relations.

Physically, the eigenfunctions correspond to modes of propagation in an unbounded medium. They all are exponential in τ. The diffusion solution which plays the central role in Chapter 5 is simply the least damped discrete mode. All other modes with discrete (if any) and continuous eigenvalues contribute to the other functions and constants introduced in Chapter 5. In the Case method they are dealt with explicitly.

Note that in the brief summary just given the thickness b or τ 0 of the slab must be specified only in stage (4). The effort to postpone as long as possible the specification of τ 0 puts this method conceptually in a quite different class from the traditional one.

However, by the time the steps of the computat ion have all been written out this difference has largely vanished. One stumbling block is that we do not have a single boundary condition specifying the intensity over the full range of direction cosines — 1 < u < 1 at one value of τ. Instead we have the radiation specified over the hemisphere (half range) corresponding to incident radiation at two values τ = 0 and τ = τ 0. This makes it necessary to introduce half-range orthogonality relations in addition to the full-range ones. It is in this stage that the X and Y functions appear.

The full potential of this method has been realized from the start but has only gradually been exploited in admitting more and more general assumptions. Among the fifty odd papers we find the following most relevant in the present context. Case and Zweifel (1967) wrote an instructive book which explains the principles of the method and works out the details mostly for isotropic and linearly anisotropic scattering. The most penetrating review, including the more recent developments, formulating the method for an arbitrary unpolarized phase function and applying it to the half-space (semi-infinite atmosphere, b = oo ) is by Kuscer and McCormick (1974). A complete formulation of the


method for scattering with polarization was presented by Domke (1975a,b; 1976). The degeneracy due to the double eigenvalue zero for conservative scattering has been discussed in detail by Lekkerkerker (1976a,b).

Complete formulas for actual computat ion with an arbitrary phase function and arbitrary thickness of the slab were presented in a mimeographed report (Kaper, 1966) up to the point where the coefficients of each mode have to be found by solving Fredholm integral equations. By tracing the computational steps in Kaper 's report we are satisfied that they are mostly identical to those of Sobolev's method shown in Display 6.13. Kaper et al (1970) report on experience obtained in the actual computations and present a collection of numerical results. See also Shultis (1973).

Final conclusion. Circumstantial evidence exists that the two methods lead to the same set of equations, but the actual identity has not been checked in detail. A complete and critical comparison of each step in the two methods, with a view to ease and accuracy, is a task which—to my knowledge—has not yet been performed.

R E F E R E N C E S

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7 • Isotropic Scattering; Solutions by

Use of the Milne Operator

Chapters 7-9 contain results based on the assumption that the elementary scattering process is isotropic with albedo a. Under this assumption, the Milne equation offers an attractive method for solving multiple scattering problems. This method is developed in a condensed notation in this chapter. We develop the method in logical rather than historical order. It may be noted, however, that the basic integral equation (for which we use the now common name "Milne equation") , is called the "Schwarzschild-Milne integral equa t ion" by Chandrasekhar (1950) after papers published in 1905-1921, and was already explicitly derived by Lommel (1889) and Chwolson (1890).

Results in formulas, tables, and graphs referring both to the reflection and transmission problem and to the escape problem, are presented in the next two chapters. Chapter 8 treats problems without boundary, or with one boundary, i.e., semi-infinite atmospheres. Chapter 9 treats problems with two boundaries, i.e., finite slabs.

7.1 MATRICES IN τ AND μ

7.1.1 Definitions

If we choose successive scattering as a method, the alternatives (Section 4.3) are either to solve the internal radiation field / (τ , u, φ) from the transfer equation, or the internal source function J (τ, w, φ) from the generalized Milne

127

128 7 Isotropic Scattering; Use of the Milne Operator

equation. For isotropic scattering the choice is obvious, because J ( T ) depends on one variable less than /(τ , u). In most classical papers this choice was made. We present the resulting formulas in this chapter. For this purpose it is useful (van de Hulst, 1963) to extend the condensed notation of Display 5.1 to include operators on or functions of optical depth τ. The most important operator in this set is the Milne operator. We shall call the basic equation in this method the Milne equation. The customary name in the literature is "auxiliary equation," but this term is misleading, because it suggests that the equation of radiative transfer is somehow more fundamental. This is not true, for it was shown in Section 4.3 that the half-steps can be combined in two different, fully equivalent, ways, one leading to the equation of transfer, the other to the Milne equation.

The two parameters albedo a and total optical thickness b characterize the atmosphere. Further independent variables are the cosines of angles μ 0 and μ and the optical depths τ and τ'. The equations and methods of solution contain integrations of two types : integration over τ or τ' from 0 to b, and integration over μ or μ 0 from 0 to 1.

In the condensed notat ion these integrations appear as matrix products. A function of one variable will be called a μ or τ vector. A function of two variables will be called a μμ, μτ, τμ, or ττ matrix. Order must be strictly observed and the physical meaning will always be that the left factor operates on the right factor. Multiplication of two matrices, a matrix and a vector, or two vectors signifies integration over the adjoining arguments, which should be both τ or both μ. We adopt the definitions

τ multiplication: FG = f F ( T ) G ( T ) άτ Jo

μ multiplication: FG'= Γ Ρ\μ)σ{μ)2μάμ Jo

where F ( T ) , G(T), Κ(μ), and β'(μ) are arbitrary functions. These multiplications obey all rules of matrix multiplication, in particular

the associative property. The factor 2μ in the definition of μ multiplication is in agreement with Section 5.1 and is necessary in order to make it possible to work mostly with symmetric matrices. N o transformation or approximation is made at this stage. We continue to work with continuous variables, changing only the notation and terminology. In machine computat ion these continuous functions may be replaced by actual matrices with a finite set of elements, but at the moment we are not concerned with the technique for doing so or with the accuracy lost in that process.

The curious fact about this notation is that it seems too economical, for the notation does not tell whether a function of μ or μ 0 is meant or whether the argument is τ or τ'. However, this is always clear from the context, i.e., from the physical meaning of the expression. N o r do the μμ and ττ matrices specify

7.1 Matrices in τ and μ 129

in what order the arguments are to be taken, for instance, in the order (μ, μ0) or ( μ 0, μ). This is usually irrelevant because of symmetry. These properties are manifestations of the reciprocity principle.

The list of the vectors and matrices which we use is given in Display 7.1. Definitions common with the earlier list in Display 5.1 are indicated by footnote a. Some symbols are used for a different quantity here when no confusion can arise.

7.1.2 Relations among Singular Matrices

The singular functions in this list are expressed in terms of.common delta functions, which are obviously indispensable in this type of calculus. A singular function may be interpreted in two ways:

(a) as an operator, i.e., by stating that matrix multiplication with these functions is an instruction to replace one variable by another variable or by a number, as specified.

(b) as a continuous function which has very large nonzero values only in a very narrow range of values of the independent variable and which, upon integration by the stated rules of matrix multiplication, yields the specified result when this interval shrinks to zero.

Definition (b) is less elegant mathematically but corresponds more closely to actual situations. For instance, sunlight falling on a planet is almost, but not quite, unidirectional because of the size of the solar disk. Similarly, a telescope does not quite view only one direction, because of its limited resolving power. In these situations the angular integration interval is small but not strictly zero.

Some products containing these singular matrices as factors have been collected in Display 7.2. These relations are mathematically trivial. For instance, verification of relation 1 proceeds by multiplying with an arbitrary τ vector S and writing the integrals out : OPS = 0(PS) = S(0) = AS.

However, the physical meaning of these relations is of interest. The meaning of relation 1 is that light incident at a grazing angle penetrates (without scattering) only into the uppermost layers of the atmosphere, or, reciprocally, that viewing an atmosphere at grazing angles we observe only the uppermost layers. The meaning of relation 5 is that radiation from a narrow layer of isotropic sources on top of the atmosphere (N) suffers scattering (P) at various optical depths, distributed as the Milne distribution function (M), measured from the top down 04) or, reciprocally that a radiation detector with isotropic sensitivity characteristic (ΛΓ) held at the top of the atmosphere records the radiation from various layers (P) weighted according to the Milne distribution function (M), measured from the top down (A). The interpretation of the other relations is similar.


D I S P L A Y 7.1

N o t a t i o n U s e d in P r o b l e m s w i t h I s o t r o p i c S c a t t e r i n g

S y m b o l F u n c t i o n a n d p h y s i c a l m e a n i n g N o t e s

μ v e c t o r s

τ v e c t o r s

μμ m a t r i c e s

τ τ m a t r i c e s

μτ ( τ μ ) m a t r i c e s

/ Ε

D

Ν

U

0

1

J s

A

B

R

T

T0

Tdiff

M

C

P(P)

Q ( Q ) K(K)

Λ η ( μ ο )

= i n c i d e n t i n t e n s i t y

Ληιΐ(μ)

= e m e r g e n t i n t e n s i t y

( 2 μ )

_1 χ a n g u l a r d e t e c t o r s e n s i t i v i t y

1 / (2μ) 1

o p e r a t o r d e f i n e d b y O F = F ( 0 ) , F a r b i t r a r y

o p e r a t o r d e f i n e d b y I F = F ( l ) , F a r b i t r a r y

J(t) = t o t a l s o u r c e f u n c t i o n

S(t) = s t a r t i n g t e r m o f s o u r c e f u n c t i o n

jp(x) = coe f f i c i en t of a

p in p o w e r e x p a n s i o n of s o u r c e

f u n c t i o n

o p e r a t o r d e f i n e d b y AF = F ( 0 ) , F a r b i t r a r y

o p e r a t o r d e f i n e d b y BF = F(b), F a r b i t r a r y

( 2 μ ) ~

1 δ(μ — μ 0) = d i a g o n a l m a t r i x , o p e r a t o r

c o n v e r t i n g t h e a r g u m e n t μ 0 i n t o μ, o r c o n v e r s e l y

/?(μ , μ 0) = r e f l e c t i o n f u n c t i o n

Τ ( μ , μ( )) = t r a n s m i s s i o n f u n c t i o n

( 2 μ ) ~

ίβ~

Ι}μ δ(μ — μ 0) = z e r o - o r d e r t r a n s m i s s i o n

f u n c t i o n , c o n v e r t s a f u n c t i o n of μ 0 i n t o t h e s a m e

f u n c t i o n of μ m u l t i p l i e d b y e x p ( — b/μ)

Τ — T0 = d i f fuse t r a n s m i s s i o n f u n c t i o n

δ(τ — τ ' ) = d i a g o n a l m a t r i x , o p e r a t o r c o n v e r t i n g t h e

a r g u m e n t τ ' i n t o τ , o r c o n v e r s e l y

ΐ £ ι ( | τ — τ ' | ) = M i l n e o p e r a t o r

(1 — aM)'

1 = c o m p l e t e r e d i s t r i b u t i o n m a t r i x

(1/μ)β-

φ-

τ)/μ

G(T, μ)/4μ = (4μ)-

a, e

a, e

e

e

e

e,f e

χ p o i n t - d i r e c t i o n g a i n a T h e s e s y m b o l s a r e a l s o u s e d in D i s p l a y 5 .1 . b S e e D i s p l a y 7.2 for r e l a t i o n s c o n t a i n i n g t h e s e v e c t o r s . c T h e s y m b o l 1 ( o r a n o t h e r v a l u e ) fo r t h e s u b s t i t u t i o n μ = 1 ( o r a n o t h e r v a l u e ) wi l l m o s t l y b e

u s e d in l e g e n d s of t a b l e s o r figures. F o r i n s t a n c e , w e m a y w r i t e 1 R U i n s t e a d of " t h e v a l u e of R U for

μ = 1." d T h e s t a r t i n g t e r m m a y b e z e r o - o r d e r o r first-order; s e e S e c t i o n 7.2. e Al l τ τ a n d μ μ m a t r i c e s in t h i s l is t a r e s y m m e t r i c , i.e., a r e t h e i r o w n t r a n s p o s e . f I n t h e a s t r o p h y s i c a l l i t e r a t u r e (e.g. , K o u r g a n o f f , 1952) t h e M i l n e o p e r a t o r is c o m m o n l y

k n o w n a s t h e A o p e r a t o r . 9 A f o r m a l d i s t i n c t i o n b e t w e e n t h e t w o m a t r i c e s Ρ a n d Ρ r e p r e s e n t i n g t h e s a m e f u n c t i o n is

n e c e s s a r y in t h e m a t r i x n o t a t i o n b u t c e a s e s t o b e v i s i b l e w h e n t h e i n t e g r a t i o n s a r e w r i t t e n o u t . Ρ is

t h e t r a n s p o s e d m a t r i x o f P ; s i m i l a r l y , Q a n d K.

7 .2 Solving the Milne Equation 131

D I S P L A Y 7 . 2

P r o d u c t s C o n t a i n i n g S i n g u l a r M a t r i c e s

1 PO = OP = A 2 QO = OQ = Β

3 ΡΑ = ΑΡ = 2Ν = μ~

ι

4 PB = BP = 2T0N = β-

ά/μ/μ

5 2AM = 2MA = NP = PN = £ , ( τ )

6 2BM = 2MB = NQ = QN = E,(b - τ) 1 2AMB = 2BMA = Ex(b)

8 2UM = 2MU = 2 - Ε2(τ) - E2(b - τ )

9 ΑΜΑ a n d BMB d i v e r g e

7.2 SOLVING THE MILNE EQUATION

Let 5(τ) be the starter part of the source function, which arises either from first-order scattering of light incident from outside (external) sources or from emission by embedded (internal) sources (see Section 4.3). If we insist—and it seems wise to do so—on using the order ρ for terms proport ional to a

p, then

we must write this par t in the form

In further applications we assume either absence of embedded sources or absence of incident radiation, so that only one term remains and the specification " ex t " or " i n t " is unnecessary. The Milne equation reads in matrix notat ion

J = S + aMJ or S = (1 - aM)J

the formal solution of which is

J = CS

where

C = (1 - aMy1

which may be developed as

C = 1 + A M + a2M

2 + a

3M

3 + · · ·

Solving the Milne equation consists of inverting the matrix 1 — aM to find C. This inversion can be made by many methods. Doing it by summing the power series is the most straightforward way but not always the most economical one in computing time. The solution of the Milne equation then takes the form

J = S + aMS + a2M

2S + a

3M

3S +

132 7 Isotropic Scattering ; Use of the Milne Operator

in which each term is a τ vector and each successive term is found by multiplying the preceding one with the matrix aM.

In reality the "matrices " are functions of a continuous variable, the equation is an integral equation and the expansion in powers of aM is its solution in the form of a Neumann series.

The physical significance of the power series expansion is that each term corresponds to a separate order of scattering. The term which has aF as a coefficient is by definition due to pth order scattering. This statement holds not only for the matrix C but also for any matrix product in which the expanded matrix C occurs as a factor. In this manner we can identify simply and unambiguously the pth order term of the reflection function, transmitted flux, point-direction gain, etc.

We now make an exception to the rule of ignoring methods that are not exact by stating a variation principle that may be used to solve the Milne equation.

Let G(T) be a trial function that somewhat resembles the function J which is sought, and let it be written as a τ vector G . Define the scalar quantity

9 = G [ 2 S + aMG - G ]

If we substitute for G the correct solution J, this quantity assumes the value

J % = JS

Upon replacing S by (1 — aM)J in both equations and using the symmetry of M, we find

_ & = (j - G)(l - aM)(J - G )

This symmetric function is always positive, as can be seen from an eigenfunction expansion (Section 7.4.2). Only if G = J is the result zero. Hence IF reaches the absolute maximum J % if G = J.

We can use this principle in practical computat ion by adopting for G a simple analytic form with some free parameters. The parameters are then varied until !F reaches a maximum. This maximum is almost the precise answer because the error in the wrong guess enters as a square.

The method has been presented here in its very simplest form. Nagirner (1973) uses it in accurate numerical computat ion and refers to it as a known principle (Mikhlin, 1970). Stokes and DeMarcus (1971) formulate it for an inhomogeneous slab with isotropic scattering in which a is an arbitrary function of depth. They point out that the method has the added advantage of giving directly the product J S = (α/4)Κ(μ, μ) and hence the limb darkening of a planet seen in opposition (cf. Sections 7.3.1 and 18.3.1).

7.3 Resulting Quantities 133

7.3 RESULTING QUANTITIES

7.3.1 Reflection and Transmission

We can derive accurate expressions for the reflection and transmission functions as well as many other quantities. The events in logical order, are the following:

First event. The radiation incident from the top / , which has an arbitrary intensity distribution with angle, penetrates into the slab and is scattered at various depths, thus establishing the original (first-order) source function

j l = s = \a?l

The factor \ appears when the integral is written fully, using the definitions. Essentially, it is due to the fact that the flux has a factor π but that scattering occurs into 4π steradians.

Second event. The radiation forthcoming from the source distribution S is scattered again and again, thus establishing the complete source function, which includes S and all higher orders and is written as

J = CS

Third event. Radiation from this source function at all depths reaches the top surface under various angles and exits, giving the emerging intensity

^ t o p

= PJ

Similarly, the radiation from these sources emerging at the bot tom is QJ but, in addition, some of the original incident radiation shines through without any scattering, giving the intensity T0I at the bottom. The combined emerging radiation intensity is

^bottom = T0I + QJ

Fourth event. If a detector is used to measure the emerging radiation (at top or bot tom) sampling the different angles as specified by the vector D, the detector reading is a number

d = DE

The combined result of these successive matrix multiplications thus assumes the form of a reflection function R and a transmission function Τ operating on J :

Etop = RI = PCiaFl

b o t t o m = TI = T0I + QCiaFl


D I S P L A Y 7.3

C o m p u t a t i o n of R e f l e c t i o n a n d T r a n s m i s s i o n b y S o l v i n g t h e M i l n e E q u a t i o n

R e f l e c t i o n

S t e p s : / i n c i d e n t i n t e n s i t y

Jx — \aPl first-order s o u r c e f u n c t i o n

J = CJX c o m p l e t e s o u r c e f u n c t i o n

Ε = Ρ J e m e r g i n g i n t e n s i t y

d = DE d e t e c t o r r e a d i n g

R e s u l t : d = DPC\aPl = DRI = (DP)C({aPl)

Dif fuse t r a n s m i s s i o n

S t e p s : first t h r e e a s for r e f l e c t i o n

Ε = QJ e m e r g i n g i n t e n s i t y

d = DE d e t e c t o r r e a d i n g

R e s u l t : d = DQC\aPl = DTdiffI = (DQ)C(iaPI)

C h o i c e s

I n c i d e n t i n t e n s i t y S t a r t i n g s o u r c e f u n c t i o n

C o n d e n s e d F u l l A s s u m e d d i s t r i b u t i o n C o n d e n s e d F u l l

( μ 0) I = ( 1 / 2 μ 0) ^ ( ^ ~ Vo) u n i d i r e c t i o n a l (iaP) = (α/4μ0)β-

τΙ»°

(U) I = 1 u n i f o r m i n t e n s i t y a s in {ïa PU) J, = ϊαΕ2(τ) L a m b e r t ' s l a w

(N) I = 1 / (2 / ι0) f r o m n a r r o w s o u r c e l a y e r (iaPN) Ji

The final equations for the reflection and transmission functions are thus

R = PCiaF

Τ = T0 + QCiaP

These may be taken as the mathematical definitions of R and T. Because of the symmetry of C, both R and Τ are symmetric matrices.

All events are summarized in Display 7.3, together with the first-order source functions that must be used for the three forms of the incident radiation field assumed in most of the tables in this book.

If a subroutine for applying the Milne operator is available, then C may always be replaced by its expansion in powers of aM, in which case Display 7.3 contains the complete procedure for a machine calculation of the reflection and transmission functions, and its moments and bimoments.

By intentionally avoiding the use of reciprocity in the computation, the accuracy can be checked by referring to the equivalent number in the reciprocal position. The functions for near-grazing angles ( μ0 or μ = 0.1) or for the narrow-layer distributed function (N) afford the most sensitive checks because they involve functions with the steepest gradients in τ. First we intended to construct Table 12 (Section 9.1.1) entirely by this method, and extensive computat ions


were made. The τ integration interval was chosen so as to permit an error not exceeding 1 0 "

5. Typically, in the range 0.1 < b < 1, integrating by steps of

0.0025 in τ appeared to suffice for such accuracy. Later, in view of reaching larger b values and also treating anisotropic

scattering, we abandoned this method and switched entirely to doubling. Checks of one method against the other generally confirmed an error not exceeding 1 0 "

5.

In the expressions for the detector reading d, shown in Display 7.3, the function D occupies a position symmetric to / . Similarly, DP (or DQ) occupies a position symmetric to PI. These formulas may, therefore, be turned around and given a reciprocal interpretation, in which / refers to the detector characteristics, and D to the incident intensity distribution. Particular choices for D and / and their reciprocal interpretations are shown in Section 9.1.1. These symmetries stand out clearly because of the condensed notation. The choice of the order of computat ion obviously may be decided on the basis of convenience. For instance, dtop may be computed from

dtop = D(PCiaP)I = DRI

or

dtop = (DP)C(\aPI\

whichever seems more convenient in a particular situation. Numerical results are presented and discussed in Chapters 8 and 9.

7.3.2 Source Function

The advantage of the method sketched is that not only the final results but also the intermediate ones have a clear physical meaning. Machine output of these intermediate quantities provides a number of other quantities as well as many useful checks. Display 7.4 summarizes the significance of the partial matrix products.

Figure 7.1 shows in a very instructive manner how the contributions of various orders to the source function J vary with depth. Both parts represent Jn(x) for perpendicular incidence and albedo 1. The curve J^x) is identical (in the interval 0 < τ < 1) in both figures but the later curves in the right-hand figure are lower, because losses occur through both boundaries. The curves for b = oo show how successive scattering consists of a slow diffusion inwards, with no obvious sign of convergence. However, the curves for b = 1 rapidly approach the symmetric form characterizing the first eigenfunction, each succeeding curve being scaled down by the eigenvalue 0.619 (Section 7.4.1).

Those interested in historical perspective may at this point wish to spend an hour (as I did) translating words, definitions, and notations to check that the function y tabulated on p. 483 of Lommel (1889), with the help of a table of the


D I S P L A Y 7.4

S i g n i f i c a n c e o f P a r t i a l M a t r i x P r o d u c t s O b t a i n e d i n C o m p u t e r P r o g r a m

P a r t i a l

p r o d u c t s o f

a DP C - PI

4

A r g u m e n t s

P h y s i c a l o n w h i c h

q u a n t i t y d e p e n d e n t

S y m b o l a n d n a m e of

f u n c t i o n t h a t m a y b e f o u n d

f r o m t h i s q u a n t i t y

0

R e f l e c t i o n fo r

u n i d i r e c t i o n a l

i n c i d e n c e

f r o m μ 0

J R d

(α, μ0,τ) {a, b, μ0,τ) (a, b, μ0,μ) (a, b, μ0)

t r i v i a l

G, p o i n t - d i r e c t i o n g a i n

R, r e f l e c t i o n f u n c t i o n

RU, RN, m o m e n t s o f R; X f u n c t i o n

R e f l e c t i o n fo r

d i s t r i b u t e d

i n c i d e n c e , U or Ν

J ι (α, τ ) t r i v i a l

J (a,b,x) 0 _ υ g o , m o m e n t s o f p o i n t -

d i r e c t i o n g a i n

Ε (a, b, μ) UR, NR, m o m e n t s o f R ; X f u n c t i o n

d (a, b) URU, URN, NRN, b i m o m e n t s o f R ;

a* !, a 0, ocl, m o m e n t s o f

X f u n c t i o n

T r a n s m i s s i o n e n t i r e l y s i m i l a r T, t r a n s m i s s i o n f u n c t i o n

w i t h m o m e n t s a n d

b i m o m e n t s

Y f u n c t i o n w i t h m o m e n t s

F i r s t o r d e r o m i t f a c t o r C f i r s t - o r d e r t e r m s

of s a m e q u a n t i t i e s

F f u n c t i o n s

G f u n c t i o n s a F o r f u n c t i o n s n o t d e p e n d e n t o n τ re fe r t o S e c t i o n 9.1 a n d D i s p l a y 9.1 for d e t a i l s . F o r f u n c t i o n s

d e p e n d e n t o n τ , i.e., g a i n a n d i t s m o m e n t s , re fe r t o S e c t i o n 9.3 a n d D i s p l a y 9.4.

integral logarithm published in 1809, is eight times the function J2 plotted in the left part of Fig. 7.1 and (in our notation)

8 J 2W = (y + In 2 + In x)e~x + Εχ(τ)

It has a maximum 0.9553 at τ = 0.2807, a result also derived by Chwolson (1890). A few lines further down Lommel regrets that he must refrain from computing the higher orders because the integrals that define them are not expressible in known functions.

7.3.3 Point-Direction Gain, X and Y Functions

The point-direction gain G(a, b, τ, μ) was defined in physical terms in Section 3.4.3. It is instructive to follow this definition carefully, write down

73 Resulting Quantities 1 3 7

F i g . 7.1. S o u r c e f u n c t i o n in s u c c e s s i v e o r d e r s se t u p in a n i s o t r o p i c a l l y s c a t t e r i n g s l a b b y p e r

p e n d i c u l a r l y i n c i d e n t r a d i a t i o n . I n t h e s e m i - i n f i n i t e s l a b ( a ) t h e r a d i a t i o n s e e p s i n w a r d a n d c o n

v e r g e n c e is s l o w . I n t h e l a y e r w i t h o p t i c a l t h i c k n e s s 1 (b ) t h e d i s t r i b u t i o n w i t h o p t i c a l d e p t h r a p i d l y

b e c o m e s s y m m e t r i c a n d r e p e t i t i v e .

all quantities in matrix notation, and express the gain as a mathematical function. We employ independently the definitions based on the two reciprocal experiments in order to check the correctness of the expressions.

D I R E C T EXPERIMENT

Assume embedded isotropic sources distributed with optical depth as 5 0( τ ) , written as the τ vector S 0, and assume absence of radiation incident from outside. The solution of the Milne equation is then

J = CS0 and the intensity emerging at the top is

£ t o p — PJ — PCS0


Now specify that S0 is a narrow source layer at depth τ 0, i.e.,

S 0( T ) = δ(τ - τ 0)

This gives

£ t 0p = PC

The same source layer in the absence of an atmosphere would give the emerging radiation intensity l/μ. Hence the point-direction gain is given by

(1//I)G(A, ft, τ0,μ) = ΡΟ

In full matrix notation (Display 7.1), this should be written as

Κ = 4PC

REVERSE EXPERIMENT

Assume unidirectional incident radiation from direction μ 0, normalized as usual to flux π on a unit area of the atmosphere. Assume absence of embedded sources. The first-order source function and the total source function are the same as in deriving the reflection function:

S = {aP, J = CS = iaCP

The radiation density at depth τ equals

q = (4n/ac)J = (n/c)CP

In the absence of the atmosphere the incident beam has the radiation density U^QC)'

1. Hence, the point direction gain is given by

(l/ft>)G(fl, ft, τ, μ 0) = CP

which in full matrix notation (Display 7.1) reads

K = ACP

Apart from the subscript 0, which we may now drop, the two results represent the same function of τ and μ, expressed first as a μτ matrix, then as a τμ matrix. We thus have found two equivalent recipes for computing G(a, ft, τ, μ).

G A I N MOMENTS

Expressions for the gain moments can be found by integrating the expression for the gain. At this point we digress to the function Φ(τ, τ 0) to which Sobolev and his co-workers attach the importance of a key function in the theory (Section 6.5; Sobolev 1975; Nagirner 1973). Consistent with the notation of this book


we write it as Φ(α, b, τ) or in condensed notat ion as a τ vector Φ. It is defined as the solution of the integral equation (arguments a and b omitted):

Φ(τ) = ±αΕχ(τ) 4- ΐ^αΕ^τ - τ'\)Φ(τ')άτ' Jo

The question of whether the same function appears in a different form in the preceding theory can be answered by rewriting this equation in the condensed notation of Section 7.1 :

Φ = aMA + αΜΦ

from which follows the solution

Φ = CaMA = ^aCPN

where we have used an identity from Display 7.2. Referring to Display 7.4 and the expression for the gain just derived, we obtain the result

Φ(α, b, τ) = \ag.x{a, b, τ)

Hence, apart from a trivial factor a/2, Sobolev's Φ function is identical to the moment of order — 1 of the point-direct ion gain.

T H E X AND Y FUNCTIONS

The X and Y functions are the point-direction gain at τ = 0 and τ = b. The preceding result immediately yields

(1/μ)Χ(α, b, μ) = PC A

(1/μ)Υ(α, b, μ) = PCE

The left-hand side is in ordinary functions, the right-hand side in matrix notation. The validity of these equations is confined to isotropic scattering.

We can also follow the definition of gain valid for an arbitrary plane-parallel atmosphere, even if it is inhomogeneous or has anisotropic scattering (Eq. (1) from Section 4.4):

(1/2μ)ΙΧ(α, b, μ) - 1] = RN

(1/2μ)Υ(α, b, μ) = TN

Either set of equations may be employed to find the X and Y functions from the computat ion sketched in the preceding section. The one set can be derived from the other set by employing relations from Display 7.2. For instance:

(1/μ)Χ(α, b, μ) = (1/μ) + 2RN = 2N + £)aPCPN

= PA + aPCMA = P ( l + aCM)A = PC A


G A I N DERIVATIVES

For future reference it may be helpful to state here without derivation (see van de Hulst, 1964) some exact relations for derivatives of the gain. These are

dG(a, b, τ, μ))^ = \ag-x{a, b,b - τ) Y(a, b, μ)

dG(a, b, τ, μ)^ + dG(a, b, τ, μ)δτ = - G(a, b, τ, μ)/μ + \aq.x{a, b, τ)Χ(α, b, μ)

where

g_ x(a9 b, τ) = f G(a, b9 τ, μ) άμ/μ Jo

The corresponding equations for the gain moments are

dgn(a, b, x)/db = i ^ - ^ a , b,b- τ)βη(α, b)

dgn(a, b, i)jdb + dgn(a, b, τ)/δτ = -gn-x(a9 b, τ)

+ jag-ifa b, τ)α„(α, ft)

where a„ and j8„ are moments of the X and Y functions.

7.4 EIGENVALUES OF THE MILNE OPERATOR

7.4.1 Numerical Values

The eigenvalues of the Milne operator are the values of η for which the equation

MJ = Y\J

has a solution. The solution corresponding to each eigenvalue Y\{ (i = 1 ,2 ,3 , . . . ) is the eigenfunction J ( T ) = C,-(T), with a normalization factor that we can leave undefined. The numbering is chosen so that η1 is highest, η2 next, etc. All eigenvalues are different and < 1, except for b = oo, a = 1, where ηί = 1 and ^ ι (τ ) is the Hopf solution (Section 8.6.2).

Writing the defining equation in the form

Y\~

1MJ = J

we see that this represents the homogeneous Milne equation with albedo a = η~

ι. With this value of the albedo the radiation field in the layer is self-

sustained, for it is maintained without any external or internal input. The new energy generated at each scattering with a > 1 inside the layer is just compensated by the energy lost by radiation from the two boundaries. In the language of nuclear reactors, the layer is then said to be critical. A table of eigenvalues ηχφ) may therefore serve as a table of critical albedos a for a given size b, or as a table of critical sizes b for a given albedo a > 1.

7.4 Eigenvalues of the Milne Operator 141

Another practical use of the eigenvalues is to judge the number of terms required to perform a computat ion by successive scattering to desired accuracy. The first eigenvalue shows how rapidly the infinite sum converges. The second eigenvalue shows how rapidly the remaining terms may be replaced by the sum of a geometric series, for if the calculation starts with a source function expandable as

J I ( T ) = C 1C 1( T ) + C 2C 2( T ) + - - -

then after ρ successive scatterings we have obtained

J p + I W = calC^x) + c2n\C2(T) + · · ·

which means that the dominant term has decreased by r\\, and the relative contribution of the nondominant terms by about {η2Μι)

ρ· It usually does not

pay to compute the coefficients cx and c2, which vary from case to case. Assuming they are of order 1, we obtain the result that for ΛΓ-figure accuracy we need very roughly either N/( — In ηχ) terms in a straight sum or N/( — In η2) terms in a straight sum plus a remainder summed as a geometric series with ratio η1.

Numerical values of ηί and η2, obtained from the Jn{x) functions computed in the main program, are presented in Table 4 and Fig. 7.2. They compare quite

T A B L E 4

E i g e n v a l u e s o f t h e M i l n e O p e r a t o r for P l a n e - P a r a l l e l S l a b s , I s o t r o p i c S c a t t e r i n g

F i r s t e i g e n v a l u e S e c o n d e i g e n v a l u e T h i r d e i g e n v a l u e

b N u m e r i c a l A p p r o x i m a t e

0 N u m e r i c a l A p p r o x i m a t e

0 N u m e r i c a l A p p r o x i m a t e

0

0.01 0 . 0 2 7 6 0 . 0 2 7 6 — _ — _ 0 .02 — 0 . 0 4 8 4 — 0 . 0 0 6 9 — —

0 .05 — 0 . 0 9 8 4 — — —

0.1 0 . 1 6 4 0 . 1 6 3 0 0 . 0 3 3 8 — —

0.2 — 0 . 2 6 0 0 . 0 6 6 — —

0 .25 0 . 3 0 0 5 0 . 2 9 9 0 . 0 9 4 5 0 .081 0 . 0 5 7 3 —

0.4 — 0 . 3 9 3 0 . 1 2 6 —

0.5 0 . 4 7 7 4 0 . 4 4 3 0 . 1 7 4 6 " — 0 . 1 0 8 4

0.8 0 . 5 5 5 - 0 .231 —

1 0 . 6 1 9 0 0 . 6 1 0 0 . 3 0 9 5 0 . 3 0 6 0 . 2 0 0 0 . 1 6 4

2 0 . 7 8 3 0 0 . 7 8 1 0 . 5 0 3 1 0 .449 0 . 3 5 2 0 . 2 8 1

4 0 . 9 0 2 1 0 .899 — 0 .691 0 . 5 6 5 * 0 . 4 9 8

8 - 0 . 9 6 4 — 0 .871 0 . 7 5 0

10 0 . 9 7 5 5 0 . 9 7 6 0 . 9 0 9 0 . 8 2 9 0 . 8 1 6

2 0 — 0 . 9 9 3 0 .979 — 0 . 9 3 9

00 1 1 1 1 1 1

° H o r i z o n t a l l i n e s e p a r a t e s t h i n - l a y e r a p p r o x i m a t i o n f r o m t h i c k - l a y e r a p p r o x i m a t i o n . b M a r g i n o f e r r o r 5 u n i t s i n l a s t w r i t t e n d e c i m a l .


I 1 . 1 I I I I I L I I 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

— • b / o + b )

F i g . 7 . 2 . E i g e n v a l u e s o f t h e h o m o g e n e o u s M i l n e e q u a t i o n f o r i s o t r o p i c s c a t t e r i n g in a s l a b o f

o p t i c a l t h i c k n e s s b.

well with the values computed by Mullikin (1962) by a different method. The corresponding first eigenfunctions C ^ T ) , normalized to C^b) = 1, are presented in Fig. 7.3. Probably many other authors have made such computations. See, for instance, Bennet (1964), Erdôs et al. (1970).

Estimates of the behavior for small and large b can be made in many ways. It seems simplest to employ the formula based on a trial function F

ηΆρρ = FMF/FF

which gives a very close approximation to ηη9 provided that F is a fair approximation to C„(T) and that eigenfunctions of lower order are absent in the expansion of F. The following results may be noted without detailed derivation.

Thin layers. Take F — 1 ; this gives

f l . a p p = 1 " β " E3{b)l/b = K - / + l)b + · · ·

take F = τ — jb; this gives

>/2 , app = 1 + 1 2 f r3[ i - £ 5( 6 ) ] - 12b-

2E4(b) - 3b~

lti + £3(ft)]

= !*> + ···


F i g . 7.3. E i g e n f u n c t i o n s b e l o n g i n g t o t h e first e i g e n v a l u e o f t h e h o m o g e n e o u s M i l n e e q u a t i o n

fo r i s o t r o p i c s c a t t e r i n g in a s l a b o f o p t i c a l t h i c k n e s s b.

or take F = 1 for τ > \b and F = — 1 for τ < jb, then the dominant term becomes

* h , aPP = Kin 2)6 + · · · = 0.35& + . . .

The approximations thus derived approach the exact values quite well for b < 0.5, as Table 4 shows.

Thick layers. If a function J{x) in an unbounded medium is developed in a Taylor expansion about τ and if, subsequently, the Milne operator is applied, the result takes the form

w r „ X

l d*

J l d 6j MJ = J ( T) +

3 ^ +

5 ^ +

7 ^+

- -

If the function is smooth enough to neglect higher order derivatives, this may be approximated by

ld*J

3dr2

This is identical to the relation underlying the opacity formula in a stellar interior and related to the Eddington approximation in a stellar atmosphere.


In this approximation the eigenvalue equation considered in the present section becomes

ld*J

3άτ2

_ _ -t %Λ, t*

which has the solution

where

J = sin ω(τ -h q)

ω2 = 3(1 - η)

The solution near both boundaries must resemble the solution for a constant net flux streaming from a semi-infinite layer (Section 8.6.2), in which the straight extrapolation of the J ( r ) curve would reach zero at optical depth = 0.71 outside the boundary. Hence we postulate for a reasonably close approximation to the correct solution that

J(x) = 0 for τ = - 0 . 7 1 , τ = b + 0.71

This fixes the constants q = 0.11 and co(b + 1.42) = mn. Consequently

ηΜ = l - m2n

2/3(b + 1.42)

2

where m = 1 gives the first eigenfunction, m = 2 the second one, etc. This formula turns out to represent the eigenvalue quite well for any b > 2, as shown again in Table 4.

It clearly is possible to introduce further refinements into this formula. In particular, the factor j equals the value of the diffusion constant D for conservative scattering (a = 1). But we know that its value for different a is different (Sections 5.2.3 and 8.2.1). It may therefore be more appropriate to write η = a~

1

and to solve for a and D from the linear equations

fl = l + Dm2n

2/(b + 1.42)

2

D = 1/3 + (4/15X1 - a)

7.4.2 Expansions in Eigenfunctions of the Milne Operator

The same expansion, of which we considered the leading terms in the preceding section in order to estimate rapidity of convergence, can be written out completely. Conceivably, this forms an attractive computing method in certain situations. We present the equations here without much comment.

We continue to use the condensed notat ion of Display 7.1. Capitals represent vectors or matrices. Assume that all eigenvalues ηί are different. We then have

CiCi = qf (qt assumed positive)

CiCj = 0 for ί φ j (orthogonality)


An arbitrary τ vector, e.g., F, can be expanded as

F = YfiCi with f = CiF/qf i

Sums here and in the following equations are taken over all eigenvalues. An arbitrary ττ matrix, e.g., A, can be similarly expanded as

A = Σ Σ ûyC , · Cj wit h a tJ = CiACj/qfq2

i J

Here a do t · signifie s a multiplication , no t a n integration , s o tha t Q · Cj i s short fo r CfàCfx'). Th e Miln e matri x M itsel f ca n als o b e expanded , bu t th e corresponding coefficient s m i} = 0 fo r i Φ j . The resulting equation is

M

This equation has some attractive properties. For instance, we may take the nth power

« - ? ( f ) C i. c ,

and by summing the Neumann series, we find the inverse matrix

C = ( l - a M ) -1 = £ 2f

l'Cl

i ft (i - and Now define the μ vectors

Pi = PCJqi = {-Vf

+iQCJql

and obtain at once from the equation in Section 7.3.1 the reflection and transmission functions

R = 5 » Ç i ^ ' « . - ï Ç t f - l V *

ι r _ i V

+ 1P . . P . 1

where the suffix η denotes the nth order of successive scattering. We did not encounter these equations in the literature but felt they should be included because they can be used for rapidly converging numerical computat ion whenever b is not too large (e.g., b < 2). Later I found that Nagirner (1973) does exactly that (for b < 1). Their simple form also makes them suitable for a discussion of absorption lines, where a must be continuously varied (Section 17.3).


7.5 THE ADDING OR DOUBLING METHOD DERIVED FROM THE MILNE EQUATION

Although the adding method (Section 4.5) is applicable to a far more general set of problems, it is of interest to verify its results by formal matrix algebra for the homogeneous atmospheres with isotropic scattering treated in the present chapters.

We write vectors and matrices for the double layer without primes, and for the separate layers with primes and double primes. We refer to Faddeeva (1959) and use a combination of the method of partitioned matrices (pp. 102-103) with that of improved convergence of iteration (pp. 127-131). The central problem (Section 7.2) is to find

C = (l -aMy1

when M and a are given. Suppose we already know

C = (1 - aMY\ C" = (1 - aM'T1

We partition the matrices for the composite layer and introduce a new one, G, as follows.

/ I 0 \ (M' L\ (C 0 1 = 0 1 ' ( L M"'

G= 0 C"

Obviously G can serve as a first approximation to C. Direct multiplication gives

G(l - aM) = 1 - H

where

0 aC'L\ H \aC"L 0

and hence

( 1 - # ) C = G or C = ( 1 - H ) -1G

Inversion of 1 — H in partitioned form gives by standard procedures

_ m- i _ / (l ~ ^C'LC'L)-

1 aC'L(l - a

2C"LC'Ly

l ( ' ~ \aC"L{\ - ctC'LC'L)-

1 (1 - a

2C"LC'Ly

l

and finally

/ (1 - a2C'LC"L)-

lC (1 - a

2C'LC"L)-

laC'LC"

'~ \ (1 - a2C"LC'L)-

laC"LC (1 - a

2C"LC'Ly

xC"

We now have reached a form for C which still contains inverse matrices, which can be solved by means of the Neumann series. For instance:

(1 - a2C'LC"LY

l = 1 + a

2C'LC"L + a

AC'LC"LC'LC"L + •••


6"·

F i g . 7.4. N o t a t i o n u s e d in d e r i v i n g t h e a d d i n g m e t h o d f r o m t h e M i l n e e q u a t i o n .

The necessity for computat ion by series expansion thus remains. However, the convergence has greatly improved in comparison with the simple Neumann series, owing to the proper choice of an approximate solution G.

The equivalence between the result obtained here expressed in ττ matrices, and the result obtained in Section 4.5 in μμ matrices, should now be established. Measure τ' from the separation layer up into the top layer to b\ and τ" from the separation layer down into the bot tom layer to b" (Fig. 7.4). According to Section 7.3.1 the reflection matrices for the separate layers can be decomposed as follows:

R' = P' C \a F (μμ) (μτ')(τ 'τ ') (τ'μ)

and

R" = Ρ" C" i« Ρ" (μμ) (μτ")(τ"τ") ( τ ' »

The matrices L and L in the present derivation have the functional form

L = L = ±Ελ(τ' + τ")

and can therefore be written as matrix products :

L = \ P" P\ L = i P' P" (TV) (τ"μ)(μτ') (τ'τ") (τ'μ) (μτ")

The general term of the iterative solution thus contains continued products as follows.

iterative matrix in earlier derivation

R' R"

• • • iP" P'C'aiP' P"C"a\P" P'Ca\ • • •

I I I I I I

iterative matrix in present derivation

f


The remaining proof consists of simple substitutions. We have

R = PCS, T = T 0 + QCS

where T0 = T'0TQ. Considering the convention of measuring τ' and τ" (Fig. 7.4), the partitioned forms are

Combining these with the partitioned form of C just found, we find after somewhat tedious multiplications that the result is fully equivalent to that derived earlier.

B e n n e t , J . H . ( 1 9 6 4 ) . Numer. Math. 6, 4 9 .


N e w Y o r k , a l s o D o v e r , N e w Y o r k , 1 9 6 0 .

C h w o l s o n , O . ( 1 8 9 0 ) . Bull. Acad. Impériale Sci. St. Petersbourg I ( 3 3 ) , 2 2 1 .

E r d ô s , P . , H a l e y , S. B . , M a r t i , J . J . , a n d M e n n i g , J . ( 1 9 7 0 ) . J. Comput. Phys. 6, 2 9 .

F a d d e e v a , V . N . ( 1 9 5 9 ) . " C o m p u t a t i o n a l M e t h o d s o f L i n e a r A l g e b r a . " D o v e r , N e w Y o r k .

K o u r g a n o f T , V . ( 1 9 5 2 ) . " B a s i c M e t h o d s in T r a n s f e r P r o b l e m s . " O x f o r d U n i v . P r e s s ( C l a r e n d o n ) ,


L o m m e l , E . ( 1 8 8 9 ) . Wiedemann's Ann. 36, 4 7 3 .

M i k h l i n , S. G . ( 1 9 7 0 ) . " V a r i a t i o n a l M e t h o d s in M a t h e m a t i c a l P h y s i c s . " N a u k a , M o s c o w ( R u s s i a n ) .

M u l l i k i n , T . W . ( 1 9 6 2 ) . J. Math. Anal. Appl. 5 , 184.

N a g i r n e r , D . I . ( 1 9 7 3 ) . Astrofizika 9, 3 4 7 [ E n g l i s h transi. : 9, 196 ( 1 9 7 5 ) ] .

S o b o l e v , V . V . ( 1 9 7 5 ) . " L i g h t S c a t t e r i n g in P l a n e t a r y A t m o s p h e r e s . " P e r g a m o n , O x f o r d . O r i g i n a l

R u s s i a n 1972 .

S t o k e s , R . Α . , a n d D e M a r c u s , W . C . ( 1 9 7 1 ) . Icarus 14 , 3 0 7 .

v a n d e H u l s t , H . C . ( 1 9 6 3 ) . A N e w L o o k a t M u l t i p l e S c a t t e r i n g . U n n u m b e r e d m i m e o g r a p h e d

r e p o r t , N A S A I n s t i t u t e f o r S p a c e S t u d i e s , N e w Y o r k ,

v a n d e H u l s t , H . C . ( 1 9 6 4 ) . Bull. Astron. Inst. Netherlands 17 , 4 9 5 .

R E F E R E N C E S

8 • Isotropic Scattering, Semi-Infinite Atmospheres

8.1 SPECIFICATIONS

Problems with isotropic scattering in a homogeneous semi-infinite a tmosphere were first tackled around 1890 (see references in Chapter 7) and rigorously solved around 1930 (see Hopf, 1934). Since such problems have now been extensively treated for anisotropic scattering with an arbitrary phase function, it is logical to regard isotropic scattering as a simple, special case. Our first task, therefore, is to specify the forms assumed by the functions and constants defined in Chapters 5 and 6 in the case of isotropic scattering. The answers are in Display 8.1. Most of these are self-evident. Some further explanations will be given in the sections which follow.

8.2 THE UNBOUNDED MEDIUM

8.2.1 Propagation in the Absence of Sources

The general form of the radiation field for nonconservative isotropic scattering in the absence of (or far from) sources can be read from Section 5.2.1., Eq. (1), with the specifications for k and P(u) from Display 8.1. The corresponding source function can be taken from Section 6.2.1. Alternatively, we can find the source function directly by solving the Milne equation (Section 7.2).

149

150 8 Isotropic Scattering, Semi-Infinite Atmospheres

In a stationary state with plane symmetry, i.e., when the source function J depends on ζ only and the albedo a is independent of z, the homogeneous Milne equation is

J ( z ) = | Γ J(z + x)E1(\x\)dx ^ J - no

Here ζ is the optical depth measured from an arbitrary reference plane. The general solution is as follows:

For a < 1 (medium with losses):

J(z) = Aek(a)z

+ Be-k{a)z

where A and Β are arbitrary constants and k(a) is the real, positive solution of

fc/a = i l n [ ( l +k)/(l - it)]

This result is reported in Chwolson (1890). For a = 1 (conservative medium):

J(z) = A + Bz

For a > 1 (medium with gains) :

J(z) = ,4 cos k\a)t + J5 sin k\a)t

D I S P L A Y 8.1

S p e c i f i c a t i o n s for I s o t r o p i c S c a t t e r i n g

Q u a n t i t y

o r

f u n c t i o n

B e c o m e s fo r i s o t r o p i c s c a t t e r i n g

N o n c o n s e r v a t i v e , a < 1 C o n s e r v a t i v e , a = 1

R e f e r e n c e t o

d e f i n i t i o n

R e d i s t r i b u t i o n f u n c t i o n

h{u, u) h(u, u) = a

S o l u t i o n in u n b o u n d e d m e d i u m

1 1 1 + k S o l u t i o n of - = — In -

a 2k 1 - k

P(u)

m

y

D

P(u) = α / (1 - ku)

m

1 - -ldk/da

y = ( i - a)/k

D = (1 - a)/k

2

h(u, u) = 1

k = 0

P(u) = 1

m = 0

y = 0

0 = 4

D i s p l a y 5.1

S e c t i o n 6.2.1

S e c t i o n 6.2.1

D i s p l a y 6.11

D i s p l a y 5.2

D i s p l a y 5.2

8.2 The Unbounded Medium

D I S P L A Y 8.1 (continued)

151

Q u a n t i t y o r

f u n c t i o n

B e c o m e s for i s o t r o p i c s c a t t e r i n g

N o n c o n s e r v a t i v e , a < 1 C o n s e r v a t i v e , a = 1

R e f e r e n c e t o d e f i n i t i o n

P o l y n o m i a l s a n d a u x i l i a r y f u n c t i o n s

9η(μ) 9ο(β) = 1

9ι(β) = (1 ~ α)μ

0200 = K3(l " α)μ2 - 1 ]

G(v, μ) G(v, μ) = α

9ο(μ) = ι qM = 0

F ( ^ , ν) F(jU, ν) = α

Ψ ( μ ) Ψ ( μ ) - i «

Γ ( ζ )

λ0(μ)

αζ Γ ( 2) = 1 - y 1η

au Λο(μ) = 1 + y l n

(£) (£)

R e f l e c t i o n f u n c t i o n a n d r e l a t e d f u n c t i o n s

Κ ( μ , ν)

<ρ„(μ)

ν) : - Η ( α , ν) 4 ( μ + ν) "

φ0(μ) = Η (α, μ)

Ψι(μ) = 1 - μ [ 1 - (1 - α )

1 / 2Η ( « , μ ) ]

E s c a p e f u n c t i o n a n d r e l a t e d q u a n t i t i e s

Κ(μ) K(0) = αΙ[τηΗ(αΛ~

ι)Λ

Κ(μ) = Κ(0)Η(α, μ)/(1 - Ιίμ)

I = 2a/mkH

2(a,k~

l)

q = -(2ky

l\nl

l0 = 2/[mH(a,k-

1ïï

/j = 2 ( 1 - a)

1,2/tkmH(a,k-

1)]

ϋο(β) = 1

ΰι(β) = 0

9ι(β) = - Ί

G(v, μ ) = 1

<?ο(μ) = ι

* ι ( μ ) = 0

FQi, ν) = 1

Ψ ( μ ) = i

n o s i m p l i f i c a t i o n



φ0(μ) = Η(Ιμ)

φΜ = ι - μ

κ(0) =

Κ(μ) = (y/î/4)H(l, μ)

ι = 1

q = q o o= 0 . 7 1 0 4

D i s p l a y s 6 .2 , 6.3

D i s p l a y 6.2

S e c t i o n 6.4.2

D i s p l a y 6.8

D i s p l a y 6.8

S e c t i o n 6.2.1

D i s p l a y 6.7

D i s p l a y 6.5

D i s p l a y 6.5

D i s p l a y 6 .11

D i s p l a y 6 .11

S e c t i o n 5.2.1

D i s p l a y 6.11


T A B L E 5

Q u a n t i t i e s G o v e r n i n g t h e D i f f u s i o n t h r o u g h U n b o u n d e d M e d i a w i t h I s o t r o p i c S c a t t e r i n g

a k k' k

2, - k '

2 D = (1 - a)/k

2 -dk/da E n e r g y

0 1 .00000 1.0000 1.0000 0

s m a l l 1 - 2e~

2/a 1 - 4e-

2/a 1 - a (4/a

2)e~

2/a

0.2 0 . 9 9 9 9 1 0 . 9 9 9 8 0 . 8 0 0 1 0 . 0 0 4 5

0 .4 0 . 9 8 5 6 2 0 . 9 7 1 5 0 . 6 1 7 6 0 . 1 8 9 4

0 .6 0 . 9 0 7 3 3 0 . 8 2 3 3 0 . 4 8 5 9 0 . 6 3 1 5 a b s o r b e d

0 .8 0 . 7 1 0 4 1 0 . 5 0 4 7 0 . 3 9 6 3 1.4436 a b s o r b e d

0 .9 0 . 5 2 5 4 3 0 . 2 7 6 1 0 . 3 6 2 2 2 . 4 0 0 3

0 .95 0 . 3 7 9 4 8 0 . 1 4 4 0 0 . 3 4 7 2 3 . 6 3 7 2

0 .99 0 . 1 7 2 5 1 0 . 0 2 9 8 0 . 3 3 6 0 8 .5560

1 - ε [ 3 ( 1 - à)Y'2

3 ( 1 - a) 1 3 + A ( l -a) i [ 3 / ( l - « ) ]

1 /2

1 0 0 0 0 . 3 3 3 3 0 0 c o n s e r v e d

1 + ε [3(fl - 1 ) ]

1 /2 3(1 - a) 1

3 + A O -a) 1.1 0 . 5 6 9 2 6 - 0 . 3 2 4 1 0 . 3 0 8 6

1.2 0 . 8 3 4 5 4 - 0 . 6 9 4 6 0 . 2 8 7 2

1.4 1 .25981 - 1 . 5 8 7 1 0 . 2 5 2 0

1.6 1 .63500 - 2 . 6 7 3 2 0 . 2 2 4 4 g e n e r a t e d

1.8 1 .98883 - 3 . 9 5 5 4 0 . 2 0 2 2 g e n e r a t e d

2 2 . 3 3 1 1 2 - 5 . 4 3 4 1 0 . 1 8 4 0

3 3 . 9 7 2 5 8 - 1 5 . 7 8 1 4 0 . 1 2 6 7

l a r g e πα/2 - π

2 a

2/4 4/π

2α

0 0 0 0 — 0 0 0

where k'(a) is the real, positive solution of

kf/a = t a n "1* '

Table 5 gives some numerical values together with the leading terms of asymptotic expressions for a small, « 1, and large. Case et al (1953) present more complete asymptotic expressions than reproduced here. A graphical presentation of k is contained in Fig. 11.1 on the line g = 0.

These general solutions can be interpreted physically as follows :

For a < 1, the medium is lossy. The flux and the source density can be maintained in a steady state by continuous input from one side or the other. They then decrease exponentially with increasing distance from the primary source plane. The length over which they drop by a factor e is the diffusion length l/k.

For a = 1, the medium is conservative. A net flux which is constant in time and space can be maintained by primary sources at one side or the other. Hence, the source function is a linear function of depth. This is the familiar situation deep inside a stellar atmosphere in radiative equilibrium.

F o r a > 1, the medium has gains, and any homogeneous distribution of sources would grow with time. However, the amplitude of a sinusoidal ripple superposed on any such distribution can just be maintained if the scattering

8.2 The Unbounded Medium 153

from the peaks into the troughs just balances the tendency for growth. This happens if the wavelength is 2n/k\a).

The corresponding time-dependent problems are discussed in some detail by Davison (1957, Chapter 3).

8.2.2 Plane Isotropic Source Layer

We consider the same medium as in the preceding section: homogeneous, with isotropic scattering, with constant albedo a < 1, and without boundaries. If layers of isotropic sources of radiation with source density J 0( z ) are embedded in this medium, the total source function J(z), describing the combined intensity of the emitted and scattered radiation forthcoming from a unit volume, is the solution of the inhomogeneous Milne equation

J(z) = J0(z) + - J J(z + x)E1(\x\)dx

The solution of this problem for J0(z) = <5(z), i.e., for a primary source layer of unit strength at ζ = 0, has the form

J(z) = δ(ζ) + aFx(z) + a2F2(z) + a

3F3(z) + · · ·

In Table 6, we tabulate for this basic problem the functions defined below. The numbers have been computed by numerical integration and summation of the defining equations:

Fn(z)

distribution function of nth order scattering centers;

Gn(z) = f Fn(x) dx J — OO

cumulative distribution of nth order scattering centers, i.e., fraction of the initial radiation which, at the nth scattering, has not exceeded a distance ζ to one side of the source layer;

F(a,z) = (a-1 - l ) [ J (z ) - δ(ζ)~]

probability density that a quantum emitted by the source layer is absorbed at a distance ζ to one side of the source layer;

G(a, z) = F(a, x) dx J — 00

fraction of the energy absorbed at any distance between — oo and z. All the energy emitted by the source layer must be absorbed somewhere

in the medium, so G(a, oo) = 1 for any a < 1. Obvious relationships are

F(a, ζ) = Σ (1 " a)an~

lFn(z\ G(a, z) = £ (1 - d)a

n-'Gn(z)

n= 1 , i = l

TA

BL

E 6

Rad

iati

on

Dif

fusi

ng

fro

m

a P

lan

e S

ou

rce

Lay

er i

nto

an

Un

bo

un

de

d M

ed

ium

P

art

1 :

Sep

arat

e o

rder

s

Va

lues

o

f F

(ζ)

η =

1 10

20

40

80

00

.91

14

6 .6

11

33

.45

28

4

.69

31

5 .5

87

52

.50

39

0 .4

35

01

.42

28

7 .4

10

28

.38

74

4 .3

61

12

.32

51

9 .3

22

05

.31

40

2 .3

02

72

.27

26

0 .2

71

29

.26

75

5 .2

61

76

,17

13

9 ,1

71

18

,17

05

7 ,

1695

6

.11

48

9 .

1148

3 ,1

14

67

, 11

441

.07

91

9 .0

79

17

.07

91

2 .0

79

04

.05

53

0 .0

55

30

.05

52

8 .0

55

25

.35

11

9 .2

79

89

.22

71

9 .1

55

30

.37

72

6 .3

28

32

.28

65

2 .2

19

67

.33

38

8 .3

07

01

.28

11

8 .2

33

93

.28

92

7 .2

74

49

.25

90

1 .2

27

60

.25

43

2 .2

45

60

.23

59

2 .2

14

79

,16

81

6 .

1663

9 .1

64

26

,15

90

2

, 11

404

.11

35

6 .

1129

9 .

1115

4

.07

89

3 .0

78

78

.07

86

0 .0

78

15

.05

52

2 .0

55

17

.05

51

1 .0

54

95

.10

96

9 .0

24

45

.00

65

2 .0

01

89

.00

05

7

. 16

958

.04

95

2 .0

15

35

.00

49

2 .0

01

61

. 19

320

.07

06

8 .0

25

13

.00

88

7 .0

03

13

. 19

734

.08

65

9 .0

34

77

.01

34

3 .0

05

08

. 19

271

.09

75

8 .0

43

58

.01

82

8 .0

07

39

.15

26

5 .

1 1

127

.07

00

8 .0

39

79

.02

09

6

.10

97

0 .0

95

73

.07

67

9 .0

57

07

.03

96

5

.07

75

7 .0

72

90

.06

57

8 .0

57

05

.04

75

9

.05

47

6 .0

53

17

.05

06

2 .0

47

26

.04

32

7

.00

01

8 .0

00

06

.00

00

2 .0

00

01

.00

00

0

.00

05

4 .0

00

18

.00

00

6 .0

00

02

.00

00

1

.00

11

1 .0

00

39

.00

01

4 .0

00

05

.00

00

2

.00

19

0 .0

00

71

.00

02

6 .0

00

10

.00

00

4

.00

29

2 .0

01

14

.00

04

4 .0

00

17

.00

00

6

.01

04

4 .0

04

99

.00

23

0 .0

01

04

.00

04

6

.02

59

7 .0

16

16

.00

96

1 .0

05

50

.00

30

4

.03

82

6 .0

29

68

.02

22

6 .0

16

16

.01

13

9

.03

88

7 .0

34

26

.02

96

4 .0

25

17

.02

09

8

.00

00

0 .0

00

00

.00

00

0 .0

00

00

.00

00

0 .0

00

00

.00

00

0 .0

00

00

.00

00

0 .0

00

00

.00

00

0 .0

00

00

.00

00

0 .0

00

00

.00

00

0 .0

00

00

.00

00

0 .0

00

00

.00

00

0 .0

00

00

.00

00

1 .0

00

00

.00

00

0 .0

00

00

.00

01

1 .0

00

00

.00

00

0 .0

00

00

.00

13

3 .0

00

09

.00

00

0 .0

00

00

.00

65

2 .0

01

35

.00

02

0 .0

00

02


2 3

4 5

ζ Ο

0. 1

0 0

.20

0.3

0

0.4

0 0

.50

0.6

0 0

.80

1.00

2

.00

3.0

0 4

.00

5.0

0

6.0

0 7

.00

8.0

0 9

.00

10

.00

15

.00

20

.00

25

.00

30

.00

ζ η

= 1

2 3

4

Ο

0.

10

0.2

0 0

.30

0.5

00

00

0.6

38

73

0.7

12

90

0.7

65

44

0.5

00

00

0.5

63

79

0.6

18

22

0.6

65

06

0.5

00

00

0.5

41

80

0.5

81

74

0.6

19

18

0.5

00

00

0.5

32

41

0.5

64

25

0.5

95

11

0.4

0 0

.50

0.6

0 0

.80

0.8

05

32

0.8

36

68

0.8

61

91

0.8

99

57

0.7

05

59

0.7

40

81

0.7

71

50

0.8

21

79

0.6

53

93

0.6

85

97

0.7

15

37

0.7

66

77

0.6

24

72

0.6

52

92

0.6

79

60

0.7

28

25

1.0

0 2

.00

3.0

0 4

.00

5.0

0

0.9

25

75

0.9

81

23

0.9

94

68

0.9

98

40

0.9

99

50

0.8

60

48

0.9

57

34

0.9

86

42

0.9

95

57

0.9

98

53

0.8

09

38

0.9

31

76

0.9

75

86

0.9

91

48

0.9

96

99

0.7

70

71

0.9

06

72

0.9

63

76

0.9

86

25

0.9

94

85

6.0

0 7

.00

8.0

0 9

.00

10

.00

0.9

99

84

0.9

99

95

0.9

99

98

0.9

99

99

1.0

00

00

99

95

1 9

99

83

99

99

4 9

99

98

99

99

9

0.9

98

94

0.9

99

62

0.9

99

87

0.9

99

95

0.9

99

98

0.9

98

09

0.9

99

29

0.9

99

74

0.9

99

90

0.9

99

96

15

.00

20

.00

25

.00

30

.00

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

Va

lue

s o

f G

(ζ

) η

5 10

2

0 4

0 8

0

0.5

00

00

0.5

27

22

0.5

54

18

0.5

80

66

0.6

06

47

0.6

31

48

0.6

55

56

0.7

00

66

0.7

41

41

0.8

83

35

0.9

50

81

0.9

80

05

0.9

92

10

0.9

96

92

0.9

98

82

0.9

99

55

0.9

99

83

0.9

99

94

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

0.5

00

00

0.5

17

13

0.5

34

22

0.5

51

23

0.5

68

12

0.5

84

85

0.6

01

39

0.6

33

74

0.6

64

92

0.7

97

69

0.8

87

69

0.9

41

59

0.9

71

12

0.9

86

28

0.9

93

68

0.9

97

17

0.9

98

76

0.9

99

46

0.9

99

99

1.0

00

00

1.0

00

00

1.0

00

00

0.5

00

00

0.5

11

49

0.5

22

96

0.5

34

42

0.5

45

84

0.5

57

22

0.5

68

55

0.5

91

01

0.6

13

14

0.7

16

45

0.8

02

94

0.8

69

78

0.9

17

87

0.9

50

35

0.9

71

11

0.9

83

75

0.9

91

14

0.9

95

30

0.9

99

86

1.0

00

00

1.0

00

00

1.0

00

00

0.5

00

00

0.5

07

92

0.5

15

83

0.5

23

74

0.5

31

64

0.5

39

53

0.5

47

40

0.5

63

07

0.5

78

65

0.6

54

11

0.7

23

63

0.7

85

14

0.8

37

48

0.8

80

37

0.9

14

26

0.9

40

12

0.9

59

22

0.9

72

89

0.9

97

46

0.9

99

85

1.0

00

00

1.0

00

00

0.5

00

00

0.5

05

53

0.5

11

06

0.5

16

59

0.5

22

11

0.5

27

63

0.5

33

14

0.5

44

15

0.5

55

12

0.6

09

17

0.6

61

14

0.7

10

13

0.7

55

44

0.7

96

54

0.8

33

12

0.8

65

06

0.8

92

44

0.9

15

49

0.9

79

53

0.9

96

46

0.9

99

55

0.9

99

96


TA

BL

E

6 (c

ontin

ued)

Par

t 2

: R

esu

lts

for

alb

edo

a

Va

lues

o

f F

(a,

z)

ζ a

=0

.01

0.1

0 0

.20

0.5

0 0

.70

0.8

0 0

.85

0.9

0

0 0

. 10

0

.20

0.3

0

GO

0.9

08

20

0.6

10

24

0.4

52

65

oo

Θ.8

77

20

0.5

99

34

0.4

50

25

00

0.8

38

79

0.5

84

51

0.4

45

77

00

0.6

89

34

0.5

14

85

0.4

14

12

00

0.5

43

63

0.4

31

36

0.3

63

55

00

0.4

44

81

0.3

67

02

0.3

18

75

oo

0.3

84

1 1

0.3

24

54

0.2

87

01

00

0.3

11

08

0.2

70

47

0.2

44

45

0.4

0 0

.50

0.6

0 0

.80

0.3

51

45

0.2

80

37

0.2

27

78

0.1

55

94

0.3

53

31

0.2

84

48

0.2

33

05

0.1

61

87

0.3

54

25

0.2

88

41

0.2

38

62

0.1

68

67

0.3

44

27

0.2

91

69

0.2

50

23

0.1

88

68

0.3

14

53

0.2

76

18

0.2

44

83

0.1

95

93

0.2

82

97

0.2

54

29

0.2

30

31

0.1

91

69

0.2

58

78

0.2

35

84

0.2

16

39

0.1

84

47

0.2

24

54

0.2

08

09

0.1

93

93

0.1

70

16

1.0

0 2

.00

3.0

0 4

.00

5.0

0

0.1

10

29

0.0

24

70

0.0

06

61

0.0

01

92

0.0

00

58

0.1

15

92

0.0

27

19

0.0

07

52

0.0

02

24

0.0

00

70

0.1

22

64

0.0

30

46

0.0

08

78

0.0

02

70

0.0

00

86

0.1

45

26

0.0

45

25

0.0

15

54

0.0

05

55

0.0

02

03

0,1

59

21

0.0

62

21

0.0

25

88

0.0

11

01

0.0

04

73

0.1

61

50

0.0

73

85

0.0

35

32

0.0

17

13

0.0

08

36

0.1

58

90

0.0

80

17

0.0

41

92

0.0

22

15

0.0

11

76

0.1

50

55

0.0

85

67

0.0

50

06

0.0

29

46

0.0

17

38

6.0

0 7

.00

8.0

0 9

.00

10

.00

0.0

00

18

0.0

00

06

0.0

00

02

0.0

00

01

0.0

00

00

0.0

00

22

0.0

00

07

0.0

00

02

0.0

00

01

0.0

00

00

0.0

00

28

0.0

00

09

0.0

00

03

0.0

00

01

0.0

00

00

0.0

00

75

0.0

00

28

0.0

00

10

0.0

00

04

0.0

00

01

0.0

02

04

0.0

00

89

0.0

00

39

0.0

00

17

0.0

00

07

0.0

04

09

0.0

02

01

0.0

00

98

0.0

00

48

0.0

00

24

0.0

06

25

0.0

03

33

0.0

01

77

0.0

00

94

0.0

00

50

0.0

10

27

0.0

06

07

0.0

03

59

0.0

02

12

0.0

01

25

15

.00

20

.00

25

.00

30

.00

0.0

00

00

0.0

00

00

0.0

00

00

0.0

00

00

0.0

00

00

0.0

00

00

0.0

00

00

0.0

00

00

0.0

00

00

0.0

00

00

0.0

00

00

0.0

00

00

0.0

00

00

0.0

00

00

0.0

00

00

0.0

00

00

0.0

00

00

0.0

00

00

0.0

00

00

0.0

00

00

0.0

00

01

0.0

00

00

0.0

00

00

0.0

00

00

0.0

00

02

0.0

00

00

0.0

00

00

0.0

00

00

0.0

00

09

0.0

00

01

0.0

00

00

0.0

00

00


Va

lue

s o

f G

(a,

z)

z a

=0

.01

0.1

0 0

.20

0.5

0 0

.70

0.8

0 0

.85

0.9

0

0 0

.50

00

0 0

.50

00

0 0

.50

00

0 0

.50

00

0 0

.50

00

0 0

.50

00

0 0

.50

00

0 0

.50

00

0 0

.10

0.6

37

98

0.6

31

00

0.6

22

78

0.5

94

10

0.5

69

84

0.5

54

94

0.5

46

31

0.5

36

41

0.2

0 0

.71

19

5 0

.70

30

5 0

.69

23

5 0

.65

33

0 0

.61

79

8 0

.59

51

3 0

.58

14

4 0

.56

52

9 0

.30

0.7

64

43

0.7

54

92

0.7

43

31

0.6

99

39

0.6

57

51

0.6

29

27

0.6

11

91

0.5

90

96

0.4

0 0

.80

43

1 0

.79

48

0 0

.78

30

4 0

.73

71

3 0

.69

13

1 0

.65

92

8 0

.63

91

4 0

.61

43

8 0

.50

0.8

35

71

0.8

26

51

0.8

15

01

0.7

68

81

0.7

20

77

0.6

86

10

0.6

63

84

0.6

35

99

0.6

0 0

.86

10

0 0

.85

22

7 0

.84

12

5 0

.79

58

3 0

.74

67

7 0

.71

03

0 0

.68

64

3 0

.65

60

7 0

.80

0.8

98

79

0.8

91

20

0.8

81

46

0.8

39

34

0.7

90

60

0.7

52

32

0.7

26

38

0.6

92

40

1.0

0 0

.92

50

9 0

.91

86

7 0

.91

02

9 0

.87

25

0 0

.82

59

5 0

.78

75

3 0

.76

06

3 0

.72

44

1 2

.00

0.9

80

99

0.9

78

56

0.9

75

19

0.9

57

10

0.9

28

24

0.8

98

79

0.8

75

08

0.8

39

00

3.0

0 0

.99

46

0 0

.99

37

3 0

.99

24

8 0

.98

47

8 0

.96

95

5 0

.95

09

4 0

.93

39

9 0

.90

52

2 4

.00

0.9

98

37

0.9

98

07

0.9

^7

62

0.9

94

46

0.9

86

92

0.9

76

05

0.9

64

96

0.9

44

06

5.0

0 0

.99

94

9 0

.99

93

9 0

.99

92

2 0

.99

79

6 0

.99

43

5 0

.98

82

8 0

.98

13

7 0

.96

69

5

6.0

0 0

.99

98

4 0

.99

98

0 0

.99

97

4 0

.99

92

4 0

.99

75

5 0

.99

42

5 0

.99

00

8 0

.98

04

6 7

.00

0.9

99

95

0.9

99

93

0.9

99

91

0.9

99

71

0.9

98

94

0.9

97

18

0.9

94

72

0.9

88

45

8.0

0 0

.99

99

8 0

.99

99

8 0

.99

99

7 0

.99

98

9 0

.99

95

4 0

.99

86

1 0

.99

71

9 0

.99

31

7 9

.00

0.9

99

99

0.9

99

99

0.9

99

99

0.9

99

96

0.9

99

80

0.9

99

32

0.9

98

50

0.9

95

96

10

.00

1.0

00

00

1.0

00

00

1.0

00

00

0.9

99

98

0.9

99

91

0.9

99

67

0.9

99

20

0.9

97

61

15

.00

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

0.9

99

99

0.9

99

97

0.9

99

82

20

.00

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

0.9

99

98

25

.00

1.0

00

00

1.0

00

00

1,0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

30

.00

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00

1.0

00

00


The step from one order to the next reduces, because of the absence of boundaries, to complete folding:

FH+1(z) = ^j Fn{x)Ex(\z-x\)dx n > 0

In the Fourier transform, defined by

Λ Ο Ο Λ QO

Γ„(ω)= et o z

F„(z) dz = cos coz F„(z) dz J — oo J — 00

/•oo /»oo

T(a9 ω) = ei(OZ

F(a, z) dz = cos ωζ F(a, z) dz

J — oo ·/ — oo

this leads by repeated multiplication to

Τη(ω) = ( t a n "1 ω/ω)" = [1 - £ ω

2 + ^ ω

4 ]

w

summable as a geometric series into

1 - a T(a, ω) = — «—

Before proceeding we note some special values:

Fx{z) = R ( | z | ) , F 1( 0 ) = o o

\\E2{-z) for z < 0 G l ( Z) l l - i £ 2W for z > 0

^ ) = i G ' n d z l ) + i G ' ^ l z l ) , F2(0 ) = In 2

expressed in the functions defined in Section 2.5. Furthermore, we have the zero-order moments

Γ„(0)= Γ ( α , 0 ) = 1

and the second-order moments

Γ F n 2, \d2Tn(w)\ 2

f00

i t / ^ 2 , Γ^Πα,ω)] 2 λ / ( α ' Ζ ) Ζ ί ί Ζ = - ί ^ Η ω = ο = 3 ( Γ ^ )

ASYMPTOTIC FORMULA FOR F ( A , z)

Two further exact expressions for the source density distribution are obtained as follows. First, by inverting the Fourier transform, we have

1 f00 . 1 - a

F(a9 z) = — el<az

— r r - r dco 2π J _ 00 (ω/tan ω) — a


Second, this can be transformed in the complex ω plane to a contour integral around the pole at ik and along the cut from i to f oo, leading to

The more modern way to derive this result is to employ a singular eigenfunction expansion (Section 6.5). The asymptotic part then arises from the one discrete eigenvalue k\ the nonasymptotic part, from the singular eigenfunctions associated with the continuum of eigenvalues.

The asymptotic term, which predominates for large |z | , may be checked directly against the expressions for general phase functions contained in earlier chapters. Take from Section 6.2.1 the equation J = s(l — ku)P(u)exp( — kx). Specify s = 4/m as in Section 5.5, 4/m = —dk/da and P(u) = a/(l — ku) as in Display 8.1, and τ = | z | ; then J = a( — dk/da)exp( —fc|z|) in agreement with J = F(a, z)l(a~

1 — 1) as defined above.

The asymptotic term, but not the full quantity F(a, z), obeys the diffusion equation d

2F/dz

2 = k

2F, if we use the correct diffusion coefficient D shown in

Display 5.2. We further note an alternative form for 4/m:

The customary expression as a derivative comes from the contour integration mentioned above. It can more rapidly be found from the defining equation

by taking the derivative of both sides with respect to k and by employing Eq. (14) from Section 5.2.2.

The asymptotic form just derived is identical to what in other chapters of this book we have called the diffusion approximation. Its practical use is confined to situations in which k differs sufficiently from 1, i.e., to values of the albedo a relatively close to 1. If a < 0.5, we have to go to extremely high values of ζ before the asymptotic term represents a useful approximation.

For those values of a another approximation can be recommended. Entirely along the lines shown below for F„(z), we may argue that the distribution in the intermediate range of ζ approaches a normal (Gaussian) distribution. Judging from the second-order moment, the dispersion is

F(a, z) = F a s y m( a , z) + F n o n a s y m( a , z)

^ a s y m f e * ) = ( ! " θ)(-dk/da)e~k^

4/m = -dk/da = k(l - k2)/a(a - 1 + k

2 )

σ2 = 2/3(1 - a)


ASYMPTOTIC FORMULA FOR F„(Z)

The asymptotic form of Fn(z) follows from the central-limit theorem. It approaches the normal (Gaussian) distribution function

Fn(z) = e-z2/2<T2

/(2n)1/2c

with σ2 = 2n/3. The integral Gn(z) approaches the corresponding error function

(Abramowitz and Stegun, 1965, Chapter 7)

Gw(z) = i + i e r f ( Z 7 3 / 2 ^ )

As an illustration, Fig. 8.1 shows Fn(x) for η < 20, χ < 5. Figure 8.2 shows Gn(x) for η = 1, 2, 3 , 4, 10, 20, 40, and 80. Dashed curves close to those for η > 10 show the corresponding Gaussian approximation. The approach to the Gaussian form is shown more directly in Fig. 8.3. Here abscissas are x, ordinates y/n9 and curves of equal Gn(x) are drawn. They approach asymptotically the dashed straight lines through the origin which correspond to the Gaussian approximation.

All these distributions are more peaked than a Gaussian, as illustrated by the fact that in Fig. 8.3 the inner curves deviate inward, the outer curves outward from their asymptotes. The crossover point, where Gn(x) exactly equals the Gaussian expression, is read from the figure to be near Gn(x) = 0.959 for η = 1 and 0.955 for η > 2.

F i g . 8.1. S o u r c e f u n c t i o n s f o r η t i m e s s c a t t e r e d r a d i a t i o n (n = 1 -20) a r i s i n g f r o m a n a r r o w l a y e r o f i s o t r o p i c s o u r c e s a t τ = 0 i n a n u n b o u n d e d m e d i u m w i t h i s o t r o p i c s c a t t e r i n g .

8.3 The H Functions and Their Moments 161

F i g . 8.3. T h e a p p r o a c h o f t h e s o u r c e f u n c t i o n s o f F i g . 8.1 a n d 8 .2 t o a G a u s s i a n d i s t r i b u t i o n

is d e m o n s t r a t e d h e r e b y p l o t t i n g t h e o p t i c a l d e p t h x , a t w h i c h a c e r t a i n p e r c e n t a g e is r e a c h e d ,

a g a i n s t a l i n e a r s c a l e o f yjn.

8.3 THE H FUNCTIONS AND THEIR MOMENTS

8.3.1 The //Functions

The physical meaning of the function H(a, μ) is the point-direct ion gain for a point at the surface of a semi-infinite, homogeneous atmosphere with isotropic


scattering. The prominent role of this function warrants a detailed discussion of its properties. H functions also occur in anisotropic scattering (Section 6.4.1), but there they do not have a simple physical meaning.

Definition. H(a, μ) is a positive function of the positive variables: albedo a, 0 < a < 1, and cosine of angle with normal μ. Usually 0 < μ < 1, but the range can be extended to include positive values of μ from 1 to oo and negative values of μ in a limited range (see end of this section).

A sensible definition of H(a, μ) is the procedure for computing X(a, b, μ) given in Section 7.3.3, here with b = oo so that τ integrations are over the interval (0, oo ). This procedure is in the form of a Neumann series, which converges for a < 1 (a = 1 is the critical value). Clearly, this procedure is not always useful as a practical method of computation, since convergence is very slow. The situation at a = 1 is discussed in a more general context in Section 12.4.1.

Integral equations

1 αμ f1 H(a, x)dx

H(a, μ) 2 J 0 μ + χ

* /ι -\i/2 . a

f1 ^ (

α>

χ)

χ dx = (1 - a)

1'2 +

2 J o Η(α,μ) 2 J0 μ + χ

The proof that H(a, μ), as defined above, satisfies these integral equations is implicit in the discussion of the invariance principle (Section 4.4) as shown in detail by Chandrasekhar (1950) or Kourganoff (1952). The first equation is usually attributed to Ambartsumian (1942), but dates back to Halpern et al. (1938). See Ivanov (1973, p. 127). Usually, the first integral equation has been adopted as the defining equation. This necessitates a discussion of existence and uniqueness which is by no means simple, but which we can omit because the definition adopted here contains no such ambiguity.

Linear integral equations. These may be taken from Display 6.7. A linear Fredholm integral equation has been derived by Domke (1976).

Some integral representations are

In H(a, μ) = ^ 2 . 2 , du π J 0 co s

2 θ + μ

2 s in

2 θ

In H(a, μ) - - Γ π J o

71/2 (θ esc

2 θ - cot θ) t a n " \μ tan θ) άθ

1 - αθ cot θ

where

Η(α, μ) = -———-expI — da Ξ(α,μ) : 1 + Κα)μ

F\ 2 j 0 J 0 '

μ' μ + μ')

~, \ Γ / ι αμ, 1 + μ \2 , (αημΥΥ

1


The first two were used by Stibbs and Weir (1959). Expressions of this type were already known to Fock in 1944. The third one was quoted to me by V. V. Ivanov (private communication). By performing the integration over a' we obtain the integral expression cited on lines 6-7 of Display 6.7 for Ψ(ί) = \a. Without performing the integration over a\ this expression is a suitable starting point for obtaining the derivatives with respect to a or (1 — a

2)112

(see below). A related explicit expression is given by Case and Zweifel (1967, p. 130, Eq. 39 combined with p. 129, Eq. 29a).

Exclusively for a = 1 we have

H ( l , μ) = ^ 3 f [τ + ς(τ)^~τ/μ άτ/μ

Jo

where q(x) is Hopf 's function (Section 8.6.2).

Numerical values. A four-decimal table for rapid orientation is given in Table 7. It is based on Stibbs and Weir (1959) and on independent computat ions by K. Grossman and has been checked against other available tables. The most important published tables are:

Placzek (1947) Chandrasekhar (1950)

Harris (1957) Stibbs and Weir (1959)

Case and Zweifel (1967)

Woodford et al. (1968)

Abhyankar and Fymat (1971)

i i / ( a , μ) for a = 1 only; μ = 0 (0.01) 1. a = 0.1 (0.1) 0.8, 0.85, 0.9 (0.025) 1; μ = 0 (0.05) 1. Extension of these tables to a = 0.975 (0.005) 1. 6-decimal table, a = 0.05 (0.05) 0.90, 0.925, 0.95, 0.96, 0.97 (0.005) 0.99 (0.0025) 1; μ = 0 (0.05) 1. In addition they give coefficients for expansions in powers of a — a0 about a0 = 0.2, 0.5, 0.7, 0.85, and 1. A function simply related to the Η function for isotropic scattering is given in Appendix L in 6 decimals for a = 0.001, 0.05, 0.1 (0.1) 0.9, 0.95, 0.99, 1: μ = 0(0.005) 1. Coefficients for rational representations of H(a, 1) up to an accuracy of 1 0 "

9.

Last column in a table (other columns refer to Rayleigh scattering) gives the Η function for isotropic scattering, 6 decimals, a = 0 (0.1) 0.6 (0.05) 0.8 (0.025) 0.9 (0.01) 0.98 (0.005) 0.995 (0.001) 0.999; μ = 0 (0.01) 1.

Expansion near a = 0. Write

H(a, μ) = 1 + α^(μ) + a2h2&) + α

3/ ι3(μ) -f

TA

BL

E 7

Th

e H

F

un

ctio

n fo

r Is

otr

op

ic S

catt

erin

g a

nd

Coe

ffic

ient

s fo

r It

s E

xp

ansi

on

s"

a t

= (1

-a)1

H(a

, μ

) e

xp

an

sio

n n

ear

μ=

1

a t

= (1

-a)1

μ=

0 μ

-.Ι

P-

.3

μ =

.5

μ-

.7

μ -

1.0

k

j(a

) k

2(a)

k3(a

0 1

.00

00

, 1

1 1

1 1

1 0

0 0

0.1

.9

48

7 1

1.0

12

4 1

.02

30

1.0

28

9 1

.03

28

1.0

35

7 1

.03

68

- .0

10

7 -.

00

7 -.

00

0.2

.8

94

4 1

1.0

25

6 1

.04

83

1.0

61

2 1

.06

98

1.0

76

1 1

.07

86

- .0

23

9 -.

01

5 -.

01

0.3

.8

36

7 1

1.0

39

9 1

.07

64

1.0

97

6 1

.11

20

1.1

22

5 1

.12

68

- .0

40

4 -.

02

4 -.

02

0.4

.7

74

6 1

1.0

55

4 1

.10

79

1.1

39

2 1

.16

08

1.1

76

8 1

.18

34

- .0

61

8 -.

03

6 -.

02

0.5

.7

07

1 1

1.0

72

4 1

.14

39

1.1

87

7 1

.21

86

1 .2

41

7 1

.25

13

- .0

90

2 -.

05

0 -.

04

0.6

.6

32

5 1

1.0

91

3 1

.18

59

1.2

45

8 1

.28

89

1.3

21

7 1

.33

54

- .1

29

6 -.

07

1 -.

05

0.7

.5

47

7 1

1.1

13

0 1

.23

64

1.3

17

9 1

.37

81

1.4

25

0 1

.44

47

- .1

87

8 -.

09

4 -.

06

0.8

.4

47

2 1

1.1

38

8 1

.30

06

1.4

13

3 1

.49

95

1.5

68

5 1

.59

82

- .2

83

1 -.

13

3 -.

08

0.9

.3

16

2 1

1.1

72

1 1

.39

14

1.5

56

0 1

.68

93

1.8

00

8 1

.85

01

- .4

74

5 -.

17

2 -.

09

0.9

5 .2

23

6 1

1.1

95

2 1

.46

04

1.6

71

8 1

.85

09

2.0

06

5 2

.07

71

- .6

84

8 -.

20

3 -.

08

0.9

6 .2

00

0 1

1.2

01

0 1

.47

87

1.7

03

5 1

.89

65

2.0

65

9 2

.14

34

- .7

53

1 -.

21

0 -.

08

0.9

7 .1

73

2 1

1.2

07

5 1

.49

97

1.7

40

6 1

.95

05

2.1

37

3 2

.22

34

- .8

39

7 -.

21

2 -.

07

0.9

8 .1

41

4 1

1.2

15

1 1

.52

51

1.7

86

3 2

.01

82

2.2

27

9 2

.32

58

- .9

57

2 -.

20

1 -.

07

0.9

9 .1

00

0 3

1.2

24

9 1

.55

87

1.8

48

6 2

.11

26

2.3

56

9 2

.47

28

-1.1

39

1 -.

19

0 -.

05

0.9

97

5 .0

50

0 1

1.2

36

3 1

.60

02

1.9

28

2 2

.23

69

2.5

31

5 2

.67

43

-1.4

13

9 -.

13

8 -.

04

1 0

J 1

.24

74

1.6

42

5 2

.01

28

2.3

74

0 2

.73

06

2.9

07

8 -1

.7

69

4 -.

02

8 -.

03

Ex

pa

nsi

on

hl 0

.11

99

.21

99

.27

47

.31

06

.33

62

.34

66

- .0

96

6 -.

06

2 -.

10

nea

r h2

0 .0

37

0 .0

95

3 .1

36

7 .1

67

4 .1

91

1 .2

01

0 -

.09

-.0

5 a

* 0

h3 0

.01

7 .0

53

.08

4 .1

08

.12

8 .1

37

- .1

Ex

pa

nsi

on

Dl 0

-0.2

16

0 -0

.85

35

-1.7

43

1 - •

2.8

78

3 -4

.25

66

-5.0

36

5 8

.10

-3.0

n

ear

D2 0

- .0

9 .1

3 1

.04

2.8

7 5

.87

7.8

8 -2

2.0

+

19.

t =

0 D

0

.10

.27

- .2

0 - 2

.33

-7.4

5 -

11

.64

+3

7.

0 Th

e la

st w

ritt

en d

ecim

als

in t

he

seco

nd

an

d th

ird

exp

ansi

on

coef

fici

ents

are

un

cert

ain

.



Integral expressions for the separate terms in the notat ion of Chapter 7 are

(2/μ)Κ(μ) = ΡΜη-χΡΝ = ΝΡΜ

η~

ιΡ

The first of these appears more convenient because the calculation for Mn~

1PN

serves for all μ simultaneously. The exact expression for hx is

Λι(μ) = i f i ( o o , -1/μ) = ίμξΛμ) = *μ1η(1 + 1/μ)

where Ft is defined in Section 2.4 and ξ1 is tabulated by Kourganoff (1952). Other functions occurring in the higher order terms may be found in the same book. Values of hl9 h2, and h3, obtained from the exact forms of hx and h2 and by numerical differentiation of the Stibbs and Weir values, with some checks by the formula above, are given in Table 7.


t = χ / (1 - a), a= 1 - t2

Then

H(a9 μ) = H(l9 μ) + D^)t + Ω2(μ)ί2 + ϋ3(μ)ί

3 + · · ·

Exact expressions presented by Yanovitskii (1968), who also mentions an independent derivation by Ivanov, are the following:

Οχ(μ)= -μ^3Η(19μ) 02(μ) = [ 3μ

2 - 2 /0 ι ) ]Η(1 , μ)

DM = ιφ\_{2Ι5)Η{\9 μ) - Ώ2{μ)-\

Here /(μ) is tabulated by Yanovitskii (1968). Values of Dl9 Dl9 and D3 are given in Table 7. Those for D2 and D 3 were first obtained by numerical differentiation from the tables of Stibbs and Weir and later checked from the exact formulas.

Behavior near μ = 0. The derivative with respect to μ has a logarithmic singularity at μ = 0. A rigorous equation is

,. \Η{α9μ) - 1 a/1 \~] a . , x

See definition and values of a* x(a) under moments, Section 8.3.3. The corresponding asymptotic formula is

H(a9 μ) = 1 + (αμ/2)[α* i(a) + 1η(1/μ + 1) + c(a9 μ)]

where c(a, μ) -> 0 for μ - • 0. The formula should be used with caution because even for μ = 0.05, the value of c is fairly high:

for a = 0.4 0.8 0.96 1 φ , 0.05) = 0.007 0.045 0.096 0.142


Expansion near μ = 1. Write

H(a, μ) = H(a, 1) + ^(α)(1 - μ) + k2(a)(l - μ)2 + /c3(a)( l - μ )

3 + · · ·

Values for kl9 k2i and k3 obtained by numerical differentiations from the tables of Stibbs and Weir are given in Table 7.

Corner domains. The combined assumptions of μ near 0, a near 0, lead to the approximation:

Η(α,μ) = 1 + \ξχμα + $\η2μα2 + {^ξ\ - 0.6168)μ

2α

2 + · · ·

where ξι = log(l + 1/μ).

The combined assumptions μ near 0, a near 1, lead to the approximation:

H(a, μ) = 1 + \μ 1η[(1/μ) + 1] - μ + ^ 3

The domain in which this approximation is useful is very small and reference to numerical data is recommended.

The combined assumptions μ near 1, a near 0, give the expansion:

H(a, μ) = 1 + 0.3466α - 0.966α(1 - μ) + 0.201α2 - 0.09α

2(1 - μ) 4- · · ·

Estimated values of some further coefficients are given in Table 7. The combined assumptions μ near 1, a near 1, give the expansion

(t = y/l - a, χ = 1 - μ)

H(a, μ) = 2.9078 - 1.769* - 0.025*2 - 5.036r + 8.102xi - 3.0x

2r

Some further coefficients are given in Table 7. They were found from exact formulas, where available, and from numerical differentiation in all other cases. Errors up to several units of the last decimal may occur.

8.3.2 The Η Function for Virtual Angles

The definition of H(a, μ) may be extended to cover real or complex values of μ outside the domain 0 < μ < 1. We shall then talk of virtual angles. The following brief discussion is confined to real values of μ.

A first method closely follows the physical definition for real angles. If radiation is incident upon a semi-infinite, homogeneous atmosphere from a narrow source layer N, the source function thus established is (see Section 8.5)

J ( T ) = iag-^a, τ)

and the intensity reflected in any direction μ is (Display 9.1)

RN = (1/2μ)[Η(α, μ) - 1] = ί J(T)e~T/fi άτ/μ

Jo

This gives Λ 00

Η(α,μ)= 1 +{a g_x(a9 z)e~T/fi

dx Jo

This equation may be used as a definition of the Η function for any virtual direction μ. The integral is convergent for real values μ~

1 > — k{a).

A second method is to use the integral equation given in Section 8.3.1 as the equation defining Η for virtual angles in terms of the known Η function for real angles. This presents no problems for μ > 0, but requires the taking of Cauchy main values for μ < 0. We shall assume that these definitions are equivalent.

Values of H(a, μ) in the domain in which these values still are > 0 are shown in Fig. 8.4. Detailed expressions useful in this domain are given below. A table for a = 1 only and μ = 1 (0.5) 100 was given by Stibbs (1962).

Near the divergence point. Fo r 0 < μ-1

-h k(a) < 1, the integral in the first definition nearly diverges. This makes it sufficient to use the asymptotic formula (Section 8.5)

g-x(a9 τ) = [ 8 Κ ( 0 ) / α > - * *

which gives [ Η ( α , μ ) ] -

1 = [ μ "

1 + fc(fl)]/4X(0)


representing the tangent at the zero point in Fig. 8 . 4 . The same derivation gives for the corresponding tangent in the case of an arbitrary phase function (Section 6 . 4 . 1 ) :

ΙΗ(μ)Τ1 =(μ~' +k)mH(y)/my)

It is possible to derive the same result by a different method. Starting from the first integral equation in Section 8 . 3 . 1 and setting μ = — y , we find R_1 = 2/a (Section 8 . 5 ) . Upon first differentiating with respect to μ and then setting μ = — y, we obtain an expression containing the integral

f1 Η (a, x)x dx mH(y) 1

J o ( 1 - kx)2 =

2a2 =

2aK(0)

which can be found by inserting the values F = G = a appropriate to isotropic scattering in I( — y, — y) in Section 6 . 4 . 4 . The resulting derivative is

{{ά/άμ)[Η{α,μ)-^}μ=.Ί = -l/y24K(0)

in agreement with the result obtained above.

Near μ~ι = 0. The expansion

1 = 1 χ | x2

μ + χ μ μ2 μ

3

applied to the integrand of the second integral equation of Section 8 . 3 . 1 gives

1 _ n _ x i / 2 , i<t*i(à) _ Tfi*i(a) , 1 ^ 3 ( 0 ) _ Η(α,μ) μ μ

1 μ

ά

The corresponding expansions of the Η function itself are

X) = 7 - Τ^α Β " 1 " " 1 + ( è - r ) » '2 +

• • · }

α Φ 1, t = (1 - a)112

and

H(l, μ) = y/3\ji + q o o - 0 .04763μ-1 + · · · ] , a = 1, q x = 0.710446

in which the second coefficient can also be expressed as

^ - \*l = ~ Φ)1 dx = 0.047633

At μ "1 = k(a). The particular function


where γ = [ £ ( α ) ] ~ \ occurs in the solution of the Milne problem (Sections 6.4.3 and 8.6). It was first derived and tabulated by Kuscer (1953). Values computed by Terhoeve are given in Table 11 (Section 8.6.1).

8.3.3 Moments of the Η Functions

Definition

oc*(a) = ClH(a, μ) - άμ (η > -1) Jo

Γ1 ι

αη(α)= Η (α, μ)μη άμ = — — + tf(a) (η > 0)

Jo η + \ Here η is an integer in usual applications, but the following expressions are valid for any real η in the indicated domain.

Exact expressions

«*-i(a) = 2 In H(a, 1) α0(α) = ( 2 / a ) [ l - ( 1 - a )

1 / 2]

<αο) = + 1 ) , a * ( 0 ) =

« o ( l ) = 2

= 2 / ^ 3

a 2( l ) = 2 ^ / 7 3

« 3( i ) =

Relations (Busbridge, 1960; Eq. 12.14)

«2(1 - "Y12 = i - Wi

« 4( i - « )1 /2

= i + ^ ( - « 3 « i

a 6( l - a )1 /2

= 4 + ^ ( - α 5α ! + a 4a 2 - | a | )

Numerical values. Numerical values for a„ are presented in Table 8. Those for ocj are given by Stibbs and Weir (1959) in six decimals. Those for a 2 were computed from the a / s by the relationship just quoted and were checked against values given by Harris (1957). Values for η = 0-20 in five decimals were computed at my request by Dr. Grossman (unpublished). The values for a* x{a) were newly computed by numerical integration. Some results with a greater than usual accuracy may be quoted: = 0.710446089800 (Kourganoff, 1952, p. 189; Huang, 1952; King et al, 1965; Grossman, private communicat ion); H(l, 1) = 2.90781050 ± 5 (Bosma, private communicat ion); N R N = £α* ^ 1 ) = l n / f ( l , 1) = 1.06740039 + 4 (improved from van de Hulst and Grossman, 1968).

TA

BL

E

8

Mo

me

nts

o

f th

e H

F

un

ctio

n fo

r Is

otr

op

ic

Sca

tter

ing

lim

a t=

/Fâ~

a

Q a

l α

2 <

* 3

α*

. 2

α*

3α

* 4

α*

5α

* 1

0α

* η

+

0 1.

0000

1.

0000

0.

1 .9

487

1.02

63

0.2

.894

4 1.

0557

0.

3 .8

367

1.08

89

0.4

.774

6 1.

1270

0.5

.707

1 1.

1716

0.

6 .6

325

1.22

51

0.7

.547

7 1.

2922

0.

8 .4

472

1.38

20

0.9

.316

2 1.

5195

0.95

.2

236

1.63

45

0.96

.2

000

1.66

67

0.97

.1

732

1.70

47

ο: 9

8 .

1414

1.

7522

0.

99

.100

0 1.

8182

0.99

75

.050

0 1.

9048

1

0 2.

0000

0.50

00

0.33

33

0.25

00

0.51

56

0.34

44

0.25

85

0.53

32

0.35

68

0.26

81

0.55

31

0.37

10

0.27

91

0.57

62

0.38

75

0.29

19

0.60

35

0.40

70

0.30

71

0.63

66

0.43

09

0.32

58

0.67

87

0.46

14

0.34

97

0.73

58

0.50

32

0.38

26

0.82

53

0.56

94

0.43

51

0.90

19

0.62

68

0.48

09

0.92

35

0.64

31

0.49

41

0.94

93

0.66

27

0.50

98

0.98

17

0.68

73

0.52

96

1.02

72

0.72

20

0.55

77

1.08

75

0.76

83

0.59

53

1.15

47

0.82

04

0.63

78

0 0

0 .0

72

.05

3 .0

47

.15

1 .1

11

.09

9 .2

39

.17

8 .1

59

.33

7 .2

54

.22

9

.448

.3

43

.310

.5

78

.450

.4

10

.736

.5

84

.536

.9

37

.764

.7

07

1.23

0 1.

039

.976

1.46

2 1.

269

1.20

6 1.

524

1.33

3 1.

271

1.59

8 1.

409

1.34

8 1.

688

1.50

4 1.

445

1.81

0 1.

636

1.58

2

1.96

7 1.

810

1.76

2 2.

134

2.00

0 1.

964

0 0

0 0

.044

.0

43

.040

.0

37

.094

.0

91

.084

.0

79

.151

.1

46

.136

.1

27

.217

.2

10

.196

.

183

.295

.2

86

.268

.2

51

.390

.3

79

.357

.3

35

.512

.4

98

.471

.4

45

.680

.6

63

.630

.5

98

.944

.9

26

.888

.8

50

1.17

4 1.

155

1.11

6 1.

077

1.23

9 1.

220

1.18

2 1.

143

1.31

7 1.

299

1.26

1 1.

223

1.41

6 1.

398

1.36

2 1.

326

1.55

4 1.

538

1.50

6 1.

473

1.74

0 1.

727

1.70

0 1.

674

1.94

8 1.

939

1.92

3 1.

908

Ex

pa

nsi

on

Α.

.25

00

.14

77

nea

r Α

2 .1

25

0 .0

78

5 a

= 0

Α3

.07

81

.05

1

1042

.0

80

3 .6

93

1 .5

00

0 0

56

8 .0

44

4 .2

81

9 .2

50

0 03

7 .0

29

.16

3 .1

56

.443

2 .4

16

7 .4

01

5 .3

728

.346

6 .2

354

.227

3 .2

222

.21

17

.20

10

.152

.1

49

.14

7 .1

42

.13

7

Exp

ansi

on

Β,

-2.0

000

-1.4

209

-1.1

047

- .9

045

near

Β

2 +

2.00

00+

1.61

7 +

1.34

8 +

1.15

3 t

= 0

Β3

-2.0

000

-1.7

7 -1

.56

-1.3

8

-3.4

641

-4.0

0 -4

.26

-4.4

2 -4

.52

-4.7

5 -5

.04

+2.

421

+4.

00

+4.

85

+5.

39

+5.

77

+6.

67

+7.

88

-2.0

8 -4

.00

-5.3

0 -6

.22

-6.9

1 -8

.75

-11

.64




a*(fl) = AUna + A2 na2 + A3,na

3 + · · ·

All coefficients for η = 0 follow from expansion of the exact expression for <x0(a). Further, for any η

Al,n

= i G 1 >w + 2 ( 0 0 ) = 1^1,n + 2

where G is in the notation of Section 2.5, and J and Ρ may be taken from Kourganoff (1952, p . 270). Estimated values for the other coefficients given in Table 8 were found by numerical differentiation of oc*(a) and by calculating the moments of Λρ(μ).

Expansion near a = 1. Write t = y/l — a,

(φ) = α„(1) + BUnt + B2,nt2 + B3fHt

3 + .··

Again, all coefficients are known for η = 0 and Βλ n for all η by

Βι,η = - > / 3 α « + ι ( 1 )

Further values were again found by differentiation of an(a) and by taking moments of Dp(p).

Moments of large order. An exact formula is

lim(n + 2)α*(α) = Η(α, 1) - 1 00

The values given in Table 8 show how this limit is approached. Approximative formulas (or upper and lower limits) for ocn(a) may be obtained by adopting an approximation (or upper and lower limit) for H(a, μ) and calculating the moments analytically. If desired, the singularity at μ = 0 may first be removed by taking out a term -(α/2)μ log μ with moment a/2(n + 2 )

2.

Among these approximations, we found the following most useful:

(η Η- 2)α*(α) = A(a) + ——- +

with

n + 2 ' (n + 2)2

A(a) = H(a, 1) - 1 exactly B(a) = A(a) + k^a) 1 convenient C(a) = 8α^(α) - 6A(a) - 2k1(a)] approximations

This formula gives the exact value for η = 0; its errors are < 0.001 for any a, and any η > 0.


8.4 MOMENTS AND BIMOMENTS OF THE REFLECTION FUNCTION

We are still discussing semi-infinite, homogeneous layers with isotropic scattering. Let α„(α) denote the nth moment of the Η function H(a, μ) (Section 8.3.3). The moments of the reflection function lead to the integrals

with the recurrence relation (n > 0)

Kn+1(a, μ) = (a/2)ccn(a) - μΚη(α, μ)

The first of these are known from the relations in Section 8.3.1 :

Κ0(α9μ) = (1/μ)1-1/Η(α,μ)+11

Kx(a, μ) = l/H(a, μ) - t

where t = (1 - a)1'2. Fur ther :

K2(a, μ) = -μ/Η(α, μ) + μί + jaa^a)

Κ3(α, μ) = μ2/Η(α, μ) - μ

2ί - μ^αοί^α)! + ia<x2(a)

The moments of the reflection function and the related functions φη(μ) (Display 6.5 with m = 0, b = oo) thus become:

RN = \H(a, μ)Κ0(α, μ) = μ) - I'M μ

RU = H(a, μ)Κχ{α, μ) = 1 - tH(a, μ)

<Ρο(β) = 1

+ 2

vRN

= Η

(α> μ)

ψχ(μ) = μ — μ / W = μίΗ(α, μ)

<Ρι(μ) = &μ2 + Ι ^ ι μ - ϊ)Η(α, μ)

The bimoments up to order 3 are systematically arranged in Table 9, which I am grateful to Dr. P. Bosma for recomputing. We have introduced the set of μ vectors

The first two of these are the vectors U and Ν used throughout (see Section 5.1 or 7.1). We have

A0 = 1/2μ = Ν, Αγ = 1 = U, A2 = \μ = \W, A3 = 2μ2, etc.

The bimoment, or symmetric matrix product onm(a) = AnRAm characterizes the nth moment of the reflected radiation if the incident radiation is proport ional to Am. The normalization has been chosen in order to give strict symmetry and unit incident flux. These bimoments exist for any m > 0, η > 0.

An = Hn+ 1)μ' .η- 1

8.4 Moments and Bimoments of the Reflection Function m

T A B L E 9

B i m o m e n t s o f R e f l e c t i o n F u n c t i o n fo r a S e m i - I n f i n i t e L a y e r w i t h I s o t r o p i c S c a t t e r i n g

F o r m u l a

V a l u e

α s m a l l a = 0 .4 a = 0.8 α = 0.9 α = 0 . 9 9 a = 1 . 0 0

NRN = o 00 = ia*_1 NRU = ο 0ί = α 0 - 1 = 1

= iaotl

toc0 0 . 3 4 6 6 a

0 . 2 5 0 0 a

0 . 1 6 8 4

0 . 1 2 7 0

0 . 4 6 8 9

0 . 3 8 2 0

0 . 6 1 5 2 0 . 9 0 5 3

0 . 5 1 9 5 0 . 8 1 8 2

1 .0674

1 .0000

O02 = f (*i - i) 0 . 2 2 1 6 α 0 . 1 1 4 3 0 . 3 5 3 7 0 . 4 8 8 0 0 . 7 9 0 8 0 . 9 8 2 1

«03 = 2(a2 - J) 0 . 2 0 8 3 α 0 . 1 0 8 3 0 . 3 3 9 8 0 . 4 7 2 2 0 . 7 7 7 2 0 . 9 7 4 0

On = 1 - 2 r a , 0 . 2 0 4 6 α 0 . 1 0 7 3 0 . 3 4 1 9 0 . 4 7 8 0 0 . 7 9 4 6 1 .0000

Οχι = 1 — 3 ί α 2 = f αα? 0 . 1 8 7 5 α 0 . 0 9 9 6 0 . 3 2 4 9 0 . 4 5 9 8 0 . 7 8 3 4 1.0000

0 1 3 = 1 - 4 ί α 3 0 . 1 7 8 8 α 0 . 0 9 5 6 0 . 3 1 5 6 0 . 4 4 9 6 0 . 7 7 6 9 1 .0000

«22 = K - i + ί α 3 + i aa^ ) 0 . 1 7 3 8 α 0 . 0 9 3 4 0 . 3 1 1 5 0 . 4 4 5 9 0 . 7 7 7 8 1.0063

023 = f û ta l 0 . 1 6 6 7 α 0 . 0 9 0 1 0 . 3 0 3 9 0 . 4 3 7 8 0 . 7 7 4 0 1 .0095

033 = 8(έ - ί α 5 - ^ « ^ 4 0 . 1 6 0 2 α 0 . 0 8 7 1 0 . 2 9 7 1 0 . 4 3 0 7 0 . 7 7 1 6 1.0142

The resulting forms are

Oo.n = on,o = - y - ( - n + α„-ι I = —j—α?-ι

οι,η = οηΛ= \ - (n+ 1)ία,

3(n + 1) 02.» = O n t2 {-WT2

+ t0C"+i +

h«")

03,n = O n, 3 = 2 (n +

^ w T + l ~ ί α

"+2

~ 2α ι 0 ί

"+1 +

2α 2 < χ

" )

( m + l)(n + 1 ) , 1 VJ 1 , α <V„ = o„,m = ^ 1 - ( - l )

m[ - ~t-^n + f « n + m- i + 2

ai

c

+ · • · + ( - 1) ' \ *i- 10,+m-t + · · · + ( - 1 Γ I «m- l«n]

A relation useful for checking calculations is

°n,m+l °«+l,m _ a a

m

an

(n + l)(m + 2) (n + 2)(m + 1) 4

with the consequence

o„,„+i = Κ» + 1)(" + 2

)αα2


If \n — m I i s od d an d > 1 , th e genera l formul a jus t give n lead s t o tw o differen t expressions fo r o nm, wit h a differen t numbe r o f term s betwee n brackets . Th e difference i s 0 b y th e quadrati c relation s i n a „ cite d i n Sectio n 8.3.3 .

For a < 1 , thes e bimoment s tak e th e for m

on,m = ïa(n + l ) ( m + l ) G n + l f m + 1( o o )

where G nm(co) i s th e functio n discusse d i n Sectio n 2.5 , whic h ca n b e furthe r reduced i n term s o f F „(oo, — 1) . Th e proo f i s no t obvious .

Several o f th e function s i n Tabl e 9 hav e a specia l physica l significance . Fo r instance, UR = 1 — tH(a, μ 0) is the reflected fraction of the incident flux for unidirectional incidence from μ 0. Similarly, URN and URU are the reflected flux fractions for incidence with distribution Ν and U. Table 15 (Section 9.2) gives full detail, including the case b = oo to which the equations above refer.

The function RW (with W = μ, Section 5.1) plays a particular role in the conservative case. We find from Eq. (49) in Section 5.4.3 that the number o 2 2( l ) in Table 9 may be expressed in terms of the extrapolation length q(cc) = 0.71044609 by

022(1) = 3

<?(°°) ~ 9/8 = 1.00633827

8.5 POINT DIRECTION GAIN IN A SEMI-INFINITE ATMOSPHERE

In this section we collect some properties of the point-direction gain G(a, τ, μ) and its moments

9Μ τ

) = f τ

> til1* d

V< (J = ~ h 0, 1,...) Jo

From a simple added-layer argument (Ambartsumian's method) we find that the gain obeys the exact differential equation (Section 7.3.3, Sobolev, 1963; van de Hulst, 1964)

(d/dT)G(a, τ , μ) = - ( l / / * ) G ( a , τ , μ) -f- ag-M τ)Η(α9 μ)

The moments obey the corresponding equation

(d/dT)gj(a, τ) = -gj-M τ) + \ag.x{a, τ)α/α)

For 7 = 0 this becomes

(d/dT)g0(a, τ) = - ( 1 - a)l/2g _,(α, τ)

By integration we obtain the rigorous equation

G(a, τ , μ) = H(a, μ > " ^ | ΐ - 2 ( ] °_ JV^ dlg0(a, x ) ] J which is suitable for numerical integration. The term added to 1 is positive.

8.5 Point-Direction Gain in a Semi-Infinite Atmosphere 175

β / α ) = — j ΓΓ J 0 1 - k(a ι)μ

can be found from the recurrence relation

kRJ(a) = Rj.1(a)-ocj(a)

starting with

R.,(a) = 2/a, R0(a) = 2(1 - a)i,2/ka

Note that

= and Rj(a) = -(2/ak)KJ+1(a, -k~l)

in terms of the function introduced in Section 8.4.

Values for small a. For small a, the expansion in powers of a corresponding to successive orders is most convenient. By a direct experiment with incident radiation from one direction with the normalization adopted throughout this book, we obtain the source function

J(T) = (α/Αμ)β{α9 τ, μ)

Values for τ = 0 {at the surface). At τ = 0 we obtain the known functions

G(a, 0, μ) = H(a, μ) (Section 8.3.1, Table 7)

g .(a, 0) = α/α) (Section 8.3.3, Table 8)

Values for large τ (in the diffusion domain). At large depths τ, the gain is very nearly proport ional to exp( — kx). This defines the diffusion domain. A practical rule for intermediate values of a is that the diffusion domain starts at τ = 3. See Fig. 8.9 (below) for more precise information. The gain in the diffusion domain copied from the equation valid for an arbitrary phase function (Display 5.3) is

G(a, τ, μ) = ΑμΚ{μ)β'^

where we should insert for Κ(μ) the specification for isotropic scattering given in Display 8.1, so that

G(a, τ, μ) = 4Χ(0)[μΗ(μ)/(1 - / φ ) > " *τ

The corresponding moments of the gain are

0/<i, τ) = 4W)Rj(a)e-k*

The definite integrals lH(a, μ)μ

]+1άμ

7 7 6 8 Isotropic Scattering, Semi-Infinite Atmospheres

any τ, any μ τ = 0, any μ

G„ - «"«·

G l - 1 β - « [ „ , o g ( l + I ) + f l ( ^ ) ]

H0 = 1

Numerical values may be derived from Table 2, Section 2.4. An example showing the terms up to the 5th order is given in the left half of Fig. 7.1.

The moments can be similarly expressed :

gj(a, τ) = g0J + aguj + a2g2J + · · ·

where

g0J = EJ+2(T\ G IJ = i G ' 1 J + 2( i ) + ^ + 2 > 1( τ )

Values at a = 1 (conservative atmosphere). The conservative case has been thoroughly studied by Sobolev (1963, 1957) and by van de Hulst (1964). The moments of the point-direct ion gain for a = 1 are

flf-ΛΙ,τ) = (2J3)(d/dz)[z + q(T)-\

g0(l, τ) = 2

Ql(l τ) = 2q(z)

g2(l τ) = 2q(cc)q(x) + 2 ί [^(oo) - q(x)] dx Jo

where q(x) is Hopf 's function (Section 8.6.2), and q(co) = 0.710446089800. These moments reach a finite limit for infinite depth :

0 / 1 , GO) = y/3ocj+1(l)

The gain itself can conveniently be derived from

G(l , τ, μ) = ν ^ Η α μ)^(τ) + μ(1 - β~τΙ») -

< ι"

χ ν μ^ ]

with the limit

G(l , GO, μ) = ^ 3 / ^ ( 1 , μ)

Table 10 gives numerical values. The expression written in brackets has been plotted for easy interpolation against Ε2(τ) in Fig. 8.5.

Separation into successive orders gives

G(a, τ, μ) = G 0 + aGx + a2G2 + · · ·

where


T A B L E 10

P o i n t - D i r e c t i o n G a i n f o r b = oo, a = 1, I s o t r o p i c S c a t t e r i n g

F I R S T PART THE 1 GAIN I T S E L F

τ μ = 0 . 0 μ = 0 . 1 μ = 0 . 3 μ = 0 . 5 μ = 0 . 7 μ . = 0 . 9 μ = 1 . 0

0 0 . 0 1 5 6 2 0 . 0 3 1 2 5 0 . 0 4 6 8 7 0 . 0 6 2 5 0

1 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 o.cooooo 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0

1 . 2 4 7 3 5 0 1 . 1 2 8 0 8 0 1 . 0 1 4 9 6 2 0 . 9 1 4 3 6 5 0 . 8 2 5 9 7 8

1 . 6 4 2 5 2 2 1 . 6 4 4 2 4 0 1 . 6 3 0 2 4 4 1 . 6 1 1 6 0 4 1 . 5 9 0 6 5 4

2 . 0 1 2 7 7 9 2 . 0 5 6 2 6 6 2 . 0 7 9 0 0 6 2 . 0 9 4 4 1 1 2 . 1 0 5 3 0 8

2 . 3 7 3 9 7 5 2 . 4 4 6 4 9 4 2 . 4 9 4 3 8 9 2 . 5 3 3 3 6 0 2 . 5 6 6 6 9 1

2 . 7 3 0 5 8 7 2 . 8 2 7 6 5 9 2 . 8 9 6 4 9 3 2 . 9 5 5 0 6 9 3 . 0 0 7 1 2 5

2 . 9 0 7 8 1 0 3 . 0 1 6 2 9 0 3 . 0 9 4 7 6 5 3 . 1 6 2 3 5 2 3 . 2 2 3 0 1 6

0 . 1 2 5 0 0 0 . 2 5 0 0 0 0 . 3 7 5 0 0 0 . 5 0 0 0 0 0 . 7 5 0 0 0

C . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0

0 . 5 7 1 9 3 2 0 . 3 4 4 3 6 3 0 . 2 6 9 1 2 8 0 . 2 4 2 3 7 6 0 . 2 2 6 4 1 0

1 . 4 9 9 7 4 7 1 . 3 3 1 2 0 3 1 . 2 0 0 0 9 9 1 . 1 0 3 4 8 0 0 . 9 8 3 6 4 7

2 . 1 2 3 8 3 9 2 . 1 0 8 6 8 0 2 . 0 7 0 0 7 3 2 . 0 2 6 3 8 9 1 . 9 4 6 0 2 6

2 . 6 6 6 7 2 9 2 . 7 8 5 9 1 7 2 . 8 5 2 7 3 6 2 . 6 9 1 9 5 2 2 . 9 2 7 3 2 3

3 . 1 7 6 2 9 1 3 . 4 1 5 7 5 7 3 . 5 8 5 5 1 8 3 . 7 1 3 6 6 9 3 . 8 9 2 6 7 0

3 . 4 2 4 1 0 6 3 . 7 2 0 1 8 7 3 . 9 3 9 9 9 8 4 . 1 1 2 9 5 7 4 . 3 6 8 3 8 6

1 . 0 0 0 0 0 1 . 2 5 0 0 0 1 . 5 0 0 0 0 1 . 7 5 0 0 0

0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0

0 . 2 2 1 6 4 6 0 . 2 1 9 4 2 7 0 . 2 1 8 1 9 0 0 . 2 1 7 4 4 7

0 . 9 2 2 3 8 3 0 . 8 9 0 9 5 8 0 . 8 7 4 5 6 0 0 . 8 6 5 7 7 6

1 . 8 8 3 9 6 8 1 . 8 3 9 3 9 5 1 . 6 0 6 4 0 6 1 . 7 8 7 2 2 4

2 . 9 3 5 1 4 6 2 . 9 3 1 8 1 9 2 . 9 2 4 4 7 6 2 . 9 1 6 3 1 4

4 . 0 0 7 9 5 0 4 . 0 8 4 7 1 8 4 . 1 3 6 8 5 6 4 . 1 7 2 7 2 6

4 . 5 4 5 3 6 3 4 . 6 7 1 7 8 9 4 . 7 6 3 7 5 7 4 . 8 3 1 4 6 6

2 . 0 0 0 0 0 2 . 2 5 0 0 0 2 . 5 0 0 0 0 2 . 7 5 0 0 0

0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0

0 . 2 1 6 9 8 2 0 . 2 1 6 6 8 2 0 . 2 1 6 4 8 4 0 . 2 1 6 3 5 1

0 . 8 6 0 9 1 8 0 . 8 5 8 1 3 5 0 . 8 5 6 4 8 1 0 . 8 5 5 4 6 4

1 . 7 7 2 8 7 9 1 . 7 6 3 2 1 2 1 . 7 5 6 7 1 0 1 . 7 5 2 3 3 8

2 . 9 0 6 7 0 6 2 . 9 0 2 1 5 8 2 . 8 9 6 7 6 9 2 . 8 9 2 4 5 6

4 . 1 9 7 6 2 3 4 . 2 1 5 0 1 5 4 . 2 2 7 2 2 1 4 . 2 3 5 8 1 8

4 . 8 8 1 7 4 3 4 . 9 1 9 3 1 4 4 . 9 4 7 5 3 0 4 . 9 6 8 8 0 2

3 . 0 0 0 0 0 3 . 2 5 0 0 0 3 . 5 0 0 0 0 3 . 7 5 0 0 0

C . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 O . 0 C 0 Û 0 O

0 . 2 1 6 2 6 0 0 . 2 1 6 1 9 6 0 . 2 1 6 1 5 4 0 . 2 1 6 1 2 4

0 . 8 5 4 8 2 0 0 . 8 5 4 3 9 9 0 . 6 5 4 1 1 9 0 . 6 5 3 9 2 9

1 . 7 4 9 3 9 5 1 . 7 4 7 4 0 9 1 . 7 4 6 0 6 5 1 . 7 4 5 1 5 2

2 . 8 8 9 0 6 9 2 . 8 8 6 4 4 5 2 . 8 6 4 4 3 2 2 . 8 8 2 8 9 9

4 . 2 4 1 8 8 7 4 . 2 4 6 1 8 0 4 . 2 4 9 2 2 1 4 . 2 5 1 3 7 5

4 . 9 8 4 8 9 2 4 . 9 9 7 0 9 4 5 . 0 0 6 3 6 9 5 . 0 1 3 4 3 1

4 . 0 0 0 0 0 5 . 0 0 0 0 0 6 . 0 0 0 0 0

CD

0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0

0 . 2 1 6 1 0 2 0 . 2 1 6 0 6 3 0 . 2 1 6 0 5 2 0 . 2 1 6 0 4 7

0 . 8 5 3 7 9 9 0 . 8 5 3 5 6 5 0 . 8 5 3 5 0 4 0 . 6 5 3 4 8 0

1 . 7 4 4 5 2 8 1 . 7 4 3 4 6 1 1 . 7 4 3 2 0 8 1 . 7 4 3 1 1 8

2 . 8 8 1 7 4 0 2 . 8 7 9 3 5 3 2 . 8 7 8 6 1 3 2 . 6 7 8 2 9 2

4 . 2 5 2 9 0 2 4 . 2 5 5 6 7 6 4 . 2 5 6 3 6 3 4 . 2 5 6 5 6 5

5 . 0 1 8 8 1 8 5 . 0 3 0 3 3 7 5 . 0 3 4 3 2 0 5 . 0 3 6 4 7 6

SECOND PAR T MOMENT S

T 0 . 5 q - i 0 . 5 qo 0 . 5 g (* q ( T ) o . 5 g 2 0

0 . 0 1 5 6 2 0 . 0 3 1 2 5 0 . 0 4 6 8 7 0 . 0 6 2 5 0

6 . 2 1 4 2 0 2 2 . 9 4 4 2 1 5 2 . 6 5 0 0 5 1 2 . 4 8 9 8 8 7 2 . 3 8 2 8 7 5

1 . 0 C 0 0 0 0 1 . 0 0 0 0 0 0 1 . 0 0 0 0 0 0 1 . 0 0 0 0 0 0 1 . 0 0 0 0 0 0

0 . 5 7 7 3 5 0 0 . 5 9 2 4 7 1 0 . 6 0 1 9 1 0 0 . 6 0 9 4 1 2 0 . 6 1 5 7 3 9

0 . 4 1 0 1 7 6 0 . 4 2 2 8 6 3 0 . 4 3 1 3 3 6 0 . 4 3 8 3 0 1 0 . 4 4 4 3 2 4

0 . 1 2 5 0 0 0 . 2 5 0 0 0 0 . 3 7 5 0 0 0 . 5 0 0 0 0 0 . 7 5 0 0 0

2 . 1 5 2 4 8 0 1 . 9 6 9 0 3 5 1 . 8 8 6 8 3 3 1 . 6 4 0 3 4 5 1 . 7 9 1 2 2 4

1 . 0 0 0 0 0 0 1 . 0 0 0 0 0 0 1 . 0 0 0 0 0 0 1 . 0 0 0 0 0 0 1 . 0 0 0 0 0 0

0 . 6 3 4 4 6 4 0 . 6 5 7 1 2 0 0 . 6 7 0 9 4 0 0 . 6 6 0 2 9 4 0 . 6 9 1 9 1 6

0 . 4 6 2 9 1 9 0 . 4 8 6 9 6 1 0 . 5 0 2 5 2 1 0 . 5 1 3 4 8 5 0 . 5 2 7 6 8 2

1 . 0 0 0 0 0 1 . 2 5 0 0 0 1 . 5 0 0 0 0 1 . 7 5 0 0 0

1 . 7 6 7 1 9 8 1 . 7 5 3 9 9 2 1 . 7 4 6 2 1 3 1 . 7 4 1 4 1 6

1 . 0 0 0 0 0 0 1 . 0 0 0 0 0 0 1 . 0 0 0 0 0 0 I . 0 0 0 0 0 0

0 . 6 9 8 5 3 9 0 . 7 0 2 5 7 1 0 . 7 0 5 1 3 0 0 . 7 0 6 8 0 1

0 . 5 3 6 1 2 0 0 . 5 4 1 4 1 8 0 . 5 4 4 8 6 1 0 . 5 4 7 1 5 4

2 . 0 0 0 0 0 2 . 2 5 0 0 0 2 . 5 0 0 0 0 2 . 7 5 0 0 0

1 . 7 3 8 3 5 9 1 . 7 3 6 3 6 2 1 . 7 3 5 0 3 2 1 . 7 3 4 1 3 3

1 . 0 0 0 0 0 0 1 . 0 0 0 0 0 0 1 . 0 0 0 0 0 0 1 . 0 0 0 0 0 0

0 . 7 0 7 9 1 7 0 . 7 0 8 6 7 3 0 . 7 0 9 1 9 3 0 . 7 0 9 5 5 4

0 . 5 4 8 7 0 9 0 . 5 4 9 7 7 9 0 . 5 5 0 5 2 2 0 . 5 5 1 0 4 4

3 . 0 0 0 0 0 3 . 2 5 0 0 0 3 . 5 0 0 0 0 3 . 7 5 0 0 0

1 . 7 3 3 5 1 7 1 . 7 3 3 0 9 0 1 . 7 3 2 7 9 3 1 . 7 3 2 5 8 3

1 . 0 0 0 0 0 0 1 . 0 0 0 0 0 0 I . 0 0 0 0 0 0 1 . 0 0 0 0 0 0

0 . 7 0 9 8 0 8 0 . 7 0 9 9 8 7 0 . 7 1 0 1 1 4 0 . 7 1 0 2 0 5

0 . 5 5 1 4 1 4 0 . 5 5 1 6 7 7 0 . 5 5 1 8 6 5 0 . 5 5 2 0 0 1

4 . 0 0 0 0 0 5 . 0 0 0 0 0 6 . 0 0 0 0 0

ω

1 . 7 3 2 4 3 4 1 . 7 3 2 1 5 8 1 . 7 3 2 0 8 2 1 . 7 3 2 0 5 1

1 . 0 0 0 0 0 0 1 . 0 0 0 0 0 0 I . 0 0 0 0 0 0 1 . 0 0 0 0 0 0

0 . 7 1 0 2 7 1 0 . 7 1 0 3 9 5 0 . 7 1 0 4 3 1 0 . 7 1 0 4 4 6

0 . 5 5 2 0 9 9 0 . 5 5 2 2 8 8 0 . 5 5 2 3 4 3 0 . 5 5 2 3 Λ 7


0 I 1 1 1 1 1 I I ' ' 1 1.0 0.5 0

linear s c a l e of E^iT)

F i g . 8 . 5 . I n t e r p o l a t i o n g r a p h p e r m i t t i n g e s t i m a t e s o f t h e p o i n t - d i r e c t i o n g a i n in a c o n s e r v a t i v e

s e m i - i n f i n i t e a t m o s p h e r e w i t h i s o t r o p i c s c a t t e r i n g f o r a n y d e p t h a n d d i r e c t i o n .

Values at a near 1 (slightly lossy atmosphere). For a nearly 1, the gain and its moments obey the equations

G(a, τ, μ) = G(l , τ, μ) - ^/3μΗ(1 μ)[τ + q(z)~]k + 0(k2)

9j(a, τ) = gj(l τ) - 73α,.+ 1(1)[τ + ^ τ ) ] * + 0(k2)

and, in particular,

g0(a, τ) = 2 - 2 /c[i + <?(τ)] -f 0(k2)

In each of these equations, the diffusion exponent k = k(a) is a small quantity and may, by neglecting third-order terms, also be replaced by where t = (1 - a)

1'2 (Section 8.2.1).

These equations describe completely what happens to the gain if we fix τ and let a move to 1, and hence k to 0. The linear approximation given here suffices if we make k small enough to have kr <ζ 1. However, the more common question is what happens to the gain as a function of τ for a fixed value of a. The situation for a very close to 1 is then as sketched in Fig. 8.6. The linear equations just given are still subject to the condition kz <ζ 1. Therefore, they do not cover the entire range of depths, but only the transition region, which links the surface to the diffusion domain.


Surface diffusion domain

transition region

τ=0 x=1 . . T=k_1 τ=οο

! region of; joverlap !

F i g . 8.6. S c h e m a t i c d o m a i n s o f v a l i d i t y o f d i f f e r e n t a p p r o x i m a t i o n s in a n e a r l y c o n s e r v a t i v e

a t m o s p h e r e .

This allows two checks against results derived earlier. First, at τ = 0, the gain becomes the Η function, and we simply recover the first terms of the expansion of H(a> μ) in t (Section 8.3.1). Second, in the overlap region 1 < τ < fc

-1, we obtain

G(a, τ, μ) = ^3μΗ(1 μ){1 - [τ + ^ ( c x ) ) ] / c + · · · }

The same result follows from the gain equation for arbitrary a in the diffusion domain (earlier in this section) by making the appropriate substitutions for a near 1, viz., H(a, μ) = H(l, μ)(1 — kμ) (Section 8.3.1) and

K(a90) = [1 - 4 ( o o ) / c ] K ( l , 0 )

(Eq. (3.6) of Section 5.4.1 with the notation adapted to show the dependence on a explicitly). We then write exp( — kx) = 1 — /CT, and the check is complete.

The overlap region is wide only if a is very close to 1. For instance, if a = 0.9999, it extends roughly from τ = 2 to τ = 20; at a = 0.99, it has shrunk to virtually nothing. Figure 8.7, adapted from Barkstrom (1972), gives the full gain curves for a = 0.99. The gain values at τ = 0 ranging from H(0A) = 1.22 to H(l) = 2.47 can be checked directly from Table 7 (Section 8.3.1). I have checked the curve f o r μ = 1 by the integration formula given at the start of this section. It has a maximum 3.31 near τ = 0.75. For τ > 0.25, it is well represented by

G(0.99, τ, 1) = 4 .51ΕΓ° 1 7 2 5τ

- 1 . 3 9 ^

but not until τ > 5 can the second term be neglected so that the strict asymptotic equation prevails. In the original paper, this computat ion was made to represent a crude model for the penetration of sunlight into packed snow or bubbly ice. The abscissa τ is then the approximate depth in centimeters, and the ordinate is proportional to the local heating by sunlight.

The gain formula for a near 1 may also be derived from a simple physical argument. In the standard experiment with flux π, incident from one direction μ 0 on a unit area of the atmosphere, we have the source function

J = (α/4μ0 (α, τ, μ0)


Τ I j ι j ι ι ι ι ι ι ι ι r

Fig . 8 . 7 . P o i n t - d i r e c t i o n g a i n fo r s i n g l e - s c a t t e r i n g a l b e d o a = 0 . 9 9 ( a d a p t e d f r o m B a r k s t r ô m ,

1972) .

This J is nearly equal to that for a = 1 near the surface. Very deep inside the atmosphere, where kx > 1, the cumulative effect of many small losses makes the deviation severe. Hence, in comparison with the conservative case, there is an energy sink deep in the atmosphere. The flux going into this sink is π F a b s, where the absorbed fraction is known to be

^ a b s = [1 ~ i a0( a ) ] i f ( a , μ)

= (k/^/3)H(l μ) + 0(k2)

(Display 9.3). Since any flux from deep inside emerges with the source function (Section 8.6.2)

j = | F [ T + q(x)-]

the nonemerging flux causes this same source function with F = Fahs to be subtracted from ^ c o n s e r v a t i v e · Multiplication by 4 μ 0 finally gives G(a, τ, μ0) , apart from terms of the order of k

2.

The same reasoning may be applied if the energy sink is due to other causes, e.g., a finite but large depth, a partly absorbing ground surface, or layers with strong absorption deep inside the atmosphere.

Values for virtual angles, μ -+ oo . Upon transition, in the standard problem, to the limit μ -> o o , the point-direct ion gain reaches a finite limit. It can be verified that this corresponds to the solution of the following problem.

Consider a homogeneous, semi-infinite atmosphere with albedo a < 1. Let there be absence of incident radiation, and let embedded sources J0 = 1


2 -^ 1 zero - order

0 I 1 ! 1

0 1 2 3 ^ τ A

F i g . 8.8. E x a m p l e o f g a i n f u n c t i o n f o r t h e v i r t u a l d i r e c t i o n μ — oo.

be homogeneously distributed at all depths. The total source density at depth τ is

G(a, τ, oo) = | \ - | # 0( a , T)J^ (1 - a)

increasing from

G(a, 0, oo) = H(a, oo) = (1 - a)~i/2

to

G(a, oo, oo) = (1 - ay1

Figure 8.8 provides an illustration for a = 0.9. The escape probability. The gain moment of order 0 has a special im

portance because ι# 0( τ) *s the escape probability of a photon released at depth

τ to escape (after any number of scatterings) through the surface. In Fig. 8.9, we have plotted this quanti ty against τ for five values of a. The following features, illustrating the formulas presented in the preceding pages, may be noted:

For a = 1, the escape probability is 1, independent of depth. For a = 0.99, 0.9, and 0.6, an approach toward the asymptotic formula in

the diffusion domain (dashed asymptotes) is seen. This occurs faster if a is closer to 1.

For a = 0.2, an approach to this asymptote is not seen because the contrast between the factor exp( — kx) and exp( —τ) is not strong enough for the τ values shown.

For a = 0, the escape probability is JE2(T).

The linear approximation in k given above has practical use only if a is very close to 1.

Fig. 8 . 9 . E s c a p e p r o b a b i l i t y f o r r a d i a t i o n p r o d u c e d b y a n i s o t r o p i c s o u r c e a t d e p t h τ in a n

a t m o s p h e r e w i t h i s o t r o p i c s c a t t e r i n g f o r v a r i o u s v a l u e s o f a.

8.6 Radiation Emerging from a Semi-Infinite Atmosphere 183

8.6 R A D I A T I O N E M E R G I N G F R O M A S E M I - I N F I N I T E A T M O S P H E R E

Suppose radiation is carried from a deeply seated source through a semi-infinite, homogeneous atmosphere toward its surface. Deep inside the a tmosphere, between the source and the surface but well away from both, the radiation field must have the same form as if the medium were unbounded. This domain is called the diffusion domain. In the diffusion domain of a conservative a tmosphere (a = 1), the flux carried forward is a constant; in the diffusion domain of a nonconservative atmosphere, it decreases exponentially by the factor e

kz. The

general theory is in Chapter 5. The coefficient k(a) for isotropic scattering is given in Table 5 (Section 8.2.1) and Table 11 (below).

Near the surface, the emerging radiation causes extra losses. Hence the solution near this surface falls below the solution extrapolated from the diffusion domain. This means that we have to apply to the diffusion solution a negative surface correction, which is sizable at the surface and becomes very small at great depths. The aim of this section is to discuss this surface correction for isotropic scattering.

In the case a = 1, this is the problem of emerging constant net flux, which forms the prototype of all radiative transfer problems in astrophysics. The surface correction to the source function is then expressed in terms of Hopf's function q(x\ the most discussed transcendental function in radiative transfer (Section 8.6.2). However, it seems more logical to start with the corresponding problem for a slightly lossy atmosphere.

8.6.1 Lossy Atmosphere, a < 1

In the illustration (Fig. 8.10), which is drawn to scale for a = 0.95, we have k = 0.370, / ( a , 0) = 0.647. The exponential term represents the diffusion solution; the term f(a, τ) is the correction term for losses at the surface. Once / ( α , τ) and hence J(a, τ) are known, it is possible to compute the intensity and the flux at arbitrary depth by the general integration procedures in Chapter 7.

The results for the diffusion domain are as follows:

The intensity of radiation at depth τ traveling in the direction u = cos 0, where θ is the angle with positive τ directions, is

We write

alek* - f(a, τ)] , τ > 0

τ < 0

a exp(/cr) 1 + ku

both for ingoing radiation (0 < u < 1 ) and for outgoing radiation (—1 < u < 0).


' ( α ) β

1 I I I 1 1

n t t f low of j t n e r g y #

5

4 / 3

e

M /

2

1

.—-

ι ι 1 1 1 1 1

(-4) (-2) 0 2 4 6 8 Τ

( b ) 0.6

r T i l l

f ( r )

- 0.4 - \ -

ι

02

ι I I 1 ^ 1 . I 0 2 4 6 8 Τ

F i g . 8 . 1 0 . T h e s o u r c e f u n c t i o n v a l i d f o r r a d i a t i o n e m e r g i n g f r o m a s e m i - i n f i n i t e a t m o s p h e r e

t h r o u g h t h e b o u n d a r y τ = 0 is c o m p a r e d t o t h e c o r r e s p o n d i n g s o u r c e f u n c t i o n in a n u n b o u n d e d

m e d i u m ( a ) . T h e d i f f e r e n c e is s h o w n o n l a r g e r s c a l e ( b ) . T h i s e x a m p l e r e f e r s t o i s o t r o p i c s c a t t e r i n g ,

a = 0 . 9 5 .

The net flux divided by π is

^ d i f f f e ^ ) = 4(a - 1) exp(kt)/k

which is negative because the energy flows towards smaller τ. Both formulas require a correction near the surface arising (1) from the term f(a, τ), and (2) from the fact that the τ integration interval for ingoing radiation is finite.

The intensity emerging at the surface (τ = 0, μ = —u) assumes the form

Ι(μ) = ηιΚ(μ) = aj[l/(l - kμ)] - JJ/fe τ)β'τίμ άτ/μ^

and for μ = 0

/(0) = J(0) = mK(0) = all - f(a, 0)]


where we have introduced the escape function and constant m defined in Section 5.2.1. The form of Κ(μ) can be derived from scratch by an invariance principle (Display 4.5) or quoted from Ambartsumian (1942) or from Chandrasekhar 's book (Section 88). We will use the result for a general phase function in Display 6.11. Inserting the specifications for isotropic scattering contained in Display 8.1, and writing y ~

1 = /c, we obtain

ηιΚ(μ) = αΗ(μ)/Η(1ϊ-ι)(\ - /ίμ)

so that

mK(0) = all - f(a, 0)] = a/Hik'1)

For a full numerical determination of the emerging radiation we therefore need a table of H(a, μ), available in Table 7, a table of k(a\ available in Table 5, and a table o f / (a , 0) or of H(k~

l\ available in Table 11. If, however, we wish

to know also the source function, intensity, and flux at any depth, we need the full table of/(α, τ). This is not available in tabular form, but the graph in Fig. 8.11 will permit accurate interpolation over the full range of the variables.

The surface correction function f(a, τ). The following discussion of the properties o f / (α , τ) largely follows van de Hulst and Terhoeve (1966). For a thorough discussion and connections with the older literature, reference may also be made to Heaslet and Warming (1968).

A rigorous method of d e t e r m i n i n g / ^ , τ) is as follows. If we should wish to restore the losses expressed by the term —/(α, τ) in the source function, we should send into the atmosphere from outside at τ = 0 the radiation which would be present in an unbounded atmosphere. This is

Ι(μ) = 1/(1 + ku\ 0 < u < 1

Reverting to the more usual notat ion μ = w, we obtain the theorem:

The correction t e rm/ (α , τ) is the source function that would be established in the atmosphere by an experiment with incident radiation of intensity

/(μ) = 1/(1 + Ιζμ) .

A more formal derivation, which will not be reproduced, gives the same theorem in the form: If J(a, τ) is to obey the homogeneous Milne equation, then /(α, τ) must be the solution of the nonhomogeneous Milne equation, the known term (starting function) of which is the primary source density established by incident radiation of the intensity stated.

Since the source function for unidirectional incidence is expressible in the point-direction gain, the source function in the experiment described is an

m 8 Isotropic Scattering, Semi-Infinite Atmospheres

F i g . 8 . 1 1 . I n t e r p o l a t i o n g r a p h p e r m i t t i n g e s t i m a t e s o f t h e c o r r e c t i o n f u n c t i o n / ( α , τ ) o v e r t h e full r a n g e o f a r g u m e n t s .

integral of this gain over the angular distribution of incident radiation. This gives

α f1 σ(α,τ,μ0)άμ0

2 J 0 1 + Κα)μ0 where the gain function may be taken from Section 8.5.

We now list a number of approximation formulas and asymptotic expressions useful in particular regions of the entire domain 0 < α < 1 , 0 < τ < ο ο . Each may be derived as a consequence of the general theorem stated above.

Small a. The Milne equation in thç shorthand notation of Chapter 7 reads

/ = fx + aMf


It can be solved by successive scattering, i.e., in the form of the usual Neumann series, starting from

* * > - ! F T ^ - IF L

' * -1

> - ' * -I

»

See Section 2.4 for the F functions. For a < 0.3, we may practically replace k by 1, and the solution takes the form

f(a9 τ) = ak^r) + a2k2(x) + · · ·

where

/ c1( T ) = M L N 2 - F 1( T , - 1 ) ]

fc1(0) = i l n 2 , k2(0) = 0.083

For a near 1. Developing the integrand in the general expression in a power series in k, we obtain

f(a, τ) = ^alg0(a, τ) - gi(a, x)k(a) + g2(a9 z)k2(a) ]

where the gain moments gj(a9 τ) may be taken from Section 8.5. Further developing a as a series in k, we may make this into a pure power series in k. The first terms are

f(a, τ) = 1 - [τ + 2$(τ)]Λ + 0(k2)

For τ = 0, any a. The gain becomes the Η function. The integral is then known and gives

1 - 0) = l/H(a, y )

with y = / c_ 1

, as stated in Section 8.3.2. Numerical values were first given by Kuscer (1953) and may also be taken from Table 11. Useful expansions are

/ ( A , 0) = K [ A 0( A ) ~ oix{a)k + <x2(a)k2 - a 3( a ) / c

3 + · · · ]

1 - f(a, 0) = aloi^k + ot3(a)k3 + oc5(a)k

5 + · · ·]

They may again be made into a pure power series in k:

f(a9 0) = 1 - (2/y/3)k + a 2( l ) / c2 - [ a 3( l ) + ( 4 / 4 5 ) 7 3 ] / c

3 + · · ·

= 1 - 1.1547* + 0.8204fc2 - 0.7918/c

3 + · · ·

Asymptotic expression for large τ. In the limit τ -* oo, the correction function itself reaches a diffusion domain in which

f(a, τ) = l(a)e-k(a)T


is the asymptotic expression. The function 1(a) is the key function in the theory for very thick layers. It was first derived for isotropic scattering by Sobolev ( 1 9 5 7 ) in the form

f1 H(a, x)x dx Ι f

1 H(a, x)x dx K a)

" J o 1 - k2x2 I J 0 ( 1 - kx)

2

We can now refer directly to the expression for an arbitrary phase function in Display 6 . 1 1 . With the specification for F from Display 8 . 1 , this gives

1(a) = 2a/mkH2(a, γ)

Yet another form is obtained by replacing G(a, τ, μ) in the integral defining /(α, τ) by its asymptotic form given in Section 8 . 5 . The result is

l(a) = 2aK(0)^ 1 - f cV

All these results are the same. With the specifications of Display 8 . 1 , both the numerator and the denominator of Sobolev's original expression have the form of Eq. ( 5 ) in Section 6 . 4 . 2 . Reducing them by Eq. ( 6 ) , Section 6 . 4 . 2 and by Eq. ( 1 0 ) , Section 6 . 4 . 4 , we find that the numerator equals [akH(a, y ) ] ~ \ and the denominator equals mH(a, y)/2a

2.

Table 1 1 shows the values of 1(a) and various related quantities, computed by F. Terhoeve. They check well with Kuscer ( 1 9 5 3 ) .

8.6.2 Conservative Atmosphere, a = 1

The solution of the problem of emerging net flux from a conservative atmosphere is in all textbooks and may be cited at once: If the net flux nF emerges from a semi-infinite conservative atmosphere with isotropic scattering (independent of wavelength), the source function at optical depth τ is

j(T) = | F [ T + q(xy]

and the emerging intensity has the angular distribution

m = (y/3/4)FH0i)

Here q(x) is Hopf's function (see Table 1 0 ; or eight-decimal table in King et al, 1 9 6 5 ) and Η(μ) = H(l, μ) (see Table 7 , Section 8 . 3 . 1 ) . The source function is shown graphically in Fig. 8 . 1 2 , which also makes clear why # ( 0 0 ) is called the extrapolation length. This constant occurs so often that we prefer to write it as q^. See the accurate value quoted in Section 8 . 3 . 3 .

TA

BL

E

11

Fu

nct

ion

s of

a R

elat

ed t

o a

Ver

y T

hic

k L

ayer

wit

h Is

otr

op

ic

Sca

tter

ing

a t

= (1

-

α)"*

k

4 dk

m

da

m

1 1

fo~H

(k->

) K

(0)

ad

~ f 0)

m

I

0 1

1 0

oo

1 0

0

0.1

0.94

87

1.00

00

0.00

00

0.96

45

0.00

00

0.00

00

0.2

0.89

44

0.99

99

0.00

45

0.92

71

0.00

02

0.00

04

0.3

0.83

67

0.99

74

0.05

83

68.6

681

0.88

74

0.00

39

0.00

69

0.4

0.77

46

0.98

56

0.18

94

21.1

240

0.84

44

0.01

60

0.02

74

0.5

0.70

71

0.95

75

0.38

22

10.4

660

0.79

67

0.03

81

0.06

33

0.6

0.63

25

0.90

73

0.63

15

6.33

42

0.74

19

0.07

03

0.11

49

0.7

0.54

77

0.82

86

0.95

94

4.16

91

0.67

57

0.11

35

0.18

50

0.8

0.44

72

0.71

04

1.44

36

2.77

08

0.58

98

0.17

03

0.28

28

0.9

0.31

62

0.52

54

2.40

03

1.66

65

0.46

06

0.24

88

0.43

62

0.92

5 0.

2739

0.

4599

2.

8714

1.

3930

0.

4134

0.

2745

0.

4935

0.95

0.

2236

0.

3795

3.

6372

1.

0997

0.

3529

0.

3048

0.

5669

0.97

5 0.

1581

0.

2711

5.

3117

0.

7531

0.

2656

0.

3439

0.

6738

0.99

0.

1000

0.

1725

8.

5560

0.

4675

0.

1783

0.

3776

0.

7807

1 0

0 oo

0

0 0.

4330

1

Do

min

an

t te

rms

nea

r a

= 0

1 -

2e~2l

a 4

a'2e

-2'a (a

2/4)e

2la

1 -

0.34

7a

a-

1 e-2

/a

2a-l e~

2la

nea

r t

= 0

1.73

2f

0.8

66

Γ-1

4.61

9i

2t(l

-

1.23

1f)

0.43

3(1

- 1.

231f

) 1

- 2.

461



- 3

X * q ( X )

- 2

\ \

1 1

- 1

\ \

\ \

\ \

\ \

\ \

\ \

2 1 0 -0.71

X

F i g . 8 . 1 2 . S o u r c e f u n c t i o n , d r a w n t o e x a c t s c a l e , in t h e c l a s s i c a l p r o b l e m o f e s c a p e o f r a d i a t i o n f r o m a s e m i - i n f i n i t e a t m o s p h e r e w i t h c o n s e r v a t i v e i s o t r o p i c s c a t t e r i n g .

We shall now verify that the formulas derived in the preceding pages for a < 1 lead to this solution in the limit a = 1. This is a consistency check rather than an independent derivation, because we have already used the Hopf solution in Section 8.5 to find the moments of the gain function.

Take a < 1 and consider the range of τ in which kx <ζ 1. In the limit a 1, this range covers all depths. Developing the general expression for the source function and emerging intensity in powers of /c, and neglecting quadratic terms, we have in this range

J(a, χ) = 1 + kx - {1 - [τ + 2q(x)-]k} = 2k\x + q(x)~\

Ι(μ) = ( 2 / V

/3 ) / c / / ( l , μ)

Integration over μ gives the emergent flux/π:

F = [1Ι(μ)2μάμ = ( 4 / ^ / 3 ) ^ ( 1 ) = ffc

Jo

Hence we may replace k by | F , and thus find the stated solution.

References 191

R E F E R E N C E S

A b h y a n k a r , K . D . , a n d F y m a t , A . L . ( 1 9 7 1 ) . Astrophys. J. Suppl. 2 3 , 3 5 .

A b r a m o w i t z , M . , a n d S t e g u n , I . A . ( 1 9 6 5 ) . " H a n d b o o k o f M a t h e m a t i c a l F u n c t i o n s . " D o v e r , N e w

Y o r k .

A m b a r t s u m y a n , V . A . ( 1 9 4 2 ) . Astron. Zh. 19 , 3 0 .

B a r k s t r o m , B . R . ( 1 9 7 2 ) . J. Glaciol. 1 1 , 3 5 7 .

B u s h b r i d g e , I . W . ( 1 9 6 0 ) . " T h e M a t h e m a t i c s o f R a d i a t i v e T r a n s f e r . " C a m b r i d g e U n i v . P r e s s ,


C a s e , Κ . M . , a n d Z w e i f e l , P . F . ( 1 9 6 7 ) .

k tL i n e a r T r a n s p o r t T h e o r y . " A d d i s o n - W e s l e y , R e a d i n g ,


C a s e , K . M . , D e H o f f m a n , F . , a n d P l a c z e k , G . P . ( 1 9 5 3 ) . " I n t r o d u c t i o n t o t h e T h e o r y o f N e u t r o n

D i f f u s i o n . " U . S . G o v t . P r i n t i n g Off ice , W a s h i n g t o n , D . C .


N e w Y o r k . A l s o D o v e r , N e w Y o r k 1960 .

D a v i s o n , B . ( 1 9 5 7 ) . " N e u t r o n T r a n s p o r t T h e o r y . " O x f o r d U n i v . P r e s s ( C l a r e n d o n ) , L o n d o n a n d

N e w Y o r k .

D o m k e , H . ( 1 9 7 8 ) . J. Quant. Spectrosc. Radiât. Transfer 16 , 9 7 3 .

H a l p e r n , O . , L u n e b e r g , R . K . , a n d C l a r k , O . ( 1 9 3 8 ) . Phys. Rev. 5 3 , 1 7 3 .

H a r r i s , D . L . ( 1 9 5 7 ) . Astrophys. J. 126 , 4 0 8 .

H e a s l e t , Μ . Α . , a n d W a r m i n g , R . F . ( 1 9 6 8 ) . Astrophys. Space Sci. 1, 4 6 0 .

H o p f , E . ( 1 9 3 4 ) . " M a t h e m a t i c a l P r o b l e m s o f R a d i a t i v e E q u i l i b r i u m . " C a m b r i d g e U n i v . P r e s s ,


H u a n g , S u - s h u ( 1 9 5 2 ) . Phys. Rev. 8 8 , 5 0 .

I v a n o v , V . V . ( 1 9 7 3 ) . T r a n s f e r o f R a d i a t i o n in S p e c t r a l L i n e s , N a t i o n a l B u r e a u S t a n d a r d s S p e c i a l

P u b l . 3 8 5 . U . S . G o v t . P r i n t i n g Off ice , W a s h i n g t o n , D . C . O r i g . R u s s i a n , 1 9 6 9 .

K i n g , J . I . F . , S i l l a r s , R . V . , a n d H a r r i s o n , R . H . ( 1 9 6 5 ) . Astrophys. J. 1 4 2 , 1 6 5 5 .

K o u r g a n o f f , V . ( 1 9 5 2 ) . " B a s i c M e t h o d s i n T r a n s f e r P r o b l e m s . " O x f o r d U n i v . P r e s s ( C l a r e n d o n )

L o n d o n a n d N e w Y o r k . A l s o D o v e r , N e w Y o r k , 1 9 6 3 .

K u s c e r , I . ( 1 9 5 3 ) . Can. J. Phys. 3 1 , 1187 .

P l a c z e k , G . ( 1 9 4 7 ) . Phys. Rev. 7 2 , 5 5 6 .

S o b o l e v , V . V . ( 1 9 5 7 ) . Astron. Zh. 3 4 , 3 3 6 [English transi. Sov. Astr.-A.J. 1, 3 3 2 ] .

S o b o l e v , V . V . ( 1 9 6 3 ) . " A T r e a t i s e o n R a d i a t i v e T r a n s f e r . " V a n N o s t r a n d - R e i n h o l d , P r i n c e t o n ,

N e w J e r s e y . O r i g . R u s s i a n , 1956 .

S t i b b s , D . W . N . ( 1 9 6 2 ) . L a p h y s i q u e d e s p l a n è t e s , Congr. Colloq. Univ. Liège 2 4 , 1 69 .

S t i b b s , D . W . N . , a n d W e i r , R . E . ( 1 9 5 9 ) . Mon. Not. R. Astron. Soc. 1 1 9 , 5 1 2 .

v a n d e H u l s t , H . C . ( 1 9 6 4 ) . Bull. Astron. Inst. Neth. 17 , 4 9 5 .

v a n d e H u l s t , H . C , a n d G r o s s m a n , K . ( 1 9 6 8 ) . I n " T h e A t m o s p h e r e s o f V e n u s a n d M a r s " ( J . C .

B r a n d t a n d M . B . M c E l r o y , e d . ) , p . 3 5 . G o r d o n a n d B r e a c h , N e w Y o r k ,

v a n d e H u l s t , H . C , a n d T e r h o e v e , F . G . ( 1 9 6 6 ) . Bull. Astron. Inst. Neth. 18 , 3 7 7 .

W o o d f o r d , C , B r i d g e m a n , T . , a n d M a k i n s o n , G . J . ( 1 9 6 8 ) . Die Farbe 17 , 2 2 4 .

Y a n o v i t s k i i , E . G . ( 1 9 6 8 ) . Astromet. Astrofiz. (Kiev) 1 , 165 .

9 • Isotropic Scattering, Finite Slabs

This chapter presents a collection of numerical results in a form convenient for consultation. All results in this chapter refer to homogeneous atmospheres with isotropic scattering. Most of the theory is found in earlier chapters. The specifications of Display 8.1 for isotropic scattering are relevant. Many of the results for semi-infinite atmospheres presented in Chapter 8 recur here in the limit b oo.

9.1 REFLECTION AND TRANSMISSION

9.1.1 The Main Table

The main table, Table 12,J refers to the reflected and transmitted radiation field. The majority of numbers given in this forty-six page table are values of the reflection function R(a, b, μ, μ 0) or the transmission function T(a, b, μ, μ0) . The arguments are arranged as follows: b and μ 0 values above each block (4 blocks per page, 4 pages for each b value); a values at the left of each line; μ values above each column. Since by reciprocity both R and Τ do not change upon interchanging μ and μ 0, the comparison of corresponding numbers gives a test of the accuracy of the integrations. The lower orders are separately given for η = 0, 1, 2, 3 and are functions of n, b, μ 0, and μ. Order 0, the unscattered radiation, is present only in the Τ tables.

ί T a b l e 12 wi l l b e f o u n d a t t h e e n d o f t h e c h a p t e r , s t a r t i n g o n p a g e 2 3 6 .

192

9.1 Reflection and Transmission m The table presents in addition to R and Τ their moments and bimoments

obtained upon matrix multiplication by U or Ν as defined in Sections 5.1 or 7.1. These moments are arranged to follow μ as the last two columns and μ 0 as the last two blocks. These functions again obey a strict reciprocity, which may be employed to make further tests of accuracy. The physical meaning of the various products so obtained can be read from the definitions in Display 5.1. For instance, NT, a, function dependent on a, b, and μ 0, expresses the average intensity below an atmosphere illuminated from above by parallel radiation from direction μ 0. It should equal (when μ 0 is replaced by μ) Τ Ν, the function dependent on a, b, and μ, which expresses the intensity emerging at the bot tom in direction μ, if a narrow source layer is placed above the atmosphere. The normalization is always to incident flux π.

Input with μ 0 = 0 was avoided, because in the first order this gives a distribution of sources which is identical, except for an added factor \ , to the zero-order distribution of sources in the case of input N. This is the physical content of matrix relations 5 and 6 in Display 7.2. However, the emerging intensity to direction μ = 0, which equals the source function in the surface layer, was recorded in Table 12, because in many applications it is desirable to see how the intensity changes over the full range of angles.

It may be noted that the matrix products with U or Ν as a factor have the character of differently weighted averages over the interval 0 < μ < 1. Hence on any one line of the table the numbers in the last two columns should be between the maximum and minimum value occurring for different μ along the . same line. The average by weight of U has a stronger bias towards steep angles of incidence (μ near 1) while the average by the weight of Ν is biased more towards the grazing angles (μ near 0).

Imperfections in the numerical integration scheme, like a too coarse grid, make themselves felt most strongly in the bimoment N R N . For that reason we quote here two numbers with increased accuracy (van de Hulst and Grossman, 1968):

b = 1, NRN = 0.6445419 ± 5 (estimated uncertainty); b = oo, NRN = In H(a, 1) = 1.0674004 (Section 8.3.3).

The zero-order term, which occurs only in the transmitted light at μ = μ 0, is indicated by the word " Peak " in the proper column. This word is not repeated in each line of the same column, which gives Τ for various values of a. Hence these lines show Tdi{{, without the zero-order part. However, in the moments and bimoments of Γ, i.e., in the last two columns or in the last two blocks, the zero-order part is not singular any more. It then has been properly evaluated and recorded in the line "zero-order," and this zero-order part is included on the lower lines in the moments and bimoments of Τ for different values of a.

As is well known (Sections 4.4 and 9.6.1), the functions R and Τ can also be expressed in terms of X and Y functions. Display 9.1 gives full detail, including moments and bimoments.

DIS

PL

AY

9.

1

Ref

lect

ion

and

Tra

nsm

issi

on

Fu

nct

ion

s,

Ex

pre

ssed

in

X

and

Y F

un

ctio

ns

and

thei

r M

om

en

ts

a ar

bit

rary

, b

finite

Z

ero

th

and

first

-ord

er

term

s" i

n a

μ, μ

0 ar

bit

rary

μ

=

0 μ,

μ 0

arb

itra

ry

μ =

0

Par

t 1

Ref

lect

ion

Fu

nct

ion

R(a,

b,

μ,

μ 0) R

=

° ίΧ

(μ)Χ

(μ0)

~ Υ{

μ)Υ(

μ 0)1

^~

XQ*o

) Λ

( °

Λ 0

~

β~^~

^)

~ 4(

μ +

μ 0) 4μ

0 4(

μ +

μ 0)

4μ0

Dia

go

nal

va

lue,

μ0

=

μ:

R =

~[

Χ2(μ)

- 7

2(μ)]

οο

-^

(1

- e~

2^)

oo

ομ

8μ

Mo

me

nt

NR

=

RN

=

~ \Χ

(μ)

- 1]

oo

~

F^b,

-

^ 1

Τ

u-

- \Γ

·/

- J

.

2μ

4μ

Κα,

b,

μ)

UR

=

RU

=

1 -

(1

- ^

αα0)*

(μ)

- ±α

β 0 Υ

(μ)

±α<

χ 0 1

2μ

'Ύ'

μ/

2

194 9 Isotropic Scattering, Finite Slabs

Bim

om

ent

r 0(a,

b)

r,(fl

, b)

Fu

nct

ion

T(a,

b,

μ,

μ 0)

Mo

men

t

t(a,

b,

μ)

Bim

om

ent

t 0(a>

b

)

h (a

, b)

NRN

=

ia*

1 NR

U

=

URN

=

a0

- 1

URU

=

1 -

2a!

+

α(α

0αι

- β 0βι

)

Par

t 2

Tra

nsm

issi

on

iaGl2(b

)

aG22(b

)

T =

ΙΧ

(β)Υ

(μ0)

~ Υ{

μ)Χ{

μοϊ]

4(/I 0

-

μ)

Dia

go

nal

va

lue*

, μ 0

=

μ

Τ=

PE

AK

+

- Χ2(μ

) —

4

θμ

ΝΤ=

ΤΝ

=

-Υ(μ

) 2μ

UT=

TU

=

&β 0Χ(

μ)

+ (1

-

>0)

7(

μ)

jVT

/V

=

NT

L/

= l/

TN

=

/i0

ϋΤϋ

=

2β1

Λ-α{

β 0α,

-α

0βχ)

4μ(

>Χ"Ο

) 4

(μ0

- μ

)

PE

AK

+

Α6

2μ

Αμ

Mo

e~

m +

α

F 2μ

'

Κ)

i£,(

fc)

+ ie

Gi,

(6)

£ 2(fe)

+

|aG

' 12

(i>

)

2£3(6

) +

aG' 2

2(6

)

4μ0

έ«£ι

(ί>

)

ία£2(6

)

" T

he

zero

th

an

d fir

st-o

rder

te

rms

are

exp

ress

ed

in f

unct

ions

de

fine

d in

Ch

apte

r 2.

No

zero

ord

er

in

refl

ecti

on.

"Th

e sy

mb

ol

"P

EA

K"

rep

rese

nts

T 0

=

(1/2

μ)δ(

μ -

μ^β'

^".

9.1 Reflection and Transmission 195


9.1.2 Terms of Order 0 and 1

The analytical expressions of the zero-order contributions are given in Display 9.1, together with the first-order terms in reflection and transmission. They can all be expressed in terms of the functions E, F, G, and G' discussed in Chapter 2.

The fact that the first-order term of the diffusely reflected light follows a simple formula has seduced many authors to compute this term precisely and leave the higher orders to plausible speculation. Two examples of this procedure may be quoted.

1. The classical law of Lommel-Seeliger, which is nothing but the first-order term for b = oo with an arbitrarily adapted coefficient, reads in our notation

ϋ(μ, μ0) = c o n s t a n t / ^ + μ0)

For half a century this law was regarded as a reasonable assumption about the diffuse reflection by planetary surfaces (cf. Section 18.1.4).

2. Lyot (1929) made the tentative suggestion that the polarization of Venus could be explained in terms of first-order scattering which should be uniformly diluted by unpolarized higher order light. This suggestion remained unchallenged and unimproved for over 35 years (see Section 18.1.5).

T A B L E 1 3

R a t i o of T o t a l t o F i r s t - O r d e r R e f l e c t i o n , I s o t r o p i c S c a t t e r i n g

1 . 0 1 . 0 1 . 0 0 . 5 0 . 5 U U U N N U U N 1 . 0 0 . 5 0 . 0 0 . 5 0 . 0 1 . 0 0 . 5 0 . 0 1 . 0 0 . 5 U Ν Ν

b = 0 . 0 3 1 2 5

α = 0 . 2 0 1 . 0 1 4 0 . 4 C 1 . 0 2 8 ύ . 6 0 1 . 0 4 3 Ο.ΘΟ 1 . 0 5 8 0 . 9 0 1 . 0 6 6 0 . 9 5 1 . 0 7 0 0 . 9 9 1 . 0 7 3 1 . 0 Û 1 . 0 7 4

1 . 0 1 4 1 . 0 1 2 1 . 0 1 4 1 . 0 2 8 1 . 0 2 5 1 . C 2 8 1 . 0 4 3 1 . 0 3 8 1 . 0 4 3 1 . 0 5 8 1 . 0 5 1 1 . 0 5 8 1 . 0 6 6 1 . 0 5 8 1 . 0 6 6 1 . 0 7 0 1 . 0 6 1 1 . 0 7 0 1 . 0 7 3 1 . 0 6 4 1 . 0 7 3 1 . 0 7 4 1 . 0 6 5 1 . 0 7 4

1 . 0 1 2 1 . 0 1 4 1 . 0 1 4 1 . 0 2 4 1 . 0 2 8 1 . 0 2 8 1 . 0 3 7 1 . 0 4 3 1 . 0 4 3 1 . 0 5 0 1 . 0 5 8 1 . 0 5 8 1 . 0 5 7 1 . 0 6 6 Ι . Ο ο ο 1 . 0 6 0 1 . 0 7 0 1 . 0 7 0 1 . 0 6 3 1 . 0 7 3 1 . 0 7 3 1 . 0 6 4 1 . 0 7 4 1 . 0 7 4

1 . 0 1 2 1 . 0 1 4 1 . 0 1 4 1 . 0 2 4 1 . 0 2 8 1 . 0 2 8 1 . 0 3 6 1 . 0 4 3 1 . 0 4 3 1 . 0 4 8 1 . 0 5 8 1 . 0 5 8 1 . 0 5 5 1 . 0 6 6 1 . 0 6 5 1 . 0 5 8 1 . 0 6 9 1 . 0 6 9 1 . 0 6 1 1 . 0 7 3 1 . 0 7 2 1 . 0 6 1 1 . 0 7 3 1 . 0 7 3

ι .014 1.014 1 .013 1 .028 1 .028 1 .C27 ι .043 1 .043 1.040 ι .058 1.058 ι . 055 ι .066 1.065 1 .0(3 2 1 .070 1.069 ι .066 ι .073 1 .072 ι .068 1 .074 1.073 ι .069

b * 0 . 0 6 2 5 0

α = 0 . 2 0 1 . 0 2 4 0 . 4 0 1 . 0 4 9 0 . 6 0 1 . 0 7 5 0 . 8 0 1 . 1 0 2 0 . 9 0 1 . 1 1 7 0 . 9 5 1 . 1 2 4 0 . 9 9 1 . 1 3 0 I . 0 0 1 . 1 3 1

b = 0 . 1 2 5 0 0

1. 024 1 .020 1. 024 1. 049 1.041 1.049 1. 075 ι .063 1. 075 1. 102 ι .087 1. 102 ι . 117 ι .099 1. 117 1. 124 1 .105 1. 124 1. 130 ι .110 1. 130 1. 131 ι .111 1 . 131

1 . 0 2 C 1 . 0 2 4 1 . 0 2 4 1 . 0 4 0 1 . 0 4 9 1 . 0 4 9 1 . 0 6 2 1 . 0 7 5 1 . 0 7 5 1 . 0 8 4 1 . 1 0 2 1 . 1 0 2 1 . 0 9 6 1 . 1 1 7 1 . 1 1 6 1 . 1 0 2 1 . 1 2 4 1 . 1 2 4 1 . 1 0 7 1 . 1 3 0 1 . 1 3 0 1 . 1 0 8 1 . 1 3 1 1 . 1 3 1

1 . 0 1 9 1 . 0 2 4 1 . 0 2 3 1 . 0 3 8 1 . 0 4 8 1 . 0 4 8 1 . 0 5 8 1 . 0 7 4 1 . 0 7 4 1 . 0 8 0 1 . 1 0 1 I . 1 0 1 1 . 0 9 1 1 . 1 1 6 1 . 1 1 5 1 . 0 9 7 1 . 1 2 3 1 . 1 2 2 1 . 1 0 1 1 . 1 2 9 1 . 1 2 8 1 . 1 0 2 1 . 1 3 0 1 . 1 3 0

ι . 024 1 .023 1.022 1. 049 1.048 1.045 1. 075 1.073 1.069 1. 102 1. 100 1.094 1. 116 1.114 1.107 1. 124 1. 122 1.114 1. 129 1.127 1.119 1. 131 1.129 1. 120

α = 0 . 2 0 1 . 0 4 0 1 . 0 3 9 1 . 0 3 2 1 . 0 3 9 1 . 0 3 0 1 . 0 3 9 1 . 0 3 9 1 . 0 2 8 1 . 0 3 9 1 . 0 3 8 1 . 0 3 9 1 . 0 3 8 1 . 0 3 5 0 . 4 0 0 . 6 0 0 . 8 0 0 . 9 0 0 . 9 5 0 . 9 9 1 . 0 0

1 . 0 8 2 1 . 1 2 9 1 . 1 7 9 1 . 2 0 6 1 . 2 2 0 1 . 2 3 2 1 . 2 3 5

1 . 0 8 2 1 . 1 2 9

1 . 0 6 6 1 . 1 0 3

1 . 1 7 9 1 . 1 4 3 1 . 2 0 6 1 . 1 6 5 1 . 2 2 0 1 . 2 3 2

1 . 1 7 6 1 . 1 8 5

1 . 2 3 5 1 . 1 8 8

1.082 1.128 1.179 1.206 1.220 1.231 1.234

1 . 0 6 3 1 . 0 9 8 1 . 1 3 6 1 . 1 5 7 I . 1 6 7 1 . 1 7 6 I . 1 7 8

0 8 2 1 . 1 2 8 1 . 1 7 9 1 . 1 7 8 1 . 2 0 6 1 . 2 0 5 1 . 2 2 0 1 . 2 1 9 1 . 2 3 1 1 . 2 3 1 1 . 2 3 4 1 . 2 3 3

1 . 0 8 2 1 . 0 5 8 1 . 1 2 8 1 . 0 9 1

1 . 1 2 6 I . 1 4 5 1 . 1 5 5 1 . 1 6 3 1 . 1 6 5

1 . 0 8 1 I . 1 2 7 1 . 1 7 6 1 . 2 0 3 1 . 2 1 7 1 . 2 2 8 1 . 2 3 1

l . O d O I . 1 2 5 1 . 1 7 5 1 . 2 0 1 1 . 2 1 5 1 . 2 2 6 1 . 2 2 9

. 0 8 2 . 1 2 8 . 1 7 8 . 2 0 4 . 2 1 8 . 2 3 0 . 2 3 3

1 . 0 7 9 1 . 0 7 3 1 . 1 2 4 1 . 1 1 3 1 . 1 7 3 1 . 1 5 8 1 . 1 9 9 1 . 1 8 2 1 . 2 1 2 1 . 1 9 4 1 . 2 2 3 1 . 2 0 4 1 . 2 2 6 1 . 2 0 6


T A B L E U (continued)

1 . 0 i . O

1 . 0 0 . 5

1 . 0 0 . 0

0 . 5 0 . 5

0 . 5 0 . 0

U 1 . 0

υ 0 . 5

U 0 . 0

Ν 1 . 0

Ν 0 . 5

b = 0 . 2 5 0 0 0

α = 0 · 2 0 1 . 0 6 3 1 . 0 6 3 1 . 0 4 7 1 . 0 6 3 1 . 0 4 3 0 . 4 0 0 . 6 0 0 . 8 0 0 . 9 0 0 . 9 5 0 . 9 9 I . 0 0

1 . 1 3 5 1 . 1 3 5 1 . 2 1 8 1 . 2 1 7 1 . 3 1 3 1 . 3 6 7 1 . 3 9 5 1 . 4 1 9 1 . 4 2 5

1 . 3 1 2 1 . 3 6 5 1 . 3 9 4 1 . 4 1 7 1 . 4 2 3

1 . 0 9 9 1 . 1 3 4 1 . 0 9 1 1 . 1 6 C 1 . 2 1 5 1 . 1 4 6 1 . 2 2 9 1 . 3 0 9 1 . 2 6 8 1 . 1 . 2 8 9 1 . 1 . 3 0 6 1 . 4 1 4

1 . 2 0 9 . 3 6 2 1 . 2 4 4 . 3 9 0 1 . 2 6 3

1 . 2 7 8 1 . 3 1 0 1 . 4 2 0 1 . 2 8 2

1 . 0 6 3 1 . 0 & 2 1 . 0 3 9 1 . 0 6 1 1 . 0 6 0 1 . 1 3 4 1 . 1 3 3 1 . 0 8 2 1 . 1 3 1 1 . 1 2 8 1 . 2 1 6 1 . 2 1 4 1 . 1 3 2 1 . 2 1 1 1 . 2 0 6 1 . 3 1 1 1 . 3 0 8 1 . 1 8 9 1 . 3 0 3 1 . 2 9 6 1 . 3 6 4 1 . 3 6 0 1 . 2 2 1 1 . 3 5 5 1 . 3 4 6 1 . 3 9 2 1 . 3 8 8 1 . 2 3 8 1 . 3 8 2 1 . 3 7 3 1 . 4 1 6 1 . 4 1 1 1 . 2 5 2 1 . 4 0 5 1 . 3 9 6 1 . 4 2 2 1 . 4 1 7 1 . 2 5 5 1 . 4 1 1 1 . 4 0 1

1 . 0 6 2 1 . 0 5 9 1 . 0 5 2 1 . 1 3 2 1 . 1 2 6 1 . 1 1 2 1 . 2 1 2 1 . 2 0 2 1 . 1 7 9 1 . 3 0 5 1 . 2 9 1 1 . 2 5 7 1 . 3 5 8 1 . 3 4 1 1 . 3 0 1 1 . 3 8 5 1 . 3 6 7 1 . 3 2 4 1 . 4 0 8 1 . 3 8 9 1 . 3 4 3 1 . 4 1 4 1 . 3 9 4 1 . 3 4 8

b = 0 . 5 0 0 0 0

a = 0 , 0 . 0 , 0 , 0 . 0 . 0 . 1 .

2 0 4 0 6 0 8 0 9 0 9 5 9 9 0 0

1 . 0 9 6 1 . 2 1 3 1 . 3 5 8 1 . 5 4 3 1 . 6 5 6 1 . 7 1 9 1 . 7 7 4 1 . 7 8 8

0 9 4 1 . 0 6 2 2 0 9 1 . 1 3 7

1 . 2 2 9 1 . 3 4 5 1 . 4 1 6 1 . 4 5 5

3 5 2 5 3 4 6 4 5 7 0 7

1 . 0 9 2 1 . 0 5 4 1 . 2 0 3 1 . 1 1 8 1 . 3 4 2 1 . 5 1 8 1 . 6 2 6 1 . 6 8 6

1 . 7 6 0 1 . 4 8 8 1 . 7 7 4 1 . 4 9 7

1 . 1 9 6 1 . 2 9 4 1 . 3 5 2 1 . 3 8 5

1 . 7 3 7 1 . 4 1 3 1 . 7 5 0 1 . 4 2 0

1 . 0 9 4 1 . 0 9 1 1 . 2 0 8 1 . 2 0 2 1 . 3 5 0 1 . 3 3 9 1 . 5 3 1 1 . 5 1 4 1 . 6 4 1 1 . 7 0 3 1 . 7 5 6

1 . 6 2 0 1 . 6 8 0 1 . 7 3 1

1 . 7 7 0 1 . 7 4 4

1 . 0 4 8 1 . 0 9 0 1 . 0 8 6 1 . 1 0 6 1 . 1 9 9 1 . 1 8 9 1 . 1 7 6 1 . 3 3 5 1 . 3 1 7 1 . 2 6 2 1 . 5 0 7 1 . 4 7 9 1 . 3 1 5 1 . 6 1 2 1 . 5 7 8 1 . 3 4 4 1 . 6 7 1 1 . 6 3 3 1 . 3 6 8 1 . 7 2 1 1 . 6 8 0 1 . 3 7 5 1 . 7 3 4 1 . 6 9 2

1 . 0 9 0 1 . 0 8 4 1 . 0 7 2 1 . 2 0 0 i . 1 8 5 1 . 1 5 8 1 . 3 3 6 1 . 3 1 0 1 . 2 6 4 1 . 5 0 9 1 . 4 6 9 1 . 3 9 7 1 . 6 1 4 1 . 5 6 6 1 . 4 7 7 1 . 6 7 3 1 . 6 2 0 1 . 5 2 2 1 . 7 2 3 1 . 6 6 6 1 . 5 6 0 1 . 7 3 6 1 . 6 7 8 1 . 5 7 0

b = 1 . 0 0 0 0 0

= 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 0 . 9 0 3 . 9 5 0 . 9 9 1 . 0 0

1 .13<> 1 . 1 2 6 1 . 0 7 4 1 . 1 1 7 1 . 0 6 0 1 . 3 0 7 1 . 2 9 2 1 . 1 6 8 1 . 2 7 0 1 . 1 3 4 1 . 5 4 8 1 . 5 2 0 1 . 2 9 4 1 . 4 7 7 1 . 2 3 2 1 . 9 0 6 1 . 8 5 6 1 . 4 7 4 1 . 7 8 1 1 . 3 6 7 2 . 1 * 9 2 . 0 9 2 1 . 5 9 7 1 . 9 9 2 1 . 4 5 8

2 3 6 1 . 6 7 2 2 . 1 2 1 1 . 5 1 2 2 . 3 1 3 2 . 4 5 4 2 . 3 6 7 1 . 7 3 9 2 . ^ 3 8 1 . 5 6 1

1 . 1 2 6 1 . 1 1 7 1 . 2 9 2 1 . 2 6 9 1 . 5 2 0 1 . 4 7 8 1 . 8 5 6 1 . 7 8 1 2 . 0 9 2 2 . 2 3 6

1 . 9 9 3 2 . 1 2 2

2 . 4 9 2 2 . 4 0 3 1 . / 5 7 2 . 2 6 9 1 . 5 7 4 2 . 3 6 6 2 . 2 3 9 2 . 4 0 3 2 . 2 7 1

1 . 0 5 4 1 . 1 2 1 1 . 2 0 9 1 . 3 3 0 1 . 4 1 2 2 . 0 0 7 1 . 8 8 1 1 . 4 6 1 2 . 1 3 8 1 . 9 9 3 1 . 5 0 5 2 . 2 5 8 2 . 0 9 5 1 . 5 1 6 2 . 2 9 0 2 . 1 2 2

1 . 1 1 8 1 . 1 0 6 1 . 2 7 2 1 . 2 4 3 1 . 4 8 2 1 . 4 2 8 1 . 7 9 1 1 . 6 9 5

1 . 1 1 7 1 . 1 0 5 1 . 0 8 6 1 . 2 6 9 1 . 2 4 0 1 . 1 9 6 1 . 4 7 7 1 . 4 2 4 1 . 3 4 3 1 . 7 8 0 1 . 6 8 8 1 . 5 5 2 1 . 9 9 2 1 . 8 7 2 1 . 6 9 5 2 . 1 2 1 1 . 9 8 3 1 . 7 8 2 2 . 2 3 9 2 . 0 8 4 1 . 8 6 0 2 . 2 7 0 2 . 1 1 2 1 . 8 8 1

b = 2 . 0 0 0 0 0

= 0 . 2 0 1 . 1 5 7 1 . 1 4 2 l . 0 7 d 1 . 1 2 6 1 . 0 6 1 1 . 1 4 5 1 . 1 2 7 0 . 4 0 1 . 3 d O I . 3 4 0 1 . 1 8 1 1 . 2 9 6 1 . 1 3 9 1 . 3 4 7 1 . 3 0 1 0 . 6 0 1 . 7 2 2 1 . 6 3 8 1 . 3 2 9 1 . 5 4 4 1 . 2 4 4 1 . 6 5 2 1 . 5 5 6 0 . 8 0 2 . 3 2 8 2 . 1 5 2 1 . 5 6 7 1 . 9 5 6 1 . 4 0 4 2 . 1 8 3 1 . 9 8 3 0 . 9 0 2 . 8 5 t > 2 . 5 9 0 1 . 7 6 1 2 . 2 9 6 1 . 5 2 7 2 . 6 3 8 2 . 3 3 9 0 . 9 5 3 . ^ 3 6 2 . 9 0 2 1 . 8 9 5 2 . 5 3 4 1 . 6 1 0 2 . 9 6 3 2 . 5 8 9 0 . 9 9 3 . 6 3 1 3 . 2 2 3 2 . 0 3 1 2 . 7 7 6 1 . 6 9 2 3 . 2 9 9 2 . 8 4 5 I . 0 0 3 . 7 4 7 3 . 3 1 8 2 . 0 7 1 2 . 8 4 7 1 . 7 1 5 3 . 3 9 8 2 . 9 1 9 1 . 6 6 2 3 . 1 0 2

1 . 0 5 6 1 . 1 3 2 1 . 1 1 3 1 . 1 2 6 1 . 3 1 5 1 . 2 6 5 1 . 2 2 3 1 . 5 8 7 1 . 4 8 4 1 . 3 7 1 2 . 0 5 3 1 . 8 4 2 1 . 4 8 5 2 . 4 4 8 2 . 1 3 5 1 . 5 6 2 2 . 7 2 8 2 . 3 3 8 1 . 6 3 9 3 . 0 1 7 2 . 5 4 6

2 . 6 0 6

1 . 1 2 9 1 . 1 1 4 1 . 0 9 2 1 . 3 0 6 1 . 2 6 6 1 . 2 1 3 1 . 5 6 7 1 . 4 8 9 1 . 3 8 4 2 . 0 0 9 1 . 8 5 6 1 . 6 5 9 2 . 3 8 1 2 . 1 5 9 1 . 8 8 1 2 . 6 4 2 2 . 3 7 1 2 . 0 3 4 2 . 9 1 1 2 . 5 8 7 2 . 1 8 9 2 . 9 9 0 2 . 6 5 0 2 . 2 3 4

b = 4 . 0 0 0 0 0

α = 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 0 . 9 0 0 . 9 5 0 . 9 9 1 . 0 0

b = 8 . 0 C 0 0 0

1 . 1 6 3 1 . 1 4 5 1 . 0 7 9 1 . 1 2 6 1 . 0 6 1 1 . 4 0 0 1 . 3 4 8 1 . 1 8 3 1 . 2 9 8 1 . 1 3 9 1 . 7 8 1 1 . 6 6 3 1 . 3 3 5 1 . 5 5 2 1 . 2 4 6 2 . 5 3 6 2 . 2 5 2 1 . 5 9 6 1 . 9 9 5 1 . 4 1 3 3 . 3 3 9 2 . 8 4 3 1 . 8 3 9 2 . 4 0 8 1 . 5 5 3 4 . 0 5 / 3 . 3 5 1 2 . 0 3 8 2 . 7 4 4 1 . 6 5 9 5 . 0 1 2 4 . 0 0 7 2 . 2 8 7 3 . 1 6 1 1 . 7 8 5 5 . 3 5 0 4 . 2 3 6 2 . 3 7 2 3 . 3 0 3 1 . 8 2 7

1 . 1 4 8 1 . 1 2 8 1 . 0 5 6 1 . 3 5 8 1 . 3 0 4 1 . 1 2 7 1 . 6 8 7 1 . 5 6 8 1 . 2 2 5 2 . 3 1 4 2 . 0 3 8 1 . 3 8 1 2 . 9 5 6 2 . 4 8 6 1 . 5 1 5 3 . 5 1 6 2 . 8 5 7 1 . 6 2 0 4 . 2 4 7 3 . 3 2 5 1 . 7 4 5 4 . 5 0 4 3 . 4 8 6 1 . 7 8 7

1 . 1 3 5 1 . 1 1 4 1 . 3 2 3 1 . 2 6 7 1 . 6 1 2 1 . 4 9 1 2 . 1 5 1 1 . 8 7 9 2 . 6 8 9 2 . 2 3 6 3 . 1 5 2 2 . 5 2 7 3 . 7 5 1 2 . 8 8 8 3 . 9 6 0 3 . 0 1 1

1 . 1 3 1 1 . 1 1 5 1 . 3 1 2 1 . 2 7 0 1 2 2 2 3

1 . 1 .

5 8 6 1 . 5 0 1 1 . 0 8 4 1 . 9 0 7 1 . 6 8 9 5 7 0 2 . 2 9 1 1 . 9 6 2

2 . 6 0 9 2 . 3 . 0 1 0 2 .

9 8 1 5 0 5

. 0 9 2

. 2 1 5 . 3 9 1

. 1 8 2 . 4 5 5

3 . 6 8 6 3 . 1 4 8 2 . 5 4 7

a = 0 . 2 0 1 . 1 6 3 1 . 1 4 5 1 . 0 7 9 1 . 1 2 6 1 . 0 6 1 0 . 4 0 1 . 4 0 0 1 . 3 4 8 1 . 1 8 3 1 . 2 9 8 1 . 1 3 9 0 . 6 0 1 . 7 8 3 1 . 6 6 4 1 . 3 3 5 1 . 5 5 2 1 . 2 4 6 0 . 8 0 2 . 5 5 4 2 . 2 5 9 1 . 5 9 8 1 . 9 9 7 1 . 4 1 3 0 . 9 0 3 . 4 2 2 2 . 8 7 8 1 . 8 5 0 2 . 4 2 1 1 . 5 5 6 0 . 9 5 4 . 3 0 2 3 . 4 6 7 2 . 0 7 5 2 . 7 9 2 1 . 6 7 1 0 . 9 9 5 . 8 6 8 4 . 4 4 6 2 . 4 3 1 3 . 3 6 1 1 . 8 3 5 1 . 0 0 6 . 6 6 0 4 . 9 2 1 2 . 5 9 9 3 . 6 2 1 1 . 9 0 6

1 . 1 4 8 1 . 1 2 8 1 . 3 5 8 1 . 3 0 4 1 . 6 8 8 1 . 5 6 9 2 . 3 2 4 2 . 0 4 1 3 . 0 0 4 2 . 5 0 4 3 . 6 6 7 2 . 9 2 2 4 . 7 9 8 3 . 5 8 1 5 . 3 5 6 3 . 8 9 0

1 . 0 5 6 1 . 1 3 5 1 . 1 1 4 1 . 1 2 7 1 . 3 2 3 1 . 2 6 7 1 . 2 2 5 1 . 6 1 3 1 . 4 9 2 1 . 3 8 2 2 . 1 5 8 1 . 8 8 1 1 . 5 1 9 2 . 7 2 5 2 . 2 4 9 1 . 6 3 4 3 . 2 6 6 2 . 5 7 2 1 . 8 0 2 4 . 1 7 2 3 . 0 6 9 1 . 8 7 7 4 . 6 1 4 3 . 2 9 9

1 . 1 3 1 1 . 1 1 5 1 . 0 9 2 1 . 3 1 2 1 . 2 7 0 1 . 2 1 5 1 . 5 8 6 1 . 5 0 1 1 . 3 9 1 2 . 0 8 9 1 . 9 1 0 1 . 6 9 1 2 . 5 9 6 2 . 3 0 9 1 . 9 7 2 3 . 0 6 6 2 . 6 6 9 2 . 2 1 8 3 . 8 3 1 3 . 2 4 0 2 . 5 9 8 4 . 1 9 6 3 . 5 1 0 2 . 7 7 4

= 00

a = 0 . 0 . 0 . 0 . 0 0. 0 1 .

2 0 4 0 6 0 8 0 9 0 9 5 9 9 0 0

1 . 1 6 3 1 . 1 4 5 l . < » 0 0 1 . 3 4 8 1 . 7 8 3 2 . 5 5 4 3 . 4 2 3 4 . ^ 1 4

1 . 6 6 4

2 . 8 7 9 3 . 4 7 3

1 . 0 7 9 1 . 1 2 6 1 . 0 6 1 1 . 1 8 3 1 . 2 9 8 1 . 1 3 9 1 . 3 3 5 1 . 5 5 2 1 . 2 4 6 1 . 5 9 d 1 . 9 9 7 1 . 4 1 3

6 . 1 1 5 4 . 5 7 1 8 . 4 5 5 5 . 8 5 3

1 . 8 5 0 2 . 4 2 1 2 . 0 7 / 2 . 7 9 5 ^ . 4 7 3 3 . 4 1 7 2 . 9 0 8 4 . 0 5 1

1 . 5 5 6 1 . 6 7 2 1 . 8 4 9 2 . 0 1 3

1 . 1 4 8 1 . 1 2 8 1 . 3 5 8 1 . 3 0 4 1 . 6 8 8 1 . 5 6 9 2 . 3 2 4 2 . 0 4 1 3 . 0 0 5 2 . 5 0 4 3 . 6 7 4 2 . 9 2 5 4 . 9 5 6 3 . 6 5 4 6 . 5 1 8 4 . 4 3 8

1 . 0 5 6 1 . 1 3 5 1 . 1 1 4 1 . 1 2 7 1 . 3 2 3 1 . 2 6 7

1 . 6 1 3 1 . 4 9 2 2 . 1 5 8 1 . 8 8 1

1 . 5 1 9 2 . 7 2 5 2 . 2 4 9 1 . 6 3 5 3 . 2 7 1 2 . 5 7 5 1 . 8 1 8 4 . 2 9 3 3 . 1 2 1 2 . 0 0 0 5 . 5 0 5 3 . 6 8 7

2 2 5 3 8 2

1 . 1 3 1 1 . 1 1 5 1 . 0 9 2 1 . 3 1 2 1 . 2 7 0 1 . 2 1 5 1 . 5 8 6 1 . 5 0 1 1 . 3 9 1 2 . 0 8 9 1 . 9 1 0 1 . 6 9 1 2 . 5 9 6 2 . 3 0 9 1 . 9 7 2 3 . 0 7 0 2 . 6 7 2 2 . 2 2 0 3 . 9 2 3 3 . 3 0 6 2 . 6 3 9 4 . 8 8 8 4 . 0 0 0 3 . 0 8 0


In view of these and other applications, it was found interesting to extract the actual ratios of the total diffuse reflection to its first-order term from Table 12. The ratios thus obtained are presented in Table 13, which shows that with small and intermediate values of the albedo a, 10 to 5 0 % correction of the first-order term suffices to give the total. Larger albedos and thick layers make the required correction factor larger. The maximum factor (for isotropic scattering) is 8.455. Understandably, the factors are smallest for grazing angles ( μ0 or μ = 0) and for distributions in which near-grazing angles have a relatively heavy weight (N). The correction factor is identically 1 for μ — μ0 = 0. This theorem was derived (or cited?) by Minnaert in a lecture given about 1938 and holds for an arbitrary primary scattering pattern. The ratio is also 1 for the combinations (N, 0) or (0, N).

9.1.3 Some Illustrations

As an example of the type of data which may be read from Table 12, we present in Fig. 9.1 a graph of the function

A*(a,b) =URU

in the entire subcritical domain. This important quantity, often called the spherical albedo (Section 12.2.3 and 18.1.2) and denoted by r* in Section 4.5.4, plays a role in the formula for the influence of ground scattering in a planetary atmosphere. The data underlying this graph are the following:

F o r a < 1, all b: Table 12. For a = aCTit = [ ihi&XT

1. all b: Table 4 (Section 7.4.1).

For 1 < a < a c r i t, all b: interpolation formulas of the form

ri = alcjil - ηγα) + c2/ ( l - η2α) + c3/ ( l - η3α)~]

where cu c 2, and c 3 depend on b. This was done after the main program was completed. It would have been simpler to obtain output of the main program also for a > 1.

A similar graph for the product UR{\\ the albedo for perpendicular incidence, is given in Fig. 9.2. This same function was used as an illustration in Fig. 4.2 to show the limits of usefulness of low-order scattering and of thick-layer theory. The upper right-hand corner of Fig. 9.2 is shown in greater detail and with an explanation of the asymptotic behavior in Fig. 19.11.

9.1.4 Thick Layers

A "very thick layer" is a layer inside which a diffusion domain exists, i.e., a layer thick enough to accommodate a range of depths in which the formulas for unbounded media hold to sufficient accuracy. This is the general definition


0 . 2 h

O I I I I I I ι ι Ι 0 0.1 0 .25 0 . 5 1 2 4 10 00 b

F i g . 9.1. B i m o m e n t o f t h e r e f l e c t i o n f u n c t i o n URU, o r s p h e r i c a l a l b e d o , f o r a s l a b o f t h i c k n e s s

b a n d s i n g l e - s c a t t e r i n g a l b e d o a. T h e c u r v e m a r k e d oo s h o w s t h e c r i t i c a l l i m i t .

b

F i g . 9.2. P l a n e a l b e d o for p e r p e n d i c u l a r i n c i d e n c e UR( 1 ) for a s l a b w i t h t h i c k n e s s b a n d s i n g l e -s c a t t e r i n g a l b e d o a.

DIS

PL

AY

9.

2

Ref

lect

ion

an

d T

ran

smis

sio

n b

y V

ery

Th

ick

Lay

ers

wit

h Is

otr

op

ic

Sca

tter

ing

0

No

nc

on

serv

ati

ve

, a

<

1 C

on

serv

ativ

e,

a =

1

aH(a

, μ)

Η(α

, μ 0)

Ref

lect

ion

fun

ctio

n R(

a,

b, μ

, μ 0)

=

—

fT(a

, b,

μ,

μ 0)

Tra

nsm

issi

on

fun

ctio

n T(

a,

b, μ

, μ 0)

••

Mo

me

nts

Bim

om

en

ts

akdf

H(a

, μ)

Η(α

, μ 0)

H(a

, μ)

-

1 kd

fH(a

, μ)

RN

=

—-^

r-

fTN

, TN

J

h

2μ

1 -

\ιμ

2tdf

H(a

, μ)

RU

=

1 -

tH(a

, μ)

-

fTU

, TU

=

1

—

kμ

NRN

=

ia

* !(

α)

- 2k

df

2/a,

NTN

=

2k

df/a

URN

=

a 0(a

) -

1 -

4tdf

2/a,

UTN

=

4t

df/a

URU

=

1

- 2t

OL x(a

) -

&t2df

2/ka

UTU

=

St

2df/k

a

Η(\,

μ)Η

(Ιμ 0)

4(μ

+

μ 0)

=

Τ(1,

ϊμ,μ0)

Η(1

,μ)-

1

R(l,

b, μ

, μ 0)

Η(Ι

μ)Η

(1,μ0)

Φ

+ 2q

J

Η(1

, μ)

2μ

RN

=

ΤΝ

=

1 -

RU

=

TU

=

1.0

67

4 -

NRN

=

N

TN

=

1 -

URN

=

U

TN

1 -

URU

=

U

TU

=

2(b

+ 2q

J

H(l

, μ)

1

b +

2qx

2/yf

i

b +

2q

œ

4/3

b +

2<

? κ a

Sy

mb

ols

: t

= (1

-

a)1

/2

; a*

.(

A),

see

Sec

tio

n 8.

3.3;

k,

I, se

e T

ab

le

11 (

Sec

tio

n 8

.6.1

);/

= le

xp(-

kb);

d

= (1

-

/

2)"



employed in Section 5.3. Using the formulas presented there with the specifications for isotropic scattering in Display 8.1, we obtain the reflection and transmission functions shown in Display 9.2. The moments and bimoments follow simply.

Historically, the equations for thick layers were first derived for isotropic scattering only (Halpern and Luneberg, 1949; Sobolev, 1957; van de Hulst, 1964; van de Hulst and Terhoeve, 1966).

The expressions for a conservative layer (a -> 1, k -> 0), given in the right-hand part of Display 9.2, can be obtained individually from the corresponding expressions at the left side. On each line the factor k in the development

d~l = (b + 2qJ2k + 0(k

2)

cancels out against the factor k in the numerator . Alternatively, the expressions for the conservative case may be derived

directly, starting from the solution in the diffusion domain, which now is a linear (not an exponential) function of τ. The results (Sobolev, 1957; van de Hulst, 1964) are, of course, identical to those obtained here.

Mathematically, the expressions given in Display 9.2 are asymptotic expressions for large b in an approximation in which e~

b can be neglected but

e~kb cannot. The distinction becomes unpractical for a < 0.8, because then k

is too close to 1. We suggest that for a < 0.8 and b > 10, it is justified to neglect e~

kb also, which reduces the equations to those for a semi-infinite layer. However,

in the important range 0.8 < a < 1, in which the medium is only slightly lossy, the expressions of Display 9.2 may be used to full advantage, with factors e~

kb

ranging anywhere from 0 to 1. This situation is illustrated in Fig. 4.2, where the 1 and 5 % error limits of the approximations are shown in one concrete example.

In all equations of Display 9.2, if b -> oo ,wege t / = 0,d = 1 and the equations for semi-infinite atmospheres are retrieved.

EXTRAPOLATION BEYOND A G I V E N (SMALL) V A L U E OF b

Among the many uses to which the formulas of Display 9.2 may be put, the following is quite striking. It often happens that some function of b is available in the published literature for relatively thin layers, say b < 1. To extrapolate from these to b = oo would seem a foolish proposition. The asymptotic formulas, which we know must hold for large ft, may be used to make this into an interpolation, which often is surprisingly secure. Two examples cited from van de Hulst (1964) may illustrate how this can be done.

First example. The transmitted fraction of incident flux i x = U Τ U for a conservative layer exposed to Lambert radiation has the asymptotic form

t, = f/(/3 + 2qJ

for b oo. Hence we plot (in Fig. 9.3) the function i f1 — |ί?. This function must

reach the value iq^ = 1.06567 for b = oo but has already reached the value


F i g . 9 . 3 . E x a m p l e s o f i n t e r p o l a t i o n g r a p h s p e r m i t t i n g a v e r y a c c u r a t e i n t e r p o l a t i o n b e t w e e n

r e s u l t s u p t o o p t i c a l t h i c k n e s s b = 1 a n d t h e k n o w n b e h a v i o r f o r v e r y t h i c k s l a b s .

1.0570 for b = 1, so that interpolation between b = 1 and b = oo presents no problem, especially if we use E2(b) for the abscissa, as we have done in the figure.

Second example. The fraction of the total flux transmitted through the atmosphere if the radiation comes from an isotropic source above the atmosphere is β0 = N TU. The asymptotic expression is

β0 = 2/JÎ{b + 2qJ

The quantity βο1 — h^fe must reach the value y/ïq^ = 1-230 5 fo r b = o o

and ha s alread y reache d th e valu e 1.201 4 fo r b = 1 . Figur e 9. 3 agai n show s that a smoot h interpolatio n i s possible , givin g four-figur e accurac y i n β0 near b = 2 and six-figure accuracy near b = 10.

The method sketched can be used for any function of b for which the asymptotic formula is known. This includes also the gain and its moments given in Display 9.5. Variations on the method are possible. For instance, upon plotting the inverse value of the reflection coefficient r j ~

1 against b~ \ we obtain a curve

for which the thick-layer theory provides the slope of the tangent at the point b'

1 = 0 .

Third example. The idea to extrapolate from numerical values for finite b to those for b = oo is systematically exploited in the method of asymptotic fitting (Section 5.6). The basic formulas, spelled out for isotropic scattering with the symbols of Display 9.2, are

R(a, oo, μ, μ0) = R(a, b, μ, μ0) + f(a, b)T(a, b, μ, μ0)

H(a, μ) = X(a, b, μ) + f(a, b)Y(a, b, μ)

9.2 Fate of Incident Energy 203

T A B L E 14

E x t r a p o l a t i o n t o b = oo, S a m p l e C o m p u t a t i o n

a = 1, μ0 = 0 .70 , μ = 0 .50 a = 0 .95 , μ 0 = 0 .70 , μ = 0 . 50

b = 8 = 16 b = S b = ie

R 0 . 8 6 8 6 6 0 . 9 2 6 8 9 0 . 6 1 1 7 2 0 . 6 1 2 4 1

T 0 . 1 2 6 8 2 0 . 0 6 8 5 7 0 . 0 2 5 5 7 0 . 0 0 1 2 3

/ 1 1 0 . 0 2 7 2 0 . 0 0 1 3

Tf 0 . 1 2 6 8 2 0 . 0 6 8 5 7 0 . 0 0 0 7 0 0 . 0 0 0 0 0

R + Tf 0 . 9 9 5 4 8 0 . 9 9 5 4 6 0 . 6 1 2 4 2 0 . 6 1 2 4 1

R(b = oo) d i r e c t l y f r o m 0 . 9 9 5 4 8 0 . 6 1 2 4 2

H f u n c t i o n s

Normally, we made this step to b = oo both from b = 8 and from b = 16, thus providing an internal check and, in addition, a sensitive check on the accuracy of the results, because tables of H functions and their moments computed by quite different methods are available in the literature. The two examples shown in Table 14 illustrate how this method works out. The values of R and Τ were taken from Table 12; those of k and / from Table 11. The Η functions reproduced in Table 7 (Section 8.3.1) are correct in four decimals, but a more accurate comparison was made by taking the Η values by the equation Η(μ) = 1 + 2μNR(μ) from Table 12 (b = oo).

9.2 FATE OF INCIDENT ENERGY

The energy that is carried into the atmosphere by the incident radiation ends up in three different parts. The values of the transmitted and reflected parts are contained in Table 12; expressions for those parts in terms of X and Y functions and their moments are in Display 9.1. The remaining (third) part is absorbed inside the atmosphere. The formulas for this part are given in Display 9.3. Numerical values obtained from those in Table 12 are presented in Table 15. J

The main features of the final destination of the energy are shown in the diagrams of Fig. 9.4. Along any vertical line in these diagrams we see the division of the total available energy 1 into the three parts mentioned, namely, reflection (below), absorption (middle), and transmission (top). This division is presented as a function of a for four values of b and for four assumptions about the angular dependence of the incident radiation. The zero-order (direct) transmission, which is independent of a, is shown separately by a dashed line. The absorption is 0 in the conservative case (a = 1). The transmission is 0 for

t T a b l e 15 wil l b e f o u n d a t t h e e n d o f t h e c h a p t e r , s t a r t i n g o n p a g e 2 8 2 .


D I S P L A Y 9.3

A b s o r b e d F r a c t i o n of I n c i d e n t F l u x

R a d i a t i o n F r a c t i o n a b s o r b e d in e n t i r e a t m o s p h e r e i n c i d e n t

f r o m M a t r i x n o t a t i o n F u n c t i o n a l n o t a t i o n

0

O n e d i r e c t i o n U - UR-- UT 1 - τ{μ) -Κμ) = 0 -• έ « α 0 -

12αβ0)ίΧ{μ0) - Υ{μ0Κ

N a r r o w s o u r c e 1 - URN - UTN ίο l a y e r = 0 -• iaoc0 -

λ

2αβ0)(*0 - β0) L a m b e r t s u r f a c e 1 - URU UTU ι - η - U

= ( 2 -• aoc0 -- «0ο)(«ι - βι) α D e p e n d e n c e o n a a n d b n o t w r i t t e n .

the semi-infinite atmosphere. In the latter case, a = 0 defines a blackbody (100% absorption).

A few examples may illustrate the use of these tables.

Problem 1. Suppose an energy of 10,000 arbitrary units falls perpendicularly on the surface of an atmosphere with isotropic scattering, albedo a = 0.90, and optical thickness b = 2. What happens to this energy?

Answer: Table 15 (for other angles we would have to go back to Table 12) shows that 1353 units go through without any scattering. To these are added 2212 units going through after one or more scattering, adding up to 3565 for the total transmission; 2819 units are absorbed in the atmosphere and 3616 units are diffusely reflected.

Problem 2. This is the same problem with a diffusely reflecting ground surface which has a reflectivity Ag = 0.20 and reflects according to Lambert 's law.

Answer : Of the 3565 units first hitting the ground, 80% (2852) is immediately absorbed in the ground and 2 0 % (713) is reflected back into the air where, by Table 15, entry U, 29 .7% is absorbed in the atmosphere, 26.6% goes back out into space, and 43.7 % is returned to the ground. This story repeats itself in a geometric progression (Section 4.5.4) with ratio Ag A* = 0.20 χ 0.437 = 0.0874 and sum (1 — AgA*)~

1 = 1.096. The final fate of all incident energy then be

comes

L o s t i n t o s p a c e f r o m A b s o r b e d in A b s o r b e d in

t o p of a t m o s p h e r e a t m o s p h e r e g r o u n d T o t a l

W i t h o u t g r o u n d r e f l e c t i o n Af te r o n e o r m o r e g r o u n d

r e f l e c t i o n s

3 6 1 6 2 0 8

2 8 1 9 2 3 2

2 8 5 2

2 7 3

9 2 8 7

7 1 3

T o t a l 3 8 2 4 3 0 5 1 3 1 2 5 10 ,000

9.2 Fate of Incident Energy

b - 0 . 5

205

b - 1

0 .2

b * o o

0 .2 . 4 .6 0 .2

1 0 0 %

K e y

- 0-ORDER > transmission

— HIGHER order absorption —

- reflection

0 — a 1

F i g . 9.4. F a t e o f i n c i d e n t e n e r g y f o r a s l a b w i t h i s o t r o p i c s c a t t e r i n g . B y c h o o s i n g t h e o p t i c a l

t h i c k n e s s o f t h e s l a b , b; t h e a l b e d o f o r s i n g l e s c a t t e r i n g , a; a n d t h e a n g u l a r d i s t r i b u t i o n o f i n c i d e n t

r a d i a t i o n , 0 ( g r a z i n g i n c i d e n c e ) , Ν ( i n c i d e n c e f r o m i s o t r o p i c s o u r c e s ) , U ( i n c i d e n c e f r o m L a m b e r t

s o u r c e s ) , 1 ( p e r p e n d i c u l a r i n c i d e n c e ) ; a v e r t i c a l l i n e w i t h t w o i n t e r s e c t i o n p o i n t s is d e f i n e d . T h e k e y

s h o w s h o w t h e s e p o i n t s d i v i d e 1 0 0 % o f t h e a v a i l a b l e e n e r g y i n t o i t s t r a n s m i t t e d , a b s o r b e d , a n d

r e f l ec t ed p a r t s .

Problem 3. This is the same as problem 1, but now we ask also how the energy deposited in the atmosphere is distributed with optical depth.

Answer: The tables in the present section do not give any functions of τ and are of no use in this problem. However, Table 17 (Section 9.3.1) gives the answer, for the difference between the net flux down at τχ and the net flux down at τ 2 is the flux absorbed between τχ and τ 2. For instance, between τχ = 0.5


and τ2 = 1.0, the absorption in the problem posed (b = 2, μ 0 = 1, a = 0.90) is 5442 - 4609 = 833 units.

If no table of net flux down is available, it is possible to use one of the relations in Display 9.4. The number at τ = 0.75 is —dF/άτ = 0.1 G = 1671 units, which, upon multiplication by τ 2 — τχ = 0.5, gives a total absorption of 836 units in a layer Δτ = 0.5 around τ = 0.75. This answer is approximate but still good enough.

The flux divergence found in this simple practice problem has—with more complicated specifications—great practical significance for the heating of clouds and, more generally, for the radiation balance of the atmosphere (Section 19.4).

9.3 POINT-DIRECTION GAIN AND ITS MOMENTS

The functions presented in this section have τ as one of the independent variables. They describe certain properties of the internal radiation field, such as may be required in discussing observations made inside a planetary atmosphere or a cloud layer.

9.3.1 Source Function, Gain, Net Flux

The function traditionally used as a measure of the internal radiation field (for isotropic scattering) is the source function J . It depends on τ, the parameters of the atmosphere a and b, and the intensity and position of the sources of radiation. If radiation is incident from one outside direction, J will depend on the direction of incidence μ 0 ; if there is an embedded source layer at one depth, it will depend, instead, on the depth of this layer τ 0. In all cases, J ( T ) is proportional to the intensity of the incident radiation or to the strength of the embedded source layers.

In Table 16,$ we present as a sample the source function for one layer thickness b = 1 and four different assumptions about the incident radiation. The intensity of the incident radiation has been normalized as usual to correspond in each case to an incident flux π on a unit area of the top surface. This table shows the expected features: the radiation is quite strong in the topmost layers for incidence Ν or μ 0 = 0.1 but penetrates more evenly for incidence U or μ 0 = 1. Figure 9.5 represents the same data graphically for the case a = 1.

The source function is proport ional to the point-direction gain G , or to one of its moments g„. The precise relationship is shown in Display 9.4, which also shows the relation to the net flux nF(z). For a more complete numerical documentation, we present in Table 17§ the values of gain (or gain moment) and flux side by side. It would have been more logical to show J and F side by

X T a b l e 16 wi l l b e f o u n d a t t h e e n d o f t h e c h a p t e r , o n p a g e 2 9 1 .

§ T a b l e 17 wi l l b e f o u n d a t t h e e n d o f t h e c h a p t e r , s t a r t i n g o n p a g e 2 9 2 .

9.3 Point-Direction Gain and Its Moments 207

0.1390

0.2 04

F i g . 9 . 5 . E x a m p l e s o f s o u r c e f u n c t i o n s se t u p in a s l a b o f o p t i c a l t h i c k n e s s 1 w i t h c o n s e r v a t i v e

i s o t r o p i c s c a t t e r i n g . T h e s o u r c e f u n c t i o n in e a c h e x a m p l e is e x p r e s s e d in t h e g a i n o r in a g a i n m o m e n t .

side, because they are always related as shown in the last line of Display 9.4. It seemed advantageous, however, to present G instead of J for the following reasons :

(a) Gain and source function differ by a trivial factor, but the gain admits of easier interpolation both with varying a and with varying μ, because it approaches 1 for all arguments if b 0.

(b) The gain has a physical significance relevant to two different situations. The first meaning (described in the "reverse experiment" of Section 7.3.3) corresponds to the situation of incident radiation described above and in Display 9.4. In its second meaning ("direct experiment" of Section 7.3.3), the gain tells


DISPLAY 9.4 R e l a t i o n s b e t w e e n S o u r c e F u n c t i o n J , G a i n G, a n d N e t F l u x D o w n F

a fo r I s o t r o p i c S c a t t e r i n g

T a b u l a t e d f u n c t i o n s

D i r e c t i o n s o f i n c i d e n t l i g h t w i t h flux π p e r

u n i t a r e a

S o u r c e f u n c t i o n a t d e p t h τ

D e r i v a t i v e o f n e t flux d o w n

T a b l e 16 b = 1

T a b l e 17 m a n y b ' s

O n e d i r e c t i o n J = ~ - G(a, b, τ, μ0) dF 1 - a

_ G δτ μ0

J G, F

F r o m n a r r o w s o u r c e l a y e r Ν J = ^flf_,(e, b, τ )

OF - — = (1 -dig-!

δτ J g-i,F

F r o m L a m b e r t s u r f a c e U a

J = ^9o(a, b, τ)

ÔF _ = 2 (1 - a)g0 δτ

J

A n y d i s t r i b u t i o n o f d i r e c t i o n s J = J(x)

dF - =4(a-

1 - 1 ) J ( T )

δτ

a T h e a c t u a l n e t flux d o w n is nF; t h e t a b u l a t e d f u n c t i o n is F.

with what intensity the radiation produced by an internal source layer escapes from the slab. In this interpretation, \g0, where g0 is the zero-order moment, is the one-sided escape probability.

(c) A further advantage of using gain is that it transforms into the very familiar X and Y functions if τ = 0 or b. Hence the left part of Table 17 is at the same time a table of X functions (top lines) and Y functions (bot tom lines), all multiplied by 10,000.

The values presented in Table 16 and in most of Table 17 were computed by the method of successive scattering described in Display 7.4. In the same computation, we obtained the gain and its two moments by applying the factors shown in Display 9.4. Furthermore, the net flux down was computed from

F(r)= ί1Ιίη(μ0)β-

τ^2μ0άμ0 Jo

+ 2 J Ε2(τ - T ' ) J (T') dx' - 2 £ £2( τ ' - T)J(T') dx'

The first term is the direct (zero-order) flux, the second plus third term, the flux of scattered radiation. This equation follows from the radiation intensity by a simple integration over all directions with a factor 2μ' in the integrand. For a check we may take the derivative with respect to a and find the last line of Display 9.4.


The τ integration scheme we employed in this computat ion was of marginal accuracy for b = 4. Some values shown for b = 4 are known to be off by 1 unit in the last decimal. The values shown in Table 17 for b = 8, 16, and 32 were obtained by the doubling method. All midlayer quantities presented in Table 17 were computed from the midlayer radiation field (matrices A and D\ as a byproduct of the doubling method, in the manner explained in Display 4.6.

For large b, unless a is very close to 1, the presence or absence of a bot tom boundary hardly influences the radiation field in higher layers. The intensity simply peters out, while the radiation diffuses down. The reaction back on the layers near the top surface, quantitatively expressed by the denominator 1 — f

2

in the theory of Section 5.3, rapidly becomes small enough to be neglected. The net flux in the conservative case, a = 1, is conserved, i.e., is independent of τ. Its value, upon any distribution of radiation incident on the top surface, is equal to the transmitted flux fraction. Therefore, its values may be read from Tables 12 or 15 as well as from Table 17.

F rom Section 7.3.3 we find that Sobolev's Φ function is related to the source function for narrow-layer incidence by

Φ(α, b, τ) = &β-ι(α, b, τ) = 2J

Nagirner's (1973) tables of this function for b = 1 and b = 2 check excellently with Tables 16 and 17. Nagirner also presents tables of moments over τ.

9.3.2 Point-Direction Gain in Thick Layers

The formulas presented in Sections 5.3 and 5.6 for very thick layers with arbitrary phase functions are at once applicable to isotropic scattering if we employ the specifications of Display 8.1. However, we can do more. The formulation of Chapter 5 gives the correct values at top and bot tom and inside the diffusion domain but leaves the equations that hold in the transition regions near the top and bot tom undiscussed. These transitions have been fully discussed for isotropic scattering in the injection into a semi-infinite atmosphere (Section 8.5) and the emergence from such an atmosphere (Section 8.6). These ingredients can now be simply combined. They yield the general gain equation (b > 1, all τ) on the left side of Display 9.5. Roughly, this equation may be trusted if the transition region from the top does not overlap with the transition region from the bottom. Taking for each an optical thickness 2j, which is about the value of τ at which / ( α , τ) reaches its asymptotic behavior, we find that the formulas of Display 9.5 are accurate for b > 5.

The further equations at the same side follow from the forms which f(a, τ) adopts if τ > 1, or τ = 0 (Section 8.5). Several of these equations may be checked in other ways. In particular, the X and Y functions for a thick layer may also be taken from Display 9.2, using the equations from Display 9.1 :

Χ(μ) = 1 + 2μΚΝ, Υ(μ) = 2μΤΝ

DIS

PL

AY

9.

5 P

oin

t-D

irec

tio

n G

ain

in L

ayer

w

ith

Op

tica

l T

hic

kn

ess

b >

1, I

sotr

op

ic

Sca

tter

ing

"

No

nco

nse

rvat

ive,

a

<

1 C

on

serv

ativ

e, a

=

1

All

τ

Ex

cep

t n

ear

bo

tto

m

Ex

cep

t n

ear

top

In d

iffu

sion

d

om

ain

(no

t n

ear

eith

er

surf

ace)

At

top

(τ

= 0)

At

bo

tto

m

(τ

= b)

All

τ,

At

top

,

At

bo

tto

m,

b, τ

) =

α/α

, b)

=

β fa

b)

=

τ' =

b

— τ

k(a)

, 1(

a)

Rj(a

)

H(a

, μ)

K(a

, μ)

=

Κ

(0)Η

(α,

μ)/(\

f(a,

τ)

f=

le

xp(-

kb),d

=

(1

-

Z(a,

b,

μ)

=

4μΚ

(α,

μ)

-kμ)

Fu

nct

ion

s E

mp

loy

ed

Tab

le

11

0, 1

, re

spec

tive

ly

—

Sec

tio

n 8.

5 α,

+ ι(

1)

Sec

tio

n 8.

3.3

Tab

le

7 H

(l,

μ)

-

(v/3

/4)H

(l,

μ)

1 -

1, o

o, r

espe

ctiv

ely

—

q(x)

T

able

10

Po

int-

Dir

ec

tio

n G

ain

Fig

. 8.

11

Dis

pla

y 9.

2

G(a

, b,

τ,

μ)

=

G(a

, τ,

μ)

- Zd

{_f(a

, x'

)e~kb

- If

(α,

τ)β~

2Ι(

^

G(a

, τ,

μ)

- Zd

le-2k\e

k* -

f(a,

τ)]

Zde~

k\e

kT' -/

(α,τ

')]

Zd(e

~kT -

l e-

2k

b +

kT)

X(a,

b,

μ)

=

H(a

, μ)

[1

- 2Ι

<μά

/2/(1

- 1<

μ)]

Y(a,

b,

μ)

=

μ^μά

[/(\

- 1<

μ)

g /α

, τ)

-

4K(0

)Rj(a

) di

f(a,

T')e

~kb -

If (a

, τ)

β~

2^

α/α

) -

2kR

jdf2

2kRj

df

G(l

, b,

τ,

μ)

=

G(l,

τ,

μ)

- ^3

μΗ(1

, μ

)[τ

+ q(

x)

+

q^

- α(

τ')]

/[*>

+

2qao}

G(l

, τ,

μ)

- [β

μΗ(\,

μ)

[τ

+ α

(τ)]

/(6

+ 2q

J

^3μΗ

(\,

μ)[ά

-

τ +

α(τ

')]/

(6

+ 2q

J

^βμΗ

(1

μ)φ

- τ

+

qj/(b

+

2g

J

Η(1,

μ)[1

-r

i(b

+ 2q

j]

H(l

μ)μ/

φ +

2g

J

Mo

me

nts

g/1

, τ)

-

^3α ]+

1(1

)1χ

+ q(

x)

+ ga

α/

1)

-α

]+

1(

1)

/(

Η2

0

aj

+1

(l)/

(fr+

2

gJ

g(i'

)]/(

fc

+ 2q

)

α Th

e ar

gu

men

t =

oo i

n G

(a,

oo,

τ, μ

) an

d g

/a,

oo,

τ) h

as

bee

n o

mit

ted

. O

the

r ar

gu

men

ts

are

om

itte

d w

hen

n

o am

big

uit

y ar

ises

.



The moments do not require any explanation. The right side of Display 9.5 shows equations for the conservative case (a = 1), exactly corresponding to those at the left side. These may be independently derived, by postulating the usual linear dependence on τ in the diffusion domain, corrected at the " injection " side (top) as explained in Section 8.5 and at the "escape" side (bottom) as in Section 8.6.2.

Alternatively, we may also find each equation at the right side by the transition a -> 1, k = 0, from the nonconservative case. It suffices to make linear approximations in k as follows:

/ = 1 - 2 9 c of c , f = l - ( b + 2qjk

d = 2k(b + 2q„\ f(a> τ) = 1 - [τ + 2^(T)]/C

Since a picture is worth a thousand words, we may refer to two examples which illustrate how the gain varies with depth. Figure 9.6 shows a logarithmic plot for a nonconservative medium (a = 0.90). The function actually plotted is source function, not gain, but the relation is clear from Display 9.4. In contrast

F i g . 9 . 6 . S o u r c e f u n c t i o n in s l a b s o f v a r y i n g t h i c k n e s s b w i t h i s o t r o p i c n o n c o n s e r v a t i v e s c a t

t e r i n g , a = 0 . 9 , μ 0 = 0 . 5 . T h e s t r a i g h t l i n e m a r k s t h e d i f f u s i o n d o m a i n .


to the linear decrease of the conservative case (Fig. 9.7, below) we see here an exponential decrease, drawn as a straight line because of the logarithmic J ( T ) scale. The curves for b > 8 all show the typical division in three domains explained in Chapter 5 :

(a) the injection region, τ ^ 0 -3 . The form of the curve in this region depends strongly on μ 0. It does not curve strongly up or down in this example because μ 0 = 0.5 is an intermediate value.

(b) the diffusion domain, represented by the exponential decrease. In this example, by Displays 9.4 and 5.3,

J(z) = aK&0)e-kT = 0 .4725e"

0 5 2 5 4t

The correction terms due to the negative diffusion stream in the opposite direction have been omitted. They are numerically insignificant on this scale and play an important role only if a is much closer to 1, and hence k much closer to 0.

(c) the escape region. The curvature down from the straight line becomes noticeable near τ = b — 2.5 and ends in a sharp bend down near the surface τ = b. At that point the flux density is a factor a~

1mK(0) = 0.461 below the

straight line relation.

Having thus explained the typical situation for large fc, we simply observe from Fig. 9.6 that the values b = 1,2, and 4 are not large enough to show these

F i g . 9 . 7 . S o u r c e f u n c t i o n J(t) se t u p in a s l a b o f o p t i c a l t h i c k n e s s 10 w i t h c o n s e r v a t i v e i s o t r o p i c

s c a t t e r i n g b y p e r p e n d i c u l a r l y i n c i d e n t r a d i a t i o n . T h e g r a p h a l s o s h o w s t h e c o r r e s p o n d i n g f u n c t i o n

f o r a s e m i - i n f i n i t e a t m o s p h e r e a n d t h e d i f f e r e n c e f u n c t i o n .


separate domains. At τ = 0, the curves nearly coincide, but the values differ by several percent, namely

b = 1 2 4 8 J(0) = 0.6560 0.6874 0.6988 0.7002

Figure 9.7 shows the situation in the conservative case. Here a linear vertical scale is used, and the diffusion domain again shows a straight line. At the right side of the figure (bot tom of the layer), radiation emerges as if from a semi-infinite atmosphere. The extrapolated straight line reaches zero at τ = b + . Also shown in the figure are the separate terms that make up this result, according to Display 9.5 (right side). The source function appropriate to reflection from a semi-infinite layer starts with a rapid rise and reaches a constant level inside. The second term, to be subtracted from this first term, clearly has the character of a diffusion stream emerging from the top level, of the same magnitude as actually emerges at the bot tom.

9.3.3 The Escape Probability

In Fig. 9.8, we have taken 6 = 2, and the value plotted is jg0(a, 2, τ). F r o m Display 9.4 we see that this quantity can be interpreted as a~

1 times the source

function set up at depth τ by radiation with uniform intensity 1 incident upon the top surface. The reciprocal meaning is more appealing, for \g0 is also the probability that a quan tum emitted by an isotropic source at depth τ will finally escape through the top surface.

A simple variation is to plot the escape probability through the top and bot tom surfaces in the same figure, thus showing graphically the complementary probability that such a quan tum does not escape at all, i.e., that it is absorbed. Two such graphs are presented in Fig. 9.9. The arrangement is rather similar to that in Fig. 9.4.

Example. An isotropic source is embedded at depth τ = \ in a layer with total optical thickness b = 2. The scattering is isotropic, with albedo a = 0.9. What percentage of the source energy escapes through the top surface, what escapes through the bo t tom surface, and what is absorbed in the layer?

Answer: The probability for escape through the top surface from Fig. 9.9 (more precisely from Fig. 9.8, and most precisely from Table 17) is

Ptop = $9o(fl> b>

τ) = ° ·

5 01

Similarly

Pbottom = ho(a, b,b - τ) = 0.228

The remaining 27.1 % is absorbed. If a were 1.0, the escape probabilities would be 0.654 through the top and 0.346 through the bot tom, and nothing would be absorbed.

F i g . 9.9. P r o b a b i l i t y o f e s c a p e a n d p r o b a b i l i t y o f a b s o r p t i o n in t h e s i t u a t i o n d e s c r i b e d b y

F i g . 9 .8 a t ( a ) a = 0 .6 a n d ( b ) a = 0 . 9 .


τ ο = 0 τ0= 0.125

0 f.125.25 .5 1 0.625

I I I

α = 1 . 0 ^ /

^ 0.6

0 -

ι I ι ι ι ι I 0 .125.25 .5 1 2 U 10 oo

I

το=0.25 τ = 0 . 5

.6 μ

.2 μ

.25 .5 1 2 U 10 oo u p p e r f o u r : l i n e a r s c a l e of 1Λ&+1! % l o w e r t w o : l i n e a r s c a l e of yg

τ = 1

a = 1 /

0.9

0.6 _

0

I I I I u 10 oo

I I ι / a=1.0/

_ / 0.9

-/ / ^ ^ ^ ^ a 6

_ ο -

I I I I I

2 U 10 oo

τ = 2

a = l / _

- / 0.99

-/ 0.9

0.6 Γ ι

0 =0 1 t 1 1

2 2.5 U 10 oo

F i g . 9 . 1 0 . O n e - s i d e d e s c a p e p r o b a b i l i t y f r o m a n i s o t r o p i c s o u r c e a t o p t i c a l d e p t h τ 0 t h r o u g h

t h e s u r f a c e a t τ = 0 o f a s l a b w i t h t h i c k n e s s b a n d s i n g l e - s c a t t e r i n g a l b e d o a.


As a further illustration, Fig. 9.10 shows the escape probability for constant τ 0 as a function of b. The values for b < 4 from Table 17 join smoothly those for ft = oo given in Fig. 8.9. The values for a = 1 correspond to those given in Fig. 9.12 (below).

9.3.4 Further Examples, Conservative Case

The gain in the conservative case is a function of three variables, G(l , ft, τ, μ), which cannot be presented in full detail in one table or graph. Some sample data are given in the following figures. Figure 9.11 shows the situation for perpendicular incidence. Among the curves shown is the curve for b = oo, corresponding to values presented in Table 10 (Section 8.5). The curve for b = 10 is explained in some detail in Section 9.3.2 (Fig. 9.7) in connection with the theory for very thick layers. The curve for ft = 1 is also seen in Fig. 9.5. For any ft, incidence at a smaller angle with the surface will deposit relatively more radiation in the upper levels and less in the lower levels, as shown most clearly in Fig. 9.5. The onsets in the range 0 < τ < 2 vary, therefore, with the angle of incidence; but at large τ, all curves show a linear decrease with τ, representing the constant gradient of radiation density necessary to carry the constant net flux along.

The corresponding set of curves for uniform incident intensity U is presented in Fig. 9.12. Since the quantity ^g0, plotted here, may also be interpreted as a one-side escape probability, we clearly must have

ko(h b, τ) + iflf0(l, ft, ft - τ) = 1

Each curve in Fig. 9.12, therefore, is symmetric with respect to the point τ = lb, \g0 = \. The accurate values at the boundaries may be taken from Table 17

G(a.b.T.jj)

6 - a = l , , . 1

0 1 2 3 4 5 T

F i g . 9.11. P o i n t - d i r e c t i o n g a i n f o r c o n s e r v a t i v e s c a t t e r i n g a n d p e r p e n d i c u l a r i n c i d e n c e , w i t h d e p t h τ a n d s l a b t h i c k n e s s b v a r y i n g .


1.00

r-

, 1

1 ,

1 1

, 1

1 |

^b

- οο

^ 0

.80

\.

—-

b-

10

~

g 0

.60

\ V

Ν.

^

S\.

b

2

1 \

\b

«0

5\

-g

ΟΑ

Ο "

\

\ \

^\

1 0.2

0 -

\ \

Ν\

^Ν

%-

^^"^

^ 0

I I

\ \l

J

1 I

JC

^ I

1 1

1

0 1

2 3

4 ^

5

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. 9

.12

. O

ne-s

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es

cap

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rob

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fro

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s o

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opti

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thic

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F i g . 9.13. E x t r a p o l a t i o n l e n g t h f o r finite l a y e r s , a s d e f i n e d in t e x t .

or 12. The tangent in this midpoint, also drawn in Fig. 9.12, has the slope, derivable from the more general formulas of Section 7.3.3

d ( k o l b , T ) / d r \ T = ib = - # ο( 1 , « 0 - ι ( 1 , * , # )

This tangent cuts the lines jg0 — 1 at τ = — q(b) and the line \θο = 0 at τ = b + q(b\ where

q(b) = W0(lb)g„l(l b9&)Yl-fr

The values of q{b\ which may be called an extrapolation length for finite layers, are shown in Fig. 9.13, and approach the Hopf extrapolation length q^ = 0.710 for large b.

9.4 INTEGRALS OF GAIN OVER OPTICAL DEPTH: HOMOGENEOUSLY EMBEDDED SOURCES

Integrals of the point-direction gain over optical depth occur in two different situations, consistent with the two reciprocal meanings of the gain itself.

(1) Radiation with net flux π per unit area falls from direction μ 0 on the top surface of a layer of thickness b consisting of isotropically scattering material with albedo a. The fraction of the incident flux which is absorbed between the levels x1 and τ 2, where 0 < xx < x2 < b is then (see Display 9.4)

1 - α Γ2

F(a, b, τΐ9 μ0) - F(a, b, τ 2, μ0) = G(a, b, τ, μ0) dx μο Λ Ι

The numbers given in the right half of Table 17 thus provide a simple way to find the energy absorbed in any layer of the atmosphere separately and, through

9.4 Integrals of Gain over Optical Depth 219

and carries the flux

UI = 2 [2g0(a,b,x)dx

Each of these integrals can thus be found from the numbers in Table 17 by referring to the formulas above.

Integrals over the full depth of the atmosphere may be found from

ί Je

b

G(a, b, τ, μ) dx = μΑ(α, b)[X{a, ft, μ) - Υ(α, b, μ)]

with

Α(α, b) = [1 - iaoc0(a, b) + $αβ0(α9 ft)]"1

= (1 - αΓ'ΙΙ - {aa0(a9 b) - ±αβ0(α, ft)] A(hb) = W0(Ub)Y

1

This follows at once from the preceding formulas by taking the flux values appropriate to bot tom and top (Display 9.1). The equivalence of both expressions for A follows from properties of the moments of the X and Y functions (Section 9.6.2). The form in brackets in the second expression may be read from Table 15 (Section 9.2). See Display 9.3.

Another interpretation is that

ί Je

b 4uu G(a, ft, τ, μ0) dx = lim R(a, ft, μ, μ0)

this, the integral of the gain over these layers. Similarly, if the incident light, again normalized to flux π, has the distributions Ν or (7, we have

N: F(a, b, τχ) - F(a, b, τ 2) = (1 - a) f V x(a , b, τ) dx

(7: ft, - F(a, b, τ 2) = 2(1 - a) f 0o(a> ^ τ

)

(2) Consider the same atmosphere, now without incident radiation. Isotropic sources are placed between depths xx and τ 2, with homogeneous source density 1 per unit optical depth. The radiation emerging at the top surface (τ = 0) in direction μ is now

1 Γ2

Ι(μ) = - G(a, b, τ, μ) dx

It has the hemispherical average

NI = [ g_l(a,b,x)dx


The physical reason is that R can be considered as an integral over the source function weighed with an exponential function, which becomes 1 for μ -> oo. The equivalence to the earlier expression is easily shown from the asymptotic expressions for the X and Y functions (Section 9.6.3).

Integrals of the gain moments over the full depth of the atmosphere follow from

ί gk_ i(a, b, τ) dx = A(a9 b)[ak(a, b) - β^α, b)] Jo

The particular moment g_l is proport ional to Sobolev's Φ function (see end of Section 9.3.1). Integrals of this function with added factor x

j or (b — x)\

where j is an integer from 0 to 5, are tabulated by Nagirner (1973). Some sample problems may illustrate the use of these formulas.

Problem 1. What is the average escape probability of photons from sources homogeneously distributed in the top 25 % (τ = 0 - | ) of a layer with b = 2, a = 0.9?

Answer: A cursory glance at the left part of Table 17, or more conveniently at Fig. 9.8, shows that the escape probability %g0 in the top quarter of the layer varies between 0.74 and 0.5 and that its average value may, therefore, be somewhat over 0.6. More precisely, from the right part of the same table and the equations given above, we find

< 1 0 O> = (0.5629 - 0.4418)/(0.5 χ 4 χ 0.1) = 0.605

The corresponding values for the second, third, and fourth quarter are 0.422, 0.287, and 0.172. The full answer to the question posed is that the average escape probability from the layer (0, is 0.605 through the top surface and 0.172 through the bot tom surface, and hence 0.777 in total.

Problem 2. Plot the intensity pattern emitted by a slab with homogeneously embedded sources.

Answer: If the source density is 1 per unit optical depth, the radiation emitted from the top or bot tom surface has the intensity

Ι(μ) = A(a, b)[X(a, b9 μ) - Y(a, b, μ)]

The corresponding direct radiation, which is all we see if a = 0, is

Ι(μ) =l-e~ blfl

Figure 9.14 shows this intensity for a slab with optical thickness b = 1. Note that the ratio between the total intensity for any a and the direct source intensity (a = 0) does not vary much with direction. Each curve goes through a maximum. The main trend is for /(μ) to become larger for smaller μ because we look through the slab along a longer path. Near grazing angles this effect is offset by the fact that the source density caused by multiple scattering always drops sharply near the surface, so that Ι(μ) must drop toward μ = 0.

Problem 3. How does the thermal emission by an isothermal slab consisting of scattering and absorbing particles depend on the angle?

Answer: By Kirchhoff's law, the thermally emitted intensity into direction μ is the blackbody intensity times the absorbed fraction for incidence from direction μ. Hence, the thermal emission for isotropic scattering is

(1 - a)A(a, b)lX(a, b, μ) - Y(a, b, μ)]

multiplied by the blackbody intensity. The solution for different scattering patterns is discussed in Section 18.6 in connection with the infrared limb darkening of planets.

9.5 THE INTENSITY INSIDE THE ATMOSPHERE

The emphasis in this book is on the properties of the radiation field outside the atmosphere. Nevertheless, situations occur in which it becomes necessary to know also the intensity in all directions somewhere inside the layer. A good


reason for not delving too deeply into this question is that again an independent variable is added. In the standard problem, with radiation incident from direction μ 0 on the top of a layer of thickness b with isotropic scattering with albedo a, the intensity at optical depth τ at angle c o s "

1 u with the downward

(increasing τ) direction is I(a, b, τ, u, μ0) . Even a fairly coarse tabular presentation of this function of five variables

would take hundreds of pages. Tables and graphs in which one variable is eliminated by assuming a special value or by integrating over it have been given in earlier sections, for instance:

Make τ = 0: I(a, b, 0, - μ , μ0) = R(a, b, μ, μ 0) Reflection, Table 12 Make τ = b: I(a, b, b, μ, μ0) = T(a, b, μ, μ0) Transmission, Table 12

Integrate over u: \a JL x I (a, b, τ, u, μ0) du = J(a, b, τ, μ0) Take u = 0: I(a, b, τ, 0, μ0) = J(a, b, τ, μ0)

To find the general function, a new computer program is needed. The relations just mentioned can then be used among the numerical checks. The variety of methods available has been reviewed in Section 4.3. We suggest that the following be considered first.

(1) For a rapid desk computation, plot the source density J(T) from any of the tables or graphs in Section 9.3, then integrate as shown in Display 4.4:

up (ii < 0): /(τ , u) = £ j ( T > "( T

' "T ) / (

"M)

dx'/(-u)

down (u > 0): /(τ , u) = ί 7 ( τ,> "

( τ _ τ ) / Μ^ τ / ι /

Jo (2) If a computer program for the adding method is available, the radiation

field at the separation layer is automatically computed. In doubling (i.e., adding equal layers) we obtain the midlayer intensity as a byproduct. See the example for an anisotropic scattering pattern in Table 40 (Section 13.3).

(3) The method of singular eigenvalue expansions gives the internal radiation field along with the reflected and transmitted radiation making it more competitive, for numerical work, for the broader question posed in this section. Similar formulas may be derived along other routes. The reader may refer to Section 4.2.1 for comments and to Section 6.5 for a detailed guide.

An example of a full set of curves showing the internal intensity is shown in Fig. 9.15. It was drawn from a table computed by singular eigenvalue expansion, made available to me by Dr. Kaper. A figure of the same format is shown in Fig. 7 of Kaper et al. (1970). The left top figure refers to isotropic scattering,

bource density, related to gain, Tables 16, 17

9.5 The Intensity inside the Atmosphere 223

g=0, b=4, μο=0.5, α=0.6 g = 0.875, b=16, μο=0.5,·α=0.9 up down up down

F i g . 9 . 1 5 . Le f t s i d e : I n t e n s i t y d i s t r i b u t i o n w i t h a n g l e a t v a r i o u s d e p t h s i n s i d e a finite s l a b w i t h

i s o t r o p i c s c a t t e r i n g . R i g h t s i d e : C o r r e s p o n d i n g d i s t r i b u t i o n s in s l a b s w i t h a n i s o t r o p i c s c a t t e r i n g

w h i c h s h o u l d p r o v i d e a n a p p r o x i m a t e m a t c h b y t h e s i m i l a r i t y r u l e s .


b = 4, a = 0.6, μ 0 = 0.5. The curves for τ = 0 and τ = 4 represent the reflected and transmitted intensity, respectively. Both these curves cover one hemisphere and must be considered to d rop to zero in the other hemisphere. All curves for intermediate τ are continuous from u = — 1 to M = + 1 . For very small τ, the downward radiation comes from a thin layer only and is roughly proport ional to l/μ. This tendency is soon changed as the maximum shifts. In a domain of τ with a downward diffusion stream, all the intensity curves plotted in this manner should have the same form and be shifted by equal amounts for equal differences in τ. Such a domain is not reached in this example, because b is too small, but the curves from τ = 1.2 to τ = 3.2 show this tendency approximately. The right-hand side refers to anisotropic scattering, and comparison of the two sides provides an illustration of the similarity relations (Section 14.12).

(4) Inside the thick layers, a diffusion domain exists, in which it is possible, with excellent accuracy, to compute the intensity from a superposition of a positive diffusion stream down and a negative diffusion stream up. The strengths of these streams are given in Section 5.3 and the corresponding pattern in Section 6.2.1, both for arbitrary phase functions. They must be used here with the specifications for isotropic scattering given in Display 8.1. The gain values inside thick layers given in Display 9.5 may be useful for checking purposes.

9.6 SOME SPECIAL FUNCTIONS

9.6.1 The X and Y Functions

The X and Y functions for homogeneous atmospheres with isotropic scattering are functions of three variables: X(a, b, μ) and Y (a, b, μ). They were first introduced (in a different notat ion) by Ambartsumian and have been employed extensively by Sobolev, Chandrasekhar, and others. See the historical note in Section 4.4.

These functions can be defined in three ways:

(1) by physical definition as the point-direction gain for a point at the top or bot tom surface (see Section 3.4.3, where the context is more general).

(2) as solutions of two simultaneous nonlinear integral equations (Section 4.4). This is the usual definition, but we do not prefer it because of the non-uniqueness in certain cases.

(3) by a power expansion in a generated by the procedure presented in matrix notation in Section 7.3.3.

The normal range of each variable is albedo of volume element: a — 0 -1 optical depth of layer: b = O-oo cosine of angle: μ = 0-1

9.6 Some Special Functions 225

F i g . 9 . 1 6 . D e p e n d e n c e o f t h e X a n d Y f u n c t i o n s f o r i s o t r o p i c s c a t t e r i n g o n t h e v a r i a b l e s b, a, a n d μ. T h e A ' f u n c t i o n f o r b = oo is c a l l e d t h e Η f u n c t i o n .

Extension to a > 1 can easily be made. The series starts to diverge at a c r it = ηϊ \ where ηχ is the eigenvalue shown in Table 4 (Section 7.4.1). Extension to μ > 1 is discussed in Section 9.6.3.

Numerical values for the normal range are contained in Table 17, top and bot tom lines. Simply related functions are shown in Table 16, top and bot tom values, and the entries with μ = 0 and the products RN and Τ Ν in Table 12. Display 9.1 may be consulted for details. This makes a separate table of X and 7 functions unnecessary. In Fig. 9.16 we present some graphs illustrating the general behavior of these functions.

Figure 9.17 shows in more detail a contour diagram of 7 (1 , ft, μ), this time including the domain of virtual angles (μ > 1). Sobouti 's work covering the lower part of the square has here been extended by the asymptotic equations for large ft of Sobolev (1957) and Ivanov (1964), which are in Display 9.5. Asymptotic expressions valid near μ = oo are given in Display 9.6.

The following tables of X and 7 functions and their moments are available in the literature. This listing is undoubtedly incomplete.

1. Chandrasekhar and Elbert (1952) give X and 7 for μ = 0 (0.01) 1; a = 0.5, 0.8, 0.9, 0.95, 1; and ft = 0.05, 0.1, 0.15, 0.2, 0.25, 0.5, 1. Moments a 0, j80, ocu βχ are given for the same combinations (in the case a = 1 they require a conversion with the help of the function Q, also tabulated).

2. Mayers (1962) gives X and 7 for μ = 0 (0.025) 1 ; the combinations (a, ft) include eleven of those in Chandrasekhar 's work and six new ones :

a 0.8 0.9 0.9 0.95 0.95 0.95 ft 2.5 2 5 2 4 10

Moments a 0 and β0 are given for the same combinations.

3. Sobouti (1962) gives X, Υ, ξ + , ξ~, Γ , and C" for μ = 0 (0.02) 1.2 (0.05) 2

(0.1) 3 (0.2) 5 (0.5) 10 (1) 20; a = 0.1 (0.1) 0.8 (0.05) 1 ; and b = 0.1 (0.1) 0.6 (0.2) 1 (0.5)3. See definitions in Section 9.6.3. This makes six functions for 117 χ 12 χ 12 combinations of the arguments: four-figure accuracy. Moments a 0, ocu a 2 and βο> βι> βι f °

r s a me te combinations are also given.

4. Carlstedt and Mullikin (1966) give X and Y for μ = 0 (0.01) 1 and moments a 0, j30 in six figures, a = 0.3 (0.1) 0.9, 0.95 (0.01) 1.0; and b = 0.2 (0.2) 3, 3.5, for all these a values. As many of the additional arguments b = 4 (0.5) 7.5 are covered as are necessary to come close to the asymptotic theory for large b.

5. Bellman et al (1966) give X and Y for seven angles from Gaussian quadrature, for a = 0.4, 0.9, 0.975, 1 and b in steps of 0.1 as far up as necessary.

6. Cohen (1969) presents a few examples with emphasis on accuracy of method.

7. Caldwell (1971) gives X and Y functions and zero-order moments for a = 1, b = 0.5 (0.5) 4.5, μ = 0 (0.2) 1. Comparison with more accurate tables shows many values to be too small by about one part in 10

5.


8, Bellman et al. (1973) give X and Y for seven angles from Gaussian quadrature for a = 1 only, by a method different from 5.

9. Loskutov (1973) gives moments α 0- α 5 and β0-β5 for a = 0.5, 0.7, 0.8, 0.9, 0.95, 0.99, 1 ; and b = 0.2, 0.5 (0.5) 3.0.

9.6.2 Moments of the X and Y Functions

DEFINITIONS AND GENERAL RELATIONS

The moments of the functions discussed in Section 9.6.1 are the following:

f1 1 ccp(a9 b) = X(a9 b9 μ)μ

ρ άμ = ——— + α*(α, b\ ρ = 0, 1, 2 , . . .

Jo Ρ + 1

fip(a9b) = \1γ(α9 μ)μ

ράμ

Jo

α*(β, b) = Γ [X(a9 b9 μ) - \~]μρ άμ

Jo

Ρ = - 1 , 0 , 1 , . . .

The following relations can be derived from the integral equations for X and Y (Chandrasekhar, 1950, pp. 187, 188):

O o - l )2 - &β0)

2 = l - a

2_ 3a

« 0 « 2 + βοβί+ΊΧΪ-Ίβί =

These and similar ones involving higher moments can be systematically derived from the expansions of Sobouti functions (Section 9.6.3).

Differential equations follow from the integro-differential equations for the X and Y functions (Chandrasekhar, 1950, pp. 185,210) or, alternatively, from the differential equations for the gain moments (Section 7.3.3):

d*p(a, b)/db = $afi-1fip δβρ(α9 b)/db= - J S , - ! + & / ? - 1α ρ

The moments of order — 1 can also be expressed by

a* x(a9 b) = lim\— [X(ay b9 μ) - 1] + In μ\

β-ι(α9 b) = dY(a9 b9 μ)

δμ μ = 0

Incident radiation with net flux π and angular distribution

/(0, - μ ) = Κρ+ 1)μρ~' (ρ = 0 , 1 , . . . )


on a slab with optical thickness b gives the reflected par t :

rp(a, b)=l-(p+ 1)[(1 - $act0)ccp + $αβ0βρ]

the transmitted part :

tp(a, b) = (p+ \)\_\αβ0*ρ + (1 - \αα0)βρ-\

and the absorbed par t :

lp(a, b) = 1 - rp - tp = (p + 1)(1 - ^αα0 - ^αβ0)(αρ - βρ)

Compare Displays 9.1 and 9.3 for cases ρ = 0 (incidence from narrow source layer) and ρ = 1 (incidence from Lambert surface). See Section 8.4 for corresponding results for b = G O .

EXPANSIONS IN SUCCESSIVE ORDERS (POWER SERIES IN a)

Integrating the corresponding expansions of X and Y, we obtain the moment expansions with terms expressed in functions defined in Sections 2.2 and 2.5:

where ρ = — 1, 0, 1, . . . . These are the first terms of the power expansion in a, useful for small a, any b, but also rapidly convergent for small b, any a. In the latter case, the first-order terms differ little, as shown by the expansions in b:

a*(a, b) = jaGUp + 2 + -'

βρ(α, b) = Ep+2(b) + \aG'ltP+2 + --

G n( 6 ) + 0(b3)

G13(b) G'13(b) + 0(b3)

where / = In b + 0.577216.

RELATIONS FOR THE CONSERVATIVE C A S E (a = 1 )

docp/db = tt _^p = da*p/db dfijdb- - j 8 p- i + i j 8 - i a p


They have the integral

«ι - βι = bfi{ 0

This relation, valid for a = 1 only, can be verified directly. I did not find it explicitly stated in the literature, but it is implicit in the way in which Chandrasekhar solves for the factor Q in his nonuniqueness problem.

By means of this integral it is possible to express the a's and their derivatives in terms of jS's and b as follows:

a 0 = 2 - β0 «ι = bβ0 + β1

0L2 = 2ββ0 - Κ^β0 + βο - β 2

da*1/db = tfll doc0/db= -άβ0/α = ίβ-1β0

dOL1/db = tt_Jl dfijdb= - β ο + τβ-^ + ίβ-Λι

NUMERICAL V A L U E S FOR THE CONSERVATIVE C A S E

Section 9.6.1 may be consulted for published tables. The moments for η = — 1,0,1, and 2 are collected in Table 18. They were obtained from the numbers in Table 12 by the relations in Display 9.1. The bot tom line in Table 18 shows the coefficient c appearing in the asymptotic expressions, which can be read from Display 9.5. Written out, we have for a = 1, b > 1 :

where s = (b + 1.42089)"1

REFLECTED FLUXES, CONSERVATIVE C A S E

Table 18 also gives the reflected fraction of the incident flux for the three simplest assumptions of angular distribution of incident radiation and thus forms an extension of some entries in Table 9 (Section 8.4) to finite layers. Incidence Ν (proportional to μο

1) gives reflected flux fraction = URN = α0 — 1.

Incidence U (uniform) gives reflected flux fraction = URU = r u with the full equation in Display 9.1 and numerical values in the next to last column of Table 18. Incidence f ^ ( p r o p o r t i o n a l to μ, written as A2 in Section 8.4) gives reflected flux fraction f URW = r 2, with numerical values in the last column. The asymptotic behavior is indicated as earlier.

a* 2.1348 2.0000 1.1547 0.8204

2.00005, 1.1547s, 0.82045, 0.63785,

2.00005 1.15475 0.82045 0.63785


T A B L E 1 8

M o m e n t s o f t h e X a n d Y F u n c t i o n s a n d o f t h e R e f l e c t i o n F u n c t i o n f o r S l a b s

w i t h I s o t r o p i c C o n s e r v a t i v e S c a t t e r i n g

b a*_, 0-, a0 00 a, 0, a2 Γι r2

0. 0, 0 . 1, .00000 1.00000 0.50000 0.50000 0 .33333 0. ,33333 0. 0, 0.001953 0. ,04459 5.70444 1, .00655 0.99345 0.50319 0.50125 0, .33546 0. 33448 0.0019 0, .0015 0.003906 0, ,07244 5.18627 1, .01171 0.98829 0.50590 0.50204 0 .33727 0. ,33532 0.0039 0, .0029

0.007812 0. ,11528 4.37936 1, .02079 0.97921 0.51054 0.50289 0 .34037 0. ,33649 0.0076 0, .0058 0.015625 0. .17913 3.73693 1, .03600 0.96400 0.51842 0.50336 0, .34565 0. ,33793 0.0150 0, .0115 0.031250 0. .27109 3.17018 1, .06143 0.93857 0.53187 0.50254 0 .35471 0. ,33943 0.0290 0, .0228

0.050000 0. ,35315 2.78231 1, .08712 0.91288 0.54578 0.50014 0, .36414 0. ,34000 0.0452 0, .03 58 0.062500 0. .39837 2.60464 1, .10234 0.89766 0.55418 0.49807 0 .36985 0. ,33993 0.0554 0, .0443 0.078125 0. ,44783 2.43314 1. .11980 0.88020 0.56393 0.49517 0, .37653 0. ,33951 0.0678 0, .0546

0.100000 0. .50764 2.25031 1, .14202 0.85798 0.57654 0.49074 0, .38520 0. ,33846 0.0843 0, .0687 0.125000 0. ,56638 2.09065 1, .16495 0.83505 0.58976 0.48538 0, .39434 0. ,33682 0.1022 0, .0842 0.156250 0. ,62953 1.93619 1. .19077 0.80923 0.60490 0.47846 0, .40489 0. ,33431 0.1233 0, .1027

0.20 0. ,70445 1.77176 1. .22287 0.77713 0.62407 0.46864 0, .41834 0. ,33025 0.1508 0, .1274 0.25 0. ,77652 1.62929 1. .25515 0.74485 0.64372 0.45751 0, .43224 0. ,32514 0.1798 0, .1538 0.3125 0. ,85242 1.49278 1. .29055 0.70945 0.66567 0.44396 0, .44791 0. ,31840 0.2128 0, .1845

0.40 0. ,94037 1.34873 1. .33320 0.66680 0.69264 0.42592 0, .46736 0. 30873 0.2541 0, .2238 0.50 1. 02286 1.22489 1. ,37466 0.62534 0.71937 0.40670 0, .48683 0. 29774 0.2958 0, .2641 0.625 1. 10751 1.10711 1. ,41852 0.58148 0.74814 0.38471 0, .50800 0. 28447 0.3413 0, .3088

0.80 1.20269 0.98390 1. ,46920 0.53080 0.78197 0.35734 0, .53316 0. 26709 0.3953 0, .3628 1.00 1.28908 0.87897 1. ,51629 0.48371 0.81390 0.33019 0, .55714 0. 24906 0.4466 0, .4151 1.25 1. ,37473 0.78020 1. ,56381 0.43619 0.84654 0.30130 0, .58186 0. 22914 0.4993 0, .4694

1.6 1.46723 0.67819 1. ,61588 0.38412 0.88270 0.26812 0, .60947 0. 20546 0.5580 0, .5305 2 .0 1.54757 0.59267 1. ,66158 0.33842 0.91472 0.23788 0, .63407 0. 18328 0.6099 0, .5851 2 .5 1.62364 0.51366 1. ,70514 0.29486 0.94541 0.20827 0, .65777 0. 16109 0.6598 0. .6378

3 .2 1.70164 0.43406 1. ,75000 0.25000 0.97715 0.17715 0, .68236 0. 13741 0.7114 0, .6926 4 . 0 1. 76576 0.36935 1.78696 0.21304 1.00337 0.15120 0, .70271 0. 11744 0.7540 0, .7379 5 .0 1. 82329 0.31159 1.82016 0.17984 1.02694 0.12772 0, .72102 0. 09927 0.7923 0, .7787

6 .4 1.87907 0.25574 1. ,85236 0.14764 1.04981 0.10489 0, .73880 0. 08154 0.8295 0, ,8183 8 .0 1.92251 0.21230 1.87743 0.12257 1.06762 0.08708 0, .75265 0. 06770 0.8585 0, .8492

10.0 1. 95968 0.17512 1. ,89890 0.10110 1.08287 0.07183 0, .76451 0. ,05585 0.8833 0, .8756

12.8 1. 99416 0.14064 1. ,91880 0.08120 1.09701 0.05769 0, .77550 0. ,04485 0.9062 0, .9001 16.0 2.02000 0.11480 1. ,93372 0.06628 1.10761 0.04709 0, .78374 0. ,03661 0.9235 0, .9184 20.0 2 . 04143 0.09337 1. ,94609 0.05391 1.11640 0.03830 0, .79058 0.02978 0.9378 0, .9337

25.6 2.06078 0.07402 1. .95727 0.04273 1.12434 0.03036 0, .79675 0. ,02360 0.9507 0, .9474 32 .0 2.07496 0.05984 1. ,96545 0.03455 1.13015 0.02455 0, .80127 0. 01908 0.9601 0, .9575 40 .0 2.08652 0.04828 1. ,97212 0.02788 1.13490 0.01981 0, .80495 0. 01540 0.9678 0, .9657

00 2. 13480 0.00000 2. ,00000 0.00000 1.15470 0.00000 0, .82035 0. 00000 1.0000 1, ,0000

c -2 .00000 2.00000 - 1 . ,15470 1.15470 -0.82035 0.82035 - 0 , .63782 0. 63782 -1 .3333 - 1 . .4209

9.6.3 Virtual Angles, Sobouti Functions

We speak of virtual angles if we mean values of μ outside the normal domain (0, 1). The X and Y functions for such values of μ can be computed from the standard nonlinear integral equations (Section 4.4, Eqs. (2)) because the argument μ 0 remains confined to the integration interval (0, 1). Sobouti has named the separate integrals over μ 0 that occur in this integration, and his notation is explained in Display 9.6. These integrals have also acquired some importance


in connection with the problem of fluorescent scattering in planetary a tmospheres (Chamberlain and Sobouti, 1962; Sobouti, 1962a,b).

The X and Y functions for virtual angles then obey two linear equations with determinant D, independent of b. The solution of this set is given in Display 9.6, along with two equivalent expressions for D.

For μ > 1, it is advantageous to use expansions in powers of μ~l. The first

few terms of these expansions are given in Display 9.6. The dominant terms are the limits for μ -» oo. Incidentally, by equating the coefficients of μ

ρ (ρ = 0,1,2, . . . )

in the expansions of both expressions of D, we obtain a systematic method of

D I S P L A Y 9 .6

F u n c t i o n s for V i r t u a l A n g l e s

F u n c t i o n G e n e r a l e x p r e s s i o n , μ > 1 E x p a n s i o n in p o w e r s o f μ

1

S o b o u t i f u n c t i o n s

ξ + (α, b, μ)

μα f

1 Χ(α,^μ0) — άμ0 2 J0 μ + μ0

ξ (a, b, μ)

ί +

(α, b, μ)

Γ (a, b, μ)

μα Γ

1 Χ(α,^μ0)

J 0 μ άμ0 2 J0 μ - μ0

μα Γ

1 Υ(α^,μ0)

2 J0 μ + μ0 μα Γ

1 Υ(α^,μ0) ι Γ Υ(α

J 0 μ μ 0 (Ιμ0

Χ a n d Υ f u n c t i o n s

1 - ξ- - C + e~

b^

Χ(α, b, μ)

Υ(α, b, μ)

D

(1 - t

+ ) e ~

m - ζ

F u n c t i o n s u s e d in t h e s e f o r m u l a s

D(a,b^) (1 - « T ) ( 1 - ξ ~ ) - ζ + ζ-

αμ 1 In

2

μ + 1

μ - 1

- ( α 0 - (χ,μ

1 + α2μ

2 )

- ( α 0 + αχμ 1

+ α 2μ

2 + · · ·)

(β0 + β ιμ - ι +β2μ-> + ...)

1 Γ α α

1 - a |_ 2 2

[α , - β , - bfio] + 2 ( 1 - ά)μ

Γ 2 6 Ί -τ- \βι ~

αι + boc0\

α)μΙ α \ 2 ( 1

1 - α - \ α μ

α / α , b) b, Vo)\ S e c t i o n 9.6.1 '!*see S e c t i o n 9.6.2

Y (a, b, μ0)\ β{α, b)\


finding the nonlinear relations between the moments of X and 7 functions (Section 9.6.1).

In the conservative case, the dominant term of X and Y leads to the very simple result

lim X(h b, μ) = lim 7 ( 1 , ft, μ) = [j80(l, ft)]"1

μ-> oo μ-*αο

In the limit b = oo we obtain from the expansion of the X function, the expansion of the Η function, which was given with one more term in Section 8.3.2 (see also Fig. 8.4).

Similar methods can be employed to derive an expression for the po in t -direction gain valid for μ > 1. The result, based on a straightforward substitution of μ = oo in expressions (Section 9.3.1) for G(a, b, τ, μ) and (d/cfy/)G(a, b, τ, μ) combined with some manipulation with the Milne equation, is

G(a, b, τ, μ) = (1 - ά)~ *[1 - (a/2)g0(a, b, τ) - (a/2)g0(a, b,b- τ)] - M l - α)Υ

ι{τ - (ab/2)g0(a, b,b - τ) + (a/2)gi(a, b, τ)

-(α/2)θί(α, Μ - τ ) ] + . . ·

where g0 and g1 are moments of the gain function (Section 9.3.1). Derivations based on the gain definition in terms of the successive scattering solution gave the same result, for both the "d i rec t" experiment and the "reverse" experiment (see Section 7.3.3). Substitution of τ = 0 or τ = b brings back the expansions of X and 7 presented in Display 9.6.

9.6.4 Eigenvalues in the Doubling Method

Solutions by means of the doubling method have the form of infinite series. The limit to which the ratio of successive terms in this series tends upon doubling from \b to b is called the first eigenvalue in the doubling method, d(a, b). The values of this limit for isotropic scattering are collected in Table 19. It is no surprise that these values are substantially smaller than those in the successive scattering method, because each suceeding term includes a large number of scatterings in the separate layers.

The numbers in Table 19 may be used in order to gain an impression of how many terms are required in the solution for any desired degree of accuracy. The numbers were obtained numerically during test runs of the doubling procedure on which the reflection and transmission tables are based. The fourth decimal may be inaccurate. Note that the last column of Table 19 is repeated in Table 42 (Section 13.4).

Physically, each succeeding term in the doubling series arises by two more reflections at the interface, one up and one down. Hence, for the case of identical layers, which is assumed here, the square root of CI is the largest eigenvalue of the equation

rli2F = RF


T A B L E 1 9

E i g e n v a l u e s i n t h e D o u b l i n g M e t h o d , I s o t r o p i c S c a t t e r i n g

D o u b l i n g F i r s t e i g e n v a l u e C i

F r o m \b T o è a = 0 . 2 a = 0 . 4 a = 0 . 6 a = 0 . 8 a = 0 . 9 a = 0 . 9 5 a = 0 . 9 9 a = 1

0 . 25 0 .5 0 . 0 2 6 6 0 . 0 4 0 9 0 . 0 4 7 1 0 .5 1 0 . 0 0 1 7 0 . 0 0 8 4 0 . 0 2 3 1 0 . 0 5 1 9 0 . 0 7 4 9 0 . 0 8 9 5 0 . 1 0 2 9 0 . 1 0 6 5

1 2 0 . 0 0 2 2 0 . 0 1 1 4 0 . 0 3 4 1 0 . 0 8 6 8 0 . 1 3 6 4 0 . 1 7 1 6 0 . 2 0 6 8 0 . 2 1 6 8 2 4 0 . 0 0 2 4 0 . 0 1 2 5 0 . 0 3 9 8 0 . 1 1 3 6 0 . 1 9 9 2 0 . 2 7 1 6 0 . 3 5 6 0 0 . 3 8 2 5

4 8 0 . 0 0 2 4 0 . 0 1 2 7 0 . 0 4 0 8 0 . 1 2 1 1 0 . 2 3 0 0 0 . 3 4 1 5 0 . 5 0 8 6 0 . 5 7 3 0

8 16 0 . 0 0 2 4 0 . 0 1 2 7 0 . 0 4 0 8 0 . 1 2 2 1 0 . 2 3 4 3 0 . 3 6 0 2 0 . 6 0 4 5 0 . 7 3 8 5 16 32 0 . 0 0 2 4 0 . 0 1 2 7 0 . 0 4 0 8 0 . 1 2 2 1 0 . 2 3 4 4 0 . 3 6 1 1 0 . 6 3 1 7 0 . 8 5 3 4

oo oo 0 . 0 0 2 4 0 . 0 1 2 7 0 . 0 4 0 8 0 . 1 2 2 1 0 . 2 3 4 4 0 . 3 6 1 2 0 . 6 3 2 0 1 .0000

where R(a, \b, μ, μ 0) is the reflection function for the half layer, Fm(a, b, μ) the mth eigenfunction, the multiplication at the right-hand side signifies a μ integration by the definition in Section 7.1.1, and £J/

2 is the mth eigenvalue.

In the limit a 1, b oo, d approaches 1 and Fm approaches the vector U (which is simply a constant), for we know that in this limit

RU = U

An impression of the behavior or ζγ near this limit has been obtained in three ways.

(1) Let b = oo, a near 1; write a = 1 — t2. Since U must still be a good

approximation to the eigenfunction, a very good approximation to the eigenvalue can be obtained from

C}(2apP = URMU/UU = 1 - 2ί^(α) = 1 - ( 4 / ^ ί + · · ·

(further detail in Section 13.4). Hence plotting y = [1 — C J/ 2

( a ) ]2 against a,

we must obtain a curve which approaches for a 1 the asymptote

y = ¥ ( i - α)

(2) Let b be very large, a = 1. The same reasoning gives

Cli2apP = URQb, l)U/UU = 1 - mb + 1.42)

(see Display 9.2). (3) Let η^φ) be the eigenvalue for successive scattering found in Section

7.4.1 for arbitrary thickness b. The layer becomes critical if a = ηϊ1. Hence, if

two layers of thickness \b are superposed and the doubling method applied, the combined layer must again be critical at the same value of a. This requires that

CI [>7 i (F>r \ f e ] = l


Linear s c a l e

of

0.2

0.4 h

0.6

0.8

1.0

0.01

0.0

a b s o r p t i o n

c r i t i c a l l i m i t

g e n e r a t i o n

0.6 1.0 1.2

F i g . 9.18. F i r s t e i g e n v a l u e ς, a p p e a r i n g in d o u b l i n g f r o m t h i c k n e s s jb t o b. F o r e x p l a n a t i o n ,

see t e x t .

These three approaches have been used in Fig. 9.18 to present a survey of CI(A, b) in the entire subcritical domain. Further refinements would be possible but hardly seem necessary in view of the intended application.

A characteristic feature of the doubling method is that the second eigenvalue, ζ2(α, b\ is extremely small. This is, of course, quite welcome in the practical computation, because it means a rapid approach to a geometric series. High accuracy can often be reached by doubling only once. The final answer is then obtained by adding to the first term the second term divided by 1 - f ι(α, b).

It would require great care to determine ζ2 with any precision, either analytically or numerically. We have not deemed it necessary to improve the accuracy of the doubling computat ion solely for this purpose. However, we shall see in Section 13.4 that the value reached in the limit a = 1, b = oo can, to a very good approximation, be linked to the extrapolation length, which gives in the present case

ζ\/2 = 2 4 ^ oo) - 17 = 0.0507, ζ2 = 0.0026

See Fig. 13.12 for more detail.

References 235

R E F E R E N C E S

B e l l m a n , R . E . , K a g i w a d a , H . H . , K a l a b a , R . E . , a n d U e n o , S. ( 1 9 6 6 ) . J. Quant. Spectrosc. Radiât.

Transfer 6 , 4 7 9 .

B e l l m a n , R . E . , P o o n , P . T . Y . , a n d U e n o , S. ( 1 9 7 3 ) . J. Quant. Spectrosc. Radiât. Transfer 1 3 , 1 2 7 3 .

C a l d w e l l , J . ( 1 9 7 1 ) . Astrophys. J. 1 6 3 , 1 1 1 .

C a r l s t e d t , J . L . , a n d M u l l i k i n , T . W . ( 1 9 6 6 ) . Astrophys. J. Suppl. 12 , 4 4 9 .

C h a m b e r l a i n , S . , a n d S o b o u t i , Y . ( 1 9 6 2 ) . Astrophys. J. 1 3 5 , 9 2 5 .


N e w Y o r k . A l s o D o v e r , N e w Y o r k 1960 .

C h a n d r a s e k h a r , S. , a n d E l b e r t , D . ( 1 9 5 2 ) . Astrophys. J. 115 , 2 4 4 , 2 6 9 .

C o h e n , H . ( 1 9 6 9 ) . J. Quant. Spectrosc. Radiât. Transfer 9 , 9 3 1 .

H a l p e r n , O . , a n d L u n e b e r g , R . K . ( 1 9 4 9 ) . Phys. Rev. 7 6 , 1 8 1 1 .

I v a n o v , V . V . ( 1 9 6 4 ) . Astron. Zh. 4 1 , 1 0 9 7 .

K a p e r , H . G . , S h u l t i s , J . K . a n d V e n i n g a , J . G . ( 1 9 7 0 ) . J. Comput. Phys. 6 , 2 8 8 .

L o s k u t o v , V . M . ( 1 9 7 3 ) . Astrofizika (Russian) 9 , 361 [English transi. : Astrophysics 9 , 2 0 5 ( 1 9 7 5 ) ] .

L y o t , B . ( 1 9 2 9 ) . Ann. Observ. Paris (Meudon) 8.

M a y e r s , D . F . ( 1 9 6 2 ) . Mon. Not. R. Astron. Soc. 1 2 3 , 4 7 1 .

N a g i r n e r , D . I . ( 1 9 7 3 ) . Astrofizika (Russian) 9 , 3 4 7 [English transi. : Astrophyics 9 , 196 ( 1 9 7 5 ) ] .

S o b o l e v , V . V . ( 1 9 5 7 ) . Astron. Zh. 3 4 , 3 3 6 [English transi: Sov. Astron.-A.J. 1, 3 3 2 ] .

S o b o u t i , Y . ( 1 9 6 2 a ) . Astrophys. J. 135 , 9 3 8 .

S o b o u t i , Y . ( 1 9 6 2 b ) . Astrophys. J. Suppl. 7, 4 1 1 .

v a n d e H u l s t , H . C . ( 1 9 6 4 ) . Icarus 3 , 3 3 6 .

v a n d e H u l s t , H . C , a n d G r o s s m a n , K . ( 1 9 6 8 ) . I n " T h e A t m o s p h e r e s o f V e n u s a n d M a r s " ( J . C .

B r a n d t a n d M . B . M c E l r o y , e d s . ) , p . 3 5 . G o r d o n a n d B r e a c h , N e w Y o r k ,

v a n d e H u l s t , H . C , a n d T e r h o e v e , F . G . ( 1 9 6 6 ) . Bull. Astron. Inst. Neth. 18 , 3 7 7 .


T A B L E 12

R e f l e c t i o n a n d T r a n s m i s s i o n b y a F i n i t e L a y e r , I s o t r o p i c S c a t t e r i n g

I n t e n s i t i e s o u t a t T o p

VECTOR

b = 0 . 0 3 1 2 5

μ = 0 . 7 /x = C . 9 μ. = \.0 AVERAGE Ν

FLUX U

F I R S T 0 R 0 E R 2 . 5 0 0 0 0 0 . 5 8 0 9 2 0 . 2 1 2 9 7 SECONO ORDER 0 . 1 3 3 5 5 THIRD ORDER 0 . 0 0 8 9 9

SUMS 0 = 0 . 2 0 0 . 5 0 5 4 2 α - 0 . 4 0 1 . 0 2 1 9 6 α = 0 . 6 0 1 . 5 5 0 1 0 α = 0 . 8 0 2 . 0 9 0 3 4

α - 0 . 9 0 2 . 3 6 5 1 6 α - 0 . 9 5 2 . 5 0 3 7 7 ο = 0 . 9 9 2 . 6 1 5 2 5 α = 1 . C 0 2 . 6 4 3 2 0

b = 0 . 0 3 1 2 5

F I R S T ORDER 0 . 8 3 3 3 3 SECOND ι ORDER C . 0 4 8 5 7 THIRD ORDER C . 0 0 3 3 1

SUMS α = 0 . 2 0 C . 1 6 8 6 4 α = 0 . 4 0 0 . 3 4 1 3 2 α = 0 . 6 0 0 . 5 1 8 2 3 α = 0 . 8 0 C . 6 9 9 5 4

α = 0 . 9 0 0 . 7 9 1 9 1 α = 0 . 9 5 0 . 8 3 8 5 4 α = 0 . 9 9 0 . 8 7 6 0 5 α = 1 . 0 0 0 . 8 8 5 4 6

0 . 1 3 0 3 0 0 . 0 9 3 8 5 0 . 0 7 3 3 4 0 . 0 6 6 1 1 0 . 2 6 7 1 1 0 . 0 0 8 9 4 0 . 0 0 6 4 4 0 . 0 0 5 0 4 0 . 0 0 4 5 4 0 . 0 1 7 9 7 0 . 0 0 0 6 2 0 . 0 0 0 4 4 0 . 0 0 C 3 5 0 . 0 0 0 3 1 0 . 0 0 1 2 3

. 0 5 3 5 9 0 . 0 3 8 6 0

. 0 8 1 5 4 0 . 0 5 8 7 3 , 0 3 0 1 6 , 0 4 5 8 9 , 0 6 2 0 8

0 1 3 4 1 0 . 0 5 4 1 5 0 2 7 1 9 0 . 1 0 9 8 0 0 4 1 3 7 0 . 1 6 7 0 1 0 5 5 9 6 0 . 2 2 5 8 6

1 2 5 3 8 0 0 8 5 9 0 0 0 5 9

0 2 5 4 2 0 5 1 5 6 0 7 8 4 5 1 0 6 1 2

0 . 2 0 4 2 9 0 . 1 2 4 9 9 0 . 0 9 0 0 3 0 . 0 7 0 3 5 0 . 0 . 2 1 6 4 3 0 . 1 3 2 4 2 0 . 0 9 5 3 8 0 . 0 7 4 5 3 0 . 0 . 2 2 6 2 0 0 . 1 3 8 4 0 0 . 0 9 9 6 9 0 . 0 7 7 9 0 0 , 0 . 2 2 8 6 6 0 . 1 3 9 9 0 0 . 1 0 0 7 7 0 . 0 7 8 7 5 0 . 0 7 0 9 9 0 . 2 8 6 4 1

0 6 3 4 2 0 . 2 5 5 9 1 0 . 1 2 0 2 6 0 6 7 1 9 0 . 2 7 U 0 0 . 1 2 7 4 1 0 7 0 2 3 0 . 2 8 3 3 4 1 3 3 1 6

1 3 4 6 0

. 0 7 8 3 6 0 . 0 4 7 9 7 0 . 0 3 4 5 7 0 . 0 2 7 0 2 , 0 0 5 3 8 0 . 0 0 3 3 0 0 . 0 0 2 3 7 0 . 0 0 1 8 6

0 2 4 3 6 0 . 0 9 7 1 5 0 . 0 4 6 1 1 0 0 1 6 7 0 . 0 0 6 6 1 0 . 0 0 3 1 7

0 . 0 0 0 1 3 0 . 0 0 0 1 2 0 . 0 0 0 4 5 0 . 0 0 0 2 2

0 . 0 1 4 2 2 O . O l l l l 0 . 0 0 4 9 4 0 . 0 1 9 7 0 0 . 0 1 0 0 2 0 . 0 3 9 9 5

0 . 0 0 9 3 5 0 . 0 1 8 9 7

0 . 1 3 3 2 7 0 . 0 4 9 0 4 0 . 0 3 0 0 2 0 . 0 2 1 6 3 0 . 0 1 6 9 1 0 . 0 1 5 2 4 0 . 0 6 0 7 7 0 . 0 2 8 8 6 0 . 1 8 0 2 7 0 . 0 6 6 3 3 0 . 0 4 0 6 1 0 . 0 2 9 2 6 0 . 0 2 2 8 7 0 . 0 2 0 6 2 0 . 0 8 2 2 0 0 . 0 3 9 0 3

0 . 2 0 4 2 9 0 . 0 7 5 1 7 0 . 0 4 6 0 2 0 . 0 3 3 1 6 0 . 0 2 5 9 2 0 . 0 2 3 3 6 0 . 0 9 3 1 4 0 . 0 4 4 2 4 0 . 2 1 6 4 3 0 . 0 7 9 6 4 0 . 0 4 8 7 6 0 . 0 3 5 1 3 0 . 0 2 7 4 6 0 . 0 2 4 7 5 0 . 0 9 8 6 7 0 . 0 4 6 8 6 0 . 2 2 6 2 C 0 . 0 8 3 2 4 0 . 0 5 0 9 6 0 . 0 3 6 7 2 0 . 0 2 8 7 0 0 . 0 2 5 8 7 0 . 1 0 3 1 3 0 . 0 4 8 9 8 0 . 2 2 8 6 6 0 . 0 8 4 1 4 0 . 0 5 1 5 1 0 . 0 3 7 1 2 0 . 0 2 9 0 1 0 . 0 2 6 1 5 0 . 1 0 4 2 5 0 . 0 4 9 5 1

b = 0 . 0 3 1 2 5

F I R S T ORDER SECOND ORDER THIRD ORDER

Mo = 0 . 5

0 . 5 0 0 0 0 0 . 1 3 0 3 0 0 . 0 4 7 9 7 0 . 0 2 9 3 8 0 . 0 2 1 1 7 0 . 0 1 6 5 4 0 . 0 1 4 9 1 0 . 0 5 9 3 4 0 . 0 2 8 2 3 0 . 0 2 9 6 7 0 . 0 0 8 9 4 0 . 0 0 3 3 0 0 . 0 0 2 0 2 0 . 0 0 1 4 5 0 . 0 0 1 1 4 0 . 0 0 1 0 2 0 . 0 0 4 0 5 0 . 0 0 1 9 4 0 . 0 0 2 0 2 0 . 0 0 0 6 2 0 . 0 0 0 2 3 0 . 0 0 0 1 4 0 . 0 0 0 1 0 0 . 0 0 0 0 8 0 . 0 0 0 0 7 0 . 0 0 0 2 8 0 . 0 0 0 1 3

SUMS α 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0

0 . 9 0 0 . 9 5 0 . 9 9 I . C O

C . 1 0 1 2 0 C . 2 0 4 8 8 0 . 3 1 1 1 4 0 . 4 2 0 0 9

C . 4 7 5 6 1 0 . 5 0 3 6 3 0 . 5 2 6 1 9 0 . 5 3 1 8 4

0 . 0 2 6 4 2 0 . 0 0 9 7 3 0 . 0 0 5 9 6 0 . 0 0 4 2 9 0 . 0 0 3 3 5 0 . 0 0 3 0 2 0 . 0 1 2 0 3 0 . 0 0 5 7 2 0 . 0 5 3 5 9 0 . 0 1 9 7 3 0 . 0 1 2 0 8 0 . 0 0 8 7 1 0 . 0 0 6 8 0 0 . 0 0 6 1 3 0 . 0 2 4 4 0 0 . 0 1 1 6 1 0 . 0 8 1 5 4 0 . 0 3 0 0 2 0 . 0 1 8 3 8 0 . 0 1 3 2 5 0 . 0 1 0 3 5 0 . 0 0 9 3 3 0 . 0 3 7 1 2 0 . 0 1 7 6 7 0 . 1 1 0 2 9 0 . 0 4 0 6 1 0 . 0 2 4 8 7 0 . 0 1 7 9 2 0 . 0 1 4 0 1 0 . 0 1 2 6 3 0 . 0 5 0 2 1 0 . 0 2 3 9 0

0 . 1 2 4 9 9 0 . 0 4 6 0 2 0 . 0 2 8 1 8 0 . 0 2 0 3 1 0 . 0 1 5 8 7 0 . 0 1 4 3 1 0 . 0 5 6 9 0 0 . 0 2 7 0 8 0 . 1 3 2 4 2 0 . 0 4 8 7 6 0 . 0 2 9 8 6 0 . 0 2 1 5 1 0 . 0 1 6 8 1 0 . 0 1 5 1 6 0 . 0 6 0 2 8 0 . 0 2 8 6 9 0 . 1 3 8 4 0 0 . 0 5 0 9 6 0 . 0 3 1 2 1 0 . 0 2 2 4 9 0 . 0 1 7 5 7 0 . 0 1 5 8 4 0 . 0 6 3 0 0 0 . 0 2 9 9 9 0 . 1 3 9 9 0 0 . 0 5 1 5 1 0 . 0 3 1 5 4 0 . 0 2 2 7 3 0 . 0 1 7 7 6 0 . 0 1 6 0 2 0 . 0 6 3 6 9 0 . 0 3 0 3 1

b = 0 . 0 3 1 2 5


SUMS α 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0

0 . 3 5 7 1 4 0 . 0 2 1 3 6 C 0 0 1 4 6

0 7 2 2 9 1 4 6 3 7 2 2 2 3 0 3 0 0 1 7

0 . 0 9 3 8 5 0 . 0 3 4 5 7 0 . 0 2 1 1 7 0 . 0 1 5 2 5 0 . 0 1 1 9 2 0 . 0 1 0 7 5 0 . 0 4 2 7 1 0 . 0 2 0 3 4 0 . 0 0 6 4 4 0 . 0 0 2 3 7 0 . 0 0 1 4 5 0 . 0 0 1 0 5 0 . 0 0 0 8 2 0 . 0 0 0 7 4 0 . 0 0 2 9 2 0 . 0 0 1 4 0 0 . 0 0 0 4 4 0 . 0 0 0 1 6 0 . 0 0 0 1 0 0 . 0 0 0 0 7 0 . 0 0 0 0 6 0 . 0 0 0 0 5 0 . 0 0 0 2 0 0 . 0 0 0 1 0

0 . 0 1 9 0 3 0 . 0 0 7 0 1 0 . 0 0 4 2 9 0 . 0 0 3 0 9 0 . 0 0 2 4 2 0 . 0 0 2 1 8 0 . 0 0 8 6 6 0 . 0 0 4 1 2 0 . 0 3 8 6 C 0 . 0 1 4 2 2 0 . 0 0 8 7 1 0 . 0 0 6 2 7 0 . 0 0 4 9 0 0 . 0 0 4 4 2 0 . 0 1 7 5 7 0 . 0 0 8 3 7 0 . 0 5 8 7 3 0 . 0 2 1 6 3 0 . 0 1 3 2 5 0 . 0 0 9 5 5 0 . 0 0 7 4 6 0 . 0 0 6 7 3 0 . 0 2 6 7 2 0 . 0 1 2 7 3 0 . 0 7 9 4 5 0 . 0 2 9 2 6 0 . 0 1 7 9 2 0 . 0 1 2 9 1 0 . 0 1 0 0 9 0 . 0 0 9 1 0 0 . 0 3 6 1 5 0 . 0 1 7 2 2

0 . 9 0 0 . 3 3 9 8 6 0 . 0 9 0 0 3 0 . 0 3 3 1 6 0 . 0 2 0 3 1 0 . 0 1 4 6 3 0 . 0 1 1 4 4 0 . 0 1 0 3 1 0 . 0 4 0 9 6 0 . 0 1 9 5 1 0 . 9 5 0 . 3 5 9 9 0 0 . 0 9 5 3 8 0 . 0 3 5 1 3 0 . 0 2 1 5 1 0 . 0 1 5 5 0 0 . 0 1 2 1 2 0 . 0 1 0 9 2 0 . 0 4 3 3 9 0 . 0 2 0 6 7 0 . 9 9 0 . 3 7 6 0 2 0 . 0 9 9 6 9 0 . 0 3 6 7 2 0 . 0 2 2 4 9 0 . 0 1 6 2 0 0 . 0 1 2 6 6 0 . 0 1 1 4 2 0 . 0 4 5 3 5 0 . 0 2 1 6 1 1 . 0 0 0 . 3 8 0 0 7 0 . 1 0 0 7 7 0 . 0 3 7 1 2 0 . 0 2 2 7 3 0 . 0 1 6 3 8 0 . 0 1 2 8 0 0 . 0 1 1 5 4 0 . 0 4 5 8 5 0 . 0 2 1 8 4

9 Isotropic Scattering, Finite Slabs 237

T A B L E 12 (continued)

I n t e n s i t i e s o u t a t B o t t o m

VECTOR t= 0 . 0 = 0 . 1 / i = 0 . 3 = 0 . 7 AVERAGE Ν

FLUX υ

b = 0 . 0 3 1 2 5 ZERO ORDER F I R S T ORDER SECOND ORDER THIRD ORDER

SOMSa 0 . 2 0 0 . 4 C 0 . 6 0 0 . 8 0

0 . 9 0 0 . 9 5 0 . 9 9 I . 0 0

μ0 = 0 . 1 0 . 0 PEAK 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 3 . 6 5 8 0 8 0 . 7 3 1 6 2 1 . 8 2 9 C 4 C . 5 7 1 5 7 0 . 2 1 1 8 2 0 . 1 2 9 8 7 0 . 0 9 3 6 3 0 . 0 7 3 2 1 0 . 0 6 6 0 0 0 . 2 5 6 7 1 0 . 1 2 4 5 6 0 . 1 2 8 3 5 0 . 0 3 9 5 4 0 . 0 1 4 5 9 0 . 0 0 8 9 4 0 . 0 0 6 4 4 0 . 0 0 5 0 3 0 . 0 0 4 5 4 0 . 0 1 7 8 6 0 . 0 0 8 5 8 0 . 0 0 8 9 3 0 . 0 0 2 7 3 0 . 0 0 1 0 0 0 . 0 0 0 6 2 0 . 0 0 0 4 4 0 . 0 0 0 3 5 0 . 0 0 0 3 1 0 . 0 0 1 2 3 0 . 0 0 0 5 9

0 . 3 7 1 0 1 0 . 1 1 5 9 2 0 . 0 4 2 9 6 0 . 0 2 6 3 4 0 . 0 1 8 9 9 0 . 0 1 4 8 5 0 . 0 1 3 3 9 3 . 7 1 0 1 4 0 . 7 5 6 8 8 0 . 7 5 2 7 4 0 . 2 3 5 1 4 0 . 0 8 7 1 3 0 . 0 5 3 4 2 0 . 0 3 8 5 1 0 . 0 3 0 1 1 0 . 0 2 7 1 5 3 . 7 6 3 7 0 0 . 7 8 2 8 5 1 . 1 4 5 6 4 0 . 3 5 7 7 9 0 . 1 3 2 5 7 0 . 0 8 1 2 8 0 . 0 5 8 6 0 0 . 0 4 5 8 1 0 . 0 4 1 3 1 3 . 8 1 8 8 1 0 . 8 0 9 5 8 1 . 5 5 0 2 2 0 . 4 8 4 0 4 0 . 1 7 9 3 4 0 . 1 0 9 9 5 0 . 0 7 9 2 7 0 . 0 6 1 9 7 0 . 0 5 5 8 8 3 . 8 7 5 5 4 0 . 8 3 7 0 8

1 . 7 5 7 0 4 0 . 5 4 8 5 6 0 . 2 0 3 2 4 0 . 1 2 4 6 0 0 . 0 8 9 8 3 0 . 0 7 0 2 3 0 . 0 6 3 3 2 3 . 9 0 4 5 4 0 . 8 5 1 1 3 1 . 8 6 1 6 2 0 . 5 8 1 1 8 0 . 2 1 5 3 2 0 . 1 3 2 0 1 0 . 0 9 5 1 7 0 . 0 7 4 4 1 0 . 0 6 7 0 9 3 . 9 1 9 2 0 0 . 8 5 8 2 4 1 . 9 4 5 8 5 0 . 6 0 7 4 5 0 . 2 2 5 0 5 0 . 1 3 7 9 7 0 . 0 9 9 4 7 0 . 0 7 7 7 7 0 . 0 7 0 1 2 3 . 9 3 1 0 0 0 . 8 6 3 9 6 1 . 9 6 6 9 8 0 . 6 1 4 0 4 0 . 2 2 7 4 9 0 . 1 3 9 4 7 0 . 1 0 0 5 5 0 . 0 7 8 6 1 0 . 0 7 0 8 8 3 . 9 3 3 9 7 0 . 8 6 5 4 0


SUMS a = 0 . 2 0 = 0 . 4 0 = 0 . 6 0 = 0 . 8 0

= 0 . 9 0 = 0 . 9 5 = 0 . 9 9 = 1 . 0 0

μο = 0 . 3 0 . 0 0 . 0 PEAK 0 . 0 0 . 0 0 . 0 0 . 0 1 . 5 0 1 7 9 0 . 9 0 1 0 8 C . 7 5 0 9 0 0 . 2 1 1 8 2 0 . 0 7 8 2 2 0 . 0 4 7 9 2 0 . 0 3 4 5 4 0 . 0 2 7 0 0 0 . 0 2 4 3 4 0 . 0 9 5 8 7 0 . 0 4 6 0 1 C . 0 4 7 9 3 0 . 0 1 4 5 9 0 . 0 0 5 3 3 0 . 0 0 3 3 0 0 . 0 0 2 3 7 0 . 0 0 1 8 6 0 . 0 0 1 6 7 0 . 0 0 6 6 0 0 . 0 0 3 1 6 0 . 0 0 3 3 0 0 . 0 0 1 0 0 0 . 0 0 0 3 7 0 . 0 0 0 2 3 0 . 0 0 0 1 6 0 . 0 0 0 1 3 0 . 0 0 0 1 2 0 . 0 0 0 4 5 0 . 0 0 0 2 2

0 . 1 5 2 1 2 0 . 0 4 2 9 6 0 . 0 1 5 8 6 0 . 0 0 9 7 2 0 . 0 0 7 0 0 0 . 0 0 5 4 8 0 . 0 0 4 9 4 1 . 5 2 1 2 3 0 . 9 1 0 4 1 C . 3 0 8 2 4 0 . 0 8 7 1 3 0 . 0 3 2 1 7 0 . 0 1 9 7 1 0 . 0 1 4 2 1 O . O l i i i 0 . 0 1 0 0 1 1 . 5 4 1 2 2 0 . 9 2 0 0 0 C . 4 6 8 5 4 0 . 1 3 2 5 7 0 . C 4 8 9 5 0 . 0 2 9 9 9 0 . 0 2 1 6 2 0 . 0 1 6 9 0 0 . 0 1 5 2 3 1 . 5 6 1 7 9 0 . 9 2 9 8 7 C . 6 3 3 1 8 0 . 1 7 9 3 4 0 . 0 6 6 2 2 0 . 0 4 0 5 7 0 . 0 2 9 2 4 0 . 0 2 2 8 6 0 . 0 2 0 6 1 1 . 5 8 2 9 5 0 . 9 4 0 0 3

0 . 7 1 7 2 0 0 . 2 0 3 2 4 0 . 0 7 5 0 4 0 . 0 4 5 9 8 0 . 0 3 3 1 4 0 . 0 2 5 9 0 0 . 0 2 3 3 5 1 . 5 9 3 7 7 0 . 9 4 5 2 2 0 . 7 5 9 6 4 0 . 2 1 5 3 2 0 . 0 7 9 5 0 0 . 0 4 8 7 1 0 . 0 3 5 1 1 0 . 0 2 7 4 4 0 . 0 2 4 7 4 1 . 5 9 9 2 4 0 . 9 4 7 8 4 0 . 7 9 3 6 0 0 . 2 2 5 0 5 0 . 0 8 3 0 9 0 . 0 5 0 9 1 0 . 0 3 6 6 9 0 . 0 2 8 6 8 0 . 0 2 5 8 6 1 . 6 0 3 6 4 0 . 9 4 9 9 6 0 . 8 0 2 3 7 0 . 2 2 7 4 9 0 . 0 8 4 0 0 0 . 0 5 1 4 6 0 . 0 3 7 0 9 0 . 0 2 8 9 9 0 . 0 2 6 1 4 1 . 6 0 4 7 4 0 . 9 5 0 4 9


SUMS a 0 . 2 0 = 0 . 4 0 = 0 . 6 0 = 0 . 8 0

= 0 . 9 0 = 0 . 9 5 = 0 . 9 9 = 1 . 0 0

μ0 = 0 . 5 C O 0 . 0 0 . 0 PEAK 0 . 0 0 . 0 0 . 0 0 . 9 3 9 4 1 0 . 9 3 9 4 1 C . 4 6 9 7 1 0 . 1 2 9 8 7 0 . 0 4 7 9 2 0 . 0 2 9 3 6 0 . 0 2 1 1 6 0 . 0 1 6 5 4 0 . 0 1 4 9 1 0 . 0 5 8 8 7 0 . 0 2 8 1 9 C . 0 2 9 4 3 0 . 0 0 8 9 4 0 . 0 0 3 3 0 0 . 0 0 2 0 2 0 . 0 0 1 4 5 0 . 0 0 1 1 4 0 . 0 0 1 0 2 0 . 0 0 4 0 4 0 . 0 0 1 9 4 0 . C 0 2 0 2 0 . 0 0 0 6 2 0 . 0 0 0 2 3 0 . 0 0 0 1 4 0 . 0 0 0 1 0 0 . 0 0 0 0 8 0 . 0 0 0 0 7 0 . 0 0 0 2 8 0 . 0 0 0 1 3

0 . 0 9 5 1 4 0 . 0 2 6 3 4 0 . 0 0 9 7 2 0 . 0 0 5 9 5 0 . 0 0 4 2 9 0 . 0 0 3 3 5 0 . 0 0 3 0 2 0 . 9 5 1 3 5 0 . 9 4 5 1 3 0 . 1 9 2 7 3 C . 0 5 3 4 2 0 . 0 1 9 7 1 0 . 0 1 2 0 7 0 . 0 0 8 7 0 0 . 0 0 6 8 0 0 . 0 0 6 1 3 0 . 9 6 3 6 3 0 . 9 5 1 0 1 C . 2 9 2 8 8 0 . 0 8 1 2 8 0 . 0 2 9 9 9 0 . 0 1 8 3 7 0 . 0 1 3 2 4 0 . 0 1 0 3 5 0 . 0 0 9 3 3 0 . 9 7 6 2 5 0 . 9 5 7 0 6 0 . 3 9 5 7 0 0 . 1 0 9 9 5 0 . 0 4 0 5 7 0 . 0 2 4 8 5 0 . 0 1 7 9 1 0 . 0 1 4 0 0 0 . 0 1 2 6 2 0 . 9 8 9 2 5 0 . 9 6 3 2 8

0 . 4 4 8 1 5 0 . 1 2 4 6 0 0 . C 4 5 9 8 0 . 0 2 8 1 6 0 . 0 2 0 3 0 0 . 0 1 5 8 7 0 . 0 1 4 3 0 0 . 9 9 5 8 9 0 . 9 6 6 4 6 C . 4 7 4 6 4 0 . 1 3 2 0 1 0 . 0 4 8 7 1 0 . 0 2 9 8 4 0 . 0 2 1 5 0 0 . 0 1 6 8 1 0 . 0 1 5 1 5 0 . 9 9 9 2 4 0 . 9 6 8 0 7 0 . 4 9 5 9 6 0 . 1 3 7 9 7 0 . 0 5 0 9 1 0 . 0 3 1 1 9 0 . 0 2 2 4 8 0 . 0 1 7 5 7 0 . 0 1 5 8 4 1 . 0 0 1 9 5 0 . 9 6 9 3 6 C . 5 0 1 3 1 0 . 1 3 9 4 7 0 . 0 5 1 4 6 0 . 0 3 1 5 2 0 . 0 2 2 7 2 0 . 0 1 7 7 6 0 . 0 1 6 0 1 1 . 0 0 2 6 2 0 . 9 6 9 6 9


SUMS a 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a ^ 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = I . 0 0

μο = 0 . 7 0 . 0 0 . 0 0 . 0 0 . 0 PEAK 0 . 0 0 . 0 0 . 6 8 3 1 0 0 . 9 5 6 3 4 C . 3 4 1 5 5 0 . 0 9 3 6 3 0 . 0 3 4 5 4 0 . 0 2 1 1 6 0 . 0 1 5 2 5 0 . 0 1 1 9 2 0 . 0 1 0 7 5 0 . 0 4 2 4 7 0 . 0 2 0 3 2 0 . 0 2 1 2 4 0 . 0 0 6 4 4 0 . 0 0 2 3 7 0 . 0 0 1 4 5 0 . 0 0 1 0 5 0 . 0 0 0 8 2 0 . 0 0 0 7 4 0 . 0 0 2 9 1 0 . 0 0 1 4 0 C . 0 0 1 4 6 0 . 0 0 0 4 4 0 . 0 0 0 1 6 0 . 0 0 0 1 0 0 . 0 0 0 0 7 0 . 0 0 0 0 6 0 . 0 0 0 0 5 0 . 0 0 0 2 0 0 . 0 0 0 1 0

C . 0 6 9 1 7 0 . 0 1 8 9 9 0 . 0 0 7 0 0 0 . 0 0 4 2 9 0 . 0 0 3 0 9 0 . 0 0 2 4 2 0 . 0 0 2 1 8 0 . 6 9 1 7 1 0 . 9 6 0 4 6 C . 1 4 0 1 1 C . 0 3 8 5 1 0 . 0 1 4 2 1 0 . 0 0 8 7 0 0 . 0 0 6 2 7 0 . 0 0 4 9 0 0 . 0 0 4 4 2 0 . 7 0 0 5 7 0 . 9 6 4 7 0 C 2 1 2 9 0 0 . 0 5 8 6 0 0 . 0 2 1 6 2 0 . 0 1 3 2 4 0 . 0 0 9 5 4 0 . 0 0 7 4 6 0 . 0 0 6 7 2 0 . 7 0 9 6 8 0 . 9 6 9 0 6 C . 2 8 7 6 2 0 . 0 7 9 2 7 0 . 0 2 9 2 4 0 . 0 1 7 9 1 0 . 0 1 2 9 1 0 . 0 1 0 0 9 0 . 0 0 9 1 0 0 . 7 1 9 0 5 0 . 9 7 3 5 4

C 3 2 5 7 3 0 . 0 8 9 8 3 0 . 0 3 3 1 4 0 . 0 2 0 3 0 0 . 0 1 4 6 3 0 . 0 1 1 4 3 0 . 0 1 0 3 1 0 . 7 2 3 8 4 0 . 9 7 5 8 3 C . 3 4 4 9 7 0 . 0 9 5 1 7 0 . 0 3 5 1 1 0 . 0 2 1 5 0 0 . 0 1 5 5 0 0 . 0 1 2 1 1 0 . 0 1 0 9 2 0 . 7 2 6 2 6 0 . 9 7 6 9 9 0 . 3 6 0 4 6 0 . 0 9 9 4 7 0 . 0 3 6 6 9 0 . 0 2 2 4 8 0 . 0 1 6 2 0 0 . 0 1 2 6 6 0 . 0 1 1 4 1 0 . 7 2 8 2 1 0 . 9 7 7 9 3 0 . 3 6 4 3 5 0 . 1 0 0 5 5 0 . 0 3 7 0 9 0 . 0 2 2 7 2 0 . 0 1 6 3 7 0 . 0 1 2 8 0 0 . 0 1 1 5 4 0 . 7 2 8 7 0 0 . 9 7 8 1 6




VECTOR

b = 0 . 0 3 1 2 5


/ x = 0 . 0 /x = C . l / x = 0 . 3 f t = 0 . 5 μ = 0 . 7 t= C . 9 μ = 1 . 0 AVERAGE Ν

FLUX U

0 . 9

C 2 7 7 7 8 0 . 0 7 3 3 4 0 . 0 2 7 0 2 0 . 0 1 6 5 4 0 . 0 1 1 9 2 0 . 0 0 9 3 2 0 . 0 0 8 4 0 0 . 0 3 3 3 6 0 . 0 1 5 9 0 0 . 0 1 6 6 8 0 . 0 0 5 0 4 0 . 0 0 1 8 6 0 . 0 0 1 1 4 0 . 0 0 0 8 2 0 . 0 0 0 6 4 0 . 0 0 0 5 8 0 . 0 0 2 2 8 0 . 0 0 1 0 9 0 . 0 0 1 1 4 0 . C 0 0 3 5 0 . 0 0 0 1 3 0 . 0 0 0 0 8 0 . 0 0 0 0 6 0 . 0 0 0 0 4 0 . 0 0 0 0 4 0 . 0 0 0 1 6 0 . 0 0 0 0 8

SUMS α = 0 . 2 0 0 . 0 5 6 2 3 α = 0 . 4 0 0 . 1 1 3 8 6 α = 0 . 6 0 0 . 1 7 2 9 3 α = 0 . 8 0 0 . 2 3 3 5 2

α = 0 . 9 0 0 . 2 6 4 4 0 α = 0 . 9 5 C 2 7 9 9 9 α = 0 . 9 9 0 . 2 9 2 5 4 α = 1 . 0 0 0 . 2 9 5 6 8

b = 0 . 0 3 1 2 5 Mo

F I R S T ORDER 0 . 2 5 0 0 0 SECOND ι ORDER 0 . 0 1 5 0 4 THIRD ORDER 0 . 0 0 1 0 3

SUMS α = 0 . 2 0 0 . 0 5 0 6 1 α = 0 . 4 0 0 . 1 0 2 4 7 α = 0 . 6 0 0 . 1 5 5 6 4 α = 0 . 8 0 0 . 2 1 0 1 8

α = 0 . 9 0 0 . 2 3 7 9 8 α = 0 . 9 5 C . 2 5 2 0 1 α = 0 . 9 9 0 . 2 6 3 3 1 α = 1 . 0 0 0 . 2 6 6 1 4

0 . 0 1 4 8 7 0 . 0 0 5 4 8 0 . 0 0 3 3 5 0 . 0 0 2 4 2 0 . 0 3 0 1 6 O . O l l i l 0 . 0 0 6 8 0 0 . 0 0 4 9 0

0 0 1 8 9 0 . 0 0 1 7 0 0 . 0 0 6 7 7 0 . 0 0 3 2 2 0 0 3 8 3 0 . 0 0 3 4 6 0 . 0 1 3 7 2 0 . 0 0 6 5 4

0 . 0 7 0 3 5 0 . 0 2 5 9 2 0 . 0 7 4 5 3 0 . 0 2 7 4 6

0 . 0 7 8 7 5

= 1 . 0

0 . 0 0 4 5 4

0 2 8 7 0 0 2 9 0 1

0 2 4 3 6 0 0 1 6 7

0 . 0 0 0 3 1 0 . 0 0 0 1 2

0 1 0 3 5 0 1 4 0 1

0 1 5 8 7 0 1 6 8 1 0 1 7 5 7 0 1 7 7 6

0 1 4 9 1 0 0 1 0 2 0 0 0 0 7

0 . 0 1 3 4 1 0 . 0 2 7 1 9 0 . 0 4 1 3 7

0 . 0 6 7 1 9 0 . 0 7 0 2 3 0 . 0 7 0 9 9

0 . 0 0 4 9 4 0 . 0 0 3 0 2 0 . 0 1 0 0 2 0 . 0 0 6 1 3 0 . 0 1 5 2 4 0 . 0 0 9 3 3

0 0 7 4 6 0 . 0 0 5 8 3 0 . 0 0 5 2 6 0 . 0 2 0 8 7 0 1 0 0 9 0 . 0 0 7 8 9 0 . 0 0 7 1 1 0 . 0 2 8 2 4

0 . 0 0 9 9 5 0 . 0 1 3 4 6

0 1 1 4 4 0 1 2 1 2

0 0 8 9 4 0 0 9 4 7

0 1 2 6 6 0 . 0 0 9 9 0

0 . 0 0 8 0 6 0 . 0 3 2 0 0 0 . 0 1 5 2 5 0 . 0 0 8 5 4 0 . 0 3 3 9 0 0 . 0 1 6 1 6 0 . 0 0 8 9 2 0 . 0 3 5 4 3 0 . 0 1 6 8 9

0 1 2 8 0 0 . 0 1 0 0 1 0 . 0 0 9 0 2 0 . 0 3 5 8 1 0 . 0 1 7 0 7

0 1 0 7 5 0 . 0 0 8 4 0 0 . 0 0 7 5 7 0 . 0 0 0 7 4 0 . 0 0 0 5 8 0 . 0 0 0 5 2 0 . 0 0 0 0 5 0 . 0 0 0 0 4 0 . 0 0 0 0 4 0 .

0 3 0 0 7 0 . 0 1 4 3 3 0 0 2 0 5 0 . 0 0 0 9 8 0 0 0 1 4 0 . 0 0 0 0 7

0 0 2 1 8 0 . 0 0 1 7 0 0 . 0 0 1 5 4 0 . 0 0 6 1 0 0 0 4 4 2 0 . 0 0 3 4 6 0 . 0 0 3 1 1 0 . 0 1 2 3 7 0 0 6 7 3 0 . 0 0 5 2 6 0 . 0 0 4 7 4 0 . 0 1 8 8 2

0 0 2 9 1 0 0 5 8 9 0 0 8 9 7 0 1 2 1 3

0 . 0 2 3 3 6 0 . 0 1 4 3 1 0 . 0 1 0 3 1 0 . 0 0 8 0 6 0 . 0 0 7 2 7 0 . 0 2 8 8 4 0 . 0 1 3 7 5 0 . 0 2 4 7 5 0 . 0 1 5 1 6 0 . 0 1 0 9 2 0 . 0 0 8 5 4 0 . 0 0 7 7 0 0 . 0 3 0 5 5 0 . 0 1 4 5 6 0 . 0 2 5 8 7 0 . 0 1 5 8 4 0 . 0 1 1 4 2 0 . 0 0 8 9 2 0 . 0 0 8 0 5 0 . 0 3 1 9 3 0 . 0 1 5 2 2 0 . 0 2 6 1 5 0 . 0 1 6 0 2 0 . 0 1 1 5 4 0 . 0 0 9 0 2 0 . 0 0 8 1 3 0 . 0 3 2 2 8 0 . 0 1 5 3 9

0 . 0 3 1 2 5 NARROW SURFACE LAYER AT TOP


I N F I N I T E 0 . 2 6 7 1 1 0 . 0 9 7 1 5 0 . 0 5 9 3 4 0 . 0 4 2 7 1 0 . 0 3 3 3 6 0 . 0 3 0 0 7 0 . 1 2 6 7 7 0 . 0 5 7 2 5 0 . 0 6 3 3 8 0 . 0 1 7 9 7 0 . 0 0 6 6 1 0 . 0 0 4 0 5 0 . 0 0 2 9 2 0 . 0 0 2 2 8 0 . 0 0 2 0 5 0 . 0 0 8 1 8 0 . 0 0 3 8 9 0 . 0 0 4 0 9 0 . 0 0 1 2 3 0 . 0 0 0 4 5 0 . 0 0 0 2 8 0 . 0 0 0 2 0 0 . 0 0 0 1 6 0 . 0 0 0 1 4 0 . 0 0 0 5 6 0 . 0 0 0 2 7

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

I N F I N I T E 0 . 0 5 4 1 5 0 . 0 1 9 7 0 0 . 0 1 2 0 3 0 . 0 0 8 6 6 0 . 0 0 6 7 7 0 . 0 0 6 1 0 0 . 0 2 5 6 9 0 . 0 1 1 6 1 I N F I N I T E 0 . 1 0 9 8 0 0 . 0 3 9 9 5 0 . 0 2 4 4 0 0 . 0 1 7 5 7 0 . 0 1 3 7 2 0 . 0 1 2 3 7 0 . 0 5 2 0 5 0 . 0 2 3 5 4 I N F I N I T E 0 . 1 6 7 0 1 0 . 0 6 0 7 7 0 . 0 3 7 1 2 0 . 0 2 6 7 2 0 . 0 2 0 8 7 0 . 0 1 8 8 2 0 . 0 7 9 1 3 0 . 0 3 5 8 1 I N F I N I T E 0 . 2 2 5 8 6 0 . 0 8 2 2 0 0 . 0 5 0 2 1 0 . 0 3 6 1 5 0 . 0 2 8 2 4 0 . 0 2 5 4 5 0 . 1 0 6 9 5 0 . 0 4 8 4 4

α = 0 . 9 0 I N F I N I T E 0 . 2 5 5 9 1 0 . 0 9 3 1 4 0 . 0 5 6 9 0 0 . 0 4 0 9 6 0 . 0 3 2 0 0 0 . 0 2 8 8 4 0 . 1 2 1 1 5 0 . 0 5 4 8 9 α = 0 . 9 5 I N F I N I T E 0 . 2 7 1 1 0 0 . 0 9 8 6 7 0 . 0 6 0 2 8 0 . 0 4 3 3 9 0 . 0 3 3 9 0 0 . 0 3 0 5 5 0 . 1 2 8 3 2 0 . 0 5 8 1 5 α = 0 . 9 9 I N F I N I T E 0 . 2 8 3 3 4 0 . 1 0 3 1 3 0 . 0 6 3 0 0 0 . 0 4 5 3 5 0 . 0 3 5 4 3 0 . 0 3 1 9 3 0 . 1 3 4 0 9 0 . 0 6 0 7 7 α = 1 . 0 0 I N F I N I T E 0 . 2 8 6 4 1 0 . 1 0 4 2 5 0 . 0 6 3 6 9 0 . 0 4 5 8 5 0 . 0 3 5 8 1 0 . 0 3 2 2 8 0 . 1 3 5 5 4 0 . 0 6 1 4 3

0 . 0 3 1 2 5 LAM8ERT SURFACE ON TOP

F I R S T ORDER SECOND ORDER T H I R D ORDER

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α * 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

0 . 5 0 0 0 0 0 . 1 2 5 3 8 0 . 0 4 6 1 1 0 . 0 2 8 2 3 0 . 0 2 0 3 4 0 . 0 1 5 9 0 0 . 0 1 4 3 3 0 . 0 5 7 2 5 0 . 0 2 7 1 4 0 . 0 2 8 6 3 0 . 0 0 8 5 9 0 . 0 0 3 1 7 0 . 0 0 1 9 4 0 . 0 0 1 4 0 0 . 0 0 1 0 9 0 . 0 0 0 9 8 0 . 0 0 3 8 9 0 . 0 0 1 8 6 0 . 0 0 1 9 5 0 . 0 0 0 5 9 0 . 0 0 0 2 2 0 . 0 0 0 1 3 0 . 0 0 0 1 0 0 . 0 0 0 0 8 0 . 0 0 0 0 7 0 . 0 0 0 2 7 0 . 0 0 0 1 3

0 . 1 0 1 1 6 0 . 0 2 5 4 2 0 . 0 0 9 3 5 0 . 0 0 5 7 2 0 . 0 0 4 1 2 0 . 0 0 3 2 2 0 . 0 0 2 9 1 0 . 0 1 1 6 1 - 0 . 0 0 5 5 0 0 . 2 0 4 7 1 0 . 0 5 1 5 6 0 . 0 1 8 9 7 0 . 0 1 1 6 1 0 . 0 0 8 3 7 0 . 0 0 6 5 4 0 . 0 0 5 8 9 0 . 0 2 3 5 4 0 . 0 1 1 1 6 0 . 3 1 0 7 4 0 . 0 7 8 4 5 0 . 0 2 8 8 6 0 . 0 1 7 6 7 0 . 0 1 2 7 3 0 . 0 0 9 9 5 0 . 0 0 8 9 7 0 . 0 3 5 8 1 0 . 0 1 6 9 8 0 . 4 1 9 3 7 0 . 1 0 6 1 2 0 . 0 3 9 0 3 0 . 0 2 3 9 0 0 . 0 1 7 2 2 0 . 0 1 3 4 6 0 . 0 1 2 1 3 0 . 0 4 8 4 4 0 . 0 2 2 9 7

0 . 4 7 4 7 0 0 . 1 2 0 2 6 0 . 0 4 4 2 4 0 . 0 2 7 0 8 0 . 0 1 9 5 1 0 . 0 1 5 2 5 0 . 0 1 3 7 5 0 . 0 5 4 8 9 0 . 0 2 6 0 3 0 . 5 0 2 6 2 0 . 1 2 7 4 1 0 . 0 4 6 8 6 0 . 0 2 8 6 9 0 . 0 2 0 6 7 0 . 0 1 6 1 6 0 . 0 1 4 5 6 0 . 0 5 8 1 5 0 . 0 2 7 5 8 0 . 5 2 5 0 8 0 . 1 3 3 1 6 0 . 0 4 8 9 8 0 . 0 2 9 9 9 0 . 0 2 1 6 1 0 . 0 1 6 8 9 0 . 0 1 5 2 2 0 . 0 6 0 7 7 0 . 0 2 8 8 3 0 . 5 3 0 7 2 0 . 1 3 4 6 0 0 . 0 4 9 5 1 0 . 0 3 0 3 1 0 . 0 2 1 8 4 0 . 0 1 7 0 7 0 . 0 1 5 3 9 0 . 0 6 1 4 3 0 . 0 2 9 1 4




V E C T u K f i = 0 . 0 μ=0.1 μ = 0 . 3 μ=0.5 AVERAGE

Ν FLUX

U

b = 0 . 0 3 1 2 5 Z E R O 0 R 0 E R F I R S T O R D E R S E C O N D O R D E R T H I R D ORDER

S U M S α = α = 0 . 2 C α - 0 . 4 0 α = 0 . 6 0 α - 0 . 8 0

α 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

μ0 = C · 9 C O 0 . 0 0 . 0 0 . 0 0 . 0 PEAK 0 . 0 0 . 5 3 6 6 0 0 . 9 6 5 8 7 0 . 2 6 8 3 0 0 . 0 7 3 2 1 0 . 0 2 7 0 0 0 . 0 1 6 5 4 0 . 0 1 1 9 2 0 . 0 0 9 3 2 0 . 0 0 8 4 0 0 . 0 3 3 2 2 0 . 0 1 5 8 8 0 . 0 l e>61 0 . 0 0 5 0 3 0 . 0 0 1 8 6 0 . 0 0 1 1 4 0 . 0 0 0 8 2 0 . 0 0 0 6 4 0 . 0 0 0 5 8 0 . 0 0 2 2 8 0 . 0 0 1 0 9 C 0 0 1 1 4 0 . 0 0 0 3 5 0 . 0 0 0 1 3 0 . 0 0 0 0 8 0 . 0 0 0 0 6 0 . 0 0 0 0 4 0 . 0 0 0 0 4 0 . 0 0 0 1 6 0 . 0 0 0 0 8

C 0 5 4 3 3 0 . 0 1 4 8 5 0 . 0 0 5 4 8 0 . 0 0 3 3 5 0 . 0 0 2 4 2 0 . 0 0 1 8 9 0 . 0 0 1 7 0 0 . 5 4 3 3 3 0 . 9 6 9 0 9 C 1 1 0 0 5 0 . 0 3 0 1 1 0 . 0 1 1 1 1 0 . 0 0 6 8 0 0 . 0 0 4 9 0 0 . 0 0 3 8 3 0 . 0 0 3 4 5 0 . 5 5 0 2 6 0 . 9 7 2 4 1 0 . 1 6 7 2 1 0 . 0 4 5 8 1 0 . 0 1 6 9 0 0 . 0 1 0 3 5 0 . 0 0 7 4 6 0 . 0 0 5 8 3 0 . 0 0 5 2 6 0 . 5 5 7 3 8 0 . 9 7 5 8 1 0 . 2 2 5 8 9 0 . 0 6 1 9 7 0 . 0 2 2 8 6 0 . 0 1 4 0 0 0 . 0 1 0 0 9 0 . 0 0 7 8 9 0 . 0 0 7 1 1 0 . 5 6 4 7 1 0 . 9 7 9 3 2

C 2 5 5 8 1 0 . 0 7 0 2 3 0 . 0 2 5 9 0 0 . 0 1 5 8 7 0 . 0 1 1 4 3 0 . 0 0 8 9 4 0 . 0 0 8 0 6 0 . 5 6 8 4 6 0 . 9 8 1 1 1 0 . 2 7 0 9 2 0 . 0 7 4 4 1 0 . 0 2 7 4 4 0 . 0 1 6 8 1 0 . 0 1 2 1 1 0 . 0 0 9 4 7 0 . 0 0 8 5 4 0 . 5 7 0 3 5 0 . 9 8 2 0 2 0 . 2 8 3 0 8 0 . 0 7 7 7 7 0 . 0 2 8 6 8 0 . 0 1 7 5 7 0 . 0 1 2 6 6 0 . 0 0 9 9 0 0 . 0 0 8 9 2 0 . 5 7 1 8 8 0 . 9 8 2 7 5 0 . 2 8 6 1 3 0 . 0 7 8 6 1 0 . 0 2 8 9 9 0 . 0 1 7 7 6 0 . 0 1 2 8 0 0 . 0 1 0 0 0 0 . 0 0 9 0 2 0 . 5 7 2 2 6 0 . 9 8 2 9 3


μο = 1 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 PEAK 0 . 4 8 4 6 2 0 . 9 6 9 2 3 0 . 2 4 2 3 1 0 . 0 6 6 0 0 0 . 0 2 4 3 4 0 . 0 1 4 9 1 0 . 0 1 0 7 5 0 . 0 0 8 4 0 0 . 0 0 7 5 7 0 . 0 2 9 9 5 0 . 0 1 4 3 2 0 . 0 1 4 9 8 0 . 0 0 4 5 4 0 . 0 0 1 6 7 0 . 0 0 1 0 2 0 . 0 0 0 7 4 0 . 0 0 0 5 8 0 . 0 0 0 5 2 0 . 0 0 2 0 5 0 . 0 0 0 9 8 C . 0 0 1 0 3 0 . 0 0 0 3 1 0 . 0 0 0 1 2 0 . 0 0 0 0 7 0 . 0 0 0 0 5 0 . 0 0 0 0 4 0 . 0 0 0 0 4 0 . 0 0 0 1 4 0 . 0 0 0 0 7

SUMS α = α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0


SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

b = 0 . 0 3 1 2 5 ZERO ORDER F I R S T ORDER SECOND ORDER T H I R D ORDER

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

C . 0 4 9 0 7 0 . 0 1 3 3 9 0 . 0 0 4 9 4 0 . 0 0 3 0 2 0 . 0 0 2 1 8 0 . 0 0 1 7 0 0 . 0 0 1 5 4 0 . 4 9 0 6 9 0 . 9 7 2 1 4 0 . 0 9 9 3 9 0 . 0 2 7 1 5 0 . 0 1 0 0 1 0 . 0 0 6 1 3 0 . 0 0 4 4 2 0 . 0 0 3 4 5 0 . 0 0 3 1 1 0 . 4 9 6 9 4 0 . 9 7 5 1 2 0 . 1 5 1 0 1 0 . 0 4 1 3 1 0 . 0 1 5 2 3 0 . 0 0 9 3 3 0 . 0 0 6 7 2 0 . 0 0 5 2 6 0 . 0 0 4 7 4 0 . 5 0 3 3 6 0 . 9 7 8 2 0 C . 2 0 3 9 9 0 . 0 5 5 8 8 0 . 0 2 0 6 1 0 . 0 1 2 6 2 0 . 0 0 9 1 0 0 . 0 0 7 1 1 0 . 0 0 6 4 1 0 . 5 0 9 9 7 0 . 9 8 1 3 6

0 . 2 3 1 0 1 0 . 0 6 3 3 2 0 . 0 2 3 3 5 0 . 0 1 4 3 0 0 . 0 1 0 3 1 0 . 0 0 8 0 6 0 . 0 0 7 2 6 0 . 5 1 3 3 5 0 . 9 8 2 9 7 0 . 2 4 4 6 5 0 . 0 6 7 0 9 0 . 0 2 4 7 4 0 . 0 1 5 1 5 0 . 0 1 0 9 2 0 . 0 0 8 5 4 0 . 0 0 7 7 0 0 . 5 1 5 0 6 0 . 9 8 3 7 9 0 . 2 5 5 6 3 0 . 0 7 0 1 2 0 . 0 2 5 8 6 0 . 0 1 5 8 4 0 . 0 1 1 4 1 0 . 0 0 8 9 2 0 . 0 0 8 0 4 0 . 5 1 6 4 3 0 . 9 8 4 4 5 C . 2 5 8 3 9 0 . 0 7 0 8 8 0 . 0 2 6 1 4 0 . 0 1 6 0 1 0 . 0 1 1 5 4 0 . 0 0 9 0 2 0 . 0 0 8 1 3 0 . 5 1 6 7 8 0 . 9 8 4 6 1

NARROW SURFACE LAYER AT TOP C O 3 . 6 5 8 0 8 1 . 5 0 1 7 9 0 . 9 3 9 4 1 0 . 6 8 3 1 0 0 . 5 3 6 6 0 0 . 4 8 4 6 2 1 . 4 5 9 7 6 0 . 8 7 8 0 0 0 . 7 3 1 2 0 0 . 2 5 6 7 1 0 . 0 9 5 8 7 0 . 0 5 8 8 7 0 . 0 4 2 4 7 0 . 0 3 3 2 2 0 . 0 2 9 9 5 0 . 1 1 4 0 4 0 . 0 5 6 3 4 0 . 0 5 7 C 2 0 . 0 1 7 8 6 0 . 0 0 6 6 0 0 . 0 0 4 0 4 0 . 0 0 2 9 1 0 . 0 0 2 2 8 0 . 0 0 2 0 5 0 . 0 0 8 0 5 0 . 0 0 3 8 8 C 0 0 4 C 2 0 . 0 0 1 2 3 0 . 0 0 0 4 5 0 . 0 0 0 2 8 0 . 0 0 0 2 0 0 . 0 0 0 1 6 0 . 0 0 0 1 4 0 . 0 0 0 5 6 0 . 0 0 0 2 7

0 . 1 4 8 5 5 3 . 7 1 0 1 4 1 . 5 2 1 2 3 0 . 9 5 1 3 5 0 . 6 9 1 7 1 0 . 5 4 3 3 3 0 . 4 9 0 6 9 1 . 4 8 5 5 4 0 . 8 8 9 4 9 0 . 3 0 1 8 7 3 . 7 6 3 7 0 1 . 5 4 1 2 2 0 . 9 6 3 6 3 0 . 7 0 0 5 7 0 . 5 5 0 2 6 0 . 4 9 6 9 4 1 . 5 0 9 3 5 0 . 9 0 1 2 3 0 . 4 6 0 1 6 3 . 8 1 8 8 1 1 . 5 6 1 7 9 0 . 9 7 6 2 5 0 . 7 0 9 6 8 0 . 5 5 7 3 8 0 . 5 0 3 3 6 1 . 5 3 3 8 5 0 . 9 1 3 3 2 C . 6 2 3 6 4 3 . 8 7 5 5 4 1 . 5 8 2 9 5 0 . 9 8 9 2 5 0 . 7 1 9 0 5 0 . 5 6 4 7 1 0 . 5 0 9 9 7 1 . 5 5 9 0 9 0 . 9 2 5 7 6

C . 7 0 7 4 0 3 . 9 0 4 5 4 1 . 5 9 3 7 7 0 . 9 9 5 8 9 0 . 7 2 3 8 4 0 . 5 6 8 4 6 0 . 5 1 3 3 5 1 . 5 7 1 9 9 0 . 9 3 2 1 2 0 . 7 4 9 8 0 3 . 9 1 9 2 0 1 . 5 9 9 2 4 0 . 9 9 9 2 4 0 . 7 2 6 2 6 0 . 5 7 0 3 5 0 . 5 1 5 0 6 1 * 5 7 8 5 2 0 . 9 3 5 3 3 C . 7 8 3 9 7 3 . 9 3 1 0 0 1 . 6 0 3 6 4 1 . 0 0 1 9 5 0 . 7 2 8 2 1 0 . 5 7 1 8 8 0 . 5 1 6 4 3 1 . 5 8 3 7 7 0 . 9 3 7 9 2 C . 7 9 2 5 5 3 . 9 3 3 9 7 1 . 6 0 4 7 4 1 . 0 0 2 6 2 0 . 7 2 8 7 0 0 . 5 7 2 2 6 0 . 5 1 6 7 8 1 . 5 8 5 0 9 0 . 9 3 8 5 7

LAMBERT SURFACE ON TOP 0 . 0 0 . 7 3 1 6 2 0 . 9 0 1 0 8 0 . 9 3 9 4 1 0 . 9 5 6 3 4 0 . 9 6 5 8 7 0 . 9 6 9 2 3 0 . 8 7 8 0 0 0 . 9 4 1 8 0 0 . 4 3 9 0 3 0 . 1 2 4 5 6 0 . 0 4 6 0 1 0 . 0 2 8 1 9 0 . 0 2 0 3 2 0 . 0 1 5 8 8 0 . 0 1 4 3 2 0 . 0 5 6 3 4 0 . 0 2 7 0 7 0 . 0 2 8 1 7 0 . 0 0 8 5 8 0 . 0 0 3 1 6 0 . 0 0 1 9 4 0 . 0 0 1 4 0 0 . 0 0 1 0 9 0 . 0 0 0 9 8 0 . 0 0 3 8 8 0 . 0 0 1 8 6 C 0 0 1 9 4 0 . 0 0 0 5 9 0 . 0 0 0 2 2 0 . 0 0 0 1 3 0 . 0 0 0 1 0 0 . 0 0 0 0 8 0 . 0 0 0 0 7 0 . 0 0 0 2 7 0 . 0 0 0 1 3

C . 0 8 8 9 5 0 . 7 5 6 8 8 0 . 9 1 0 4 1 0 . 9 4 5 1 3 0 . 9 6 0 4 6 0 . 9 6 9 0 9 0 . 9 7 2 1 4 0 . 8 8 9 4 9 0 . 9 4 7 2 8 0 . 1 8 0 2 5 0 . 7 8 2 8 5 0 . 9 2 0 0 0 0 . 9 5 1 0 1 0 . 9 6 4 7 0 0 . 9 7 2 4 1 0 . 9 7 5 1 2 0 . 9 0 1 2 3 0 . 9 5 2 9 3 0 . 2 7 4 0 0 0 . 8 0 9 5 8 0 . 9 2 9 8 7 0 . 9 5 7 0 6 0 . 9 6 9 0 6 0 . 9 7 5 8 1 0 . 9 7 8 2 0 0 . 9 1 3 3 2 0 . 9 5 8 7 3 C 3 7 0 3 0 0 . 8 3 7 0 8 . 0 . 9 4 0 0 3 0 . 9 6 3 2 8 0 . 9 7 3 5 4 0 . 9 7 9 3 2 0 . 9 8 1 3 6 0 . 9 2 5 7 6 0 . 9 6 4 7 1

0 . 4 1 9 4 5 0 . 8 5 1 1 3 0 . 9 4 5 2 2 0 . 9 6 6 4 6 0 . 9 7 5 8 3 0 . 9 8 1 1 1 0 . 9 8 2 9 7 0 . 9 3 2 1 2 0 . 9 6 7 7 6 0 . 4 4 4 2 8 0 . 8 5 8 2 4 0 . 9 4 7 8 4 0 . 9 6 8 0 7 0 . 9 7 6 9 9 0 . 9 8 2 0 2 0 . 9 8 3 7 9 0 . 9 3 5 3 3 0 . 9 6 9 3 1 0 . 4 6 4 2 7 0 . 8 6 3 9 6 0 . 9 4 9 9 6 0 . 9 6 9 3 6 0 . 9 7 7 9 3 0 . 9 8 2 7 5 0 . 9 8 4 4 5 0 . 9 3 7 9 2 0 . 9 7 0 5 5 0 . 4 6 9 2 8 0 . 8 6 5 4 0 0 . 9 5 0 4 9 0 . 9 6 9 6 9 0 . 9 7 8 1 6 0 . 9 8 2 9 3 0 . 9 8 4 6 1 0 . 9 3 8 5 7 0 . 9 7 0 8 6


T A B L E 12 {continued)


/ x = 0 . 0 = 0 . 7 AVERAGE Ν

FLUX U

b» 0 . 0 6 2 5 0


2 . 5 0 0 0 0 0 . 1 9 6 5 9 0 . 0 2 1 8 2

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

C 5 0 8 0 4 1 . 0 3 2 9 2 1 . 5 7 5 8 4 2 . 1 3 8 1 3

α = 0 . 9 0 α = 0 . 9 5 α - 0 . 9 9 α = 1 . 0 0

2 . 4 2 6 9 9 2 . 5 7 3 4 4 2 . 6 9 1 5 9 2 . 7 2 1 2 6

b= 0 . 0 6 2 5 0 Mo


0 . 8 3 3 3 3 0 . 0 7 7 0 6 0 . 0 0 8 7 8

SUMS α = 0 . 2 0 α = 0 . 4 0 α * 0 . 6 0 α - 0 . 8 0

0 . 1 6 9 8 2 0 . 3 4 6 2 5 0 . 5 2 9 7 * 8 0 . 7 2 0 9 4

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

0 . 8 1 9 5 7 0 . 8 6 9 6 8 0 . 9 1 0 1 6 0 . 9 2 0 3 3

b= 0 . 0 6 2 5 0 Mo


0 . 5 0 0 0 0 C 0 4 7 8 4 0 . 0 0 5 4 8

SUMS a = 0 . 2 0 a * 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

0 . 1 0 1 9 6 C . 2 0 8 0 2 0 . 3 1 8 5 0 0 . 4 3 3 7 1

a * 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

0 . 4 9 3 2 2 0 . 5 2 3 4 6 0 . 5 4 7 9 0 0 . 5 5 4 0 5

b = 0 . 0 6 2 5 0 Mo


0 . 3 5 7 1 4 0 . 0 3 4 6 8 C . 0 0 3 9 8

SUMS a * 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

0 . 0 7 2 8 5 0 . 1 4 8 6 7 C . 2 2 7 7 0 0 . 3 1 0 1 6

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 α » 1 . 0 0

0 . 3 5 2 7 7 0 . 3 7 4 4 3 C . 3 9 1 9 3 0 . 3 9 6 3 3

0 . 3 5 3 3 8 0 . 2 1 9 8 5 0 . 1 5 9 5 2 0 . 1 2 5 1 6 0 . 0 4 0 6 5 0 . 0 2 5 3 8 0 . 0 1 8 4 4 0 . 0 1 4 4 8 0 . 0 0 4 7 2 0 . 0 0 2 9 5 0 . 0 0 2 1 5 0 . 0 0 1 6 8

0 . 1 1 2 9 9 0 . 3 9 3 1 8 0 . 2 0 7 7 0 0 . 0 1 3 0 8 0 . 0 4 3 6 4 0 . 0 2 3 8 8 0 . 0 0 1 5 2 0 . 0 0 5 0 4 0 . 0 0 2 7 7

3 7 3 6 6 5 7 4 1 3

0 7 2 3 4 1 4 8 1 7 2 2 7 7 6 3 1 1 3 9

0 4 5 0 1 0 . 0 3 2 6 6 0 . 0 2 5 6 3 0 . 0 2 3 1 3 0 . 0 8 0 4 2 0 . 0 4 2 5 2 0 9 2 2 0 1 4 1 7 3 1 9 3 7 9

0 . 3 5 4 8 1 0 . 2 2 0 8 2

9 9 4 5 7 0 0 5 9 0

3 9 4 8 7 3 9 9 3 7

2 4 5 7 6 2 4 8 5 7

0 . 1 4 1 9 8 0 . 0 8 8 5 8

0 . 0 0 4 7 2 0 . 0 0 1 9 1 0 . 0 0 1 1 9

0 . 0 2 9 0 7 0 . 0 5 9 5 5 0 . 0 9 1 5 5 0 . 1 2 5 1 9

0 1 8 1 4 0 3 7 1 6 0 5 7 1 2 0 7 8 1 1

0 6 6 9 0 1 0 2 8 5

0 5 2 5 0 0 8 0 7 0

1 4 0 6 3 0 . 1 1 0 3 5

0 . 0 4 7 3 9 0 . 1 6 4 5 9 0 . 0 8 7 0 9 0 . 0 7 2 8 6 0 . 2 5 2 7 9 0 . 1 3 3 8 6 0 . 0 9 9 6 2 0 . 3 4 5 3 2 0 . 1 8 3 0 1

1 6 0 2 5 0 . 1 2 5 7 5 0 . 1 1 3 5 3 0 . 3 9 3 3 2 0 . 2 0 8 5 3 1 7 0 2 5 0 . 1 3 3 6 0 0 . 1 2 0 6 1 0 . 4 1 7 7 7 0 . 2 2 1 5 4 1 7 8 3 5 0 . 1 3 9 9 5 0 . 1 2 6 3 5 0 . 4 3 7 5 5 0 . 2 3 2 0 7 1 8 0 3 9 0 . 1 4 1 5 5 0 . 1 2 7 7 9 0 . 4 4 2 5 3 0 . 2 3 4 7 2

0 6 4 3 5 0 0 7 4 6 0 0 0 8 7

0 1 3 1 8 0 2 6 9 9

0 5 0 5 3 0 0 5 8 6 0 0 0 6 8

0 4 5 6 3 0 . 1 5 4 1 3 0 0 5 2 9 0 . 0 1 7 5 6 0 0 0 6 2 0 . 0 0 2 0 4

0 4 1 5 0 0 . 0 3 2 5 9 0 5 6 7 5 0 . 0 4 4 5 6

0 1 0 3 5 0 . 0 0 9 3 4 0 2 1 1 9 0 . 0 1 9 1 4

0 . 0 2 9 4 2 0 . 0 4 0 2 4

0 8 3 4 3 0 0 9 6 5 0 0 1 1 2

0 1 7 0 8 0 3 4 9 9

0 3 1 5 4 0 6 4 6 0 0 9 9 2 7 0 . 0 5 3 7 9 1 3 5 6 9 0 . 0 7 3 5 5

0 . 1 4 2 6 6 0 . 0 8 9 0 2 0 . 0 6 4 6 7 0 . 0 5 0 7 8 0 . 1 5 1 5 6 0 . 0 9 4 5 7 0 . 0 6 8 7 1 0 . 0 5 3 9 5 0 . 1 5 8 7 7 0 . 0 9 9 0 7 0 . 0 7 1 9 8 0 0 . 1 6 0 5 8 0 . 1 0 0 2 0 0 . 0 7 2 8 0 0

0 . 0 4 5 8 5 0 . 1 5 4 6 0 0 . 0 4 8 7 2 0 . 1 6 4 2 4

0 5 6 5 2 0 . 0 5 1 0 3 0 . 1 7 2 0 3 0 5 7 1 6 0 . 0 5 1 6 2 0 . 1 7 4 0 0

0 . 0 8 3 8 2 0 . 0 8 9 0 5 0 . 0 9 3 2 9 0 . 0 9 4 3 5

0 . 2 1 9 8 5 0 . 0 8 8 5 8 0 . 0 5 5 3 0 0 . 0 4 0 1 8 . 0 2 8 5 0 0 . 0 9 5 6 9 0 0 3 3 1 0 . 0 1 0 9 6

0 . 0 0 2 9 5 0 . 0 0 1 1 9 0 . 0 0 0 7 5 0 . 0 0 0 5 4 0 . 0 0 0 4 3 0 . 0 0 0 3 8 0 . 0 0 1 2 7

0 3 1 5 6 0 0 3 6 6

0 5 2 0 5 0 0 6 0 3 0 0 0 7 0

0 . 1 4 1 7 3

0 . 0 1 8 1 4 0 . 0 3 7 1 6 0 . 0 5 7 1 2

0 1 1 3 2 0 2 3 2 0 0 3 5 6 6 0 4 8 7 7

0 . 0 0 8 2 3 0 . 0 0 6 4 6 0 . 0 0 5 8 3 0 . 0 1 9 5 9 0 . 0 1 6 8 6 0 . 0 1 3 2 4 0 . 0 1 1 9 5 0 . 0 4 0 1 1

0 2 5 9 1 0 . 0 2 0 3 5 0 . 0 1 8 3 8 0 3 5 4 4 0 . 0 2 7 8 3 0 . 0 2 5 1 3

0 . 0 1 0 6 6 0 . 0 2 1 8 3

0 . 0 6 1 6 6 0 . 0 3 3 5 6 0 . 0 8 4 2 9 0 . 0 4 5 8 9

0 . 2 2 0 8 2 0 . 0 8 9 0 2 0 . 0 5 5 5 7 0 . 0 4 0 3 8 0 . 0 3 1 7 1 0 . 0 2 8 6 4 0 . 0 9 6 0 4 0 . 0 5 2 3 0 0 . 0 3 0 4 3 0 . 1 0 2 0 2 0 . 0 5 5 5 6 0 . 0 3 1 8 7 0 . 1 0 6 8 7 0 . 0 5 8 2 1 0 . 0 3 2 2 4 0 . 1 0 8 0 9 0 . 0 5 8 8 7

0 . 7

0 9 9 0 7 1 0 0 2 0

0 6 1 8 5 0 . 0 4 4 9 5 0 . 0 3 5 3 0 0 6 2 5 6 0 . 0 4 5 4 6 0 . 0 3 5 7 0

0 . 1 5 9 5 2 0 . 0 6 4 3 5 0 . 0 4 0 1 8 0 . 0 2 9 2 0 0 . 0 2 2 9 3 0 . 0 2 0 7 1 0 . 0 6 9 3 7 0 . 0 3 7 8 1 0 . 0 1 8 4 4 0 . 0 0 7 4 6 0 . 0 0 4 6 6 0 . 0 0 3 3 9 0 . 0 0 2 6 6 0 . 0 0 2 4 0 0 . 0 0 7 9 6 0 . 0 0 4 3 8 0 . 0 0 2 1 5 0 . 0 0 0 8 7 0 . 0 0 0 5 4 0 . 0 0 0 3 9 0 . 0 0 0 3 1 0 . 0 0 0 2 8 0 . 0 0 0 9 3 0 . 0 0 0 5 1

. 0 6 6 9 0 0 . 0 2 6 9 9

. 1 0 2 8 5 0 . 0 4 1 5 0

. 1 4 0 6 3 0 . 0 5 6 7 5

0 . 0 1 6 8 6 0 . 0 2 5 9 1 0 . 0 3 5 4 4

0 1 2 2 5 0 1 8 8 3

0 . 0 0 9 6 2 0 . 0 1 4 7 9

0 0 8 6 9 0 1 3 3 6

. 0 1 4 2 0 0 . 0 0 7 7 4 , 0 2 9 0 8 0 . 0 1 5 8 6 , 0 4 4 7 0 0 . 0 2 4 3 8

0 2 5 7 5 0 . 0 2 0 2 3 0 . 0 1 8 2 6 0 . 0 6 1 1 2 0 . 0 3 3 3 4

0 6 4 6 7 0 . 0 4 0 3 8 0 . 0 2 9 3 5 0 . 0 2 3 0 5 0 . 0 2 0 8 1 0 6 8 7 1 0 . 0 4 2 9 1 0 . 0 3 1 1 8 0 . 0 2 4 4 9 0 . 0 2 2 1 1

0 3 2 6 7 0 . 0 2 5 6 5 0 . 0 2 3 1 7 1 7 0 2 5 0 1 7 8 3 5 0 . 0 7 1 9 8 0 . 0 4 4 9 5

0 . 0 6 9 6 4 0 . 0 3 7 9 9 0 . 0 7 3 9 8 0 . 0 4 0 3 7 0 . 0 7 7 5 0 0 . 0 4 2 2 9

0 . 0 2 5 9 5 0 . 0 2 3 4 3 0 . 0 7 8 3 8 0 . 0 4 2 7 7




V E C T O R / x = 0 . C / x = 0 . i = 0 . 3 = 0 . 5 AVERAGE

Ν FLUX

υ b = 0 . 0 6 2 5 0 Z E R O O R O E R F I R S T O R O E R S E C O N D O R D E R T H I R D O R D E R

S U M S α : 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0

0 . 9 0 C . 9 5 0 . 9 9 1 . 0 0

μ0 = 0 . 1 C . C P E A K 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 2 . 6 7 6 3 1 0 . 5 3 5 2 6 1 . 3 3 8 1 5 0 . 8 3 6 3 5 0 . 3 4 5 8 4 0 . 2 1 7 0 2 0 . 1 5 8 0 5 0 . 1 2 4 2 7 0 . 1 1 2 2 6 0 . 3 5 7 8 4 0 . 2 0 3 1 1 C . 1 7 8 9 2 0 . C 9 9 5 4 0 . 0 4 0 4 8 0 . 0 2 5 3 1 0 . 0 1 8 4 1 0 . 0 1 4 4 6 0 . 0 1 3 0 6 0 . 0 4 2 8 7 0 . 0 2 3 7 7 C . C 2 1 4 3 0 . 0 1 1 6 4 0 . 0 0 4 7 2 0 . 0 0 2 9 5 0 . 0 0 2 1 4 0 . 0 0 1 6 8 0 . 0 0 1 5 2 0 . 0 0 5 0 2 0 . 0 0 2 7 7

0 . 2 7 4 9 6 0 . 1 7 1 3 5 0 . 0 7 0 8 3 0 . 0 4 4 4 4 0 . 0 3 2 3 6 0 . 0 2 5 4 5 0 . 0 2 2 9 9 2 . 7 4 9 6 3 0 . 5 7 6 8 6 C . 5 6 5 3 3 0 . 3 5 1 2 5 0 . 1 4 5 1 3 0 . 0 9 1 0 6 0 . 0 6 6 3 1 0 . 0 5 2 1 3 0 . 0 4 7 1 0 2 . 8 2 6 6 4 0 . 6 2 0 4 9 C . 8 7 2 2 8 C . 5 4 0 3 5 0 . 2 2 3 1 7 0 . 1 4 0 0 1 0 . 1 0 1 9 6 0 . 0 8 0 1 6 0 . 0 7 2 4 1 2 . 9 0 7 6 1 0 . 6 6 6 3 3 1 . 1 9 7 1 4 0 . 7 3 9 3 6 0 . 3 0 5 2 5 0 . 1 9 1 4 8 0 . 1 3 9 4 3 0 . 1 0 9 6 2 0 . 0 9 9 0 3 2 . 9 9 2 8 5 0 . 7 1 4 5 3

1 . 3 6 6 7 3 0 . 8 4 2 8 3 0 . 3 4 7 8 9 0 . 2 1 8 2 3 0 . 1 5 8 9 0 0 . 1 2 4 9 2 0 . 1 1 2 8 6 3 . 0 3 7 1 7 0 . 7 3 9 5 7 1 . 4 5 3 4 0 0 . 8 9 5 5 9 0 . 3 6 9 6 3 0 . 2 3 1 8 6 0 . 1 6 8 8 3 0 . 1 3 2 7 3 0 . 1 1 9 9 1 3 . 0 5 9 7 8 0 . 7 5 2 3 4 1 . 5 2 3 6 5 0 . 9 3 8 3 2 0 . 3 8 7 2 3 0 . 2 4 2 9 0 0 . 1 7 6 8 6 0 . 1 3 9 0 4 0 . 1 2 5 6 1 3 . 0 7 8 0 9 0 . 7 6 2 6 8 1 . 5 4 1 3 5 0 . 9 4 9 0 7 0 . 3 9 1 6 6 0 . 2 4 5 6 7 0 . 1 7 8 8 9 0 . 1 4 0 6 3 0 . 1 2 7 0 5 3 . 0 8 2 7 0 0 . 7 6 5 2 8

b = 0 . 0 6 2 5 0 Z E R O O R D E R F I R S T O R D E R S E C O N D O R D E R T H I R D O R D E R

α = 0 . 2 0 0 - 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

μ0 = 0 . 3 0 .0 0 . 0 P E A K 0 . 0 0 . 0 0 . 0 0 . 0 1 . 3 5 3 2 3 0 . 8 1 1 9 4 C . 6 7 6 6 1 0 . 3 4 5 8 4 0 . 1 4 0 9 6 0 . 0 8 8 2 0 0 . 0 6 4 1 5 0 . 0 5 0 4 1 0 . 0 4 5 5 3 0 . 1 4 9 3 4 0 . 0 8 2 8 0 Û . G 7 4 6 7 0 . 0 4 0 4 8 0 . 0 1 6 4 1 0 . 0 1 0 2 6 0 . 0 0 7 4 6 0 . 0 0 5 8 6 0 . 0 0 5 2 9 0 . 0 1 7 4 6 0 . 0 0 9 6 4 0 . 0 0 8 7 3 0 . 0 0 4 7 2 0 . 0 0 1 9 1 0 . 0 0 1 1 9 0 . 0 0 0 8 7 0 . 0 0 0 6 8 0 . 0 0 0 6 2 0 . 0 0 2 0 4 0 . 0 0 1 1 2

0 . 1 3 8 3 8 0 . 0 7 0 8 3 0 . 0 2 8 8 6 0 . 0 1 8 0 6 0 . 0 1 3 1 4 0 . 0 1 0 3 2 0 . 0 0 9 3 2 1 . 3 8 3 8 1 0 . 8 2 8 8 9 C . 2 8 3 1 8 0 . 1 4 5 1 3 0 . 0 5 9 1 4 0 . 0 3 7 0 0 0 . 0 2 6 9 1 0 . 0 2 1 1 5 0 . 0 1 9 1 0 1 . 4 1 5 8 9 0 . 8 4 6 6 7 C . 4 3 4 8 8 C . 2 2 3 1 7 0 . 0 9 0 9 3 0 . 0 5 6 8 9 0 . 0 4 1 3 8 0 . 0 3 2 5 1 0 . 0 2 9 3 6 1 . 4 4 9 5 9 0 . 8 6 5 3 5 C . 5 9 4 0 1 0 . 3 0 5 2 5 0 . 1 2 4 3 5 0 . 0 7 7 8 0 0 . 0 5 6 5 9 0 . 0 4 4 4 6 0 . 0 4 0 1 6 1 . 4 8 5 0 2 0 . 8 8 4 9 8

C . 6 7 6 5 4 0 . 3 4 7 8 9 0 . 1 4 1 7 2 0 . 0 8 8 6 6 0 . 0 6 4 4 9 0 . 0 5 0 6 7 0 . 0 4 5 7 6 1 . 5 0 3 4 3 0 . 8 9 5 1 8 0 . 7 1 8 5 9 0 . 3 6 9 6 3 0 . 1 5 0 5 7 0 . 0 9 4 2 0 0 . 0 6 8 5 2 0 . 0 5 3 8 3 0 . 0 4 8 6 2 1 . 5 1 2 8 2 0 . 9 0 0 3 8 0 . 7 5 2 6 1 0 . 3 8 - 7 2 3 0 . 1 5 7 7 4 0 . 0 9 8 6 8 0 . 0 7 1 7 8 0 . 0 5 6 3 9 0 . 0 5 0 9 3 1 . 5 2 0 4 1 0 . 9 0 4 5 9 C . 7 6 1 1 6 0 . 3 9 1 6 6 0 . 1 5 9 5 4 0 . 0 9 9 8 1 0 . 0 7 2 6 0 0 . 0 5 7 0 4 0 . 0 5 1 5 2 1 . 5 2 2 3 3 0 . 9 0 5 6 5

b = O . Û 6 2 5 0 Z E R O O R D E R F I R S T O R D E R S E C O N D O R D E R T H I R D O R D E R

S U M S α = C . 2 0 α = C . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

μ0 « 0 . 5 0 . 0 C O 0 . 0 PEAK 0 . 0 0 . 0 0 . 0 0 . 8 8 2 5 0 0 . 8 8 2 5 0 C . 4 4 1 2 5 0 . 2 1 7 0 2 0 . 0 8 8 2 0 0 . 0 5 5 1 6 0 . 0 4 0 1 1 0 . 0 3 1 5 1 0 . 0 2 8 4 6 0 . 0 9 3 8 9 0 . 0 5 1 8 1 C 0 4 6 9 5 0 . 0 2 5 3 1 0 . 0 1 0 2 6 0 . 0 0 6 4 1 0 . 0 0 4 6 6 0 . 0 0 3 6 6 0 . 0 0 3 3 1 0 . 0 1 0 9 2 0 . 0 0 6 0 3 0 . 0 0 5 4 6 0 . 0 0 2 9 5 0 . 0 0 1 1 9 0 . 0 0 0 7 5 0 . 0 0 0 5 4 0 . 0 0 0 4 3 0 . 0 0 0 3 8 0 . 0 0 1 2 7 0 . 0 0 0 7 0

C . 0 9 0 1 7 0 . 0 4 4 4 4 0 . 0 1 8 0 6 0 . 0 1 1 2 9 0 . 0 0 8 2 1 0 . 0 0 6 4 5 0 . 0 0 5 8 3 0 . 9 0 1 7 2 0 . 8 9 3 1 1 C . 1 8 4 3 8 0 . 0 9 1 0 6 0 . 0 3 7 0 0 0 . 0 2 3 1 4 0 . 0 1 6 8 3 0 . 0 1 3 2 2 0 . 0 1 1 9 4 0 . 9 2 1 8 9 0 . 9 0 4 2 3 C . 2 8 2 9 2 0 . 1 4 0 0 1 0 . 0 5 6 8 9 0 . 0 3 5 5 8 0 . 0 2 5 8 7 0 . 0 2 0 3 2 0 . 0 1 8 3 5 0 . 9 4 3 0 6 0 . 9 1 5 9 2 0 . 3 8 6 1 3 0 . 1 9 1 4 8 0 . 0 7 7 8 0 0 . 0 4 8 6 5 0 . 0 3 5 3 8 0 . 0 2 7 7 9 0 . 0 2 5 1 0 0 . 9 6 5 3 2 0 . 9 2 8 2 0

C . 4 3 9 6 0 0 . 2 1 8 2 3 0 . 0 8 8 6 6 0 . 0 5 5 4 4 0 . 0 4 0 3 2 0 . 0 3 1 6 7 0 . 0 2 8 6 0 0 . 9 7 6 8 8 0 . 9 3 4 5 8 C . 4 6 6 3 2 0 . 2 3 1 8 6 0 . 0 9 4 2 0 0 . 0 5 8 9 0 0 . 0 4 2 8 3 0 . 0 3 3 6 5 0 . 0 3 0 3 9 0 . 9 8 2 7 8 0 . 9 3 7 8 3 C 4 8 8 8 4 0 . 2 4 2 9 C 0 . 0 9 8 6 8 0 . 0 6 1 7 1 0 . 0 4 4 8 7 0 . 0 3 5 2 5 0 . 0 3 1 8 4 0 . 9 8 7 5 5 0 . 9 4 0 4 7 0 . 4 9 4 3 8 0 . 2 4 5 6 7 0 . 0 9 9 8 1 0 . 0 6 2 4 1 0 . 0 4 5 3 8 0 . 0 3 5 6 5 0 . 0 3 2 2 0 0 . 9 8 8 7 5 0 . 9 4 1 1 3

b = 0 . 0 6 2 5 0 Z E R O O R D E R F I R S T O R D E R S E C O N D O R D E R T H I R D O R D E R

S U M S α = 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0

0 . 9 0 0 . 9 5 0 . 9 9 1 . 0 0

μ0 = 0 . 7 C O 0 . 0 C O 0 . 0 PEAK 0 . 0 0 . 0 0 . 6 5 3 2 7 0 . 9 1 4 5 8 C . 3 2 6 6 4 0 . 1 5 8 0 5 0 . 0 6 4 1 5 0 . 0 4 0 1 1 0 . 0 2 9 1 6 0 . 0 2 2 9 1 0 . 0 2 0 6 9 0 . 0 6 8 4 4 0 . 0 3 7 6 9 C 0 3 4 2 2 0 . 0 1 8 4 1 0 . 0 0 7 4 6 0 . 0 0 4 6 6 0 . 0 0 3 3 9 0 . 0 0 2 6 6 0 . 0 0 2 4 0 0 . 0 0 7 9 4 0 . 0 0 4 3 8 0 . 0 0 3 9 7 0 . 0 0 2 1 4 0 . 0 0 0 8 7 0 . 0 0 0 5 4 0 . 0 0 0 3 9 0 . 0 0 0 3 1 0 . 0 0 0 2 8 0 . 0 0 0 9 3 0 . 0 0 0 5 1

0 . 0 6 6 7 3 0 . 0 3 2 3 6 0 . 0 1 3 1 4 0 . 0 0 8 2 1 0 . 0 0 5 9 7 0 . 0 0 4 6 9 0 . 0 0 4 2 4 0 . 6 6 7 2 9 0 . 9 2 2 3 0 0 . 1 3 6 4 0 0 . 0 6 6 3 1 0 . 0 2 6 9 1 0 . 0 1 6 8 3 0 . 0 1 2 2 3 0 . 0 0 9 6 1 0 . 0 0 8 6 8 0 . 6 8 1 9 8 0 . 9 3 0 3 9 0 . 2 0 9 2 2 0 . 1 0 1 9 6 0 . 0 4 1 3 8 0 . 0 2 5 8 7 0 . 0 1 8 8 1 0 . 0 1 4 7 8 0 . 0 1 3 3 4 0 . 6 9 7 4 1 0 . 9 3 8 8 9 C 2 8 5 4 5 0 . 1 3 9 4 3 0 . 0 5 6 5 9 0 . 0 3 5 3 8 0 . 0 2 5 7 2 0 . 0 2 0 2 1 0 . 0 1 8 2 5 0 . 7 1 3 6 3 0 . 9 4 7 8 3

0 . 3 2 4 9 2 0 . 1 5 8 9 0 0 . 0 6 4 4 9 0 . 0 4 0 3 2 0 . 0 2 9 3 1 0 . 0 2 3 0 3 0 . 0 2 0 8 0 0 . 7 2 2 0 6 0 . 9 5 2 4 7 C . 3 4 5 C 2 0 . 1 6 8 8 3 0 . 0 6 8 5 2 0 . 0 4 2 8 3 0 . 0 3 1 1 4 0 . 0 2 4 4 6 0 . 0 2 2 1 0 0 . 7 2 6 3 5 0 . 9 5 4 8 3 0 . 3 6 1 2 6 0 . 1 7 6 8 6 0 . 0 7 1 7 8 0 . 0 4 4 8 7 0 . 0 3 2 6 3 0 . 0 2 5 6 3 0 . 0 2 3 1 5 0 . 7 2 9 8 3 0 . 9 5 6 7 5 0 . 3 6 5 3 5 0 . 1 7 8 8 9 0 . 0 7 2 6 0 0 . 0 4 5 3 8 0 . 0 3 3 0 0 0 . 0 2 5 9 2 0 . 0 2 3 4 1 0 . 7 3 0 7 0 0 . 9 5 7 2 3




VECTOR

b = 0 . 0 6 2 5 0


SUMS α α 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α - 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

0 . 0 6 2 5 C


/ x = 0 . 3 / x = 0 . 5 μ. = 0.Ί / x = C . 9 /x = 1 . 0 AVERAGE Ν

FLUX U

0 . 9

C . 2 7 7 7 8 0 . 1 2 5 1 6 0 . 0 5 0 5 3 0 . 0 3 1 5 6 0 . 0 2 2 9 3 0 . 0 1 8 0 1 0 . 0 1 6 2 6 0 . 0 5 4 4 0 0 . 0 2 9 6 9 C . C 2 7 2 0 0 . C 1 4 4 8 0 . 0 0 5 8 6 0 . 0 0 3 6 6 0 . 0 0 2 6 6 0 . 0 0 2 0 9 0 . 0 0 1 8 9 0 . 0 0 6 2 5 0 . 0 0 3 4 4 0 . 0 0 3 1 3 C . 0 0 1 6 8 C . 0 0 0 6 8 0 . 0 0 0 4 3 0 . 0 0 0 3 1 0 . 0 0 0 2 4 0 . 0 0 0 2 2 0 . 0 0 0 7 3 0 . 0 0 0 4 0

0 . 0 5 6 6 7 0 . 0 2 5 6 3 0 . 0 1 0 3 5 0 . 0 0 6 4 6 0 . 0 0 4 7 0 0 . 0 0 3 6 9 0 . 0 0 3 3 3 0 . 0 1 1 1 4 0 . 0 0 6 0 8 0 . 1 1 5 6 7 0 . 0 5 2 5 G 0 . 0 2 1 1 9 0 . 0 1 3 2 4 0 . 0 0 9 6 2 0 . 0 0 7 5 6 0 . 0 0 6 8 2 0 . 0 2 2 8 1 0 . 0 1 2 4 5 C . 1 7 7 1 9 0 . 0 8 0 7 0 0 . 0 3 2 5 9 0 . 0 2 0 3 5 0 . 0 1 4 7 9 0 . 0 1 1 6 2 0 . 0 1 0 4 9 0 . 0 3 5 0 6 0 . 0 1 9 1 4 C . 2 4 1 4 0 0 . 1 1 0 3 5 0 . 0 4 4 5 6 0 . 0 2 7 8 3 0 . 0 2 0 2 3 0 . 0 1 5 8 8 0 . 0 1 4 3 4 0 . 0 4 7 9 4 0 . 0 2 6 1 8

C . 2 7 4 5 8 0 . 1 2 5 7 5 C . 0 5 0 7 8 0 . 0 3 1 7 1 0 . 0 2 3 0 5 0 . 0 1 8 1 0 0 . 0 1 6 3 5 0 . 0 5 4 6 2 0 . 0 2 9 8 3 C . 2 9 1 4 5 0 . 1 3 3 6 0 0 . 0 5 3 9 5 0 . 0 3 3 6 9 0 . 0 2 4 4 9 0 . 0 1 9 2 3 0 . 0 1 7 3 7 0 . 0 5 8 0 3 0 . 0 3 1 6 9 0 . 3 0 5 C 9 0 . 1 3 9 9 5 0 . 0 5 6 5 2 0 . 0 3 5 3 0 0 . 0 2 5 6 5 0 . 0 2 0 1 5 0 . 0 1 8 1 9 0 . 0 6 0 7 9 0 . 0 3 3 2 0 C . 3 0 8 5 2 0 . 1 4 1 5 5 0 . 0 5 7 1 6 0 . 0 3 5 7 0 0 . 0 2 5 9 5 0 . 0 2 0 3 8 0 . 0 1 8 4 0 0 . 0 6 1 4 8 0 . 0 3 3 5 8

1 . 0

C . 2 5 0 0 0 0 . 1 1 2 9 9 0 . 0 4 5 6 3 0 . 0 2 8 5 0 0 . 0 2 0 7 1 0 . 0 1 6 2 6 0 . 0 1 4 6 9 0 . 0 4 9 1 1 0 . 0 2 6 8 1 0 . 0 2 4 5 5 0 . 0 1 3 0 8 0 . 0 0 5 2 9 0 . 0 0 3 3 1 0 . 0 0 2 4 0 0 . 0 0 1 8 9 0 . 0 0 1 7 1 0 . 0 0 5 6 5 0 . 0 0 3 1 1 0 . 0 0 2 8 2 0 . 0 0 1 5 2 0 . 0 0 0 6 2 0 . 0 0 0 3 8 0 . 0 0 0 2 8 0 . 0 0 0 2 2 0 . 0 0 0 2 0 0 . 0 0 0 6 6 0 . 0 0 0 3 6

α = 0 . 2 0 α = 0 . 4 0 α - 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

b = 0 . 0 6 2 5 0

C . 0 5 1 0 1 0 . 0 2 3 1 3 0 . 0 0 9 3 4 0 . 0 0 5 8 3 0 . 0 0 4 2 4 0 . 0 0 3 3 3 0 . 0 0 3 0 1 0 . 0 1 0 0 5 0 . 0 0 5 4 9 0 . 1 0 4 1 2 0 . 0 4 7 3 9 0 . 0 1 9 1 4 0 . 0 1 1 9 5 0 . 0 0 8 6 9 0 . 0 0 6 8 2 0 . 0 0 6 1 6 0 . 0 2 0 5 9 0 . 0 1 1 2 4 0 . 1 5 9 4 9 0 . C 7 2 8 6 0 . 0 2 9 4 2 0 . 0 1 8 3 8 0 . 0 1 3 3 6 0 . 0 1 0 4 9 0 . 0 0 9 4 7 0 . 0 3 1 6 5 0 . 0 1 7 2 9 0 . 2 1 7 3 1 0 . 0 9 9 6 2 0 . 0 4 0 2 4 0 . 0 2 5 1 3 0 . 0 1 8 2 6 0 . 0 1 4 3 4 0 . 0 1 2 9 5 0 . 0 4 3 2 7 0 . 0 2 3 6 4

0 . 2 4 7 1 9 0 . 1 1 3 5 3 0 . 0 4 5 8 5 0 . 0 2 8 6 4 0 . 0 2 0 8 1 0 . 0 1 6 3 5 0 . 0 1 4 7 6 0 . 0 4 9 3 0 0 . 0 2 6 9 4 C . 2 6 2 3 8 0 . 1 2 0 6 1 0 . 0 4 8 7 2 0 . 0 3 0 4 3 0 . 0 2 2 1 1 0 . 0 1 7 3 7 0 . 0 1 5 6 8 0 . 0 5 2 3 8 0 . 0 2 8 6 2 0 . 2 7 4 6 6 C . 1 2 6 3 5 0 . 0 5 1 0 3 0 . 0 3 1 8 7 0 . 0 2 3 1 7 0 . 0 1 8 1 9 0 . 0 1 6 4 3 0 . 0 5 4 8 7 0 . 0 2 9 9 8 0 . 2 7 7 7 5 0 . 1 2 7 7 9 0 . 0 5 1 6 2 0 . 0 3 2 2 4 0 . 0 2 3 4 3 0 . 0 1 8 4 0 0 . 0 1 6 6 2 0 . 0 5 5 5 0 0 . 0 3 0 3 2

NARROW SURFACE LAYER AT TOP


SUMS α = α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α 0 . 9 0 α - 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

I N F I N I T E 0 . 3 9 3 1 8 0 . 1 5 4 1 3 0 . 0 9 5 6 9 0 . 0 6 9 3 7 0 . 0 5 4 4 0 0 . 0 4 9 1 1 0 . 1 7 7 7 9 0 . 0 9 0 6 6 C . 0 8 8 8 9 0 . 0 4 3 6 4 0 . 0 1 7 5 6 0 . 0 1 0 9 6 0 . 0 0 7 9 6 0 . 0 0 6 2 5 0 . 0 0 5 6 5 0 . 0 1 8 9 3 0 . 0 1 0 3 2 0 . 0 0 9 4 7 0 . 0 0 5 0 4 0 . 0 0 2 0 4 0 . 0 0 1 2 7 0 . 0 0 0 9 3 0 . 0 0 0 7 3 0 . 0 0 0 6 6 0 . 0 0 2 1 8 0 . 0 0 1 2 0



b = 0 . 0 6 2 5 0


α 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α 0 . 9 0 α = 0 . 9 5 α - 0 . 9 9 α = 1 . 0 0

LAMBERT SURFACE ON TOP

C . 5 0 0 0 0 0 . 2 0 7 7 0 0 . 0 8 3 4 3 0 . 0 5 2 0 5 0 . 0 3 7 8 1 0 . 0 2 9 6 9 0 . 0 2 6 8 1 0 . 0 9 0 6 6 0 . 0 4 9 0 2 0 . 0 4 5 3 3 0 . 0 2 3 8 8 0 . 0 0 9 6 5 0 . 0 0 6 0 3 0 . 0 0 4 3 8 0 . 0 0 3 4 4 0 . 0 0 3 1 1 0 . 0 1 0 3 2 0 . 0 0 5 6 7 C . 0 0 5 1 6 0 . 0 0 2 7 7 0 . 0 0 1 1 2 0 . 0 0 0 7 0 0 . 0 0 0 5 1 0 . 0 0 0 4 0 0 . 0 0 0 3 6 0 . 0 0 1 2 0 0 . 0 0 0 6 6

C . 1 0 1 8 6 0 . 0 4 2 5 2 0 . 0 1 7 0 8 0 . 0 1 0 6 6 0 . 0 0 7 7 4 0 . 0 0 6 0 8 0 . 0 0 5 4 9 0 . 0 1 8 5 6 0 . 0 1 0 0 4 C . 2 0 7 6 0 0 . 0 8 7 0 9 0 . 0 3 4 9 9 0 . 0 2 1 8 3 0 . 0 1 5 8 6 0 . 0 1 2 4 5 0 . 0 1 1 2 4 0 . 0 3 8 0 0 0 . 0 2 0 5 6 0 . 3 1 7 5 2 0 . 1 3 3 8 6 0 . 0 5 3 7 9 0 . 0 3 3 5 6 0 . 0 2 4 3 8 0 . 0 1 9 1 4 0 . 0 1 7 2 9 0 . 0 5 8 3 9 0 . 0 3 1 6 1 C . 4 3 1 9 2 0 . 1 8 3 0 1 0 . 0 7 3 5 5 0 . 0 4 5 8 9 0 . 0 3 3 3 4 0 . 0 2 6 1 8 0 . 0 2 3 6 4 0 . 0 7 9 8 1 0 . 0 4 3 2 2

C . 4 9 0 9 2 0 . 2 0 8 5 3 0 . 0 8 3 8 2 0 . 0 5 2 3 0 0 . 0 3 7 9 9 0 . 0 2 9 8 3 0 . 0 2 6 9 4 0 . 0 9 0 9 3 0 . 0 4 9 2 5 C . 5 2 0 8 8 0 . 2 2 1 5 4 0 . 0 8 9 0 5 0 . 0 5 5 5 6 0 . 0 4 0 3 7 0 . 0 3 1 6 9 0 . 0 2 8 6 2 0 . 0 9 6 6 0 0 . 0 5 2 3 2 C . 5 4 5 0 9 0 . 2 3 2 0 7 0 . 0 9 3 2 9 0 . 0 5 8 2 1 0 . 0 4 2 2 9 0 . 0 3 3 2 0 0 . 0 2 9 9 8 0 . 1 0 1 1 8 0 . 0 5 4 8 1 0 . 5 5 1 1 7 0 . 2 3 4 7 2 0 . C 9 4 3 5 0 . 0 5 8 8 7 0 . 0 4 2 7 7 0 . 0 3 3 5 8 0 . 0 3 0 3 2 0 . 1 0 2 3 4 0 . 0 5 5 4 4




VECTOR j . = Q. 5 /χ = 0 . 7 μ = 0.9 μ = 1.0 AVERAGE

Ν FLUX

U


α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = Ο.ΘΟ

α = 0 . 9 0 α s 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

μ0 = 0 . 9 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 PEAK 0 . 0 0 . 5 1 8 2 8 0 . 9 3 2 9 1 C . 2 5 9 1 4 0 . 1 2 4 2 7 0 . 0 5 0 4 1 0 . 0 3 1 5 1 0 . 0 2 2 9 1 0 . 0 1 8 0 0 0 . 0 1 6 2 5 0 . 0 5 3 8 3 0 . 0 2 9 6 1 0 . 0 2 6 9 2 0 . 0 1 4 4 6 0 . 0 0 5 8 6 0 . 0 0 3 6 6 0 . 0 0 2 6 6 0 . 0 0 2 0 9 0 . 0 0 1 8 9 0 . 0 0 6 2 4 0 . 0 0 3 4 4 C . 0 0 3 1 2 0 . C 0 1 6 8 0 . 0 0 0 6 8 0 . 0 0 0 4 3 0 . 0 0 0 3 1 0 . 0 0 0 2 4 0 . 0 0 0 2 2 0 . 0 0 0 7 3 0 . 0 0 0 4 0

C . 0 5 2 9 3 0 . 0 2 5 4 5 0 . 0 1 0 3 2 0 . 0 0 6 4 5 0 . 0 0 4 6 9 0 . 0 0 3 6 8 0 . 0 0 3 3 3 0 . 5 2 9 3 1 0 . 9 3 8 9 8 C . 1 0 8 1 7 0 . 0 5 2 1 3 0 . 0 2 1 1 5 0 . 0 1 3 2 2 0 . 0 0 9 6 1 0 . 0 0 7 5 5 0 . 0 0 6 8 2 0 . 5 4 0 8 6 0 . 9 4 5 3 3 0 . 1 6 5 9 0 0 . 0 8 0 1 6 0 . 0 3 2 5 1 0 . 0 2 0 3 2 0 . 0 1 4 7 8 0 . 0 1 1 6 1 0 . 0 1 0 4 8 0 . 5 5 3 0 0 0 . 9 5 2 0 1 0 . 2 2 6 3 0 0 . 1 0 9 6 2 0 . 0 4 4 4 6 0 . 0 2 7 7 9 0 . 0 2 0 2 1 0 . 0 1 5 8 7 0 . 0 1 4 3 3 0 . 5 6 5 7 6 0 . 9 5 9 0 3

0 . 2 5 7 5 7 0 . 1 2 4 9 2 0 . 0 5 0 6 7 0 . 0 3 1 6 7 0 . 0 2 3 0 3 0 . 0 1 8 0 9 0 . 0 1 6 3 4 0 . 5 7 2 3 8 0 . 9 6 2 6 8 0 . 2 7 3 4 9 0 . 1 3 2 7 3 0 . 0 5 3 8 3 0 . 0 3 3 6 5 0 . 0 2 4 4 6 0 . 0 1 9 2 2 0 . 0 1 7 3 6 0 . 5 7 5 7 6 0 . 9 6 4 5 3 C . 2 8 6 3 5 0 . 1 3 9 0 4 0 . 0 5 6 3 9 0 . 0 3 5 2 5 0 . 0 2 5 6 3 0 . 0 2 0 1 3 0 . 0 1 8 1 8 0 . 5 7 8 4 9 0 . 9 6 6 0 4 0 . 2 8 9 5 9 0 . 1 4 0 6 3 0 . 0 5 7 0 4 0 . 0 3 5 6 5 0 . 0 2 5 9 2 0 . 0 2 0 3 6 0 . 0 1 8 3 9 0 . 5 7 9 1 8 0 . 9 6 6 4 2

b = 0 . 0 6 2 5 0 2ERO ORDER F I R S T ORDER SECOND ORDER THIRD ORDER

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

Μο = 1 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 PEAK 0 . 4 6 9 7 1 0 . 9 3 9 4 1 0 . 2 3 4 8 5 0 . 1 1 2 2 6 0 . 0 4 5 5 3 0 . 0 2 8 4 6 0 . 0 2 0 6 9 0 . 0 1 6 2 5 0 . 0 1 4 6 8 C . 0 4 8 6 4 0 . 0 2 6 7 5 0 . 0 2 4 3 2 0 . 0 1 3 0 6 0 . 0 0 5 2 9 0 . 0 0 3 3 1 0 . 0 0 2 4 0 0 . 0 0 1 8 9 0 . 0 0 1 7 0 0 . 0 0 5 6 4 0 . 0 0 3 1 1 0 . 0 0 2 8 2 0 . 0 0 1 5 2 0 . 0 0 0 6 2 0 . 0 0 0 3 8 0 . 0 0 0 2 8 0 . 0 0 0 2 2 0 . 0 0 0 2 0 0 . 0 0 0 6 6 C . 0 0 0 3 6

0 . 0 4 7 9 7 0 . 0 2 2 9 9 0 . 0 0 9 3 2 0 . 0 0 5 8 3 0 . 0 0 4 2 4 0 . 0 0 3 3 3 0 . 0 0 3 0 1 0 . 4 7 9 6 7 0 . 9 4 4 8 9 0 . 0 9 8 0 2 0 . 0 4 7 1 0 0 . 0 1 9 1 0 0 . 0 1 1 9 4 0 . 0 0 8 6 8 0 . 0 0 6 8 2 0 . 0 0 6 1 6 0 . 4 9 0 1 1 0 . 9 5 0 6 3 C . 1 5 0 3 2 0 . 0 7 2 4 1 0 . 0 2 9 3 6 0 . 0 1 8 3 5 0 . 0 1 3 3 4 0 . 0 1 0 4 8 0 . 0 0 9 4 7 0 . 5 0 1 0 7 0 . 9 5 6 6 6 C . 2 0 5 0 4 0 . 0 9 9 0 3 0 . 0 4 0 1 6 0 . 0 2 5 1 0 0 . 0 1 8 2 5 0 . 0 1 4 3 3 0 . 0 1 2 9 5 0 . 5 1 2 6 0 0 . 9 6 3 0 0

0 . 2 3 3 3 6 0 . 1 1 2 8 6 0 . 0 4 5 7 6 0 . 0 2 8 6 0 0 . 0 2 0 8 0 0 . 0 1 6 3 4 0 . 0 1 4 7 5 0 . 5 1 8 5 8 0 . 9 6 6 3 0 0 . 2 4 7 7 8 0 . 1 1 9 9 1 0 . 0 4 8 6 2 0 . 0 3 0 3 9 0 . 0 2 2 1 0 0 . 0 1 7 3 6 0 . 0 1 5 6 7 0 . 5 2 1 6 4 0 . 9 6 7 9 7 0 . 2 5 9 4 3 0 . 1 2 5 6 1 0 . 0 5 0 9 3 0 . 0 3 1 8 4 0 . 0 2 3 1 5 0 . 0 1 8 1 8 0 . 0 1 6 4 2 0 . 5 2 4 1 1 0 . 9 6 9 3 3 0 . 2 6 2 3 6 0 . 1 2 7 0 5 0 . 0 5 1 5 2 0 . 0 3 2 2 0 0 . 0 2 3 4 1 0 . 0 1 8 3 9 0 . 0 1 6 6 1 0 . 5 2 4 7 3 0 . 9 6 9 6 8


SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = I . 0 0


SUMS α - 0 . 2 0 o = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α » 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

NARROW SURFACE LAYER AT TOP C O 2 . 6 7 6 3 1 1 . 3 5 3 2 3 0 . 8 8 2 5 0 0 . 6 5 3 2 7 0 . 5 1 8 2 8 0 . 4 6 9 7 1 1 . 1 2 8 4 5 0 . 7 9 8 3 6 0 . 5 6 4 1 6 0 . 3 5 7 8 4 0 . 1 4 9 3 4 0 . 0 9 3 8 9 0 . 0 6 8 4 4 0 . 0 5 3 8 3 0 . 0 4 8 6 4 0 . 1 5 3 0 9 0 . 0 8 7 7 2 0 . 0 7 6 5 4 0 . 0 4 2 8 7 0 . 0 1 7 4 6 0 . 0 1 0 9 2 0 . 0 0 7 9 4 0 . 0 0 6 2 4 0 . 0 0 5 6 4 0 . 0 1 8 4 5 0 . 0 1 0 2 5 0 . 0 0 9 2 3 0 . 0 0 5 0 2 0 . 0 0 2 0 4 0 . 0 0 1 2 7 0 . 0 0 0 9 3 0 . 0 0 0 7 3 0 . 0 0 0 6 6 0 . 0 0 2 1 7 0 . 0 0 1 2 0

0 . 1 1 5 9 7 2 . 7 4 9 6 3 1 . 3 8 3 8 1 0 . 9 0 1 7 2 0 . 6 6 7 2 9 0 . 5 2 9 3 1 0 . 4 7 9 6 7 1 . 1 5 9 7 0 0 . 8 1 6 3 0 C . 2 3 8 5 3 2 . 8 2 6 6 4 1 . 4 1 5 8 9 0 . 9 2 1 8 9 0 . 6 8 1 9 8 0 . 5 4 0 8 6 0 . 4 9 0 1 1 1 . 1 9 2 6 6 0 . 8 3 5 1 4 0 . 3 6 8 2 0 2 . 9 0 7 6 1 1 . 4 4 9 5 9 0 . 9 4 3 0 6 0 . 6 9 7 4 1 0 . 5 5 3 0 0 0 . 5 0 1 0 7 1 . 2 2 7 3 3 0 . 8 5 4 9 4 C . 5 0 5 5 3 2 . 9 9 2 8 5 1 . 4 8 5 0 2 0 . 9 6 5 3 2 0 . 7 1 3 6 3 0 . 5 6 5 7 6 0 . 5 1 2 6 0 1 . 2 6 3 8 3 0 . 8 7 5 7 5

0 . 5 7 7 2 7 3 . 0 3 7 1 7 1 . 5 0 3 4 3 0 . 9 7 6 8 8 0 . 7 2 2 0 6 0 . 5 7 2 3 8 0 . 5 1 8 5 8 1 . 2 8 2 8 2 0 . 8 8 6 5 6 C . 6 1 3 9 4 3 . 0 5 9 7 8 1 . 5 1 2 8 2 0 . 9 8 2 7 8 0 . 7 2 6 3 5 0 . 5 7 5 7 6 0 . 5 2 1 6 4 1 . 2 9 2 5 0 0 . 8 9 2 0 8 0 . 6 4 3 6 7 3 . 0 7 8 0 9 1 . 5 2 0 4 1 0 . 9 8 7 5 5 0 . 7 2 9 8 3 0 . 5 7 8 4 9 0 . 5 2 4 1 1 1 . 3 0 0 3 4 0 . 8 9 6 5 4 0 . 6 5 1 1 6 3 . C 8 2 7 0 1 . 5 2 2 3 3 0 . 9 8 8 7 5 0 . 7 3 0 7 0 0 . 5 7 9 1 8 0 . 5 2 4 7 3 1 . 3 0 2 3 2 0 . 8 9 7 6 6

LAMBERT SURFACE ON TOP C O 0 . 5 3 5 2 6 0 . 8 1 1 9 4 0 . 8 8 2 5 0 0 . 9 1 4 5 8 0 . 9 3 2 9 1 0 . 9 3 9 4 1 0 . 7 9 8 3 6 0 . 8 8 9 5 2 C . 3 9 9 1 7 0 . 2 0 3 1 1 0 . 0 8 2 8 0 0 . 0 5 1 8 1 0 . 0 3 7 6 9 0 . 0 2 9 6 1 0 . 0 2 6 7 5 0 . 0 8 7 7 2 0 . 0 4 8 6 4 0 . 0 4 3 8 6 0 . 0 2 3 7 7 0 . 0 0 9 6 4 0 . 0 0 6 0 3 0 . 0 0 4 3 8 0 . 0 0 3 4 4 0 . 0 0 3 1 1 0 . 0 1 0 2 5 0 . 0 0 5 6 6 C 0 0 5 1 3 0 . 0 0 2 7 7 0 . 0 0 1 1 2 0 . 0 0 0 7 0 0 . 0 0 0 5 1 0 . 0 0 0 4 0 0 . 0 0 0 3 6 0 . 0 0 1 2 0 0 . 0 0 0 6 6

0 . 0 8 1 6 3 0 . 5 7 6 8 6 0 . 8 2 8 8 9 0 . 8 9 3 1 1 0 . 9 2 2 3 0 0 . 9 3 8 9 8 0 . 9 4 4 8 9 0 . 8 1 6 3 0 0 . 8 9 9 4 8 C . 1 6 7 0 3 0 . 6 2 0 4 9 0 . 8 4 6 6 7 0 . 9 0 4 2 3 0 . 9 3 0 3 9 0 . 9 4 5 3 3 0 . 9 5 0 6 3 0 . 8 3 5 1 4 0 . 9 0 9 9 2 0 . 2 5 6 4 8 0 . 6 6 6 3 3 0 . 8 6 5 3 5 0 . 9 1 5 9 2 0 . 9 3 8 8 9 0 . 9 5 2 0 1 0 . 9 5 6 6 6 0 . 8 5 4 9 4 0 . 9 2 0 8 9 0 . 3 5 0 3 0 0 . 7 1 4 5 3 0 . 8 8 4 9 8 0 . 9 2 8 2 0 0 . 9 4 7 8 3 0 . 9 5 9 0 3 0 . 9 6 3 0 0 0 . 8 7 5 7 5 0 . 9 3 2 4 2

C . 3 9 8 9 5 0 . 7 3 9 5 7 0 . 8 9 5 1 8 0 . 9 3 4 5 8 0 . 9 5 2 4 7 0 . 9 6 2 6 8 0 . 9 6 6 3 0 0 . 8 8 6 5 6 0 . 9 3 8 4 1 0 . 4 2 3 7 4 0 . 7 5 2 3 4 0 . 9 0 0 3 8 0 . 9 3 7 8 3 0 . 9 5 4 8 3 0 . 9 6 4 5 3 0 . 9 6 7 9 7 0 . 8 9 2 0 8 0 . 9 4 1 4 7 0 . 4 4 3 7 9 0 . 7 6 2 6 8 0 . 9 0 4 5 9 0 . 9 4 0 4 7 0 . 9 5 6 7 5 0 . 9 6 6 0 4 0 . 9 6 9 3 3 0 . 8 9 6 5 4 0 . 9 4 3 9 4 0 . 4 4 8 8 3 0 . 7 6 5 2 8 0 . 9 0 5 6 5 0 . 9 4 1 1 3 0 . 9 5 7 2 3 0 . 9 6 6 4 2 0 . 9 6 9 6 8 0 . 8 9 7 6 6 0 . 9 4 4 5 6




/ * = 0 . 0 i = 0 . 1 / i . e O . 3 f t = 0 . 5 f t = 0 . 7 f t = 0 . 9 AVERAGE Ν

FLUX

υ b 0 . 1 2 5 0 0 Μο

F I R S T OROER 2 . 5 0 0 0 0 SECOND ORDER C . 2 5 6 6 9 THIRD OROER C . 0 4 4 1 2

SUMS α = 0 . 2 0 C . 5 1 0 6 3 α = 0 . 4 0 1 . 0 4 4 1 2 α = 0 . 6 0 1 . 6 0 3 1 4 α = Ο.ΘΟ 2 . 1 9 0 8 5

α = 0 . 9 0 2 . 4 9 6 6 0 α = 0 . 9 5 2 . 6 5 2 6 8 α = 0 . 9 9 2 . 7 7 9 1 4 α = 1 . 0 0 2 . 8 1 0 9 8

1 . 1 4 7 3 9 0 . 5 0 6 9 5 0 . 3 2 3 7 0 0 . 1 9 8 1 5 0 . 0 9 3 0 8 0 . 0 6 0 2 0

0 . 2 3 7 6 1 0 . 1 8 7 6 6 0 . 1 6 9 8 1 0 . 0 . 0 4 4 4 3 0 . 0 3 5 2 0 0 . 0 3 1 8 8 0 ,

0 . 0 1 1 4 3 0 . 0 0 8 4 5 0 . 0 0 6 7 0 0 . 0 0 6 0 7 0 . 0 1 6 4 8 0 . 0 1 0 4 6

5 1 3 3 7 0 . 2 9 9 9 8 0 8 8 2 5 0 . 0 5 5 2 2

. 2 3 7 7 1 0 . 1 0 5 2 6

. 4 9 3 2 2 0 . 2 1 8 9 0

. 7 6 8 7 8 0 . 3 4 1 9 8

. 0 6 7 C 4 C . 4 7 5 7 8

. 2 2 5 6 6 C . 5 4 7 1 6

. 3 0 7 5 2 0 . 5 8 4 0 6

. 3 7 4 3 0 0 . 6 1 4 1 9

. 3 9 1 1 7 0 . 6 2 1 8 1

0 6 7 2 4 0 . 0 4 9 3 7 0 . 0 3 9 0 0 1 3 9 9 0 0 . 1 0 2 7 4 0 . 0 8 1 1 6 2 1 8 6 8 0 . 1 6 0 6 2 0 . 1 2 6 9 0 3 0 4 3 9 0 . 2 2 3 6 3 0 . 1 7 6 7 0

0 . 0 3 5 2 9 0 . 1 0 6 3 4 0 . 0 6 2 2 9 0 . 0 7 3 4 5 0 . 2 2 0 6 1 0 . 1 2 9 5 5 0 . 1 1 4 8 4 0 . 3 4 3 8 1 0 . 2 0 2 4 2 0 . 1 5 9 9 2 0 . 4 7 7 1 3 0 . 2 8 1 6 4

3 5 0 1 5 3 7 3 8 1 3 9 3 1 3 3 9 8 0 2

, 2 5 7 2 7 2 7 4 6 7

, 2 8 8 8 9 , 2 9 2 4 8

. 2 0 3 3 0 0 . 1 8 4 0 0 0 . 5 4 8 0 1 0 . 3 2 3 9 1 , 2 1 7 0 6 0 . 1 9 6 4 5 0 . 5 8 4 5 9 0 . 3 4 5 7 6 . 2 2 8 3 0 0 . 2 0 6 6 2 0 . 6 1 4 4 3 0 . 3 6 3 6 1 , 2 3 1 1 4 0 . 2 0 9 1 9 0 . 6 2 1 9 7 0 . 3 6 8 1 2

b = 0 . 1 2 5 0 0


Mo 0 . 3

0 . 8 3 3 3 3 0 . 5 0 6 9 5 0 . 2 3 5 5 8 0 . 1 5 2 0 6 0 . 1 1 2 1 4 0 . 0 8 8 8 0 0 . 0 8 0 4 3 0 . 2 2 7 0 8 0 . 1 3 9 7 5 0 . 1 1 3 5 4 0 . 0 9 3 0 8 0 . 0 4 4 2 9 0 . 0 2 8 7 2 0 . 0 2 1 2 2 0 . 0 1 6 8 2 0 . 0 1 5 2 4 0 . 0 4 1 5 2 0 . 0 2 6 2 9 C . 0 2 0 7 6 0 . 0 1 7 6 2 0 . 0 0 8 4 3 0 . 0 0 5 4 7 0 . 0 0 4 0 5 0 . 0 0 3 2 1 0 . 0 0 2 9 1 0 . 0 0 7 8 6 0 . 0 0 5 0 1

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α - C . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . C O

0 . 1 7 1 3 8 0 . 1 0 5 2 6 0 . 0 4 8 9 6 0 . 0 3 1 6 1 0 . 0 2 3 3 1 0 . 0 1 8 4 6 0 . 0 1 6 7 2 0 . 0 4 7 1 4 0 . 0 2 9 0 4 0 . 3 5 2 9 4 0 . 2 1 8 9 0 0 . 1 0 1 9 0 0 . 0 6 5 8 0 0 . 0 4 8 5 3 0 . 0 3 8 4 3 0 . 0 3 4 8 1 0 . 0 9 8 0 2 0 . 0 6 0 4 5 0 . 5 4 5 9 3 0 . 3 4 1 9 8 0 . 1 5 9 3 5 0 . 1 0 2 9 1 0 . 0 7 5 9 1 0 . 0 6 0 1 2 0 . 0 5 4 4 5 0 . 1 5 3 1 1 0 . 0 9 4 5 4 C . 7 5 1 8 6 0 . 4 7 5 7 8 0 . 2 2 1 9 1 0 . 1 4 3 3 3 0 . 1 0 5 7 4 0 . 0 8 3 7 5 0 . 0 7 5 8 5 0 . 2 1 2 9 9 0 . 1 3 1 6 5

0 . 8 6 0 2 2 0 . 5 4 7 1 6 0 . 2 5 5 3 2 0 . 1 6 4 9 3 0 . 1 2 1 6 8 0 . 0 9 6 3 7 0 . 0 8 7 2 9 0 . 2 4 4 9 2 0 . 1 5 1 4 8 0 . 9 1 5 8 5 0 . 5 8 4 0 6 0 . 2 7 2 6 1 0 . 1 7 6 1 0 0 . 1 2 9 9 2 0 . 1 0 2 9 0 0 . 0 9 3 2 1 0 . 2 6 1 4 3 0 . 1 6 1 7 4 0 . 9 6 1 C 8 0 . 6 1 4 1 9 0 . 2 8 6 7 2 0 . 1 8 5 2 3 0 . 1 3 6 6 6 0 . 1 0 8 2 4 0 . 0 9 8 0 4 0 . 2 7 4 9 1 0 . 1 7 0 1 1 0 . 9 7 2 4 9 0 . 6 2 1 - 8 1 C . 2 9 0 2 9 0 . 1 8 7 5 4 0 . 1 3 8 3 6 0 . 1 0 9 5 9 0 . 0 9 9 2 6 0 . 2 7 8 3 2 0 . 1 7 2 2 3

b = 0 . 1 2 5 0 0


Mo = 0 . 5

0 . 5 0 G 0 0 0 . 3 2 3 7 C 0 . 1 5 2 0 6 0 . 0 9 8 3 7 0 . 0 7 2 6 2 0 . 0 5 7 5 3 0 . 0 5 2 1 2 0 . 1 4 5 1 2 0 . 0 9 0 2 6 0 . 0 7 2 5 6 0 . 0 6 0 2 0 0 . 0 2 8 7 2 0 . 0 1 8 6 3 0 . 0 1 3 7 7 0 . 0 1 0 9 2 0 . 0 0 9 8 9 0 . 0 2 6 8 6 0 . 0 1 7 0 5 C . O 1 3 4 3 0 . 0 1 1 4 3 0 . 0 0 5 4 7 0 . 0 0 3 5 5 0 . 0 0 2 6 3 0 . 0 0 2 0 8 0 . 0 0 1 8 9 0 . 0 0 5 1 0 0 . 0 0 3 2 5

SUMS α α 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

C . 1 0 3 0 1 0 . 0 6 7 2 4 0 . 0 3 1 6 1 0 . 0 2 0 4 5 0 . 0 1 5 1 0 0 . 0 1 1 9 6 0 . 0 1 0 8 4 0 . 0 3 0 1 4 0 . 0 1 8 7 6 0 . 2 1 2 5 4 0 . 1 3 9 9 0 0 . 0 6 5 8 0 0 . 0 4 2 5 7 0 . 0 3 1 4 3 0 . 0 2 4 9 0 0 . 0 2 2 5 6 0 . 0 6 2 7 0 0 . 0 3 9 0 6 C . 3 2 9 4 0 0 . 2 1 8 6 8 0 . 1 0 2 9 1 0 . 0 6 6 5 9 0 . 0 4 9 1 7 0 . 0 3 8 9 6 0 . 0 3 5 2 9 0 . 0 9 7 9 9 0 . 0 6 1 0 8 0 . 4 5 4 5 5 0 . 3 0 4 3 9 0 . 1 4 3 3 3 0 . 0 9 2 7 6 0 . 0 6 8 4 9 0 . 0 5 4 2 7 0 . 0 4 9 1 7 0 . 1 3 6 3 7 0 . 0 8 5 0 8

C . 5 2 0 5 9 0 . 3 5 0 1 5 0 . 1 6 4 9 3 0 . 1 0 6 7 5 0 . 0 7 8 8 2 0 . 0 6 2 4 6 0 . 0 5 6 5 8 0 . 1 5 6 8 6 0 . 0 9 7 9 0 0 . 5 5 4 5 4 0 . 3 7 3 8 1 0 . 1 7 6 1 0 0 . 1 1 3 9 8 0 . 0 8 4 1 6 0 . 0 6 6 6 9 0 . 0 6 0 4 2 0 . 1 6 7 4 5 0 . 1 0 4 5 3 C . 5 8 2 1 7 0 . 3 9 3 1 3 0 . 1 8 5 2 3 0 . 1 1 9 8 9 0 . 0 8 8 5 3 0 . 0 7 0 1 5 0 . 0 6 3 5 5 0 . 1 7 6 1 0 0 . 1 0 9 9 5 C . 5 8 9 1 4 0 . 3 9 8 0 2 0 . 1 8 7 5 4 0 . 1 2 1 3 9 0 . 0 8 9 6 3 0 . 0 7 1 0 3 0 . 0 6 4 3 4 0 . 1 7 8 2 9 0 . 1 1 1 3 2

b = 0 . 1 2 5 0 0


Mo 0 . 7

0 . 3 5 7 1 4 0 . 2 3 7 6 1 0 . 1 1 2 1 4 0 . 0 7 2 6 2 0 . 0 5 3 6 3 0 . 0 4 2 5 0 0 . 0 3 8 5 0 0 . 1 0 6 5 7 0 . 0 6 6 5 8 0 . 0 5 3 2 9 0 . 0 4 4 4 3 0 . 0 2 1 2 2 0 . 0 1 3 7 7 0 . 0 1 0 1 8 0 . 0 0 8 0 7 0 . 0 0 7 3 1 0 . 0 1 9 8 3 0 . 0 1 2 6 0 C . 0 0 9 9 2 0 . 0 0 8 4 5 0 . 0 0 4 0 5 0 . 0 0 2 6 3 0 . 0 0 1 9 4 0 . 0 0 1 5 4 0 . 0 0 1 4 0 0 . 0 0 3 7 7 0 . 0 0 2 4 0

SUMS α = 0 . 2 0 C . 0 7 3 6 4 0 . 0 4 9 3 7 0 . 0 2 3 3 1 0 . 0 1 5 1 0 0 . 0 1 1 1 5 0 . 0 0 8 8 4 0 . 0 0 8 0 0 0 . 0 2 2 1 4 0 . 0 1 3 8 4 a = 0 . 4 0 C . 1 5 2 0 7 0 . 1 0 2 7 4 0 . 0 4 8 5 3 0 . 0 3 1 4 3 0 . 0 2 3 2 2 0 . 0 1 8 4 0 0 . 0 1 6 6 7 0 . 0 4 6 0 6 0 . 0 2 8 8 1 a = 0 . 6 0 C . 2 3 5 8 9 0 . 1 6 0 6 2 0 . 0 7 5 9 1 0 . 0 4 9 1 7 0 . 0 3 6 3 2 0 . 0 2 8 7 8 0 . 0 2 6 0 7 0 . 0 7 2 0 0 0 . 0 4 5 0 7 a = 0 . 8 0 0 . 3 2 5 8 1 0 . 2 2 3 6 3 0 . 1 0 5 7 4 0 . 0 6 8 4 9 0 . 0 5 0 5 9 0 . 0 4 0 1 0 0 . 0 3 6 3 3 0 . 1 0 0 2 3 0 . 0 6 2 7 8

0 . 9 0 C . 3 7 3 3 1 0 . 2 5 7 2 7 0 . 1 2 1 6 8 0 . 0 7 8 8 2 0 . 0 5 8 2 2 0 . 0 4 6 1 4 0 . 0 4 1 8 0 0 . 1 1 5 3 0 0 . 0 7 2 2 4 0 . 9 5 0 . 3 9 7 7 5 0 . 2 7 4 6 7 0 . 1 2 9 9 2 0 . 0 8 4 1 6 0 . 0 6 2 1 7 0 . 0 4 9 2 7 0 . 0 4 4 6 4 0 . 1 2 3 0 9 0 . 0 7 7 1 4 0 . 9 9 0 . 4 1 7 6 5 0 . 2 8 8 8 9 0 . 1 3 6 6 6 0 . 0 8 8 5 3 0 . 0 6 5 3 9 0 . 0 5 1 8 3 0 . 0 4 6 9 5 0 . 1 2 9 4 5 0 . 0 8 1 1 4 1 . 0 0 0 . 4 2 2 6 7 0 . 2 9 2 4 8 0 . 1 3 8 3 6 0 . 0 8 9 6 3 0 . 0 6 6 2 1 0 . 0 5 2 4 7 0 . 0 4 7 5 4 0 . 1 3 1 0 6 0 . 0 8 2 1 5




VECTOR / χ = 0 . 0 / χ = 0 . 1 / i * 0 . 3 AVERAGE

Ν FLUX

U

b= 0 . 1 2 5 0 0 ZERO ORDER F I R S T ORDER SECOND ORDER THIRD ORDER

α 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α 0 . 8 0

α - 0 . 9 0 α * 0 . 9 5 α = 0 . 9 9 α 1 . 0 0

Mo = 0 . 1 C O PEAK 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 1 . 4 3 2 5 2 0 . 2 8 6 5 0 0 . 7 1 6 2 6 0 . 8 9 5 3 3 0 . 4 6 5 9 2 0 . 3 0 7 6 9 0 . 2 2 9 1 5 0 . 1 8 2 4 4 0 . 1 6 5 5 5 0 . 4 0 8 6 0 0 . 2 7 8 2 5 C . 2 0 4 3 0 0 . 1 8 6 7 9 0 . 0 9 1 2 3 0 . 0 5 9 4 7 0 . 0 4 4 0 5 0 . 0 3 4 9 6 0 . 0 3 1 6 9 0 . 0 8 3 7 6 0 . 0 5 4 2 4 0 . 0 4 1 8 8 0 . 0 3 6 4 3 0 . 0 1 7 5 4 0 . 0 1 1 4 0 0 . 0 0 8 4 3 0 . 0 0 6 6 9 0 . 0 0 6 0 6 0 . 0 1 6 2 7 0 . 0 1 0 4 2

C . 1 5 1 7 7 0 . 1 8 6 8 4 0 . 0 9 6 9 8 0 . 0 6 4 0 1 0 . 0 4 7 6 6 0 . 0 3 7 9 4 0 . 0 3 4 4 3 1 . 5 1 7 7 3 0 . 3 4 4 4 1 0 . 3 2 2 1 0 0 . 3 9 0 5 4 0 . 2 0 2 1 8 0 . 1 3 3 3 8 0 . 0 9 9 2 9 0 . 0 7 9 0 3 0 . 0 7 1 7 1 1 . 6 1 0 4 9 0 . 4 0 7 2 1 0 . 5 1 3 5 4 0 . 6 1 3 3 3 0 . 3 1 6 6 7 0 . 2 0 8 8 0 0 . 1 5 5 4 0 0 . 1 2 3 6 8 0 . 1 1 2 2 2 1 . 7 1 1 8 1 0 . 4 7 5 5 3 0 . 7 2 9 1 4 0 . 8 5 7 8 4 0 . 4 4 1 7 2 0 . 2 9 1 1 0 0 . 2 1 6 6 1 0 . 1 7 2 3 7 0 . 1 5 6 3 9 1 . 8 2 2 8 5 0 . 5 5 0 1 2

0 . 8 4 7 1 0 0 . 9 8 9 1 9 0 . 5 0 8 6 6 0 . 3 3 5 1 2 0 . 2 4 9 3 4 0 . 1 9 8 4 1 0 . 1 8 0 0 0 1 . 8 8 2 4 5 0 . 5 9 0 0 4 C . 9 0 8 8 4 1 . 0 5 7 3 3 0 . 5 4 3 3 3 0 . 3 5 7 9 1 0 . 2 6 6 2 8 0 . 2 1 1 8 8 0 . 1 9 2 2 2 1 . 9 1 3 3 5 0 . 6 1 0 7 1 C . 9 5 9 6 2 1 . 1 1 3 0 7 0 . 5 7 1 6 6 0 . 3 7 6 5 4 0 . 2 8 0 1 2 0 . 2 2 2 8 9 0 . 2 0 2 2 1 1 . 9 3 8 6 2 0 . 6 2 7 6 1 0 . 9 7 2 5 1 1 . 1 2 7 1 8 0 . 5 7 8 8 3 0 . 3 8 1 2 5 0 . 2 8 3 6 2 0 . 2 2 5 6 7 0 . 2 0 4 7 3 1 . 9 4 5 0 2 0 . 6 3 1 8 8

b= 0 . 1 2 5 0 0 ZERO ORDER F I R S T ORDER SECOND ORDER T H I R D ORDER

μο = 0 . 3 C O 0 . 0 PEAK 0 . 0 0 . 0 0 . 0 0 . 0 1 . 0 9 8 7 3 0 . 6 5 9 2 4 0 . 5 4 9 3 7 0 . 4 6 5 9 2 0 . 2 2 8 9 0 0 . 1 4 9 4 5 0 . 1 1 0 7 6 0 . 0 8 7 9 5 0 . 0 7 9 7 3 0 . 2 1 0 0 9 0 . 1 3 6 2 2 0 . 1 0 5 0 4 0 . 0 9 1 2 3 0 . 0 4 3 9 9 0 . 0 2 8 6 0 0 . 0 2 1 1 6 0 . 0 1 6 7 8 0 . 0 1 5 2 1 0 . 0 4 0 7 9 0 . 0 2 6 1 3 0 . 0 2 0 4 0 0 . 0 1 7 5 4 0 . 0 0 8 4 2 0 . 0 0 5 4 7 0 . 0 0 4 0 4 0 . 0 0 3 2 1 0 . 0 0 2 9 0 0 . 0 0 7 8 3 0 . 0 0 5 0 0

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

C . 1 1 4 2 4 0 . 0 9 6 9 8 0 . 0 4 7 6 1 0 . 0 3 1 0 8 0 . 0 2 3 0 3 0 . 0 1 8 2 9 0 . 0 1 6 5 8 1 . 1 4 2 4 5 0 . 6 8 7 5 7 0 . 2 3 7 9 7 0 . 2 0 2 1 8 0 . 0 9 9 1 8 0 . 0 6 4 7 3 0 . 0 4 7 9 7 0 . 0 3 8 0 9 0 . 0 3 4 5 3 1 . 1 8 9 8 4 0 . 7 1 8 2 6 0 . 3 7 2 4 1 0 . 3 1 6 6 7 0 . 1 5 5 2 3 0 . 1 0 1 3 0 0 . 0 7 5 0 6 0 . 0 5 9 6 0 0 . 0 5 4 0 2 1 . 2 4 1 3 8 0 . 7 5 1 6 0 0 . 5 1 9 0 6 0 . 4 4 1 7 2 0 . 2 1 6 3 6 0 . 1 4 1 1 7 0 . 1 0 4 6 0 0 . 0 8 3 0 4 0 . 0 7 5 2 8 1 . 2 9 7 6 4 0 . 7 8 7 9 6

α = 0 . 9 0 0 . 5 9 7 4 9 0 . 5 0 8 6 6 0 . 2 4 9 0 5 0 . 1 6 2 4 8 0 . 1 2 0 3 8 0 . 0 9 5 5 7 0 . 0 8 6 6 4 1 . 3 2 7 7 5 0 . 8 0 7 4 0 α = 0 . 9 5 0 . 6 3 8 0 8 0 . 5 4 3 3 3 0 . 2 6 5 9 7 0 . 1 7 3 5 1 0 . 1 2 8 5 6 0 . 1 0 2 0 6 0 . 0 9 2 5 2 1 . 3 4 3 3 3 0 . 8 1 7 4 7 α = 0 . 9 9 0 . 6 7 1 2 5 0 . 5 7 1 6 6 0 . 2 7 9 8 0 0 . 1 8 2 5 3 0 . 1 3 5 2 3 0 . 1 0 7 3 6 0 . 0 9 7 3 2 1 . 3 5 6 0 7 0 . 8 2 5 6 9 α = I . 0 0 C . 6 7 9 6 5 0 . 5 7 8 8 3 0 . 2 8 3 2 9 0 . 1 8 4 8 1 0 . 1 3 6 9 2 0 . 1 0 8 7 0 0 . 0 9 8 5 3 1 . 3 5 9 2 9 0 . 8 2 7 7 7

0 . 1 2 5 0 0 ZERO ORDER F I R S T ORDER SECOND ORDER THIRD ORDER

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

μ0 = 0 . 5 C O 0 . 0 0 . 0 PEAK 0 . 0 0 . 0 0 . 0 0 . 7 7 8 8 0 0 . 7 7 8 8 0 C . 3 8 9 4 0 0 . 3 0 7 6 8 0 . 1 4 9 4 5 0 . 0 9 7 3 5 0 . 0 7 2 0 8 0 . 0 5 7 2 0 0 . 0 5 1 8 5 0 . 1 3 8 4 9 0 . 0 8 8 8 8 0 . 0 6 9 2 5 0 . 0 5 9 4 7 0 . 0 2 8 6 0 0 . 0 1 8 5 8 0 . 0 1 3 7 5 0 . 0 1 C 9 0 0 . 0 0 9 8 8 0 . 0 2 6 5 8 0 . 0 1 6 9 9 0 . 0 1 3 2 9 0 . 0 1 1 4 0 0 . 0 0 5 4 7 0 . 0 0 3 5 5 0 . 0 0 2 6 3 0 . 0 0 2 0 8 0 . 0 0 1 8 9 0 . 0 0 5 0 9 0 . 0 0 3 2 5

C . 0 8 0 7 6 0 . 0 6 4 0 1 0 . 0 3 1 0 8 0 . 0 2 0 2 4 0 . 0 1 4 9 9 0 . 0 1 1 8 9 0 . 0 1 0 7 8 0 . 8 0 7 6 0 0 . 7 9 7 2 8 0 . 1 6 7 7 6 0 . 1 3 3 3 8 0 . 0 6 4 7 3 0 . 0 4 2 1 6 0 . 0 3 1 2 1 0 . 0 2 4 7 7 0 . 0 2 2 4 5 0 . 8 3 8 8 0 0 . 8 1 7 2 9 0 . 2 6 1 8 1 0 . 2 0 8 8 0 0 . 1 0 1 3 0 0 . 0 6 5 9 7 0 . 0 4 8 8 4 0 . 0 3 8 7 5 0 . 0 3 5 1 3 0 . 8 7 2 7 0 0 . 8 3 9 0 3 0 . 3 6 3 8 7 0 . 2 9 1 1 0 0 . 1 4 1 1 7 0 . 0 9 1 9 2 0 . 0 6 8 0 5 0 . 0 5 4 0 0 0 . 0 4 8 9 4 0 . 9 0 9 6 8 0 . 8 6 2 7 4

α = 0 . 9 0 C . 4 1 8 2 5 0 . 3 3 5 1 2 0 . 1 6 2 4 8 0 . 1 0 5 7 9 0 . 0 7 8 3 2 0 . 0 6 2 1 5 0 . 0 5 6 3 3 0 . 9 2 9 4 5 0 . 8 7 5 4 1 a = 0 . 9 5 0 . 4 4 6 3 5 0 . 3 5 7 9 1 0 . 1 7 3 5 1 0 . 1 1 2 9 7 0 . 0 8 3 6 3 0 . 0 6 6 3 6 0 . 0 6 0 1 5 0 . 9 3 9 6 8 0 . 8 8 1 9 7 a = 0 . 9 9 C . 4 6 9 2 8 0 . 3 7 6 5 4 0 . 1 8 2 5 3 0 . 1 1 8 8 4 0 . 0 8 7 9 7 0 . 0 6 9 8 1 0 . 0 6 3 2 7 0 . 9 4 8 0 4 0 . 8 8 7 3 2 a = 1 . 0 0 C . 4 7 5 0 8 0 . 3 8 1 2 5 0 . 1 8 4 8 1 0 . 1 2 0 3 2 0 . 0 8 9 0 7 0 . 0 7 0 6 8 0 . 0 6 4 0 6 0 . 9 5 0 1 6 0 . 8 8 8 6 8

b= 0 . 1 2 5 0 0 ZERU ORDER F I R S T ORDER SECOND ORDER THIRD ORDER

μ0 = 0 . 7 C O 0 . 0 0 . 0 0 . 0 PEAK 0 . 0 0 . 0 0 . 5 9 7 4 7 0 . 8 3 6 4 6 0 . 2 9 8 7 4 0 . 2 2 9 1 5 0 . 1 1 0 7 6 0 . 0 7 2 0 8 0 . 0 5 3 3 5 0 . 0 4 2 3 3 0 . 0 3 8 3 6 0 . 1 0 3 0 7 0 . 0 6 5 8 5 C . 0 5 1 5 3 0 . 0 4 4 0 5 0 . 0 2 1 1 6 0 . 0 1 3 7 5 0 . 0 1 0 1 7 0 . 0 0 8 0 6 0 . 0 0 7 3 1 0 . 0 1 9 6 8 0 . 0 1 2 5 7 0 . 0 0 9 8 4 0 . 0 0 8 4 3 0 . 0 0 4 0 4 0 . 0 0 2 6 3 6 . 0 0 1 9 4 0 . 0 0 1 5 4 0 . 0 0 1 3 9 0 . 0 0 3 7 6 0 . 0 0 2 4 0

SUMS α - 0 . 2 0 0 . 0 6 1 8 9 0 . 0 4 7 6 6 0 . 0 2 3 0 3 0 . 0 1 4 9 9 0 . 0 1 1 0 9 0 . 0 0 8 8 0 0 . 0 0 7 9 8 0 . 6 1 8 9 1 0 . 8 5 0 1 6 α = 0 . 4 0 0 . 1 2 8 4 2 0 . 0 9 9 2 9 0 . 0 4 7 9 7 0 . 0 3 1 2 1 0 . 0 2 3 1 0 0 . 0 1 8 3 3 0 . 0 1 6 6 1 0 . 6 4 2 1 1 0 . 8 6 4 9 8 α = 0 . 6 0 0 . 2 0 0 2 0 0 . 1 5 5 4 0 0 . 0 7 5 0 6 0 . 0 4 8 8 4 0 . 0 3 6 1 4 0 . 0 2 8 6 7 0 . 0 2 5 9 9 0 . 6 6 7 3 2 0 . 8 8 1 0 9 α = 0 . 8 0 C 2 7 7 9 2 0 . 2 1 6 6 1 0 . 1 0 4 6 0 0 . 0 6 8 0 5 0 . 0 5 0 3 6 0 . 0 3 9 9 5 0 . 0 3 6 2 1 0 . 6 9 4 8 0 0 . 8 9 8 6 4

α = 0 . 9 0 C . 3 1 9 2 7 0 . 2 4 9 3 4 0 . 1 2 0 3 8 0 . 0 7 8 3 2 0 . 0 5 7 9 5 0 . 0 4 5 9 8 0 . 0 4 1 6 7 0 . 7 0 9 4 9 0 . 9 0 8 0 2 α = 0 . 9 5 0 . 3 4 0 6 2 0 . 2 6 6 2 8 0 . 1 2 8 5 6 0 . 0 8 3 6 3 0 . 0 6 1 8 9 0 . 0 4 9 1 0 0 . 0 4 4 5 0 0 . 7 1 7 0 9 0 . 9 1 2 8 8 α = 0 . 9 9 0 . 3 5 8 0 4 0 . 2 8 0 1 2 0 . 1 3 5 2 3 0 . 0 8 7 9 7 0 . 0 6 5 1 0 0 . 0 5 1 6 5 0 . 0 4 6 8 1 0 . 7 2 3 3 1 0 . 9 1 6 8 5 α = 1 . 0 0 0 . 3 6 2 4 4 0 . 2 8 3 6 2 0 . 1 3 6 9 2 0 . 0 8 9 0 7 0 . 0 6 5 9 1 0 . 0 5 2 2 9 0 . 0 4 7 3 9 0 . 7 2 4 8 8 0 . 9 1 7 8 5




L = 0 . 0 = 0 . 5 AVERAGE Ν

FLUX U

b = 0 . 1 2 5 0 0


Mo 0 . 9

C . 2 7 7 7 8 0 . 1 8 7 6 6 0 . 0 8 8 8 0 0 . 0 5 7 5 3 0 . 0 4 2 5 0 0 . 0 3 3 6 9 0 . 0 3 0 5 2 0 . 0 8 4 1 9 0 . 0 5 2 7 3 G . 0 4 2 1 0 0 . 0 3 5 2 0 0 . 0 1 6 8 2 0 . 0 1 0 9 2 0 . 0 0 8 0 7 0 . 0 0 6 4 0 0 . 0 0 5 8 0 0 . 0 1 5 7 1 0 . 0 0 9 9 9 0 . 0 0 7 8 6 0 . C 0 6 7 C 0 . 0 0 3 2 1 0 . 0 0 2 0 8 0 . 0 0 1 5 4 0 . 0 0 1 2 2 0 . 0 0 1 1 1 0 . 0 0 2 9 9 0 . 0 0 1 9 0

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

C . C 5 7 3 0 0 . C 3 9 0 0 0 . 0 1 8 4 6 0 . 0 1 1 9 6 0 . 0 0 8 8 4 0 . 0 0 7 0 0 0 . 0 0 6 3 4 0 . 0 1 7 4 9 0 . 0 1 0 9 6 C . 1 1 8 3 9 0 . 0 8 1 1 6 0 . 0 3 8 4 3 0 . 0 2 4 9 0 0 . 0 1 8 4 0 0 . 0 1 4 5 8 0 . 0 1 3 2 1 0 . 0 3 6 4 0 0 . 0 2 2 8 2 C . 1 8 3 7 4 0 . 1 2 6 9 C 0 . 0 6 0 1 2 0 . 0 3 8 9 6 0 . 0 2 8 7 8 0 . 0 2 2 8 1 0 . 0 2 0 6 7 0 . 0 5 6 9 0 0 . 0 3 5 7 0 0 . 2 5 3 9 1 0 . 1 7 6 7 0 0 . 0 8 3 7 5 0 . 0 5 4 2 7 0 . 0 4 0 1 0 0 . 0 3 1 7 8 0 . 0 2 8 7 9 0 . 0 7 9 2 2 0 . 0 4 9 7 3

α = 0 . 9 0 C . 2 9 1 0 1 0 . 2 0 3 3 0 0 . 0 9 6 3 7 0 . 0 6 2 4 6 0 . 0 4 6 1 4 0 . 0 3 6 5 7 0 . 0 3 3 1 4 0 . 0 9 1 1 3 0 . 0 5 7 2 2 α = 0 . 9 5 C . 3 1 0 1 0 0 . 2 1 7 0 6 0 . 1 0 2 9 0 0 . 0 6 6 6 9 0 . 0 4 9 2 7 0 . 0 3 9 0 5 0 . 0 3 5 3 8 0 . 0 9 7 2 9 0 . 0 6 1 1 0 α = 0 . 9 9 0 . 3 2 5 6 5 0 . 2 2 8 3 C 0 . 1 0 8 2 4 0 . 0 7 0 1 5 0 . 0 5 1 8 3 0 . 0 4 1 0 8 0 . 0 3 7 2 2 0 . 1 0 2 3 3 0 . 0 6 4 2 7 α = 1 . 0 0 0 . 3 2 9 5 8 0 . 2 3 1 1 4 0 . 1 0 9 5 9 0 . 0 7 1 0 3 0 . 0 5 2 4 7 0 . 0 4 1 5 9 0 . 0 3 7 6 8 0 . 1 0 3 6 0 0 . 0 6 5 0 7

b = 0 . 1 2 5 0 0


Mo 1 . 0

C . 2 5 0 0 0 C . 1 6 9 8 1 0 . 0 8 0 4 3 0 . 0 5 2 1 2 0 . 0 3 8 5 0 0 . 0 3 0 5 2 0 . 0 2 7 6 5 0 . 0 7 6 1 9 0 . 0 4 7 7 6 C . 0 3 8 0 9 0 . 0 3 1 8 8 0 . 0 1 5 2 4 0 . 0 0 9 8 9 0 . 0 0 7 3 1 0 . 0 0 5 8 0 0 . 0 0 5 2 5 0 . 0 1 4 2 3 0 . 0 0 9 0 5 0 . 0 0 7 1 2 0 . C 0 b 0 7 0 . 0 0 2 9 1 0 . 0 0 1 8 9 0 . 0 0 1 4 0 0 . 0 0 1 1 1 0 . 0 0 1 0 0 0 . 0 0 2 7 1 0 . 0 0 1 7 3

SUMS α = 0 . 2 0 α = 0 . 4 0 α c ϋ . 6 0 α = 0 . 8 0

C . 0 5 1 5 8 0 . C 3 5 2 9 0 . 0 1 6 7 2 0 . 0 1 0 8 4 0 . 0 0 8 0 0 0 . 0 0 6 3 4 0 . 0 0 5 7 5 0 . 0 1 5 8 3 0 . 0 0 9 9 3 C . 1 0 6 5 9 0 . C 7 3 4 5 0 . 0 3 4 8 1 0 . 0 2 2 5 6 0 . 0 1 6 6 7 0 . 0 1 3 2 1 0 . 0 1 1 9 7 0 . 0 3 2 9 4 0 . 0 2 0 6 7 0 . 1 6 5 4 5 0 . 1 1 4 8 4 0 . 0 5 4 4 5 0 . 0 3 5 2 9 0 . 0 2 6 0 7 0 . 0 2 0 6 7 0 . 0 1 8 7 3 0 . 0 5 1 5 0 0 . 0 3 2 3 4 0 . 2 2 8 6 8 0 . 1 5 9 9 2 0 . 0 7 5 8 5 0 . 0 4 9 1 7 0 . 0 3 6 3 3 0 . 0 2 8 7 9 0 . 0 2 6 0 9 0 . 0 7 1 7 0 0 . 0 4 5 0 4

α = 0 . 9 0 0 . 2 6 2 1 2 0 . 1 8 4 0 0 C . 0 8 7 2 9 0 . 0 5 6 5 8 0 . 0 4 1 8 0 0 . 0 3 3 1 4 0 . 0 3 0 0 2 0 . 0 8 2 4 8 0 . 0 5 1 8 3 α = 0 . 9 5 0 . 2 7 9 3 3 0 . 1 9 6 4 5 0 . 0 9 3 2 1 0 . 0 6 0 4 2 0 . 0 4 4 6 4 0 . 0 3 5 3 8 0 . 0 3 2 0 6 0 . 0 8 8 0 6 0 . 0 5 5 3 5 α = 0 . 9 9 C . 2 9 3 3 5 0 . 2 0 6 6 . 2 0 . 0 9 8 0 4 0 . 0 6 3 5 5 0 . 0 4 6 9 5 0 . 0 3 7 2 2 0 . 0 3 3 7 2 0 . 0 9 2 6 2 0 . 0 5 8 2 2 α = 1 . 0 0 0 . 2 9 6 8 9 0 . 2 0 9 1 9 0 . 0 9 9 2 6 0 . 0 6 4 3 4 0 . 0 4 7 5 4 0 . 0 3 7 6 8 0 . 0 3 4 1 4 0 . 0 9 3 7 7 0 . 0 5 8 9 4

0 . 1 2 5 0 0 NARROW SURFACE LAYER AT TOP


I N F I N I T E 0 . 5 1 3 3 7 0 . 2 2 7 0 8 0 . 1 4 5 1 2 0 . 1 0 6 5 7 0 . 0 8 4 1 9 0 . 0 7 6 1 9 0 . 2 3 4 7 4 0 . 1 3 4 5 5 0 . 1 1 7 3 7 0 . 0 8 8 2 5 0 . 0 4 1 5 2 0 . 0 2 6 8 6 0 . 0 1 9 8 3 0 . 0 1 5 7 1 0 . 0 1 4 2 3 0 . 0 3 9 3 7 0 . 0 2 4 6 4 C . 0 1 9 6 8 0 . 0 1 6 4 8 0 . 0 0 7 8 6 0 . 0 0 5 1 0 0 . 0 0 3 7 7 0 . 0 0 2 9 9 0 . 0 0 2 7 1 0 . 0 0 7 3 5 0 . 0 0 4 6 7

SUMS α a = 0 . 2 0 α = Û . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α 0 . 9 9 α = 1 . 0 0

I N F I N I T E C . 1 0 6 3 4 0 . 0 4 7 1 4 0 . I N F I N I T E 0 . 2 2 0 6 1 0 . 0 9 8 0 2 Ο, I N F I N I T E 0 . 3 4 3 8 1 0 . 1 5 3 1 1 0 . I N F I N I T E 0 . 4 7 7 1 3 0 . 2 1 2 9 9 0 .

I N F I N I T E 0 . 5 4 8 0 1 0 . 2 4 4 9 2 0 . I N F I N I T E 0 . 5 8 4 5 9 0 . 2 6 1 4 3 0 . I N F I N I T E 0 . 6 1 4 4 3 0 . 2 7 4 9 1 0 , I N F I N I T E 0 . 6 2 1 9 7 0 . 2 7 8 3 2 0 ,

0 3 0 1 4 0 . 0 2 2 1 4 0 . 0 1 7 4 9 0 . 0 1 5 8 3 0 . 0 4 8 5 8 0 . 0 2 7 9 3 0 6 2 7 0 0 . 0 4 6 0 6 0 . 0 3 6 4 0 0 . 0 3 2 9 4 0 . 1 0 0 7 0 0 . 0 5 8 0 8 0 9 7 9 9 0 . 0 7 2 0 0 0 . 0 5 6 9 0 0 . 0 5 1 5 0 0 . 1 5 6 8 1 0 . 0 9 0 7 4 1 3 6 3 7 0 . 1 0 0 2 3 0 . 0 7 9 2 2 0 . 0 7 1 7 0 0 . 2 1 7 4 3 0 . 1 2 6 2 2

1 5 6 8 6 0 . 1 1 5 3 0 0 . 0 9 1 1 3 0 . 0 8 2 4 8 0 . 2 4 9 6 2 0 . 1 4 5 1 6 1 6 7 4 5 0 . 1 2 3 0 9 0 . 0 9 7 2 9 0 . 0 8 8 0 6 0 . 2 6 6 2 3 0 . 1 5 4 9 4 1 7 6 1 0 0 . 1 2 9 4 5 0 . 1 0 2 3 3 0 . 0 9 2 6 2 0 . 2 7 9 7 7 0 . 1 6 2 9 3 1 7 8 2 9 0 . 1 3 1 0 6 0 . 1 0 3 6 0 0 . 0 9 3 7 7 0 . 2 8 3 1 9 0 . 1 6 4 9 5

b = 0 . 1 2 5 0 0 LAMBERT SURFACE ON TOP

F I R S T ORDER 0 . 5 0 0 C 0 0 . 2 9 9 9 8 0 . 1 3 9 7 5 0 , SECOND ORDER 0 . 0 6 7 2 7 0 . 0 5 5 2 2 0 . 0 2 6 2 9 0 . THIRD ORDER C . 0 1 2 3 2 0 . 0 1 0 4 6 0 . 0 0 5 0 1 0 ,

0 9 0 2 6 0 . 0 6 6 5 8 0 . 0 5 2 7 3 0 . 0 4 7 7 6 0 . 1 3 4 5 5 0 . 0 8 2 9 2 0 1 7 0 5 0 . 0 1 2 6 0 0 . 0 0 9 9 9 0 . 0 0 9 0 5 0 . 0 2 4 6 4 0 . 0 1 5 6 1 0 0 3 2 5 0 . 0 0 2 4 0 0 . 0 0 1 9 0 0 . 0 0 1 7 3 0 . 0 0 4 6 7 0 . 0 0 2 9 7

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

0 . 1 0 2 7 9 0 . 0 6 2 2 9 0 . 0 2 9 0 4 0 . C . 2 1 1 6 2 0 . 1 2 9 5 5 0 . 0 6 0 4 5 Ο. C . 3 2 7 2 2 0 . 2 0 2 4 2 0 . 0 9 4 5 4 0 . C . 4 5 0 4 9 0 . 2 8 1 6 4 0 . 1 3 1 6 5 0 .

0 1 8 7 6 0 . 0 1 3 8 4 0 . 0 1 0 9 6 0 . 0 0 9 9 3 0 . 0 2 7 9 3 0 . 0 1 7 2 3 0 3 9 0 6 0 . 0 2 8 8 1 0 . 0 2 2 8 2 0 . 0 2 0 6 7 0 . 0 5 8 0 8 0 . 0 3 5 8 7 0 6 1 0 8 0 . 0 4 5 0 7 0 . 0 3 5 7 0 0 . 0 3 2 3 4 0 . 0 9 0 7 4 0 . 0 5 6 1 0 0 8 5 0 8 0 . 0 6 2 7 8 0 . 0 4 9 7 3 0 . 0 4 5 0 4 0 . 1 2 6 2 2 0 . 0 7 8 1 2

0 . 9 0 0 . 5 1 5 3 2 0 . 3 2 3 9 1 0 . 1 5 1 4 8 Ο, 0 . 9 5 0 . 5 4 8 6 0 0 . 3 4 5 7 6 0 . 1 6 1 7 4 0 . 0 . 9 9 C . 5 7 5 6 5 0 . 3 6 3 6 1 0 . 1 7 0 1 1 Ο, 1 . 0 0 C . 5 8 2 4 8 0 . 3 6 8 1 2 0 . 1 7 2 2 3 0 .

0 9 7 9 0 0 . 0 7 2 2 4 0 . 0 5 7 2 2 0 . 0 5 1 8 3 0 . 1 4 5 1 6 0 . 0 8 9 8 9 1 0 4 5 3 0 . 0 7 7 1 4 0 . 0 6 1 1 0 0 . 0 5 5 3 5 0 . 1 5 4 9 4 0 . 0 9 5 9 8 1 0 9 9 5 0 . 0 8 1 1 4 0 . 0 6 4 2 7 0 . 0 5 8 2 2 0 . 1 6 2 9 3 0 . 1 0 0 9 5 1 1 1 3 2 0 . 0 8 2 1 5 0 . 0 6 5 0 7 0 . 0 5 8 9 4 0 . 1 6 4 9 5 0 . 1 0 2 2 0


T A B L E 12 (continued) I n t e n s i t i e s o u t a t B o t t o m

VECTOR / ι = 0 . 0 / χ = 0 . 1 μ = 0 . 3 / x = 0 . 5 /χ = 0 . 7 μ . = 0 . 9 A= l . O AVERAGE

Ν FLUX

U


SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

μ0 = 0 . 9 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 PEAK 0 . 0 0 . 4 8 3 5 1 0 . 8 7 0 3 2 C . 2 4 1 7 6 0 . 1 8 2 4 4 0 . 0 8 7 9 5 0 . 0 5 7 2 0 0 . 0 4 2 3 3 0 . 0 3 3 5 8 0 . 0 3 0 4 3 0 . 0 8 2 0 3 0 . 0 5 2 2 8 C . 0 4 1 0 2 0 . 0 3 4 9 6 0 . 0 1 6 7 8 0 . 0 1 0 9 0 0 . 0 0 8 0 6 0 . 0 0 6 3 9 0 . 0 0 5 7 9 0 . 0 1 5 6 2 0 . 0 0 9 9 7 C . 0 0 7 8 1 0 . 0 0 6 6 9 0 . 0 0 3 2 1 0 . 0 0 2 0 8 0 . 0 0 1 5 4 0 . 0 0 1 2 2 O . O O l i l 0 . 0 0 2 9 8 0 . 0 0 1 9 0

0 . 0 5 0 0 6 0 . 0 3 7 9 4 0 . C 1 8 2 9 0 . 0 1 1 8 9 0 . 0 0 8 8 0 0 . 0 0 6 9 8 0 . 0 0 6 3 3 0 . 5 0 0 5 7 0 . 8 8 1 2 0 0 . 1 0 3 8 1 0 . 0 7 9 0 3 0 . 0 3 8 0 9 0 . 0 2 4 7 7 0 . 0 1 8 3 3 0 . 0 1 4 5 4 0 . 0 1 3 1 8 0 . 5 1 9 0 3 0 . 8 9 2 9 6 0 . 1 6 1 7 3 0 . 1 2 3 6 8 0 . 0 5 9 6 0 0 . 0 3 8 7 5 0 . 0 2 8 6 7 0 . 0 2 2 7 5 0 . 0 2 0 6 1 0 . 5 3 9 0 8 0 . 9 0 5 7 5 0 . 2 2 4 3 8 0 . 1 7 2 3 7 0 . 0 8 3 0 4 0 . 0 5 4 0 0 0 . 0 3 9 9 5 0 . 0 3 1 6 9 0 . 0 2 8 7 2 0 . 5 6 0 9 4 0 . 9 1 9 6 8

0 . 2 5 7 6 8 0 . 1 9 8 4 1 0 . 0 9 5 5 7 0 . 0 6 2 1 5 0 . 0 4 5 9 8 0 . 0 3 6 4 7 0 . 0 3 3 0 5 0 . 5 7 2 6 2 0 . 9 2 7 1 3 0 . 2 7 4 8 7 0 . 2 1 1 8 8 0 . 1 0 2 0 6 0 . 0 6 6 3 6 0 . 0 4 9 1 0 0 . 0 3 8 9 5 0 . 0 3 5 3 0 0 . 5 7 8 6 7 0 . 9 3 0 9 8 C . 2 8 8 8 8 0 . 2 2 2 8 9 0 . 1 0 7 3 6 0 . 0 6 9 8 1 0 . 0 5 1 6 5 0 . 0 4 0 9 7 0 . 0 3 7 1 3 0 . 5 8 3 6 1 0 . 9 3 4 1 3 0 . 2 9 2 4 3 0 . 2 2 5 6 7 0 . 1 0 8 7 0 0 . 0 7 0 6 8 0 . 0 5 2 2 9 0 . 0 4 1 4 8 0 . 0 3 7 5 9 0 . 5 8 4 8 5 0 . 9 3 4 9 3


Mo = 1 · 0 C O C O 0 . 0 0 . 0 0 . 0 0 . 0 PEAK 0 . 4 4 1 2 5 0 . 8 8 2 5 0 C . 2 2 0 6 2 0 . 1 6 5 5 5 0 . 0 7 9 7 3 0 . 0 5 1 8 5 0 . 0 3 8 3 6 0 . 0 3 0 4 3 0 . 0 2 7 5 8 0 . 0 7 4 4 3 0 . 0 4 7 3 9 0 . 0 3 7 2 1 0 . 0 3 1 6 9 0 . 0 1 5 2 1 0 . 0 0 9 8 8 0 . 0 0 7 3 1 0 . 0 0 5 7 9 0 . 0 0 5 2 5 0 . 0 1 4 1 6 0 . 0 0 9 0 3 0 . 0 0 7 0 8 0 . 0 0 6 0 6 0 . 0 0 2 9 0 0 . 0 0 1 8 9 0 . 0 0 1 3 9 O . O O l i l 0 . 0 0 1 0 0 0 . 0 0 2 7 0 0 . 0 0 1 7 3

SUMS a = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 C α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

0 . 0 4 5 6 7 0 . 0 3 4 4 3 0 . 0 1 6 5 8 0 . 0 1 0 7 8 0 . 0 0 7 9 8 0 . 0 0 6 3 3 0 . 0 0 5 7 3 0 . 4 5 6 7 2 0 . 8 9 2 3 5 0 . 0 9 4 6 9 0 . C 7 1 7 1 0 . 0 3 4 5 3 0 . 0 2 2 4 5 0 . 0 1 6 6 1 0 . 0 1 3 1 8 0 . C U 9 4 0 . 4 7 3 4 7 0 . 9 0 3 0 2 0 . 1 4 7 5 0 0 . 1 1 2 2 2 0 . 0 5 4 0 2 0 . 0 3 5 1 3 0 . 0 2 5 9 9 0 . 0 2 0 6 1 0 . 0 1 8 6 8 0 . 4 9 1 6 6 0 . 9 1 4 6 1 0 . 2 0 4 5 9 . C . 1 5 6 3 9 0 . 0 7 5 2 8 0 . 0 4 8 9 4 0 . 0 3 6 2 1 0 . 0 2 8 7 2 0 . 0 2 6 0 3 0 . 5 1 1 4 9 0 . 9 2 7 2 4

C . 2 3 4 9 4 0 . 1 8 0 0 0 0 . 0 8 6 6 4 0 . 0 5 6 3 3 0 . 0 4 1 6 7 0 . 0 3 3 0 5 0 . 0 2 9 9 6 0 . 5 2 2 0 8 0 . 9 3 3 9 9 0 . 2 5 0 5 9 0 . 1 9 2 2 2 0 . 0 9 2 5 2 0 . 0 6 0 1 5 0 . 0 4 4 5 0 0 . 0 3 5 3 0 0 . 0 3 1 9 9 0 . 5 2 7 5 7 0 . 9 3 7 4 8 C 2 6 3 3 6 0 . 2 0 2 2 1 0 . 0 9 7 3 2 0 . 0 6 3 2 7 0 . 0 4 6 8 1 0 . 0 3 7 1 3 0 . 0 3 3 6 5 0 . 5 3 2 0 4 0 . 9 4 0 3 3 0 . 2 6 6 5 9 0 . 2 0 4 7 3 0 . 0 9 8 5 3 0 . 0 6 4 0 6 0 . 0 4 7 3 9 0 . 0 3 7 5 9 0 . 0 3 4 0 7 0 . 5 3 3 1 8 0 . 9 4 1 0 6


SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0


SUMS a= 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

NARROW SURFACE LAYER AT TOP C O 1 . 4 3 2 5 2 1 . 0 9 8 7 3 0 . 7 7 8 8 0 0 . 5 9 7 4 7 0 . 4 8 3 5 1 0 . 4 4 1 2 5 0 . 8 1 1 7 1 0 . 6 7 9 5 7 0 . 4 0 5 8 7 0 . 4 0 8 6 0 0 . 2 1 0 0 9 0 . 1 3 8 4 9 0 . 1 0 3 0 7 C 0 8 2 0 3 0 . 0 7 4 4 3 0 . 1 8 7 0 4 0 . 1 2 5 4 7 C 0 9 3 5 2 0 . C 8 3 7 6 0 . 0 4 0 7 9 0 . 0 2 6 5 8 0 . 0 1 9 6 8 0 . 0 1 5 6 2 0 . 0 1 4 1 6 0 . 0 3 7 5 6 0 . 0 2 4 2 5 0 . 0 1 8 7 8 0 . 0 1 6 2 7 0 . 0 0 7 8 3 0 . 0 0 5 0 9 0 . 0 0 3 7 6 0 . 0 0 2 9 8 0 . 0 0 2 7 0 0 . 0 0 7 2 7 0 . 0 0 4 6 5

0 . 0 8 5 0 7 1 . 5 1 7 7 3 1 . 1 4 2 4 5 0 . 8 0 7 6 0 0 . 6 1 8 9 1 0 . 5 0 0 5 7 0 . 4 5 6 7 2 0 . 8 5 0 7 1 0 . 7 0 5 6 8 C . 1 7 8 6 1 1 . 6 1 0 4 9 1 . 1 8 9 8 4 0 . 8 3 8 8 0 0 . 6 4 2 1 1 0 . 5 1 9 0 3 0 . 4 7 3 4 7 0 . 8 9 3 0 6 0 . 7 3 3 9 7 C . 2 8 1 7 8 1 . 7 1 1 8 1 1 . 2 4 1 3 8 0 . 8 7 2 7 0 0 . 6 6 7 3 2 0 . 5 3 9 0 8 0 . 4 9 1 6 6 0 . 9 3 9 2 5 0 . 7 6 4 7 2 0 . 3 9 5 9 2 1 . 8 2 2 8 5 1 . 2 9 7 6 4 0 . 9 0 9 6 8 0 . 6 9 4 8 0 0 . 5 6 0 9 4 0 . 5 1 1 4 9 0 . 9 8 9 8 0 0 . 7 9 8 2 8

0 . 4 5 7 6 0 1 . 8 8 2 4 5 1 . 3 2 7 7 5 0 . 9 2 9 4 5 0 . 7 0 9 4 9 0 . 5 7 2 6 2 0 . 5 2 2 0 8 1 . 0 1 6 8 9 0 . 8 1 6 2 4 0 . 4 8 9 6 9 1 . 9 1 3 3 5 1 . 3 4 3 3 3 0 . 9 3 9 6 8 0 . 7 1 7 0 9 0 . 5 7 8 6 7 0 . 5 2 7 5 7 1 . 0 3 0 9 4 0 . 8 2 5 5 3 0 . 5 1 6 0 0 1 . 9 3 8 6 2 1 . 3 5 6 0 7 0 . 9 4 8 0 4 0 . 7 2 3 3 1 0 . 5 8 3 6 1 0 . 5 3 2 0 4 1 . 0 4 2 4 2 0 . 8 3 3 1 3 0 . 5 2 2 6 6 1 . 9 4 5 0 2 1 . 3 5 9 2 9 0 . 9 5 0 1 6 0 . 7 2 4 8 8 0 . 5 8 4 8 5 0 . 5 3 3 1 8 1 . 0 4 5 3 2 0 . 8 3 5 0 5

LAMBERT SURFACE ON TOP 0 . 0 0 . 2 8 6 5 0 0 . 6 5 9 2 4 0 . 7 7 8 8 0 0 . 8 3 6 4 6 0 . 8 7 0 3 2 0 . 8 8 2 5 0 0 . 6 7 9 5 7 0 . 7 9 7 5 5 0 . 3 3 9 7 9 0 . 2 7 8 2 5 0 . 1 3 6 2 2 0 . 0 8 8 8 8 0 . 0 6 5 8 5 0 . 0 5 2 2 8 0 . 0 4 7 3 9 0 . 1 2 5 4 7 0 . 0 8 1 0 5 0 . 0 6 2 7 4 0 . 0 5 4 2 4 0 . 0 2 6 1 3 0 . 0 1 6 9 9 0 . 0 1 2 5 7 0 . 0 0 9 9 7 0 . 0 0 9 0 3 0 . 0 2 4 2 5 0 . 0 1 5 5 2 0 . 0 1 2 1 3 0 . 0 1 0 4 2 0 . 0 0 5 0 0 0 . 0 0 3 2 5 0 . 0 0 2 4 0 0 . 0 0 1 9 0 0 . 0 0 1 7 3 0 . 0 0 4 6 5 0 . 0 0 2 9 7

0 . 0 7 0 5 7 0 . 3 4 4 4 1 0 . 6 8 7 5 7 0 . 7 9 7 2 8 0 . 8 5 0 1 6 0 . 8 8 1 2 0 0 . 8 9 2 3 5 0 . 7 0 5 6 8 0 . 8 1 4 4 1 0 . 1 4 6 7 9 0 . 4 0 7 2 1 0 . 7 1 8 2 6 0 . 8 1 7 2 9 0 . 8 6 4 9 8 0 . 8 9 2 9 6 0 . 9 0 3 0 2 0 . 7 3 3 9 7 0 . 8 3 2 6 6 0 . 2 2 9 4 2 0 . 4 7 5 5 3 0 . 7 5 1 6 0 0 . 8 3 9 0 3 0 . 8 8 1 0 9 0 . 9 0 5 7 5 0 . 9 1 4 6 1 0 . 7 6 4 7 2 0 . 8 5 2 4 9 0 . 3 1 9 3 1 0 . 5 5 0 1 2 0 . 7 8 7 9 6 0 . 8 6 2 7 4 0 . 8 9 8 6 4 0 . 9 1 9 6 8 0 . 9 2 7 2 4 0 . 7 9 8 2 8 0 . 8 7 4 1 2

0 . 3 6 7 3 1 0 . 5 9 0 0 4 0 . 8 0 7 4 0 0 . 8 7 5 4 1 0 . 9 0 8 0 2 0 . 9 2 7 1 3 0 . 9 3 3 9 9 0 . 8 1 6 2 4 0 . 8 8 5 6 8 0 . 3 9 2 1 3 0 . 6 1 0 7 1 0 . 8 1 7 4 7 0 . 8 8 1 9 7 0 . 9 1 2 8 8 0 . 9 3 C 9 8 0 . 9 3 7 4 8 0 . 8 2 5 5 3 0 . 8 9 1 6 7 0 . 4 1 2 4 0 0 . 6 2 7 6 1 0 . 8 2 5 6 9 0 . 8 8 7 3 2 0 . 9 1 6 8 5 0 . 9 3 4 1 3 0 . 9 4 0 3 3 0 . 8 3 3 1 3 0 . 8 9 6 5 6 0 . 4 1 7 5 2 0 . 6 3 1 8 8 0 . 8 2 7 7 7 0 . 8 8 8 6 8 0 . 9 1 7 8 5 0 . 9 3 4 9 3 0 . 9 4 1 0 6 0 . 8 3 5 0 5 0 . 8 9 7 8 0




^ = 0 . 1 μ=0 . 3 x = G . 9 AVERAGE Ν

FLUX U

b = 0 . 2 5 0 0 0


SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = I . C O

Mo 0 . 1

2 . 5 0 0 0 0 1 . 2 4 1 5 8 0 . 6 0 2 7 0 0 . 3 9 5 9 2 0 . 2 9 4 5 5 0 . 2 3 4 4 6 0 . 2 1 2 7 4 0 . 5 8 2 5 4 0 . 3 6 2 2 4 0 . 2 9 1 2 7 0 . 2 7 7 4 C 0 . 1 5 9 6 9 0 . 1 0 9 2 6 0 . 0 8 2 7 7 0 . 0 6 6 5 6 0 . 0 6 0 6 1 0 . 1 3 9 2 0 0 . 0 9 7 9 6 C . 0 6 9 6 0 0 . 0 7 5 9 6 0 . 0 4 6 3 2 0 . 0 3 2 1 0 0 . 0 2 4 4 5 0 . 0 1 9 7 2 0 . 0 1 7 9 8 0 . 0 3 9 3 3 0 . 0 2 8 6 2

C . 5 1 2 2 4 0 . 2 6 0 0 6 C . 1 2 7 3 2 0 . 0 8 3 8 3 0 . 0 6 2 4 3 0 . 0 4 9 7 2 0 . 0 4 5 1 3 0 . 1 2 2 4 1 0 . 0 7 6 6 1 1 . 0 5 1 6 3 0 . 5 4 6 5 2 0 . 2 7 0 0 0 0 . 1 7 8 1 8 0 . 1 3 2 8 4 0 . 1 0 5 8 7 0 . 0 9 6 1 0 0 . 2 5 8 1 4 0 . 1 6 2 6 5 1 . 6 2 2 9 9 0 . 8 6 4 7 2 0 . 4 3 1 2 9 0 . 2 8 5 3 4 0 . 2 1 2 9 7 0 . 1 6 9 8 3 0 . 1 5 4 2 0 0 . 4 0 9 9 6 0 . 2 6 0 1 4 2 . 2 3 2 6 0 1 . 2 2 1 6 4 0 . 6 1 5 5 1 0 . 4 0 8 2 7 0 . 3 0 5 0 8 0 . 2 4 3 4 4 0 . 2 2 1 1 0 0 . 5 8 1 5 1 0 . 3 7 1 7 4

2 . 5 5 4 2 6 1 . 4 1 7 4 1 0 . 7 1 7 9 3 0 . 4 7 6 8 5 0 . 3 5 6 5 4 0 . 2 8 4 6 1 0 . 2 5 8 5 1 0 . 6 7 6 1 4 0 . 4 3 3 9 0 2 . 7 1 9 8 4 1 . 5 2 0 2 2 0 . 7 7 2 1 0 0 . 5 1 3 1 8 0 . 3 8 3 8 2 0 . 3 0 6 4 4 0 . 2 7 8 3 6 0 . 7 2 5 9 8 0 . 4 6 6 8 0 2 . 8 5 4 7 4 1 . 6 0 5 0 2 0 . 8 1 6 9 6 0 . 5 4 3 3 0 0 . 4 0 6 4 5 0 . 3 2 4 5 5 0 . 2 9 4 8 3 0 . 7 6 7 1 6 0 . 4 9 4 0 6 2 . 8 8 8 8 2 1 . 6 2 6 5 9 0 . 8 2 8 3 9 0 . 5 5 0 9 8 0 . 4 1 2 2 3 0 . 3 2 9 1 8 0 . 2 9 9 0 3 0 . 7 7 7 6 4 0 . 5 0 1 0 1

b = 0 . 2 5 0 0 0


SUMS

Mo = 0 . 3

a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

0 . 8 3 3 3 3 0 . 6 0 2 7 0 0 . 3 3 7 9 7 0 . 2 3 0 1 3 0 . 1 7 3 9 8 0 . 1 3 9 7 5 0 . 1 2 7 2 2 0 . 3 0 0 9 0 0 . 2 0 6 9 6 0 . 1 5 0 4 5 0 . 1 5 9 6 9 0 . 0 9 6 7 5 0 . 0 6 6 9 7 0 . 0 5 0 9 8 0 . 0 4 1 1 1 0 . 0 3 7 4 8 0 . 0 8 2 4 4 0 . 0 5 9 7 4 0 . 0 4 1 2 2 0 . 0 4 6 3 2 0 . 0 2 8 7 0 0 . 0 1 9 9 6 0 . 0 1 5 2 3 0 . 0 1 2 2 9 0 . 0 1 1 2 1 0 . 0 2 4 2 1 0 . 0 1 7 7 7

G . 1 7 3 0 4 0 . 1 2 7 3 2 0 . 0 7 1 7 1 0 . 0 4 8 8 7 0 . 0 3 6 9 7 0 . 0 2 9 7 0 0 . 0 2 7 0 4 0 . 0 6 3 6 8 0 . 0 4 3 9 3 0 . 3 6 0 4 0 0 . 2 7 0 0 0 0 . 1 5 2 7 5 0 . 1 0 4 2 2 0 . 0 7 8 8 6 0 . 0 6 3 3 7 0 . 0 5 7 7 0 0 . 1 3 5 3 1 0 . 0 9 3 6 3 0 . 5 6 * 9 8 0 . 4 3 1 2 9 0 . 2 4 5 1 7 0 . 1 6 7 4 4 0 . 1 2 6 7 5 0 . 1 0 1 8 9 0 . 0 9 2 7 8 0 . 2 1 6 5 9 0 . 1 5 0 3 6 0 . 7 9 0 5 8 C . 6 1 5 5 1 0 . 3 5 1 6 2 0 . 2 4 0 4 0 0 . 1 8 2 0 7 0 . 1 4 6 4 0 0 . 1 3 3 3 1 0 . 3 0 9 7 8 0 . 2 1 5 7 7

0 . 9 1 2 7 8 0 . 7 1 7 9 3 0 . 4 1 1 1 8 0 . 2 8 1 2 9 0 . 2 1 3 0 9 0 . 1 7 1 3 6 0 . 1 5 6 0 5 0 . 3 6 1 7 4 0 . 2 5 2 3 9 0 . 9 7 6 5 6 0 . 7 7 2 1 C 0 . 4 4 2 7 8 0 . 3 0 2 9 9 0 . 2 2 9 5 6 0 . 1 8 4 6 1 0 . 1 6 8 1 3 0 . 3 8 9 2 6 0 . 2 7 1 8 3 1 . 0 2 8 9 7 0 . 8 1 6 9 6 0 . 4 6 9 0 0 0 . 3 2 1 0 0 0 . 2 4 3 2 3 0 . 1 9 5 6 2 0 . 1 7 8 1 5 0 . 4 1 2 0 7 0 . 2 8 7 9 6 1 . 0 4 2 2 8 0 . 8 2 8 3 9 0 . 4 7 5 7 0 0 . 3 2 5 6 0 0 . 2 4 6 7 2 0 . 1 9 8 4 3 0 . 1 8 0 7 1 0 . 4 1 7 8 9 0 . 2 9 2 0 8

b = 0 . 2 5 0 0 0


Mo 0 . 5

0 . 5 0 0 0 0 0 . 3 9 5 9 2 0 . 2 3 0 1 3 0 . 1 5 8 0 3 0 . 1 1 9 9 2 0 . 0 9 6 5 3 0 . 0 8 7 9 4 0 . 2 0 1 3 9 0 . 1 4 1 6 0 C . 1 0 0 7 0 0 . 1 0 9 2 6 0 . 0 6 6 9 7 0 . 0 4 6 4 7 0 . 0 3 5 4 2 0 . 0 2 8 5 8 0 . 0 2 6 0 6 0 . 0 5 6 8 0 0 . 0 4 1 4 1 C . 0 2 8 4 0 0 . 0 3 2 1 C 0 . 0 1 9 9 6 0 . 0 1 3 8 9 0 . 0 1 0 6 0 0 . 0 0 8 5 6 0 . 0 0 7 8 1 0 . 0 1 6 8 2 0 . 0 1 2 3 6

SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

C . 1 0 4 2 7 0 . 0 8 3 8 3 0 . 0 4 8 8 7 0 . 0 3 3 5 8 0 . 0 2 5 4 9 0 . 0 2 C 5 2 0 . 0 1 8 7 0 0 . 0 4 2 6 9 0 . 0 3 0 0 8 0 . 2 1 8 1 7 0 . 1 7 8 1 8 0 . 1 0 4 2 2 0 . 0 7 1 6 6 0 . 0 5 4 4 1 0 . 0 4 3 8 1 0 . 0 3 9 9 1 0 . 0 9 0 8 7 0 . 0 6 4 1 7 0 . 3 4 3 7 1 0 . 2 8 5 3 4 0 . 1 6 7 4 4 0 . 1 1 5 2 1 0 . 0 8 7 5 0 0 . 0 7 0 4 6 0 . 0 6 4 2 0 0 . 1 4 5 7 1 0 . 1 0 3 1 3 C . 4 8 3 5 1 0 . 4 0 8 2 7 0 . 2 4 0 4 0 0 . 1 6 5 5 2 0 . 1 2 5 7 5 0 . 1 0 1 2 8 0 . 0 9 2 2 9 0 . 2 0 8 7 9 0 . 1 4 8 1 1

0 . 5 5 9 8 2 0 . 4 7 6 8 5 0 . 2 8 1 2 8 0 . 1 9 3 7 4 0 . 1 4 7 2 1 0 . 1 1 8 5 8 0 . 1 0 8 0 5 0 . 2 4 4 0 4 0 . 1 7 3 3 3 0 . 5 9 9 8 0 0 . 5 1 3 1 8 0 . 3 0 2 9 9 0 . 2 0 8 7 3 0 . 1 5 8 6 1 0 . 1 2 7 7 7 0 . 1 1 6 4 2 0 . 2 6 2 7 3 0 . 1 8 6 7 2 C . 6 3 2 7 3 0 . 5 4 3 3 C 0 . 3 2 1 0 0 0 . 2 2 1 1 7 0 . 1 6 8 0 8 0 . 1 3 5 4 0 0 . 1 2 3 3 8 0 . 2 7 8 2 4 0 . 1 9 7 8 4 0 . 6 4 1 1 0 0 . 5 5 0 9 8 0 . 3 2 5 6 0 0 . 2 2 4 3 5 0 . 1 7 0 4 9 0 . 1 3 7 3 5 0 . 1 2 5 1 5 0 . 2 8 2 2 0 0 . 2 0 0 6 8

b = 0 . 2 5 0 0 0


SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

MO

3 5 7 1 4 0 7 5 5 7 0 2 1 5 8

0 7 4 6 3 1 5 6 5 2 2 4 7 1 6 3 4 8 5 8

0 . 4 0 4 1 3 0 C . 4 3 3 2 9 0 0 . 4 5 7 3 3 C 0 . 4 6 3 4 5 0

0 . 7

, 2 9 4 5 5 , 0 8 2 7 7 , 0 2 4 4 5

, 0 6 2 4 3 , 1 3 2 8 4 , 2 1 2 9 7 , 3 0 5 0 8

, 3 5 6 5 4 , 3 8 3 8 2 , 4 0 6 4 5 . 4 1 2 2 3

0 . 1 7 3 9 8 0 . 1 1 9 9 2 0 . 0 9 1 1 5 0 . 0 7 3 4 4 0 . 0 6 6 9 3 0 . 1 5 1 1 4 0 . 1 0 7 2 8 0 . 0 5 0 9 8 0 . 0 3 5 4 2 0 . 0 2 7 0 1 0 . 0 2 1 8 0 0 . 0 1 9 8 8 0 . 0 4 3 1 6 0 . 0 3 1 5 5 0 . 0 1 5 2 3 0 . 0 1 0 6 0 0 . 0 0 8 0 9 0 . 0 0 6 5 3 0 . 0 0 5 9 6 0 . 0 1 2 8 2 0 . 0 0 9 4 3

0 . 0 3 6 9 7 0 . 0 2 5 4 9 0 . 0 1 9 3 8 0 . 0 1 5 6 2 0 . 0 1 4 2 3 0 . 0 3 2 0 6 0 . 0 2 2 8 0 0 . 0 7 8 8 6 0 . 0 5 4 4 1 0 . 0 4 1 3 7 0 . 0 3 3 3 4 0 . 0 3 0 3 8 0 . 0 6 8 2 9 0 . 0 4 8 6 5 0 . 1 2 6 7 5 0 . 0 8 7 5 0 0 . 0 6 6 5 5 0 . 0 5 3 6 3 0 . 0 4 8 8 8 0 . 1 0 9 6 C 0 . 0 7 8 2 1 0 . 1 8 2 0 7 0 . 1 2 5 7 5 0 . 0 9 5 6 6 0 . 0 7 7 1 1 0 . 0 7 0 2 8 0 . 1 5 7 1 7 0 . 1 1 2 3 8

0 . 2 1 3 0 9 0 . 1 4 7 2 1 0 . 1 1 2 0 0 0 . 0 9 0 2 8 0 . 0 8 2 2 9 0 . 1 8 3 7 8 0 . 1 3 1 5 4 0 . 2 2 9 5 6 0 . 1 5 8 6 1 0 . 1 2 0 6 8 0 . 0 9 7 2 8 0 . 0 8 8 6 7 0 . 1 9 7 9 0 0 . 1 4 1 7 1 0 . 2 4 3 2 3 0 . 1 6 8 0 8 0 . 1 2 7 8 9 0 . 1 0 3 1 0 0 . 0 9 3 9 7 0 . 2 0 9 6 2 0 . 1 5 0 1 6 0 . 2 4 6 7 2 0 . 1 7 0 4 9 0 . 1 2 9 7 3 0 . 1 0 4 5 8 0 . 0 9 5 3 2 0 . 2 1 2 6 1 0 . 1 5 2 3 2




V E C T OR / i = 0 . 0 AVERAGE

Ν FLUX

U


Mo = C . l C O PEAK 0 . 0 C O 0 . 0 0 . 0 0 . 0 0 . 4 1 0 4 2 0 . 0 8 2 0 8 G . 2 0 5 2 1 0 . 5 1 3 0 3 0 . 4 4 0 6 4 0 . 3 2 7 7 8 0 . 2 5 7 3 3 0 . 2 1 1 0 6 0 . 1 9 3 5 3 0 . 3 3 7 8 9 0 . 2 8 5 4 0 0 . 1 6 8 9 4 0 . 2 1 4 9 9 0 . 1 4 5 6 4 0 . 1 0 3 3 5 0 . 0 7 9 5 4 0 . 0 6 4 5 3 0 . 0 5 8 9 4 0 . 1 1 9 2 9 0 . 0 9 1 3 5 C . 0 5 9 6 5 0 . C 7 0 3 6 0 . 0 4 5 0 5 0 . 0 3 1 5 7 0 . 0 2 4 1 6 0 . 0 1 9 5 4 0 . 0 1 7 8 3 0 . 0 3 7 5 7 0 . 0 2 8 0 2

SUMS α = 0 . 2 0 α = C 4 0 α = 0 . 6 0 α = 0 . 8 0

C 0 4 8 3 1 0 . 1 1 1 8 1 0 . 0 9 4 3 4 0 . 0 6 9 9 6 0 . 0 5 4 8 5 0 . 0 4 4 9 6 0 . 0 4 1 2 2 0 . 4 8 3 0 9 0 . 1 4 3 0 6 C . 1 1 3 4 8 0 . 2 4 4 7 5 0 . 2 0 2 8 4 0 . 1 4 9 9 5 0 . 1 1 7 4 2 0 . 0 9 6 1 7 0 . 0 8 8 1 4 0 . 5 6 7 4 1 0 . 2 1 2 9 0 G . 1 9 9 8 1 0 . 4 0 3 8 5 0 . 3 2 8 7 1 0 . 2 4 2 2 0 0 . 1 8 9 4 0 0 . 1 5 5 0 2 0 . 1 4 2 0 4 0 . 6 6 6 0 3 0 . 2 9 3 6 1 C . 3 1 3 C 0 0 . 5 9 5 7 7 0 . 4 7 6 1 6 0 . 3 4 9 6 7 0 . 2 7 3 0 7 0 . 2 2 3 3 2 0 . 2 0 4 5 8 0 . 7 8 2 5 0 0 . 3 8 7 7 9

α = 0 . 9 0 α = 0 . 9 5 α = C . 9 9 α = 1 . 0 0


0 . 3 8 1 9 8 0 . 7 0 6 7 2 0 . 5 5 9 6 9 0 . 4 1 0 3 1 0 . 3 2 0 1 9 0 . 2 6 1 7 6 0 . 2 3 9 7 5 0 . 8 4 8 8 5 0 . 4 4 1 0 0 0 . 4 2 0 0 7 0 . 7 6 6 5 3 0 . 6 0 4 2 7 0 . 4 4 2 6 0 0 . 3 4 5 2 7 C 2 8 2 2 1 0 . 2 5 8 4 6 0 . 8 8 4 3 7 0 . 4 6 9 3 6 C . 4 5 2 4 3 0 . 8 1 6 6 3 0 . 6 4 1 3 9 0 . 4 6 9 4 7 0 . 3 6 6 1 2 0 . 2 9 9 2 0 0 . 2 7 4 0 2 0 . 9 1 3 9 9 0 . 4 9 2 9 6 0 . 4 6 0 7 9 0 . 8 2 9 4 8 0 . 6 5 0 8 9 0 . 4 7 6 3 4 0 . 3 7 1 4 5 0 . 3 0 3 5 4 0 . 2 7 7 9 9 0 . 9 2 1 5 7 0 . 4 9 8 9 9

Mo = C . 3 C O C O PEAK C O 0 . 0 0 . 0 0 . 0 0 . 7 2 4 3 3 0 . 4 3 4 6 0 C . 3 6 2 1 7 0 . 4 4 0 6 4 0 . 3 0 1 8 0 0 . 2 1 4 9 2 0 . 1 6 5 6 7 0 . 1 3 4 5 3 0 . 1 2 2 9 3 0 . 2 4 7 1 3 0 . 1 8 9 8 4 0 . 1 2 3 5 6 0 . 1 4 5 6 4 0 . 0 9 3 5 8 0 . 0 6 5 6 3 0 . 0 5 0 2 6 0 . 0 4 0 6 6 0 . 0 3 7 1 0 0 . 0 7 7 9 8 0 . 0 5 8 2 5 0 . 0 3 8 9 9 0 . 0 4 5 0 5 0 . 0 2 8 4 1 0 . 0 1 9 8 4 0 . 0 1 5 1 6 0 . 0 1 2 2 5 0 . 0 1 1 1 8 0 . 0 2 3 8 1 0 . 0 1 7 6 3

SUMS α = α = 0 . 2 0 α = 0 . 4 0 α - C . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α - 1 . 0 0


C . 0 7 7 7 1 0 . 0 9 4 3 4 0 . 0 6 4 3 5 0 . 0 4 5 7 8 0 . 0 3 5 2 7 0 . 0 2 8 6 4 0 . 0 2 6 1 7 0 . 7 7 7 0 8 0 . 4 7 5 0 5 C . 1 6 7 4 8 0 . 2 0 2 8 4 0 . 1 3 7 7 6 0 . 0 9 7 9 1 0 . 0 7 5 4 1 0 . 0 6 1 2 1 0 . C 5 5 9 2 0 . 8 3 7 3 9 0 . 5 2 1 1 4 C . 2 7 2 0 9 0 . 3 2 8 7 1 0 . 2 2 2 2 6 0 . 1 5 7 8 1 0 . 1 2 1 4 9 0 . 0 9 8 5 8 0 . 0 9 0 0 6 0 . 9 0 6 9 6 0 . 5 7 4 1 2 C . 3 9 5 2 1 0 . 4 7 6 1 6 0 . 3 2 0 5 0 0 . 2 2 7 3 2 0 . 1 7 4 9 2 0 . 1 4 1 9 0 0 . 1 2 9 6 2 0 . 9 8 8 0 1 0 . 6 3 5 6 5

C . 4 6 5 1 8 0 . 5 5 9 6 9 0 . 3 7 5 8 4 0 . 2 6 6 4 2 0 . 2 0 4 9 7 0 . 1 6 6 2 5 0 . 1 5 1 8 6 1 . 0 3 3 7 4 0 . 6 7 0 2 8 0 . 5 0 2 6 0 0 . 6 0 4 2 7 0 . 4 0 5 3 0 0 . 2 8 7 2 2 0 . 2 2 0 9 4 0 . 1 7 9 2 0 0 . 1 6 3 6 8 1 . 0 5 8 1 0 0 . 6 8 8 7 0 C . 5 3 3 7 9 0 . 6 4 1 3 9 0 . 4 2 9 7 9 0 . 3 0 4 5 1 0 . 2 3 4 2 2 0 . 1 8 9 9 5 0 . 1 7 3 5 0 1 . 0 7 8 3 6 0 . 7 0 4 0 1 0 . 5 4 1 7 7 0 . 6 5 0 8 9 0 . 4 3 6 0 5 0 . 3 0 8 9 3 0 . 2 3 7 6 1 0 . 1 9 2 7 0 0 . 1 7 6 0 1 1 . 0 8 3 5 4 0 . 7 0 7 9 2

Mo = 0 . 5 C O 0 . 0 0 . 0 PEAK 0 . 0 0 . 0 0 . 0 0 . 6 0 6 5 3 0 . 6 0 6 5 3 C . 3 0 3 2 7 0 . 3 2 7 7 8 0 . 2 1 4 9 2 0 . 1 5 1 6 3 0 . 1 1 6 4 3 0 . 0 9 4 3 3 0 . 0 8 6 1 4 0 . 1 7 8 8 0 0 . 1 3 4 4 0 C . 0 8 9 4 0 0 . 1 0 3 3 5 0 . 0 6 5 6 3 0 . 0 4 5 9 1 0 . 0 3 5 1 1 0 . 0 2 8 3 9 0 . 0 2 5 9 0 0 . 0 5 4 9 2 0 . 0 4 0 7 9 0 . 0 2 7 4 6 0 . 0 3 1 5 7 0 . 0 1 9 8 4 0 . 0 1 3 8 4 0 . 0 1 0 5 7 0 . 0 0 8 5 4 0 . 0 0 7 7 9 0 . 0 1 6 6 5 0 . 0 1 2 3 1

SUMS α = α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α - 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0


0 . 0 6 4 4 6 0 . 0 6 9 9 6 0 . 0 4 5 7 8 0 . 0 3 2 2 8 0 . 0 2 4 7 8 0 . 0 2 0 0 8 0 . 0 1 8 3 3 0 . 6 4 4 6 3 0 . 6 3 5 1 5 0 . 1 3 7 6 1 0 . 1 4 9 9 5 0 . 0 9 7 9 1 0 . 0 6 9 0 1 0 . 0 5 2 9 6 0 . 0 4 2 9 0 0 . 0 3 9 1 6 0 . 6 8 8 0 5 0 . 6 6 7 7 1 C 2 2 1 3 9 0 . 2 4 2 2 0 0 . 1 5 7 8 1 0 . 1 1 1 1 5 0 . 0 8 5 2 8 0 . 0 6 9 0 7 0 . 0 6 3 0 6 0 . 7 3 7 9 7 0 . 7 0 5 1 0 C . 3 1 8 3 8 0 . 3 4 9 6 7 0 . 2 2 7 3 2 0 . 1 6 0 0 2 0 . 1 2 2 7 4 0 . 0 9 9 3 9 0 . 0 9 0 7 4 0 . 7 9 5 9 5 0 . 7 4 8 4 6

C . 3 7 2 8 6 0 . 4 1 C 3 1 0 . 2 6 6 4 2 0 . 1 8 7 4 9 0 . 1 4 3 7 9 0 . 1 1 6 4 3 0 . 1 0 6 2 9 0 . 8 2 8 5 9 0 . 7 7 2 8 3 0 . 4 0 1 8 3 0 . 4 4 2 6 0 0 . 2 8 7 2 2 0 . 2 0 2 1 0 0 . 1 5 4 9 9 0 . 1 2 5 4 9 0 . 1 1 4 5 5 0 . 8 4 5 9 5 0 . 7 8 5 8 0 C . 4 2 5 8 9 0 . 4 6 9 4 7 0 . 3 0 4 5 1 0 . 2 1 4 2 4 0 . 1 6 4 2 9 0 . 1 3 3 0 1 0 . 1 2 1 4 2 0 . 8 6 0 3 9 0 . 7 9 6 5 7 C 4 3 2 0 4 0 . 4 7 6 3 4 0 . 3 0 8 9 3 0 . 2 1 7 3 4 0 . 1 6 6 6 6 0 . 1 3 4 9 4 0 . 1 2 3 1 8 0 . 8 6 4 0 8 0 . 7 9 9 3 2

Mo = C . 7 C O C O 0 . 0 0 . 0 PEAK 0 . 0 0 . 0 0 . 4 9 9 7 7 0 . 6 9 9 6 7 C . 2 4 9 8 8 0 . 2 5 7 3 3 0 . 1 6 5 6 7 0 . 1 1 6 4 3 0 . 0 8 9 2 4 0 . 0 7 2 2 4 0 . 0 6 5 9 4 0 . 1 3 8 8 0 0 . 1 0 3 3 5 0 . 0 6 9 4 0 0 . 0 7 9 5 4 0 . 0 5 0 2 6 0 . 0 3 5 1 1 0 . 0 2 6 8 4 0 . 0 2 1 7 0 0 . 0 1 9 7 9 0 . 0 4 2 1 3 0 . 0 3 1 2 1 0 . 0 2 1 0 6 0 . 0 2 4 1 6 0 . 0 1 5 1 6 0 . 0 1 0 5 7 0 . 0 0 8 0 8 0 . 0 0 6 5 2 0 . 0 0 5 9 5 0 . 0 1 2 7 3 0 . 0 0 9 4 0

SUMS α α 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α 0 . 8 0

α 0 . 9 0 α 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

C . 0 5 2 9 3 0 . 0 5 4 8 5 0 . 0 3 5 2 7 0 . 0 2 4 7 8 0 . 0 1 8 9 9 0 . 0 1 5 3 7 0 . 0 1 4 0 3 0 . 5 2 9 3 2 0 . 7 2 1 6 7 0 . 1 1 2 5 9 0 . 1 1 7 4 2 0 . 0 7 5 4 1 0 . 0 5 2 9 6 0 . 0 4 0 5 8 0 . 0 3 2 8 4 0 . 0 2 9 9 8 0 . 5 6 2 9 5 0 . 7 4 6 6 9 0 . 1 8 0 4 7 0 . 1 8 9 4 0 0 . 1 2 1 4 9 0 . 0 8 5 2 8 0 . 0 6 5 3 4 0 . 0 5 2 8 7 0 . 0 4 8 2 6 0 . 6 0 1 5 7 0 . 7 7 5 4 0 C . 2 5 8 5 4 0 . 2 7 3 0 7 0 . 1 7 4 9 2 0 . 1 2 2 7 4 0 . 0 9 4 0 2 0 . 0 7 6 0 8 0 . 0 6 9 4 3 0 . 6 4 6 3 6 0 . 8 0 8 6 7

C . 3 0 2 1 9 0 . 3 2 0 1 9 0 . 2 0 4 9 7 0 . 1 4 3 7 9 0 . 1 1 0 1 3 0 . 0 8 9 1 1 0 . 0 8 1 3 3 0 . 6 7 1 5 4 0 . 8 2 7 3 7 0 . 3 2 5 3 4 0 . 3 4 5 2 7 ' 0 . 2 2 0 9 4 0 . 1 5 4 9 9 0 . 1 1 8 7 0 0 . 0 9 6 0 4 0 . 0 8 7 6 5 0 . 6 8 4 9 4 0 . 8 3 7 3 1 C . 3 4 4 5 5 0 . 3 6 6 1 2 0 . 2 3 4 2 2 0 . 1 6 4 2 9 0 . 1 2 5 8 2 0 . 1 0 1 8 0 0 . 0 9 2 9 0 0 . 6 9 6 0 7 0 . 8 4 5 5 7 C . 3 4 9 4 6 C . 3 7 1 4 5 0 . 2 3 7 6 1 0 . 1 6 6 6 6 0 . 1 2 7 6 4 0 . 1 0 3 2 6 0 . 0 9 4 2 4 0 . 6 9 8 9 1 0 . 8 4 7 6 8


T A B L E 12 (continued) I n t e n s i t i e s o u t a t T o p

VECTOR / x = 0 . 0 μ-0.1 μ =0,3 = 0 . 7 AVERAGE

Ν FLUX

U

b = 0 . 2 5 0 0 0 μ0 0 . 9

F I R S T ORDER 0 . 2 7 7 7 8 0 . 2 3 4 4 6 0 . 1 3 9 7 5 SECOND ORDER 0 . 0 6 0 4 5 0 . 0 6 6 5 6 0 . 0 4 1 1 1 THIRD ORDER C . 0 1 7 3 8 0 . 0 1 9 7 2 0 . 0 1 2 2 9

SUMS α = 0 . 2 0 C . 0 5 8 1 2 0 . 0 4 9 7 2 0 . 0 2 9 7 0 α = 0 . 4 0 C . 1 2 2 C 5 0 . 1 0 5 8 7 0 . 0 6 3 3 7 α = 0 . 6 0 0 . 1 9 3 0 0 0 . 1 6 9 8 3 0 . 1 0 1 8 9 α = 0 . 8 0 C . 2 7 2 6 0 0 . 2 4 3 4 4 0 . 1 4 6 4 0

α = 0 . 9 0 0 . 3 1 6 2 9 0 . 2 8 4 6 1 0 . 1 7 1 3 6 α = 0 . 9 5 C . 3 3 9 2 4 0 . 3 0 6 4 4 0 . 1 8 4 6 1 α = 0 . 9 9 C . 3 5 8 1 9 0 . 3 2 4 5 5 0 . 1 9 5 6 2 α = 1 . 0 0 C . 3 6 3 C 0 0 . 3 2 9 1 8 0 . 1 9 8 4 3

b = 0 . 2 5 0 0 0 μ0 = 1 . 0

0 . 0 9 6 5 3 0 . 0 7 3 4 4 0 5 9 2 0 0 . 0 5 3 9 6 0 1 7 6 0 0 . 0 1 6 0 5

0 . 0 4 3 8 1 0 . 0 3 3 3 4 0 . 0 2 6 8 8 0 . 0 7 0 4 6 0 . 0 5 3 6 3 0 . 0 4 3 2 5 0 . 1 0 1 2 8 0 . 0 7 7 1 1 0 . 0 6 2 1 8

0 . 1 2 7 7 7 0 . 0 9 7 2 8 0 . 0 7 8 4 5 0 . 1 3 5 4 0 0 . 1 0 3 1 0 0 . 0 8 3 1 4 0 . 1 3 7 3 5 0 . 1 0 4 5 8 0 . 0 8 4 3 4

0 . 0 2 4 5 0 0 . 0 3 9 4 2 0 . 0 5 6 6 8

0 . 1 2 0 9 1 0 , 0 . 0 3 4 7 6 0 .

0 8 6 2 8 0 2 5 4 5

0 . 0 5 4 6 8 0 . 0 8 7 7 9 0 . 1 2 5 9 5

0 . 0 3 9 1 4 0 . 0 6 2 9 4 0 . 0 9 0 4 5

0 . 0 6 6 3 7 0 . 1 4 7 3 1 0 . 1 0 5 8 8 0 . 0 7 1 5 2 0 . 1 5 8 6 4 0 . 1 1 4 0 8 0 . 0 7 5 7 9 0 . 1 6 8 0 5 0 . 1 2 0 8 8 0 . 0 7 6 8 9 0 . 1 7 0 4 5 0 . 1 2 2 6 2


0 . 2 5 0 0 0 0 . 2 1 2 7 4 0 . 1 2 7 2 2 0 . 0 8 7 9 4 0 . 0 6 6 9 3 0 . 0 5 3 9 6 0 . 0 5 4 9 5 0 . 0 6 0 6 1 0 . 0 3 7 4 8 0 . 0 2 6 0 6 0 . 0 1 9 8 8 0 . 0 1 6 0 5 0 . 0 1 5 8 4 0 . 0 1 7 9 8 0 . 0 1 1 2 1 0 . 0 0 7 8 1 0 . 0 0 5 9 6 0 . 0 0 4 8 1

0 . 0 4 9 1 8 0 . 1 0 9 9 1 0 . 0 7 8 5 8 0 . 0 1 4 6 3 0 . 0 3 1 6 8 0 . 0 2 3 2 0 0 . 0 0 4 3 9 0 . 0 0 9 4 3 0 . 0 0 6 9 5

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

C . 0 5 2 3 3 C . 0 4 5 1 3 0 . 0 2 7 0 4 0 . 0 1 8 7 0 0 . 0 1 4 2 3 0 . 0 1 1 4 7 0 . 0 1 0 4 6 0 . 0 2 3 3 3 0 . 0 1 6 7 0 0 . 1 0 9 9 4 0 . C 9 6 1 0 0 . 0 5 7 7 0 0 . 0 3 9 9 1 0 . 0 3 0 3 8 0 . 0 2 4 5 0 0 . 0 2 2 3 3 0 . 0 4 9 7 2 0 . 0 3 5 6 5 0 . 1 7 3 9 5 0 . 1 5 4 2 0 0 . 0 9 2 7 8 0 . 0 6 4 2 0 0 . 0 4 8 8 8 0 . 0 3 9 4 2 0 . 0 3 5 9 4 0 . 0 7 9 8 3 0 . 0 5 7 3 3 C . 2 4 5 8 2 0 . 2 2 1 1 0 0 . 1 3 3 3 1 0 . 0 9 2 2 9 0 . 0 7 0 2 8 0 . 0 5 6 6 8 0 . 0 5 1 6 7 0 . 1 1 4 5 6 0 . 0 8 2 4 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

b = 0 . 2 5 0 0 0

C . 2 8 5 3 0 0 . 2 5 8 5 1 0 . 1 5 6 0 5 0 . 1 0 8 0 5 0 . 0 8 2 2 9 0 . 0 6 6 3 7 0 . 0 6 0 5 0 0 . 1 3 4 0 0 0 . 0 9 6 4 6 0 . 3 0 6 0 5 0 . 2 7 8 3 6 0 . 1 6 8 1 3 0 . 1 1 6 4 2 0 . 0 8 8 6 7 0 . 0 7 1 5 2 0 . 0 6 5 2 0 0 . 1 4 4 3 1 0 . 1 0 3 9 3 C . 3 2 3 1 7 0 . 2 9 4 8 3 0 . 1 7 8 1 5 0 . 1 2 3 3 8 0 . 0 9 3 9 7 0 . 0 7 5 7 9 0 . 0 6 9 1 0 0 . 1 5 2 8 8 0 . 1 1 0 1 3 0 . 3 2 7 5 3 0 . 2 9 9 0 3 0 . 1 8 0 7 1 0 . 1 2 5 1 5 0 . 0 9 5 3 2 0 . 0 7 6 8 9 0 . 0 7 0 0 9 0 . 1 5 5 0 6 0 . 1 1 1 7 1


F I R S T ORDER I N F I N I T E SECOND ORDER 0 . 1 4 3 9 9 T H I R D ORDER 0 . 0 3 5 5 5

SUMS α = 0 . 2 0 I N F I N I T E α = 0 . 4 0 I N F I N I T E α = 0 . 6 0 I N F I N I T E α * 0 . 8 0 I N F I N I T E

α = 0 . 9 0 I N F I N I T E ο = 0 . 9 5 I N F I N I T E α = 0 . 9 9 I N F I N I T E α = 1 . 0 0 I N F I N I T E

0 . 0 8 2 4 4 0 . 0 5 6 8 0 1 5 1 1 4 0 . 1 2 0 9 1 0 4 3 1 6 0 . 0 3 4 7 6

0 . 1 0 9 9 1 0 . 2 8 7 9 7 0 . 1 8 2 9 9 0 . 0 3 1 6 8 0 . 0 7 1 1 1 0 . 0 5 0 7 8 0 . 0 0 9 4 3 0 . 0 2 0 4 8 0 . 0 1 4 9 8

4 0 9 9 6 0 . 2 1 6 5 9 5 8 1 5 1 0 . 3 0 9 7 8

0 . 0 9 0 8 7 0 . 0 6 8 2 9 0 . 0 5 4 6 8 0 . 0 4 9 7 2 0 . 1 2 8 0 5 0 . 0 8 2 4 1 0 . 1 4 5 7 1 0 . 1 0 9 6 0 0 . 0 8 7 7 9 0 . 0 7 9 8 3 0 . 2 0 3 7 7 0 . 1 3 2 0 2 0 . 2 0 8 7 9 0 . 1 5 7 1 7 0 . 1 2 5 9 5 0 . 1 1 4 5 6 0 . 2 8 9 6 6 0 . 1 8 8 9 8

0 . 6 7 6 1 4 0 . 3 6 1 7 4 0 . 2 4 4 0 4 0 . 1 8 3 7 8 0 . 1 4 7 3 1 0 . 1 3 4 0 0 0 . 3 3 7 1 8 0 . 2 2 0 7 7 0 . 7 2 5 9 8 0 . 3 8 9 2 6 0 . 2 6 2 7 3 0 . 1 9 7 9 0 0 . 1 5 8 6 4 0 . 1 4 4 3 1 0 . 3 6 2 2 4 0 . 2 3 7 6 2 0 . 7 6 7 1 6 0 . 4 1 2 0 7 0 . 2 7 8 2 4 0 . 2 0 9 6 2 0 . 1 6 8 0 5 0 . 1 5 2 8 8 0 . 3 8 2 9 8 0 . 2 5 1 5 9 0 . 7 7 7 6 4 0 . 4 1 7 8 9 0 . 2 8 2 2 0 0 . 2 1 2 6 1 0 . 1 7 0 4 5 0 . 1 5 5 0 6 0 . 3 8 8 2 6 0 . 2 5 5 1 5

b = 0 . 2 5 0 0 0 LAMBERT SURFACE ON TOP


0 . 5 0 0 C 0 0 . 3 6 2 2 4 0 . 2 0 6 9 6 0 . 1 4 1 6 0 0 . 1 0 7 2 8 0 . 0 8 6 2 8 0 . 0 7 8 5 8 0 . 1 8 2 9 9 0 . 1 2 7 1 1 0 . 0 9 1 4 9 0 . 0 9 7 9 6 0 . 0 5 9 7 4 0 . 0 4 1 4 1 0 . 0 3 1 5 5 0 . 0 2 5 4 5 0 . 0 2 3 2 0 0 . 0 5 0 7 8 0 . 0 3 6 9 3 0 . 0 2 5 3 9 0 . 0 2 8 6 2 0 . 0 1 7 7 7 0 . 0 1 2 3 6 0 . 0 0 9 4 3 0 . 0 0 7 6 2 0 . 0 0 6 9 5 0 . 0 1 4 9 8 0 . 0 1 1 0 0

SUMS α = 0 . 2 0 α * 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

0 . 1 0 3 8 8 0 . C 7 6 6 1 0 . 0 4 3 9 3 0 . 0 3 0 0 8 0 . 0 2 2 8 0 0 . 0 1 8 3 4 0 . 0 1 6 7 0 0 . 0 3 8 7 6 0 . 0 2 6 9 9 C . 2 1 6 4 8 0 . 1 6 2 6 5 0 . 0 9 3 6 3 0 . 0 6 4 1 7 0 . 0 4 8 6 5 0 . 0 3 9 1 4 0 . 0 3 5 6 5 0 . 0 8 2 4 1 0 . 0 5 7 5 5 0 . 3 3 9 6 1 0 . 2 6 0 1 4 0 . 1 5 0 3 6 0 . 1 0 3 1 3 0 . 0 7 8 2 1 0 . 0 6 2 9 4 0 . 0 5 7 3 3 0 . 1 3 2 0 2 0 . 0 9 2 4 6 0 . 4 7 5 5 9 0 . 3 7 1 7 4 0 . 2 1 5 7 7 0 . 1 4 8 1 1 0 . 1 1 2 3 8 0 . 0 9 0 4 5 0 . 0 8 2 4 0 0 . 1 8 8 9 8 0 . 1 3 2 7 3

α = 0 . 9 0 0 . 5 4 9 3 5 0 . 4 3 3 9 0 0 . 2 5 2 3 9 0 . 1 7 3 3 3 0 . 1 3 1 5 4 0 . 1 0 5 8 8 0 . 0 9 6 4 6 0 . 2 2 0 7 7 0 . 1 5 5 3 0 α = 0 . 9 5 0 . 5 8 7 8 7 0 . 4 6 6 8 0 0 . 2 7 1 8 3 0 . 1 8 6 7 2 0 . 1 4 1 7 1 0 . 1 1 4 0 8 0 . 1 0 3 9 3 0 . 2 3 7 6 2 0 . 1 6 7 2 7 α = 0 . 9 9 C . 6 1 9 5 4 0 . 4 9 4 0 6 0 . 2 8 7 9 6 0 . 1 9 7 8 4 0 . 1 5 0 1 6 0 . 1 2 0 8 8 0 . 1 1 0 1 3 0 . 2 5 1 5 9 0 . 1 7 7 2 2 α = I . 0 0 0 . 6 2 7 5 8 0 . 5 0 1 0 1 0 . 2 9 2 0 8 0 . 2 0 0 6 8 0 . 1 5 2 3 2 0 . 1 2 2 6 2 0 . 1 1 1 7 1 0 . 2 5 5 1 5 0 . 1 7 9 7 6




V E C T O R μ=0 .0 = 0 . 5 Μ* 1 . 0 A V E R A G E

Ν F L U X

υ b * 0 . 2 5 0 0 0 Z E R O O R O E R F I R S T O R D E R S E C O N O OROER T H I R D O R D E R

S U M S α = 0 . 2 0 α = 0 . 4 0 α « 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α * 0 . 9 5 α « 0 . 9 9 α » 1 . 0 0

Μο = ° ·

9

0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 P E A K 0 . 0 0 . 4 2 0 8 1 0 . 7 5 7 4 7 0 . 2 1 0 4 1 0 . 2 1 1 0 6 0 . 1 3 4 5 3 0 . 0 9 4 3 3 0 . 0 7 2 2 4 0 . 0 5 8 4 5 0 . 0 5 3 3 4 0 . 1 1 3 1 6 0 . 0 8 3 8 1 0 . 0 5 6 5 8 0 . 0 6 4 5 3 0 . 0 4 0 6 6 0 . 0 2 8 3 9 0 . 0 2 1 7 0 0 . 0 1 7 5 3 0 . 0 1 5 9 9 0 . 0 3 4 1 2 0 . 0 2 5 2 4 0 . 0 1 7 0 6 0 . 0 1 9 5 4 0 . 0 1 2 2 5 0 . 0 0 8 5 4 0 . 0 0 6 5 2 0 . 0 0 5 2 7 0 . 0 0 4 8 1 0 . 0 1 0 2 9 0 . 0 0 7 6 0

0 . 0 4 4 4 9 0 . 0 4 4 9 6 0 . 0 2 8 6 4 0 . 0 2 0 0 8 0 . 0 1 5 3 7 0 . 0 1 2 4 4 0 . 0 1 1 3 5 0 . 4 4 4 9 0 0 . 7 7 5 3 0 0 . 0 9 4 4 6 0 . 0 9 6 1 7 0 . 0 6 1 2 1 0 . 0 4 2 9 0 0 . 0 3 2 8 4 0 . 0 2 6 5 7 0 . 0 2 4 2 4 0 . 4 7 2 2 8 0 . 7 9 5 5 8 0 . 1 5 1 1 1 0 . 1 5 5 0 2 0 . 0 9 8 5 8 0 . 0 6 9 0 7 0 . 0 5 2 8 7 0 . 0 4 2 7 7 0 . 0 3 9 0 3 0 . 5 0 3 7 0 0 . 8 1 8 8 4 0 . 2 1 6 0 4 0 . 2 2 3 3 2 0 . 1 4 1 9 0 0 . 0 9 9 3 9 0 . 0 7 6 0 8 0 . 0 6 1 5 3 0 . 0 5 6 1 5 0 . 5 4 0 1 1 0 . 8 4 5 7 9

0 . 2 5 2 2 6 0 . 2 6 1 7 6 0 . 1 6 6 2 5 0 0 . 2 7 1 4 4 0 . 2 8 2 2 1 0 . 1 7 9 2 0 0 0 . 2 8 7 3 5 0 . 2 9 9 2 0 0 . 1 8 9 9 5 0 0 . 2 9 1 4 0 0 . 3 0 3 5 4 0 . 1 9 2 7 0 0

, 1 1 6 4 3 0 . 0 8 9 1 1 0 . 0 7 2 0 7 0 . 0 6 5 7 6 0 . 5 6 0 5 8 0 . 8 6 0 9 3 1 2 5 4 9 0 . 0 9 6 0 4 0 . 0 7 7 6 7 0 . 0 7 0 8 8 0 . 5 7 1 4 6 0 . 8 6 8 9 8

, 1 3 3 0 1 0 . 1 0 1 8 0 0 . 0 8 2 3 3 0 . 0 7 5 1 2 0 . 5 8 0 5 0 0 . 8 7 5 6 7 , 1 3 4 9 4 0 . 1 0 3 2 6 0 . 0 8 3 5 1 0 . 0 7 6 2 1 0 . 5 8 2 8 0 0 . 8 7 7 3 8

b e 0 . 2 5 0 0 0 Z E R O O R D E R F I R S T O R D E R S E C O N D O R D E R T H I R D O R D E R

S U M S 0 ; 0 . 2 0 α = 0 . 4 0 α * 0 . 6 0 α » 0 . 8 0

α » 0 . 9 0 α m 0 . 9 5 α = 0 . 9 9 α * 1 . 0 0

μο * 1 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 P E A K 0 . 3 8 9 4 0 0 . 7 7 8 8 0 0 . 1 9 4 7 0 0 . 1 9 3 5 3 0 . 1 2 2 9 3 0 . 0 8 6 1 4 0 . 0 6 5 9 4 0 . 0 5 3 3 4 0 . 0 4 8 6 8 0 . 1 0 3 5 4 0 . 0 7 6 5 5 0 . 0 5 1 7 7 0 . 0 5 8 9 4 0 . 0 3 7 1 0 0 . 0 2 5 9 0 0 . 0 1 9 7 9 0 . 0 1 5 9 9 0 . 0 1 4 5 9 0 . 0 3 1 1 5 0 . 0 2 3 0 3 0 . 0 1 5 5 7 0 . 0 1 7 8 3 0 . 0 1 1 1 8 0 . 0 0 7 7 9 0 . 0 0 5 9 5 0 . 0 0 4 8 1 0 . 0 0 4 3 8 0 . 0 0 9 3 9 0 . 0 0 6 9 3

0 . 0 4 1 1 4 0 . 0 4 1 2 2 0 . 0 2 6 1 7 0 . 0 1 8 3 3 0 . 0 1 4 0 3 0 . 0 1 1 3 5 0 . 0 1 0 3 6 0 . 4 1 1 4 3 0 . 7 9 5 0 9 0 . 0 8 7 3 0 0 . 0 8 8 1 4 0 . 0 5 5 9 2 0 . 0 3 9 1 6 0 . 0 2 9 9 8 0 . 0 2 4 2 4 0 . 0 2 2 1 2 0 . 4 3 6 4 8 0 . 8 1 3 6 1 0 . 1 3 9 5 6 0 . 1 4 2 0 4 0 . 0 9 0 0 6 0 . 0 6 3 0 6 0 . 0 4 8 2 6 0 . 0 3 9 0 3 0 . 0 3 5 6 1 0 . 4 6 5 2 1 0 . 8 3 4 8 5 0 . 1 9 9 4 0 0 . 2 0 4 5 8 0 . 1 2 9 6 2 0 . 0 9 0 7 4 0 . 0 6 9 4 3 0 . 0 5 6 1 5 0 . 0 5 1 2 3 0 . 4 9 8 5 0 0 . 8 5 9 4 5

0 . 2 3 2 7 4 0 . 2 3 9 7 5 0 . 1 5 1 8 6 0 . 1 0 6 2 9 0 . 0 8 1 3 3 0 . 0 6 5 7 6 0 . 0 6 0 0 1 0 . 5 1 7 2 0 0 . 8 7 3 2 8 0 . 2 5 0 3 9 0 . 2 5 8 4 6 0 . 1 6 3 6 8 0 . 1 1 4 5 5 0 . 0 8 7 6 5 0 . 0 7 0 8 8 0 . 0 6 4 6 7 0 . 5 2 7 1 4 0 . 8 8 0 6 2 0 . 2 6 5 0 3 0 . 2 7 4 0 2 0 . 1 7 3 5 0 0 . 1 2 1 4 2 0 . 0 9 2 9 0 0 . 0 7 5 1 2 0 . 0 6 8 5 4 0 . 5 3 5 4 1 0 . 8 8 6 7 3 0 . 2 6 8 7 6 0 . 2 7 7 9 9 0 . 1 7 6 0 1 0 . 1 2 3 1 8 0 . 0 9 4 2 4 0 . 0 7 6 2 1 0 . 0 6 9 5 3 0 . 5 3 7 5 2 0 . 8 8 8 2 9

b«= 0 . 2 5 0 0 0 Z E R O O R O E R F I R S T O R D E R S E C O N D O R D E R T H I R D O R O E R

S U M S α * 0 . 2 0 α s 0 . 4 0 α = 0 . 6 0 α « 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α * 1 . 0 0

b « 0 . 2 5 0 0 0 Z E R O O R D E R F I R S T O R D E R S E C O N D O R D E R T H I R D O R D E R

S U N S α • 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0

0 . 9 0 0 . 9 5 0 . 9 9 1 . 0 0

NARROW S U R F A C E L A Y E R AT TOP 0 . 0 0 . 4 1 0 4 2 0 . 7 2 4 3 3 0 . 6 0 6 5 3 0 . 4 9 9 7 7 0 . 4 2 0 8 1 0 . 3 8 9 4 0 0 . 5 2 2 1 4 0 . 5 1 7 7 3 0 . 2 6 1 0 7 0 . 3 3 7 8 9 0 . 2 4 7 1 3 0 . 1 7 8 8 0 0 . 1 3 8 8 0 0 . 1 1 3 1 6 0 . 1 0 3 5 4 0 . 1 9 9 2 9 0 . 1 5 7 2 6 0 . 0 9 9 6 4 0 . 1 1 9 2 9 0 . 0 7 7 9 8 0 . 0 5 4 9 2 0 . 0 4 2 1 3 0 . 0 3 4 1 2 0 . 0 3 1 1 5 0 . 0 6 4 6 6 0 . 0 4 8 6 8 0 . 0 3 2 3 3 0 . 0 3 7 5 7 0 . 0 2 3 8 1 0 . 0 1 6 6 5 0 . 0 1 2 7 3 0 . 0 1 0 2 9 0 . 0 0 9 3 9 0 . 0 1 9 9 3 0 . 0 1 4 7 9

0 . 0 5 6 4 7 0 . 4 8 3 0 9 0 . 7 7 7 0 8 0 . 6 4 4 6 3 0 . 5 2 9 3 2 0 . 4 4 4 9 0 0 . 4 1 1 4 3 0 . 5 6 4 7 5 0 . 5 5 1 2 6 0 . 1 2 2 7 3 0 . 5 6 7 4 1 0 . 8 3 7 3 9 0 . 6 8 8 0 5 0 . 5 6 2 9 5 0 . 4 7 2 2 8 0 . 4 3 6 4 8 0 . 6 1 3 6 4 0 . 5 8 9 5 0 0 . 2 0 1 0 7 0 . 6 6 6 0 3 0 . 9 0 6 9 6 0 . 7 3 7 9 7 0 . 6 0 1 5 7 0 . 5 0 3 7 0 0 . 4 6 5 2 1 0 . 6 7 0 2 4 0 . 6 3 3 5 1 0 . 2 9 4 5 6 0 . 7 8 2 5 0 0 . 9 8 8 0 1 0 . 7 9 5 9 5 0 . 6 4 6 3 6 0 . 5 4 0 1 1 0 . 4 9 8 5 0 0 . 7 3 6 4 0 0 . 6 8 4 6 8

0 . 3 4 8 2 2 0 . 8 4 8 8 5 1 . 0 3 3 7 4 0 . 0 . 3 7 7 0 5 0 . 8 8 4 3 7 1 . 0 5 8 1 0 0 . 0 . 4 0 1 1 5 0 . 9 1 3 9 9 1 . 0 7 8 3 6 0 . 0 . 4 0 7 3 2 0 . 9 2 1 5 7 1 . 0 8 3 5 4 0 ,

L A M 8 E R T S U R F A C E ON TOP 0 . 0 0 . 0 8 2 0 8 0 . 4 3 4 6 0 0 . 0 . 2 5 8 8 6 0 . 2 8 5 4 0 0 . 1 8 9 8 4 0 . 0 . 0 7 8 6 3 0 . 0 9 1 3 5 0 . 0 5 8 2 5 0 . 0 . 0 2 4 3 4 0 . 0 2 8 0 2 0 . 0 1 7 6 3 0 .

0 . 0 5 5 1 3 0 . 1 4 3 0 6 0 . 4 7 5 0 5 0 . 0 . 1 1 7 9 0 0 . 2 1 2 9 0 0 . 5 2 1 1 4 0 . 0 . 1 9 0 0 5 0 . 2 9 3 6 1 0 . 5 7 4 1 2 0 . 0 . 2 7 3 8 7 0 . 3 8 7 7 9 0 . 6 3 5 6 5 0 ,

0 . 3 2 1 0 7 0 . 4 4 1 0 0 0 . 6 7 0 2 8 0 . 0 . 3 4 6 2 0 0 . 4 6 9 3 6 0 . 6 8 8 7 0 0 . 0 . 3 6 7 0 9 0 . 4 9 2 9 6 0 . 7 0 4 0 1 0 , 0 . 3 7 2 4 2 0 . 4 9 8 9 9 0 . 7 0 7 9 2 0 ,

, 8 2 8 5 9 0 . 6 7 1 5 4 0 . 5 6 0 5 8 0 . 5 1 7 2 0 0 . 7 7 3 8 2 0 . 7 1 3 5 0 , 8 4 5 9 5 0 . 6 8 4 9 4 0 . 5 7 1 4 6 0 . 5 2 7 1 4 0 . 7 9 3 7 8 0 . 7 2 8 8 3 , 8 6 0 3 9 0 . 6 9 6 0 7 0 . 5 8 0 5 0 0 . 5 3 5 4 1 0 . 8 1 0 4 0 0 . 7 4 1 5 9 , 8 6 4 0 8 0 . 6 9 8 9 1 0 . 5 8 2 8 0 0 . 5 3 7 5 2 0 . 8 1 4 6 4 0 . 7 4 4 8 5

6 0 6 5 3 0 . 6 9 9 6 7 0 . 7 5 7 4 7 0 . 7 7 8 8 0 0 . 5 1 7 7 3 0 . 6 4 9 3 7 1 3 4 4 0 0 . 1 0 3 3 5 0 . 0 8 3 8 1 0 . 0 7 6 5 5 0 . 1 5 7 2 6 0 . 1 1 9 0 0 0 4 0 7 9 0 . 0 3 1 2 1 0 . 0 2 5 2 4 0 . 0 2 3 0 3 0 . 0 4 8 6 8 0 . 0 3 6 2 2 0 1 2 3 1 0 . 0 0 9 4 0 0 . 0 0 7 6 0 0 . 0 0 6 9 3 0 . 0 1 4 7 9 0 . 0 1 0 9 4

, 6 3 5 1 5 0 . 7 2 1 6 7 0 . 7 7 5 3 0 0 . 7 9 5 0 9 0 . 5 5 1 2 6 0 . 6 7 4 7 1 6 6 7 7 1 0 . 7 4 6 6 9 0 . 7 9 5 5 8 0 . 8 1 3 6 1 0 . 5 8 9 5 0 0 . 7 0 3 5 6

, 7 0 5 1 0 0 . 7 7 5 4 0 0 . 8 1 8 8 4 0 . 8 3 4 8 5 0 . 6 3 3 5 1 0 . 7 3 6 6 9 , 7 4 8 4 6 0 . 8 0 8 6 7 0 . 8 4 5 7 9 0 . 8 5 9 4 5 0 . 6 8 4 6 8 0 . 7 7 5 1 3

, 7 7 2 8 3 0 . 8 2 7 3 7 0 . 8 6 0 9 3 0 . 8 7 3 2 8 0 . 7 1 3 5 0 0 . 7 9 6 7 5 , 7 8 5 8 0 0 . 8 3 7 3 1 0 . 8 6 8 9 8 0 . 8 8 0 6 2 0 . 7 2 8 8 3 0 . 8 0 8 2 5 , 7 9 6 5 7 0 . 8 4 5 5 7 0 . 8 7 5 6 7 0 . 8 8 6 7 3 0 . 7 4 1 5 9 0 . 8 1 7 8 0 , 7 9 9 3 2 0 . 8 4 7 6 8 0 . 8 7 7 3 8 0 . 8 8 8 2 9 0 . 7 4 4 8 5 0 . 8 2 0 2 4




VECTOR μ = 0. 1 /x = 0 . 3 /x = 0 . 5 μ=0.Ί μ =0.9 μ = 1.0 AVERAGE

Ν FLUX

U

b = 0 . 5 0 0 0 0


Mo 0 . 1

SUMS α •• 0 . 2 0 α » 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α » 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

2 . 5 0 0 0 0 1 . 2 4 9 9 4 0 . 6 2 4 2 0 0 . 4 1 5 6 3 0 . 3 1 1 4 7 0 . 2 4 9 0 3 0 . 2 2 6 3 4 0 . 5 9 8 6 9 0 . 3 7 9 1 6 C . 2 9 9 3 5 0 . 2 9 8 7 3 0 . 2 0 1 1 8 0 . 1 4 7 1 7 0 . 1 1 5 3 7 0 . 0 9 4 7 1 0 . 0 8 6 9 0 0 . 1 7 1 4 0 0 . 1 3 0 7 3 0 . 0 8 5 7 0 0 . 1 0 1 5 7 0 . 0 7 9 4 5 0 . 0 6 0 6 4 0 . 0 4 8 4 6 0 . 0 4 0 2 2 0 . 0 3 7 0 5 0 . 0 6 5 5 2 0 . 0 5 3 4 3

0 . 5 1 2 7 2 0 . 2 6 2 8 2 0 . 1 3 3 5 8 0 . 0 8 9 5 4 0 . 0 6 7 3 3 0 . 0 5 3 9 5 0 . 0 4 9 0 7 0 . 1 2 7 1 7 0 . 0 8 1 5 3 1 . 0 5 4 3 9 C . 5 5 5 5 4 0 . 2 8 8 0 1 0 . 1 9 4 5 0 0 . 1 4 6 8 1 0 . 1 1 7 8 9 0 . 1 0 7 3 2 0 . 2 7 1 9 5 0 . 1 7 6 7 2 1 . 6 3 1 9 9 C . 8 8 6 5 9 0 . 4 7 0 0 9 0 . 3 2 0 1 1 0 . 2 4 2 6 3 0 . 1 9 5 3 3 0 . 1 7 7 9 8 0 . 4 3 9 9 5 0 . 2 9 0 1 9 2 . 2 5 5 8 9 1 . 2 6 8 7 3 0 . 6 9 0 2 9 0 . 4 7 4 4 7 0 . 3 6 1 3 1 0 . 2 9 1 6 8 0 . 2 6 6 0 4 0 . 6 3 9 7 3 0 . 4 2 9 0 5

2 . 5 9 0 0 1 1 . 4 8 4 9 1 2 . 7 6 3 7 8 1 . 6 0 0 7 4 2 . 9 0 6 4 2 1 . 6 9 7 6 1 2 . 9 4 2 6 2 1 . 7 2 2 4 6

0 . 8 1 9 6 2 0 . 5 6 6 2 6 0 . 4 3 2 3 1 0 . 3 4 9 5 3 0 . 0 . 8 9 0 3 2 0 . 6 1 6 7 7 0 . 4 7 1 5 0 0 . 3 8 1 5 1 0 . 0 . 9 5 0 2 0 0 . 6 5 9 7 1 0 . 5 0 4 8 7 0 . 4 0 8 7 8 0 . 0 . 9 6 5 6 6 0 . 6 7 0 8 2 0 . 5 1 3 5 2 0 . 4 1 5 8 5 0 .

3 1 8 9 7 0 . 7 5 5 5 8 C . 5 1 1 3 7 3 4 8 2 5 0 . 8 1 8 4 9 0 . 5 5 6 5 9 3 7 3 2 3 0 . 8 7 1 5 6 0 . 5 9 5 0 0 3 7 9 7 1 0 . 8 8 5 2 4 0 . 6 0 4 9 4

b = 0 . 5 0 0 0 0


Mo = 0 . 3

0 . 8 3 3 3 3 0 . 6 2 4 2 0 0 . 4 0 1 8 0 0 . 2 9 0 7 9 0 . 2 2 6 8 8 0 . 1 8 5 7 6 0 . 1 7 0 2 8 0 . 3 4 9 8 8 0 . 2 5 9 2 2 0 . 1 7 4 9 4 0 . 2 0 1 1 8 0 . 1 5 5 0 2 0 . 1 1 7 8 8 0 . 0 9 4 0 6 0 . 0 7 8 0 0 0 . 0 7 1 8 2 0 . 1 2 8 3 2 0 . 1 0 3 9 5 0 . 0 6 4 1 6 0 . 0 7 9 4 5 0 . 0 6 5 7 5 0 . 0 5 0 9 7 0 . 0 4 1 0 2 0 . 0 3 4 1 8 0 . 0 3 1 5 3 0 . 0 5 3 7 7 0 . 0 4 4 8 1

SUMS a * 0 . 2 0 0 . 1 7 4 2 2 0 . 1 3 3 5 8 0 . 0 8 7 1 4 0 . 0 6 3 3 2 0 . 0 4 9 5 0 0 . 0 4 0 5 7 0 . 0 3 7 2 1 0 . 0 7 5 5 8 0 . 0 5 6 4 0 a = 0 . 4 0 0 . 3 6 6 2 6 0 . 2 8 8 0 1 0 . 1 9 0 6 3 0 . 1 3 9 1 4 0 . 1 0 9 0 0 0 . 0 8 9 4 4 0 . 0 8 2 0 6 0 . 1 6 4 6 6 0 . 1 2 3 8 1 a = 0 . 6 0 0 . 5 8 1 5 7 0 . 4 7 0 0 9 0 . 3 1 6 1 9 0 . 2 3 1 9 1 0 . 1 8 2 0 7 0 . 1 4 9 6 0 0 . 1 3 7 3 1 0 . 2 7 1 8 9 0 . 2 0 6 1 3 o = 0 . 8 0 0 . 8 2 8 4 7 0 . 6 9 0 2 9 0 . 4 7 2 6 6 0 . 3 4 8 5 2 0 . 2 7 4 3 0 0 . 2 2 5 7 0 0 . 2 0 7 2 6 0 . 4 0 4 5 0 0 . 3 0 9 4 4

a » 0 . 9 0 0 . 9 6 7 6 8 0 . 8 1 9 6 2 0 . 5 6 6 6 9 0 . 4 1 9 0 4 0 . 3 3 0 2 4 0 . 2 7 1 9 3 0 . 2 4 9 7 8 0 . 4 8 3 7 4 0 . 3 7 1 8 4 a = 0 . 9 5 1 . 0 4 2 2 0 0 . 8 9 0 3 2 0 . 6 1 8 7 0 0 . 4 5 8 1 8 0 . 3 6 1 3 3 0 . 2 9 7 6 5 0 . 2 7 3 4 4 0 . 5 2 7 4 5 0 . 4 0 6 4 5 a * 0 . 9 9 1 . 1 0 4 5 1 0 . 9 5 0 2 0 0 . 6 6 3 0 6 0 . 4 9 1 6 3 0 . 3 8 7 9 2 0 . 3 1 9 6 6 0 . 2 9 3 6 9 0 . 5 6 4 6 6 0 . 4 3 6 0 1 a = 1 . 0 0 1 . 1 2 0 4 9 0 . 9 6 5 6 6 0 . 6 7 4 5 7 0 . 5 0 0 3 1 0 . 3 9 4 8 3 0 . 3 2 5 3 7 0 . 2 9 8 9 5 0 . 5 7 4 3 0 0 . 4 4 3 6 8

b = 0 . 5 0 0 0 0


Mo 0 . 5

0 . 5 0 0 0 0 0 . 4 1 5 6 3 0 . 2 9 0 7 9 0 . 2 1 6 1 7 0 . 1 7 0 8 1 0 . 1 4 0 8 8 0 . 1 2 9 4 8 0 . 2 4 8 1 8 0 . 1 9 1 7 4 0 . 1 2 4 0 9 0 . 1 4 7 1 7 0 . 1 1 7 8 8 0 . 0 9 0 6 6 0 . 0 7 2 7 1 - 0 . 0 6 0 4 7 0 . 0 5 5 7 4 0 . 0 9 7 0 1 0 . 0 7 9 8 1 0 . 0 4 8 5 0 0 . 0 6 0 6 4 0 . 0 5 0 9 7 0 . 0 3 9 6 9 0 . 0 3 2 0 1 0 . 0 2 6 7 0 0 . 0 2 4 6 4 0 . 0 4 1 6 0 0 . 0 3 4 8 7

SUMS 0 . 2 0 a * 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a * 0 . 9 0 a » 0 . 9 5 a * 0 . 9 9 a » I . 0 0

0 . 1 0 5 3 9 0 . 2 2 3 6 1 0 . 3 5 8 8 2 0 . 5 1 7 4 4

0 . 6 0 8 5 6 0 . 6 5 7 8 3 0 . 6 9 9 2 8 0 . 7 0 9 9 4

0 . 0 8 9 5 4 0 . 0 6 3 3 2 0 . 0 4 7 2 1 0 . 0 3 7 3 5 0 . 0 3 0 8 3 0 . 0 2 8 3 4 0 . 0 5 3 8 8 0 . 0 4 1 8 5 0 . 1 9 4 5 0 0 . 1 3 9 1 4 0 . 1 0 4 0 6 0 . 0 8 2 4 5 0 . 0 6 8 1 1 0 . 0 6 2 6 3 0 . 1 1 8 0 3 0 . 0 9 2 1 8 0 . 3 2 0 1 1 0 . 2 3 1 9 1 0 . 1 7 4 0 3 0 . 1 3 8 1 0 0 . 1 1 4 1 7 0 . 1 0 5 0 2 0 . 1 9 6 0 6 0 . 1 5 4 0 5 0 . 4 7 4 4 7 0 . 3 4 8 5 2 0 . 2 6 2 5 1 0 . 2 0 8 6 6 0 . 1 7 2 6 6 0 . 1 5 8 8 7 0 . 2 9 3 6 0 0 . 2 3 2 1 9

0 . 5 6 6 2 6 0 . 4 1 9 0 4 0 . 3 1 6 2 5 0 . 2 5 1 5 9 0 . 2 0 8 2 9 0 . 1 9 1 6 9 0 . 3 5 2 3 6 0 . 2 7 9 6 1 0 . 6 1 6 7 7 0 . 4 5 8 1 8 0 . 3 4 6 1 4 0 . 2 7 5 5 0 0 . 2 2 8 1 4 0 . 2 0 9 9 7 0 . 3 8 4 9 0 0 . 3 0 5 9 7 0 . 6 5 9 7 1 0 . 4 9 1 6 3 0 . 3 7 1 7 2 0 . 2 9 5 9 6 0 . 2 4 5 1 4 0 . 2 2 5 6 4 0 . 4 1 2 6 8 0 . 3 2 8 5 2 0 . 6 7 0 8 2 0 . 5 0 0 3 1 0 . 3 7 8 3 7 0 . 3 0 1 2 8 0 . 2 4 9 5 6 0 . 2 2 9 7 1 0 . 4 1 9 8 9 0 . 3 3 4 3 8

b = 0 . 5 0 0 0 0 Mo

F I R S T ORDER 0 . 3 5 7 1 4 SECOND ORDER 0 . 0 9 6 0 2 T H I R D ORDER 0 . 0 3 8 6 1

SUMS a * 0 . 2 0 C . 0 7 5 6 1 a s 0 . 4 0 0 . 1 6 1 2 1 a - 0 . 6 0 0 . 2 6 0 1 5 a - 0 . 8 0 0 . 3 7 7 5 6

a » 0 . 9 0 0 . 4 4 5 6 3 a » 0 . 9 5 0 . 4 8 2 6 1 a * 0 . 9 9 0 . 5 1 3 8 2 α » 1 . 0 0 0 . 5 2 1 8 7

= 0 . 7

0 . 3 1 1 4 7 0 . 2 2 6 8 8 0 . 1 7 0 8 1 0 . 1 3 5 7 8 0 . 1 1 2 3 6 0 . 1 0 3 3 9 0 . 1 9 2 0 3 0 . 1 5 1 2 0 0 . 1 1 5 3 7 0 . 0 9 4 0 6 0 . 0 7 2 7 1 0 . 0 5 8 4 6 0 . 0 4 8 6 8 0 . 0 4 4 8 9 0 . 0 7 7 2 2 0 . 0 6 3 9 7 0 . 0 4 8 4 6 0 . 0 4 1 0 2 0 . 0 3 2 0 1 0 . 0 2 5 8 4 0 . 0 2 1 5 6 0 . 0 1 9 9 0 0 . 0 3 3 4 6 0 . 0 2 8 1 2

0 . 0 6 7 3 3 0 . 0 4 9 5 0 0 . 0 3 7 3 5 0 . 0 2 9 7 2 0 . 0 2 4 6 1 0 . 0 2 2 6 5 0 . 0 4 1 7 9 0 . 0 3 3 0 5 0 . 1 4 6 8 1 0 . 1 0 9 0 0 0 . 0 8 2 4 5 0 . 0 6 5 6 8 0 . 0 5 4 4 1 0 . 0 5 0 0 9 0 . 0 9 1 7 7 0 . 0 7 2 9 0 0 . 2 4 2 6 3 0 . 1 8 2 0 7 0 . 1 3 8 1 0 0 . 1 1 0 1 3 0 . 0 9 1 3 0 0 . 0 8 4 0 7 0 . 1 5 2 8 7 0 . 1 2 2 0 3 0 . 3 6 1 3 1 0 . 2 7 4 3 0 0 . 2 0 8 6 6 0 . 1 6 6 6 0 0 . 1 3 8 2 2 0 . 1 2 7 2 9 0 . 2 2 9 6 1 0 . 1 8 4 2 6

0 . 4 3 2 3 1 0 . 3 3 0 2 4 0 . 2 5 1 5 9 0 . 2 0 1 0 2 0 . 1 6 6 8 3 0 . 1 5 3 6 7 0 . 2 7 6 0 0 0 . 2 2 2 1 1 0 . 4 7 1 5 0 0 . 3 6 1 3 3 0 . 2 7 5 5 0 0 . 2 2 0 1 9 0 . 1 8 2 7 8 0 . 1 6 8 3 7 0 . 3 0 1 7 4 0 . 2 4 3 1 7 0 . 5 0 4 8 7 0 . 3 8 7 9 2 0 . 2 9 5 9 6 0 . 2 3 6 6 2 0 . 1 9 6 4 4 0 . 1 8 0 9 6 0 . 3 2 3 7 4 0 . 2 6 1 2 0 0 . 5 1 3 5 2 0 . 3 9 4 8 3 0 . 3 0 1 2 8 0 . 2 4 0 8 9 0 . 1 9 9 9 9 0 . 1 8 4 2 4 0 . 3 2 9 4 5 0 . 2 6 5 8 8



t= o . o VECTOR


1 . - 0 . 3 μ*0·5 / x « 0 . 7 μ*0.9 μ*1.0

SUMS 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = Ο .ΘΟ

α * 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = I . 0 0

AVERAGE Ν

FLUX U

μ0 = 0 . 1 0 . 0 PEAK 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 3 3 6 9 0 . 0 0 6 7 4 0 . 0 1 6 8 4 0 . 0 8 4 2 2 0 . 2 2 7 6 7 0 . 2 2 5 7 1 0 . 2 0 1 1 7 0 . 1 7 7 1 9 0 . 1 6 6 6 1 0 . 1 8 3 3 0 0 . 1 9 6 6 1 0 . 0 9 1 6 5 0 . 1 2 5 5 7 0 . 1 3 8 8 6 0 . 1 1 7 1 5 0 . 0 9 7 9 1 0 . 0 8 3 3 3 0 . 0 7 7 4 3 0 . 1 1 1 8 1 0 . 1 0 2 3 5 0 . 0 5 5 9 0 0 . 0 7 3 2 5 0 . 0 6 9 0 3 0 . 0 5 5 6 1 0 . 0 4 5 5 4 0 . 0 3 8 3 1 0 . 0 3 5 4 6 0 . 0 5 5 8 4 0 . 0 4 8 7 0

0 . 0 0 7 5 3 0 . 0 2 2 5 2 0 . 0 5 1 7 0 0 . 0 5 0 3 2 0 . 0 4 4 5 5 0 . 0 3 9 1 1 0 . 0 3 6 7 3 0 . 0 7 5 3 1 0 . 0 5 0 5 8 C . 0 2 5 8 6 0 . 0 5 9 6 C 0 . 1 1 8 7 1 0 . 1 1 3 3 8 0 . 0 9 9 6 9 0 . 0 8 7 2 0 0 . 0 8 1 8 0 0 . 1 2 9 2 9 0 . 1 0 5 5 7 0 . 0 6 0 1 9 0 . 1 1 8 0 1 0 . 2 0 7 2 2 0 . 1 9 4 1 4 0 . 1 6 9 4 6 0 . 1 4 7 6 7 0 . 1 3 8 3 5 0 . 2 0 0 6 3 0 . 1 7 6 0 4 0 . 1 1 8 8 9 0 . 2 0 8 5 6 0 . 3 2 6 9 9 0 . 3 0 0 3 3 0 . 2 6 0 1 6 0 . 2 2 5 7 9 0 . 2 1 1 2 5 0 . 2 9 7 2 4 0 . 2 6 8 7 8

0 . 1 6 1 4 7 0 . 2 7 1 1 4 0 . 4 0 3 2 8 0 . 3 6 6 6 8 0 . 3 1 6 3 7 0 . 2 7 4 0 1 0 . 2 5 6 1 7 0 . 3 5 8 8 2 0 . 3 2 6 7 6 C . 1 8 7 1 0 0 . 3 0 8 0 9 0 . 4 4 6 7 0 0 . 4 0 4 1 0 0 . 3 4 7 9 5 0 . 3 0 1 0 3 0 . 2 8 1 3 4 0 . 3 9 3 8 9 0 . 3 5 9 4 6 C . 2 1 0 C 4 0 . 3 4 0 8 2 0 . 4 8 4 3 7 0 . 4 3 6 3 9 0 . 3 7 5 1 4 0 . 3 2 4 2 7 0 . 3 0 2 9 6 0 . 4 2 4 3 2 0 . 3 8 7 6 9 û . 2 1 6 1 4 0 . 3 4 9 4 8 0 . 4 9 4 2 4 0 . 4 4 4 8 2 0 . 3 8 2 2 3 0 . 3 3 0 3 2 0 . 3 0 8 5 9 0 . 4 3 2 2 9 0 . 3 9 5 0 6

b * 0 . 5 0 0 0 0 ZERO ORDE R F I R S T ORDE R SECOND ORDE R T H I R D ORDE R

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α * 0 . 9 9 α = 1 . 0 0

μ0 = 0 . 3 0 . 0 0 . 0 PEAK 0 . 0 0 . 0 0 . 0 0 . 0 0 . 3 1 4 7 9 0 . 1 8 8 8 8 0 . 1 5 7 4 0 0 . 2 2 7 6 7 0 . 2 6 2 3 3 0 . 2 2 3 7 5 0 . 1 8 7 9 2 0 . 1 6 0 3 7 0 . 1 4 9 1 6 0 . 2 1 0 9 0 0 . 1 9 5 4 3 0 . 1 0 5 4 5 0 . 1 3 8 8 6 0 . 1 3 2 2 7 0 . 1 0 6 9 1 0 . 0 8 7 6 8 0 . 0 7 3 8 3 0 . 0 6 8 3 6 0 . 1 0 6 9 5 0 . 0 9 3 6 2 0 . 0 5 3 4 8 0 . 0 6 9 0 3 0 . 0 6 1 9 0 0 . 0 4 9 1 1 0 . 0 3 9 9 4 0 . 0 3 3 4 8 0 . 0 3 0 9 4 0 . 0 5 0 2 1 0 . 0 4 3 0 6

0 . 0 3 6 1 7 0 . 0 5 1 7 0 0 . 0 5 8 3 0 0 . 0 4 9 4 6 0 . 0 4 1 4 4 0 . 0 3 5 3 2 0 . 0 3 2 8 4 0 . 3 6 1 6 9 0 . 2 3 2 0 8 0 . 0 8 4 0 4 0 . 1 1 3 7 1 0 . 1 3 0 9 4 0 . 1 1 0 4 4 0 . 0 9 2 3 1 0 . 0 7 8 5 7 0 . 0 7 3 0 2 0 . 4 2 0 2 0 0 . 2 8 5 3 9 C . 1 4 8 4 2 0 . 2 0 7 2 2 0 . 2 2 3 3 9 0 . 1 8 7 2 9 0 . 1 5 6 1 3 0 . 1 3 2 7 0 0 . 1 2 3 2 6 0 . 4 9 4 7 5 0 . 3 5 2 5 9 0 . 2 3 6 9 3 0 . 3 2 6 9 9 0 . 3 4 4 2 4 0 . 2 8 6 7 6 0 . 2 3 8 4 0 0 . 2 0 2 3 0 0 . 1 8 7 8 0 0 . 5 9 2 3 3 0 . 4 3 9 6 3

0 . 2 9 3 8 9 0 . 4 0 3 2 8 0 . 4 1 9 4 1 0 . 3 4 8 2 2 0 . 2 8 9 0 6 0 . 2 4 5 1 0 0 . 2 2 7 4 7 0 . 6 5 3 0 8 0 . 4 9 3 4 2 0 . 3 2 6 4 6 0 . 4 4 6 7 0 0 . 4 6 1 7 2 0 . 3 8 2 6 9 0 . 3 1 7 4 4 0 . 2 6 9 0 5 0 . 2 4 9 6 5 0 . 6 8 7 2 9 0 . 5 2 3 6 0 0 . 3 5 4 8 0 0 . 4 8 4 3 7 0 . 4 9 8 1 8 0 . 4 1 2 3 3 0 . 3 4 1 8 2 0 . 2 8 9 6 1 0 . 2 6 8 7 1 0 . 7 1 6 7 7 0 . 5 4 9 5 6 C . 3 6 2 2 3 0 . 4 9 4 2 4 0 . 5 0 7 6 9 0 . 4 2 0 0 6 0 . 3 4 8 1 7 0 . 2 9 4 9 7 0 . 2 7 3 6 7 0 . 7 2 4 4 6 0 . 5 5 6 3 2

b = 0 . 5 0 0 0 0 ZERO ORDER F I R S T ORDER SECOND OROER T H I R D ORDER

SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0


SUMS a * 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a s 0 . 9 9 o » 1 . 0 0

Mo = 0 . 5 0 . 0 0 . 0 0 . 0 PEAK 0 . 0 0 . 0 0 . 0 0 . 3 6 7 8 8 0 . 3 6 7 8 8 0 . 1 8 3 9 4 0 . 2 2 5 7 1 0 . 2 2 3 7 5 0 . 1 8 3 9 4 0 . 1 5 2 0 8 0 . 1 2 8 6 7 0 . 1 1 9 3 3 0 . 1 8 1 7 2 0 . 1 6 1 1 0 0 . 0 9 0 8 6 0 . 1 1 7 1 5 0 . 1 0 6 9 1 0 . 0 8 5 3 7 0 . 0 6 9 6 3 0 . 0 5 8 4 6 0 . 0 5 4 0 7 0 . 0 8 6 7 2 0 . 0 7 4 8 3 0 . 0 4 3 3 6 0 . 0 5 5 6 1 0 . 0 4 9 1 1 0 . 0 3 8 8 0 0 . 0 3 1 4 9 0 . 0 2 6 3 6 0 . 0 2 4 3 5 0 . 0 3 9 8 9 0 . 0 3 4 0 3

C . 0 4 0 8 0 0 . 0 5 0 3 2 0 . 0 4 9 4 6 0 . 0 4 0 5 4 0 . 0 3 3 4 8 0 . 0 2 8 3 0 0 . 0 2 6 2 4 0 . 4 0 8 0 4 0 . 4 0 3 3 9 0 . 0 9 1 5 1 0 . 1 1 3 3 8 0 . 1 1 0 4 4 0 . 0 9 0 2 6 0 . 0 7 4 4 3 0 . 0 6 2 8 8 0 . 0 5 8 2 8 0 . 4 5 7 5 6 0 . 4 4 6 9 5 0 . 1 5 5 9 8 0 . 1 9 4 1 4 0 . 1 8 7 2 9 0 . 1 5 2 5 7 0 . 1 2 5 6 2 0 . 1 0 6 0 4 0 . 0 9 8 2 6 0 . 5 1 9 9 5 0 . 5 0 1 5 5 0 . 2 4 0 2 8 0 . 3 0 0 3 3 0 . 2 8 6 7 6 0 . 2 3 2 8 1 0 . 1 9 1 3 8 0 . 1 6 1 4 0 0 . 1 4 9 5 1 0 . 6 0 0 7 1 0 . 5 7 1 8 6

0 . 2 9 2 7 8 0 . 3 6 6 6 8 0 . 3 4 8 2 2 0 . 2 8 2 1 9 0 . 2 3 1 7 9 0 . 1 9 5 3 8 0 . 1 8 0 9 6 0 . 6 5 0 6 1 0 . 6 1 5 1 5 0 . 3 2 2 3 4 0 . 4 0 4 1 0 0 . 3 8 2 6 9 0 . 3 0 9 8 3 0 . 2 5 4 3 8 0 . 2 1 4 3 8 0 . 1 9 8 5 3 0 . 6 7 8 6 0 0 . 6 3 9 3 8 0 . 3 4 7 8 2 0 . 4 3 6 3 9 0 . 4 1 2 3 3 0 . 3 3 3 5 8 0 . 2 7 3 7 8 0 . 2 3 0 6 8 0 . 2 1 3 6 2 0 . 7 0 2 6 8 0 . 6 6 0 2 0 C . 3 5 4 4 7 0 . 4 4 4 8 2 0 . 4 2 0 0 6 0 . 3 3 9 7 6 0 . 2 7 8 8 3 0 . 2 3 4 9 3 0 . 2 1 7 5 4 0 . 7 0 8 9 5 0 . 6 6 5 6 2

Mo = 0 . 7 0 . 0 0 . 0 0 . 0 0 . 0 PEAK 0 . 0 0 . 0 0 . 3 4 9 6 7 0 . 4 8 9 5 4 0 . 1 7 4 8 4 0 . 2 0 1 1 7 0 . 1 8 7 9 2 0 . 1 5 2 0 8 0 . 1 2 4 8 8 0 . 1 0 5 2 6 0 . 0 9 7 4 9 0 . 1 5 3 4 6 0 . 1 3 3 3 9 0 . 0 7 6 7 3 0 . 0 9 7 9 1 0 . 0 8 7 6 8 0 . 0 6 9 6 3 0 . 0 5 6 6 6 0 . 0 4 7 5 1 0 . 0 4 3 9 2 0 . 0 7 1 2 3 0 . 0 6 1 0 7 0 . 0 3 5 6 2 0 . 0 4 5 5 4 0 . 0 3 9 9 4 0 . 0 3 1 4 9 0 . 0 2 5 5 3 0 . 0 2 1 3 7 0 . 0 1 9 7 4 0 . 0 3 2 4 6 0 . 0 2 7 6 2

C . 0 3 8 3 5 0 . 0 4 4 5 5 0 . 0 4 1 4 4 0 . 0 3 3 4 8 0 . 0 2 7 4 7 0 . 0 2 3 1 4 0 . 0 2 1 4 3 0 . 3 8 3 5 0 0 . 5 1 8 9 0 0 . 0 8 5 0 0 0 . 0 9 9 6 9 0 . 0 9 2 3 1 0 . 0 7 4 4 3 0 . 0 6 1 0 1 0 . 0 5 1 3 7 0 . 0 4 7 5 6 0 . 4 2 4 9 9 0 . 5 5 4 8 2 0 . 1 4 3 1 0 0 . 1 6 9 4 6 0 . 1 5 6 1 3 0 . 1 2 5 6 2 0 . 1 0 2 8 7 0 . 0 8 6 5 7 0 . 0 8 0 1 4 0 . 4 7 7 0 0 0 . 5 9 9 7 3 0 . 2 1 7 6 0 0 . 2 6 0 1 6 0 . 2 3 8 4 0 0 . 1 9 1 3 8 0 . 1 5 6 5 6 0 . 1 3 1 6 7 0 . 1 2 1 8 5 0 . 5 4 3 9 9 0 . 6 5 7 4 1

C . 2 6 3 3 6 0 . 3 1 6 3 7 0 . 2 8 9 0 6 0 . 2 3 1 7 9 0 . 1 8 9 5 0 0 . 1 5 9 3 2 0 . 1 4 7 4 2 0 . 5 8 5 2 4 0 . 6 9 2 8 5 0 . 2 8 8 9 6 0 . 3 4 7 9 5 0 . 3 1 7 4 4 0 . 2 5 4 3 8 0 . 2 0 7 9 1 0 . 1 7 4 7 8 0 . 1 6 1 7 1 0 . 6 0 8 3 4 0 . 7 1 2 6 7 0 . 3 1 0 9 5 0 . 3 7 5 1 4 0 . 3 4 1 8 2 0 . 2 7 3 7 8 0 . 2 2 3 7 2 0 . 1 8 8 0 4 0 . 1 7 3 9 7 0 . 6 2 8 1 8 0 . 7 2 9 6 9 G . 3 1 6 6 8 0 . 3 8 2 2 3 0 . 3 4 8 1 7 0 . 2 7 8 8 3 0 . 2 2 7 8 3 0 . 1 9 1 4 9 0 . 1 7 7 1 6 0 . 6 3 3 3 5 0 . 7 3 4 1 2



VECTOR

b = 0 . 5 0 0 0 0


SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

/A =0.0 / i . = 0 . 3 ^ . = 0 . 5 / x = 0 . 7 / x = C 9 AVERAGE Ν

FLUX U

μο = 0 . 9

0 . 2 7 7 7 8 0 . 2 4 9 0 3 0 . 1 8 5 7 6 0 . L 4 0 8 8 0 . 1 1 2 3 6 0 . 0 9 3 1 7 0 . 0 8 5 7 9 0 . 1 5 6 5 2 0 . 1 2 4 5 6 C . 0 7 8 2 6 0 . 0 9 4 7 1 0 . 0 7 8 0 0 0 . 0 6 0 4 7 0 . 0 4 8 6 8 0 . 0 4 0 5 6 0 . 0 3 7 4 2 0 . 0 6 3 9 4 0 . 0 5 3 1 8 C . 0 3 1 9 7 0 . 0 4 0 2 2 0 . 0 3 4 1 8 0 . 0 2 6 7 0 0 . 0 2 1 5 6 0 . 0 1 8 0 0 0 . 0 1 6 6 2 0 . 0 2 7 8 6 0 . 0 2 3 4 5

0 . 0 5 8 9 7 0 . 0 5 3 9 5 0 . 0 4 0 5 7 0 . 0 3 0 8 3 0 . 0 2 4 6 1 0 . 0 2 0 4 1 0 . 0 1 8 8 0 0 . 0 3 4 1 1 0 . 0 2 7 2 5 0 . 1 2 6 1 1 0 . 1 1 7 8 9 0 . 0 8 9 4 4 0 . 0 6 8 1 1 0 . 0 5 4 4 1 0 . 0 4 5 1 6 0 . 0 4 1 6 0 0 . 0 7 5 0 1 0 . 0 6 0 1 6 0 . 2 G 4 2 1 0 . 1 9 5 3 3 0 . 1 4 9 6 0 0 . 1 1 4 1 7 0 . 0 9 1 3 0 0 . 0 7 5 8 2 0 . 0 6 9 8 5 0 . 1 2 5 1 4 0 . 1 0 0 7 9 0 . 2 9 7 5 3 0 . 2 9 1 6 8 0 . 2 2 5 7 0 0 . 1 7 2 6 6 0 . 1 3 8 2 2 0 . 1 1 4 8 3 0 . 1 0 5 8 1 0 . 1 8 8 2 8 0 . 1 5 2 3 4

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

b = 0 . 5 0 0 0 0


SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

b = 0 . 5 0 0 0 0

C . 3 5 1 9 4 0 . 3 4 9 5 3 0 . 2 7 1 9 3 0 . 2 0 8 2 9 0 . 1 6 6 8 3 0 . 1 3 8 6 5 0 . 1 2 7 7 7 0 . 2 2 6 5 3 C . 1 8 3 7 3 0 . 3 8 1 5 8 0 . 3 8 1 5 1 0 . 2 9 7 6 5 0 . 2 2 8 1 4 0 . 1 8 2 7 8 0 . 1 5 1 9 3 0 . 1 4 0 0 2 0 . 2 4 7 7 7 0 . 2 0 1 2 1 0 . 4 C 6 6 4 0 . 4 0 8 7 8 0 . 3 1 9 6 6 0 . 2 4 5 1 4 0 . 1 9 6 4 4 0 . 1 6 3 3 1 0 . 1 5 0 5 1 0 . 2 6 5 9 4 0 . 2 1 6 1 7 0 . 4 1 3 1 1 0 . 4 1 5 8 5 0 . 3 2 5 3 7 0 . 2 4 9 5 6 0 . 1 9 9 9 9 0 . 1 6 6 2 6 0 . 1 5 3 2 4 0 . 2 7 0 6 6 0 . 2 2 0 0 6

μο 1 . 0

0 . 2 5 0 0 0 0 . 2 2 6 3 4 0 . 1 7 0 2 8 0 . 1 2 9 4 8 0 . 1 0 3 3 9 0 . 0 8 5 7 9 0 . 0 7 9 0 2 0 . 1 4 3 2 5 0 . 1 1 4 4 3 0 . 0 7 1 6 3 0 . 0 8 6 9 0 0 . 0 7 1 8 2 0 . 0 5 5 7 4 0 . 0 4 4 8 9 0 . 0 3 7 4 2 0 . 0 3 4 5 2 0 . 0 5 8 8 5 0 . 0 4 9 0 1 0 . 0 2 9 4 3 C . 0 3 7 0 5 0 . 0 3 1 5 3 0 . 0 2 4 6 4 0 . 0 1 9 9 0 0 . 0 1 6 6 2 0 . 0 1 5 3 4 0 . 0 2 5 7 0 0 . 0 2 1 6 4

C . 0 5 3 1 2 0 . 0 4 9 0 7 0 . 0 3 7 2 1 0 . 0 2 8 3 4 0 . 0 2 2 6 5 0 . 0 1 8 8 0 0 . 0 1 7 3 2 0 . 0 3 1 2 3 0 . 0 2 5 0 4 0 . 1 1 3 7 4 0 . 1 0 7 3 2 0 . 0 8 2 0 6 0 . 0 6 2 6 3 0 . 0 5 0 0 9 0 . 0 4 1 6 0 0 . 0 3 8 3 2 0 . 0 6 8 7 2 0 . 0 5 5 3 0 0 . 1 8 4 4 1 0 . 1 7 7 9 8 0 . 1 3 7 3 1 0 . 1 0 5 0 2 0 . 0 8 4 0 7 0 . 0 6 9 8 5 0 . 0 6 4 3 6 0 . 1 1 4 7 1 0 . 0 9 2 6 8 0 . 2 6 9 0 8 0 . 2 6 6 0 4 0 . 2 0 7 2 6 0 . 1 5 8 8 7 0 . 1 2 7 2 9 0 . 1 0 5 8 1 0 . 0 9 7 5 2 0 . 1 7 2 6 9 0 . 1 4 0 1 4

0 . 3 1 8 5 3 0 . 3 1 8 9 7 0 . 2 4 9 7 8 0 . 1 9 1 6 9 0 . 1 5 3 6 7 0 . 0 . 3 4 5 5 0 0 . 3 4 8 2 5 0 . 2 7 3 4 4 0 . 2 0 9 9 7 0 . 1 6 8 3 7 0 . C . 3 6 8 3 2 0 . 3 7 3 2 3 0 . 2 9 3 6 9 0 . 2 2 5 6 4 0 . 1 8 0 9 6 0 . C 3 7 4 2 1 0 . 3 7 9 7 1 0 . 2 9 8 9 5 0 . 2 2 9 7 1 0 . 1 8 4 2 4 0 .


1 2 7 7 7 0 . 1 1 7 7 7 0 . 2 0 7 8 4 0 . 1 6 9 0 4 1 4 0 0 2 0 . 1 2 9 0 6 0 . 2 2 7 3 7 0 . 1 8 5 1 4 1 5 0 5 1 0 . 1 3 8 7 4 0 . 2 4 4 0 7 0 . 1 9 8 9 2 1 5 3 2 4 0 . 1 4 1 2 5 0 . 2 4 8 4 1 0 . 2 0 2 5 1


I N F I N I T E 0 . 5 9 8 6 9 0 . 3 4 9 8 8 0 . 2 4 8 1 8 0 . 1 9 2 0 4 0 . 1 5 6 5 2 0 . 1 4 3 2 5 0 . 3 2 5 6 7 0 . 2 2 3 3 3 0 . 1 6 2 8 4 0 . 1 7 1 4 0 0 . 1 2 8 3 2 0 . 0 9 7 0 1 0 . 0 7 7 2 2 0 . 0 6 3 9 4 0 . 0 5 8 8 5 0 . 1 0 7 2 2 0 . 0 8 5 6 9 C . 0 5 3 6 1 0 . 0 6 5 5 2 0 . 0 5 3 7 7 0 . 0 4 1 6 0 0 . 0 3 3 4 6 0 . 0 2 7 8 6 0 . 0 2 5 7 0 0 . 0 4 4 0 5 0 . 0 3 6 5 9

SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

b = 0 . 5 0 0 0 0

I N F I N I T E 0 . 1 2 7 1 7 0 . 0 7 5 5 8 I N F I N I T E 0 . 2 7 1 9 5 0 . 1 6 4 6 6 I N F I N I T E 0 . 4 3 9 9 5 0 . 2 7 1 8 9 I N F I N I T E 0 . 6 3 9 7 3 0 . 4 0 4 5 0

I N F I N I T E 0 . 7 5 5 5 8 0 . 4 8 3 7 4 I N F I N I T E 0 . 8 1 8 4 9 0 . 5 2 7 4 5 I N F I N I T E 0 . 6 7 1 5 6 0 . 5 6 4 6 6 I N F I N I T E 0 . 8 8 5 2 4 0 . 5 7 4 3 0


0 . 0 5 3 8 8 0 . 0 4 1 7 9 0 . 0 3 4 1 1 0 . 0 3 1 2 3 0 . 0 6 9 8 1 0 . 0 4 8 4 1 0 . 1 1 8 0 3 0 . 0 9 1 7 7 0 . 0 7 5 0 1 0 . 0 6 8 7 2 0 . 1 5 0 8 4 0 . 1 0 5 8 8 0 . 1 9 6 0 6 0 . 1 5 2 8 7 0 . 1 2 5 1 4 0 . 1 1 4 7 1 0 . 2 4 6 9 0 0 . 1 7 5 6 0 0 . 2 9 3 6 0 0 . 2 2 9 6 1 0 . 1 8 8 2 8 0 . 1 7 2 6 9 0 . 3 6 3 8 9 0 . 2 6 2 4 9

0 . 3 5 2 3 6 0 . 2 7 6 0 0 0 . 2 2 6 5 3 0 . 2 0 7 8 4 0 . 4 3 3 0 1 0 . 3 1 4 7 2 0 . 3 8 4 9 0 0 . 3 0 1 7 4 0 . 2 4 7 7 7 0 . 2 2 7 3 7 0 . 4 7 0 9 3 0 . 3 4 3 6 2 0 . 4 1 2 6 8 0 . 3 2 3 7 4 0 . 2 6 5 9 4 0 . 2 4 4 0 7 0 . 5 0 3 1 1 0 . 3 6 8 2 7 0 . 4 1 9 8 9 0 . 3 2 9 4 5 0 . 2 7 0 6 6 0 . 2 4 8 4 1 0 . 5 1 1 4 3 0 . 3 7 4 6 6


SUMS a = 0 . 2 0 a = 0 . 4 0 a » 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a * 0 . 9 5 a « 0 . 9 9 a » 1 . 0 0

0 . 5 0 0 0 0 0 . 3 7 9 1 6 0 . 2 5 9 2 2 0 . 1 1 1 6 6 0 . 1 3 0 7 3 0 . 1 0 3 9 5 0 . 0 4 2 8 4 0 . 0 5 3 4 3 0 . 0 4 4 8 1

0 . 1 0 4 8 4 0 . 0 8 1 5 3 0 . 0 5 6 4 0 0 . 2 2 1 1 8 0 . 1 7 6 7 2 0 . 1 2 3 8 1 0 . 3 5 2 6 8 0 . 2 9 0 1 9 0 . 2 0 6 1 3 C . 5 0 4 9 9 0 . 4 2 9 0 5 0 . 3 0 9 4 4

0 . 5 9 1 6 2 0 . 5 1 1 3 7 0 . 3 7 1 8 4 0 . 6 3 8 2 2 0 . 5 5 6 5 9 0 . 4 0 6 4 5 0 . 6 7 7 2 9 0 . 5 9 5 0 0 0 . 4 3 6 0 1 0 . 6 8 7 3 3 0 . 6 0 4 9 4 0 . 4 4 3 6 8

0 . 1 9 1 7 4 0 . 1 5 1 2 0 0 . 1 2 4 5 6 0 . 1 1 4 4 3 0 . 2 2 3 3 3 0 . 1 7 0 3 6 0 . 0 7 9 8 1 0 . 0 6 3 9 7 0 . 0 5 3 1 8 0 . 0 4 9 0 1 0 . 0 8 5 6 9 0 . 0 7 0 2 9 0 . 0 3 4 8 7 0 . 0 2 8 1 2 0 . 0 2 3 4 5 0 . 0 2 1 6 4 0 . 0 3 6 5 9 0 . 0 3 0 6 4

0 . 0 4 1 8 5 0 . 0 3 3 0 5 0 . 0 2 7 2 5 0 . 0 2 5 0 4 0 . 0 4 8 4 1 0 . 0 3 7 1 5 0 . 0 9 2 1 8 0 . 0 7 2 9 0 0 . 0 6 0 1 6 0 . 0 5 5 3 0 0 . 1 0 5 8 8 0 . 0 8 1 7 8 0 . 1 5 4 0 5 0 . 1 2 2 0 3 0 . 1 0 0 7 9 0 . 0 9 2 6 8 0 . 1 7 5 6 0 0 . 1 3 6 5 4 0 . 2 3 2 1 9 0 . 1 8 4 2 6 0 . 1 5 2 3 4 0 . 1 4 0 1 4 0 . 2 6 2 4 9 0 . 2 0 5 6 2

0 . 2 7 9 6 1 0 . 2 2 2 1 1 0 . 1 8 3 7 3 0 . 1 6 9 0 4 0 . 3 1 4 7 2 0 . 2 4 7 4 9 0 . 3 0 5 9 7 0 . 2 4 3 1 7 0 . 2 0 1 2 1 0 . 1 8 5 1 4 0 . 3 4 3 6 2 0 . 2 7 0 7 6 0 . 3 2 8 5 2 0 . 2 6 1 2 0 0 . 2 1 6 1 7 0 . 1 9 8 9 2 0 . 3 6 8 2 7 0 . 2 9 0 6 6 0 . 3 3 4 3 8 0 . 2 6 5 8 8 0 . 2 2 0 0 6 0 . 2 0 2 5 1 0 . 3 7 4 6 6 0 . 2 9 5 8 3



VECTOR μ = 0 . 0 μ=0.1 μ=0.3 μ=0·5 μ = 0. 7 / χ = 0 . 9 μ=1.0 AVERAGE

Ν FLUX

U

b = 0 . 5 0 0 0 0 ZERO OROER F I R S T OROER SECOND ORDER THIRD ORDER

μ0 = 0 . 9 O . û 0 . 0 0 . 0 0 . 0 0 . 0 PEA K 0 . 0 0 . 3 1 8 7 5 0 . 5 7 3 7 5 0 . 1 5 9 3 8 0 . 1 7 7 1 9 0 . 1 6 0 3 7 0 . 1 2 8 6 7 0 . 1 0 5 2 6 0 . 0 8 8 5 4 0 . 0 8 1 9 4 0 . 1 3 1 4 0 0 . 1 1 2 9 6 0 . 0 6 5 7 0 0 . 0 8 3 3 3 0 . 0 7 3 8 3 0 . 0 5 8 4 6 0 . 0 4 7 5 1 0 . 0 3 9 8 0 0 . 0 3 6 7 8 0 . 0 6 0 0 4 0 . 0 5 1 2 9 0 . 0 3 0 0 2 0 . 0 3 8 3 1 0 . 0 3 3 4 8 0 . 0 2 6 3 6 0 . 0 2 1 3 7 0 . 0 1 7 8 7 0 . 0 1 6 5 1 0 . 0 2 7 2 1 0 . 0 2 3 1 3

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 C α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

C . 0 3 4 7 7 0 . 0 3 9 1 1 0 . 0 3 5 3 2 0 . 0 2 8 3 0 0 . 0 2 3 1 4 0 . 0 1 9 4 6 0 . 0 1 8 0 1 0 . 3 4 7 6 7 0 . 5 9 8 6 0 0 . 0 7 6 6 1 0 . 0 8 7 2 0 0 . 0 7 8 5 7 0 . 0 6 2 8 8 0 . 0 5 1 3 7 0 . 0 4 3 1 8 0 . 0 3 9 9 5 0 . 3 8 3 0 4 0 . 6 2 8 9 5 0 . 1 2 8 1 8 0 . 1 4 7 6 7 0 . 1 3 2 7 0 0 . 1 0 6 0 4 0 . 0 8 6 5 7 0 . 0 7 2 7 3 0 . 0 6 7 2 8 0 . 4 2 7 2 5 0 . 6 6 6 8 3 0 . 1 9 3 6 2 0 . 2 2 5 7 9 0 . 2 0 2 3 0 0 . 1 6 1 4 0 0 . 1 3 1 6 7 0 . 1 1 0 5 7 0 . 1 0 2 2 7 0 . 4 8 4 0 5 0 . 7 1 5 4 1

0 . 2 3 3 5 3 0 . 2 7 4 0 1 0 . 2 4 5 1 0 0 . 1 9 5 3 8 0 . 1 5 9 3 2 0 . 1 3 3 7 6 0 . 1 2 3 7 1 0 . 5 1 8 9 5 0 . 7 4 5 2 3 0 . 2 5 5 7 7 0 . 3 0 1 0 3 0 . 2 6 9 C 5 0 . 2 1 4 3 8 0 . 1 7 4 7 8 0 . 1 4 6 7 1 0 . 1 3 5 6 8 0 . 5 3 8 4 7 0 . 7 6 1 9 0 C . 2 7 4 8 4 0 . 3 2 4 2 7 0 . 2 8 9 6 1 0 . 2 3 0 6 8 0 . 1 8 8 0 4 0 . 1 5 7 8 3 0 . 1 4 5 9 5 0 . 5 5 5 2 3 0 . 7 7 6 2 1 C . 2 7 9 8 0 0 . 3 3 0 3 2 0 . 2 9 4 9 7 0 . 2 3 4 9 3 0 . 1 9 1 4 9 0 . 1 6 0 7 2 0 . 1 4 8 6 3 0 . 5 5 9 6 0 0 . 7 7 9 9 4


SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

μο = 1 . 0 C O 0 . 0 0 . 0 0 . 0 0 . 0 0 . 1 5 1 6 3 0 . 1 6 6 6 1 0 . 1 4 9 1 6 0 . 1 1 9 3 3 0 . 0 9 7 4 9 0 . 0 6 1 1 8 0 . 0 7 7 4 3 0 . 0 6 8 3 6 0 . C 5 4 0 7 0 . 0 4 3 9 2 0 . 0 2 7 8 0 0 . 0 3 5 4 6 0 . 0 3 0 9 4 0 . 0 2 4 3 5 0 . 0 1 9 7 4

0 . 0 3 3 C 2 0 . 0 3 6 7 3 0 . 0 3 2 8 4 0 . 0 2 6 2 4 0 . C . 0 7 2 6 1 0 . 0 8 1 8 0 0 . 0 7 3 0 2 0 . 0 5 8 2 8 0 . C . 1 2 L 2 4 0 . 1 3 8 3 5 0 . 1 2 3 2 6 0 . 0 9 8 2 6 0 . C 1 8 2 7 4 0 . 2 1 1 2 5 0 . 1 8 7 8 0 0 . 1 4 9 5 1 0 .

0 2 1 4 3 0 4 7 5 6 0 8 0 1 4 1 2 1 8 5

0 PEAK . 0 8 1 9 4 0 . 0 7 5 8 2 , 0 3 6 7 8 0 . 0 3 3 9 9 . 0 1 6 5 1 0 . 0 1 5 2 5

, 0 1 8 0 1 0 . 0 1 6 6 6 , 0 3 9 9 5 0 . 0 3 6 9 5 . 0 6 7 2 8 0 . 0 6 2 2 3 . 1 0 2 2 7 0 . 0 9 4 5 7

0 . 3 0 3 2 7 0 . 6 0 6 5 3 0 . 1 2 2 3 6 0 . 1 0 4 7 9 0 . 0 5 5 6 1 0 . 0 4 7 4 4 0 . 0 2 5 1 6 0 . 0 2 1 3 7

0 . 3 3 0 1 8 0 . 6 2 9 5 7 0 . 3 6 3 0 7 0 . 6 5 7 7 0 0 . 4 0 4 1 4 0 . 6 9 2 8 0 0 . 4 5 6 8 5 0 . 7 3 7 7 8

α = 0 . 9 0 0 . 2 2 0 1 5 0 . 2 5 6 1 7 0 . 2 2 7 4 7 0 . 1 8 0 9 6 0 . 1 4 7 4 2 0 . 1 2 3 7 1 0 . 1 1 4 3 9 0 . 4 8 9 2 1 0 . 7 6 5 3 8 α = 0 . 9 5 C . 2 4 0 9 7 0 . 2 8 1 3 4 0 . 2 4 9 6 5 0 . 1 9 8 5 3 0 . 1 6 1 7 1 0 . 1 3 5 6 8 0 . 1 2 5 4 5 0 . 5 0 7 3 1 0 . 7 8 0 8 1 α = 0 . 9 9 0 . 2 5 8 8 1 0 . 3 0 2 9 6 0 . 2 6 8 7 1 0 . 2 1 3 6 2 0 . 1 7 3 9 7 0 . 1 4 5 9 5 0 . 1 3 4 9 5 0 . 5 2 2 8 5 0 . 7 9 4 0 5 α = 1 . 0 0 C . 2 6 3 4 5 0 . 3 0 8 5 9 0 . 2 7 3 6 7 0 . 2 1 7 5 4 0 . 1 7 7 1 6 0 . 1 4 8 6 3 0 . 1 3 7 4 2 0 . 5 2 6 8 9 0 . 7 9 7 4 9

b= C . 5 0 0 0 0 ZERO ORDER F I R S T ORDER SECOND ORDER THIRD ORDER

NARROW SURFACE LAYER AT TOP C O 0 . 0 3 3 6 9 0 . 3 1 4 7 9 0 . 3 6 7 8 8 0 . 3 4 9 6 7 0 . 3 1 8 7 5 0 . 3 0 3 2 7 0 . 2 7 9 8 9 0 . 3 2 6 6 4 C 1 3 9 9 4 0 . 1 8 3 3 0 0 . 2 1 0 9 0 0 . 1 8 1 7 2 0 . 1 5 3 4 6 0 . 1 3 1 4 0 0 . 1 2 2 3 6 0 . 1 7 1 3 8 0 . 1 5 9 0 4 0 . 0 8 5 6 9 0 . 1 1 1 8 1 0 . 1 0 6 9 5 0 . 0 8 6 7 2 0 . 0 7 1 2 3 0 . 0 6 0 0 4 0 . 0 5 5 6 1 0 . 0 8 6 6 2 0 . 0 7 5 9 5 0 . 0 4 3 3 1 0 . 0 5 5 8 4 0 . 0 5 0 2 1 0 . 0 3 9 8 9 0 . 0 3 2 4 6 0 . 0 2 7 2 1 0 . 0 2 5 1 6 0 . 0 4 0 7 4 0 . 0 3 4 9 7

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

C . 0 3 1 8 0 0 . 0 7 5 3 1 0 . 3 6 1 6 9 0 . 0 7 3 1 0 0 . 1 2 9 2 9 0 . 4 2 0 2 0 0 . 1 2 7 8 0 0 . 2 0 0 6 3 0 . 4 9 4 7 5 0 . 2 0 2 0 8 0 . 2 9 7 2 4 0 . 5 9 2 3 3

C . 2 4 9 5 3 0 . 3 5 8 8 2 0 . 6 5 3 0 8 C . 2 7 6 5 8 0 . 3 9 3 8 9 0 . 6 8 7 2 9 0 . 3 0 0 0 7 0 . 4 2 4 3 2 0 . 7 1 6 7 7 0 . 3 0 6 2 2 0 . 4 3 2 2 9 0 . 7 2 4 4 6

0 . 4 0 8 0 4 0 . 3 8 3 5 0 0 . 3 4 7 6 7 0 . 3 3 0 1 8 0 . 3 1 7 9 9 0 . 3 6 1 8 0 0 . 4 5 7 5 6 0 . 4 2 4 9 9 0 . 3 8 3 0 4 0 . 3 6 3 0 7 0 . 3 6 5 4 9 0 . 4 0 5 1 4 0 . 5 1 9 9 5 0 . 4 7 7 0 0 0 . 4 2 7 2 5 0 . 4 0 4 1 4 0 . 4 2 6 0 0 0 . 4 5 9 7 7 0 . 6 0 0 7 1 0 . 5 4 3 9 9 0 . 4 8 4 0 5 0 . 4 5 6 8 5 0 . 5 0 5 2 0 0 . 5 3 0 5 1

0 . 6 5 0 6 1 0 . 5 8 5 2 4 0 . 5 1 8 9 5 0 . 4 8 9 2 1 0 . 5 5 4 5 1 0 . 5 7 4 2 3 0 . 6 7 8 6 0 0 . 6 0 8 3 4 0 . 5 3 8 4 7 0 . 5 0 7 3 1 0 . 5 8 2 2 7 0 . 5 9 8 7 5 0 . 7 0 2 6 8 0 . 6 2 8 1 8 0 . 5 5 5 2 3 0 . 5 2 2 8 5 0 . 6 0 6 2 0 0 . 6 1 9 8 4 0 . 7 0 8 9 5 0 . 6 3 3 3 5 0 . 5 5 9 6 0 0 . 5 2 6 8 9 0 . 6 1 2 4 4 0 . 6 2 5 3 4


LAMBERT SURFACE ON TOP C O 0 . 0 0 6 7 4 0 . 1 8 8 8 8 0 . 1 6 3 3 2 0 . 1 9 6 6 1 0 . 1 9 5 4 3 0 . 0 7 9 5 2 0 . 1 0 2 3 5 0 . 0 9 3 6 2 C 0 3 7 9 8 0 . 0 4 8 7 0 0 . 0 4 3 0 6

0 . 3 6 7 8 8 0 . 4 8 9 5 4 0 . 5 7 3 7 5 0 . 6 0 6 5 3 0 . 3 2 6 6 4 0 . 4 4 3 2 1 0 . 1 6 1 1 0 0 . 1 3 3 3 9 0 . 1 1 2 9 6 0 . 1 0 4 7 9 0 . 1 5 9 0 4 0 . 1 4 1 1 4 0 . 0 7 4 8 3 0 . 0 6 1 0 7 0 . 0 5 1 2 9 0 . 0 4 7 4 4 0 . 0 7 5 9 5 0 . 0 6 5 5 9 0 . 0 3 4 0 3 0 . 0 2 7 6 2 0 . 0 2 3 1 3 0 . 0 2 1 3 7 0 . 0 3 4 9 7 0 . 0 2 9 8 5

SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = I . 0 0

C . 0 3 6 1 8 0 . 0 5 0 5 8 0 . 2 3 2 0 8 C 0 8 1 0 3 C 1 0 5 5 7 0 . 2 8 5 3 9 C . 1 3 7 9 3 0 . 1 7 6 0 4 0 . 3 5 2 5 9 0 . 2 1 2 2 0 0 . 2 6 8 7 8 0 . 4 3 9 6 3

0 . 2 5 8 4 0 0 . 3 2 6 7 6 0 . 4 9 3 4 2 0 . 2 8 4 4 1 0 . 3 5 9 4 6 C . 5 2 3 6 0 C 3 0 6 8 2 0 . 3 8 7 6 9 0 . 5 4 9 5 6 C 3 1 2 6 7 0 . 3 9 5 0 6 0 . 5 5 6 3 2

0 . 4 0 3 3 9 0 . 5 1 8 9 0 0 . 5 9 8 6 0 0 . 6 2 9 5 7 0 . 3 6 1 8 0 0 . 4 7 4 3 2 0 . 4 4 6 9 5 0 . 5 5 4 8 2 0 . 6 2 8 9 5 0 . 6 5 7 7 0 0 . 4 0 5 1 4 0 . 5 1 2 4 9 0 . 5 0 1 5 5 0 . 5 9 9 7 3 0 . 6 6 6 8 3 0 . 6 9 2 8 0 0 . 4 5 9 7 7 0 . 5 6 0 3 4 0 . 5 7 1 8 6 0 . 6 5 7 4 1 0 . 7 1 5 4 1 0 . 7 3 7 7 8 0 . 5 3 0 5 1 0 . 6 2 1 9 7

0 . 6 1 5 1 5 0 . 6 9 2 8 5 0 . 7 4 5 2 3 0 . 7 6 5 3 8 0 . 5 7 4 2 3 0 . 6 5 9 9 2 0 . 6 3 9 3 8 0 . 7 1 2 6 7 0 . 7 6 1 9 0 0 . 7 8 0 8 1 0 . 5 9 8 7 5 0 . 6 8 1 1 6 0 . 6 6 0 2 0 0 . 7 2 9 6 9 0 . 7 7 6 2 1 0 . 7 9 4 0 5 0 . 6 1 9 8 4 0 . 6 9 9 4 2 0 . 6 6 5 6 2 0 . 7 3 4 1 2 0 . 7 7 9 9 4 0 . 7 9 7 4 9 0 . 6 2 5 3 4 0 . 7 0 4 1 7




VECTOR μ=0 . 0

b = 1 . 0 0 0 0 0 Mo


2 . 5 0 0 0 0 C . 2 9 9 7 4 0 . 0 9 1 3 7

SUMS α = 0 . 2 0 α « 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

0 . 5 1 2 7 9 I . C 5 5 1 5 1 . 6 3 5 5 8 2 . 2 6 8 9 9

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

2 . 6 1 4 0 0 2 . 7 9 6 1 3 2 . 9 4 7 4 9 2 . 9 8 6 2 2

b = 1 . 0 0 0 0 0 Mo


0 . 8 3 3 3 3 0 . 1 8 2 6 3 0 . 0 7 6 9 3

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

0 . 1 7 4 6 6 0 . 3 6 8 8 1 0 . 5 9 0 3 5 0 . 8 5 3 8 8

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

ι.οιοοι 1 . 0 9 6 8 4 i . 1 7 1 6 2 1 . 1 9 1 1 6

b = 1 . 0 0 0 0 0 MO


0 . 5 0 0 0 0 0 . 1 3 5 2 5 0 . 0 6 3 7 1

SUMS α » 0 . 2 0 α * 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

0 . 1 0 5 9 8 0 . 2 2 6 8 9 0 . 3 6 9 5 2 0 . 5 4 6 7 3

α = 0 . 9 0 α * 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

0 . 6 5 6 0 1 0 . 7 1 8 2 3 0 . 7 7 2 6 6 0 . 7 8 7 0 2

b = 1 . 0 0 0 0 0 Mo


0 . 3 5 7 1 4 0 . 1 0 7 7 6 0 . 0 5 3 6 4

SUMS α * 0 . 2 0 α * 0 . 4 0 α * 0 . 6 0 α = 0 . 8 0

0 . 0 7 6 2 2 0 . 1 6 4 5 5 0 . 2 7 0 8 2 0 . 4 0 6 1 8

α « 0 . 9 0 α » 0 . 9 5 ο * 0 . 9 9 ο = 1 . 0 0

0 . 4 9 1 5 1 0 . 5 4 0 7 1 0 . 5 8 4 1 0 0 . 5 9 5 6 0

M - O . l Μ * 0 · 3 M = 0 . 5 μ = 0·Ί μ=0.9

2 5 0 0 C 2 9 9 7 3

2 6 2 9 5 5 5 6 6 3 8 9 1 5 0

1 . 5 1 6 5 8

6 2 5 0 0 0 . 4 1 6 6 6 0 . 3 1 2 5 0 0 . 2 5 0 0 0 2 1 1 6 1 0 . 1 6 1 9 1 0 . 1 3 0 7 6 0 . 1 0 9 5 3 0 9 6 1 5 0 . 0 8 0 4 3 0 . 0 6 7 9 9 0 . 0 5 8 5 3

= 1 . 0

2 2 7 2 7 1 0 1 2 9 0 5 4 6 6

AVERAGE Ν

FLUX U

5 9 9 4 7 0 . 1 8 2 7 3 Ο, 0 8 1 9 2 0 . 0 7 1 7 7

3 8 0 1 0 1 4 4 6 5

1 3 4 3 3 0 . 0 9 0 5 3 0 . 0 6 8 3 4 0 . 0 5 4 9 1 0 . C 5 0 Û 0 2 9 1 7 1 0 . 1 9 9 2 2 0 . 1 5 1 5 8 0 . 1 2 2 4 1 0 . 1 1 1 6 9 4 8 2 1 4 0 . 3 3 4 7 1 0 . 2 5 7 1 4 0 . 2 0 8 9 8 0 . 7 2 4 1 9 0 . 5 1 3 4 1 0 . 3 9 9 2 9 0 . 3 2 7 0 8 0 .

0 . 1 2 7 9 4 0 . 0 8 2 4 5 0 . 2 7 5 7 3 0 . 1 8 1 1 3

1 9 1 1 2 0 . 4 5 1 9 2 0 . 3 0 3 , 7 4 3 0 0 0 3 0 . 6 7 2 4 8 0 . 4 6 5 0 1

8 7 5 5 7 9 6 2 2 8

7 5 1 5 0 1 . 0 3 8 3 5 7 7 9 6 4 1 . 0 5 8 4 4

0 . 6 2 8 8 3 0 . 4 9 2 7 2 0 . 4 0 5 5 5 0 . 6 9 6 1 6 0 . 5 4 7 7 5 0 . 4 5 2 0 4 0 . 7 5 5 9 3 0 . 5 9 6 9 1 0 . 4 9 3 7 2 0 . 7 7 1 8 2 0 . 6 1 0 0 2 0 . 5 0 4 8 6

0 . 6 2 5 0 0 0 . 4 1 6 1 4 0 . 3 1 C 9 9 0 . 2 4 7 8 6 0 . 2 0 5 8 9 0 . 2 1 1 6 1 0 . 1 8 0 7 8 0 . 1 4 9 0 6 0 . 1 2 5 0 3 0 . 1 0 7 1 2 0 . 0 9 6 1 5 0 . 0 9 3 8 1 0 . 0 8 1 7 5 0 . 0 7 0 5 1 0 . 0 6 1 4 2

0 . 3 7 2 6 7 0 . 8 0 8 8 9 0 . 5 6 8 9 9 0 . 4 1 5 8 0 0 . 8 8 6 5 8 0 . 6 2 9 5 9 0 . 4 5 4 5 2 0 . 9 5 4 5 3 0 . 6 8 3 3 7 0 . 4 6 4 8 8 0 . 9 7 2 4 4 0 . 6 9 7 6 6

0 . 1 8 9 7 8 0 . 3 6 5 2 6 0 . 2 7 8 2 0 0 . 0 9 9 8 6 0 . 1 5 3 8 6 0 . 1 3 2 8 9 0 . 0 5 7 6 0 0 . 0 8 0 2 0 0 . 0 7 3 0 8

0 . 0 6 8 9 0 0 . 0 5 5 2 1 0 . 0 4 6 0 2 0 . 0 4 2 4 7 0 . 0 7 9 9 3

3 4 5 8 0 5 3 8 6 6

2 6 7 6 4 4 2 4 0 9

2 1 7 4 9 0 . 1 8 2 8 7 0 . 1 6 9 3 3 3 4 7 8 5 0 . 2 9 4 1 7 0 . 2 7 2 9 7

. 8 7 5 5 7 0 . 6 6 5 7 0 0 . 5 2 9 4 6 0 . 4 3 6 6 7 0 . 3 7 0 5 2 , 9 6 2 2 8 0 . 7 4 0 5 7 0 . 5 9 2 3 2 0 . 4 8 9 9 7 0 . 4 1 6 5 1 , 0 3 8 3 5 0 . 8 0 7 4 7 0 . 6 4 8 9 0 Ο, . 0 5 8 4 4 0 . 8 2 5 3 1 0 . 6 6 4 0 6 0 .

0 . 3 4 4 2 4 0 . 3 8 7 2 2

5 3 8 1 3 0 . 4 5 8 1 6 0 . 4 2 6 1 8 5 5 1 0 6 0 . 4 6 9 3 5 0 . 4 3 6 6 6

3 0 1 1 6 4 6 8 0 2

5 7 7 7 9 6 4 2 4 6 7 0 0 2 5 7 1 5 6 6

0 . 0 6 1 6 2 0 . 1 3 8 6 6 0 . 2 3 9 2 6 0 . 3 7 9 1 5

0 . 4 7 3 4 1 0 . 5 2 9 6 5 0 . 5 8 0 3 0 0 . 5 9 3 8 7

0 . 4 1 6 6 6 0 . 3 1 0 9 9 0 . 2 4 5 4 2 0 . 2 0 1 5 8 0 . 1 7 0 6 2 0 . 1 5 8 3 7 0 . 2 7 0 4 9 0 . 2 1 9 4 6 0 . 1 6 1 9 1 0 . 1 4 9 0 6 0 . 1 2 7 2 0 0 . 1 0 8 6 1 0 . 0 9 4 0 6 0 . 0 8 8 0 3 0 . 1 2 7 4 2 0 . 1 1 3 6 4 0 . 0 8 0 4 3 0 . 0 8 1 7 5 0 . 0 7 2 5 9 0 . 0 6 3 2 1 0 . 0 5 5 3 8 0 . 0 5 2 0 4 0 . 0 7 0 2 3 0 . 0 6 5 0 0

0 9 0 5 3 0 . 0 6 8 9 0 0 . 0 5 4 8 3 0 . 0 4 5 2 3 0 . 0 3 8 3 9 0 . 0 3 5 6 7 0 . 0 5 9 8 3 0 . 0 4 9 0 3 1 9 9 2 2 0 . 1 5 5 0 9 0 . 1 2 4 6 3 0 . 1 0 3 3 4 0 . 0 8 7 9 8 0 . 0 8 1 8 3 0 . 1 3 4 4 7 0 . 1 1 1 4 4 3 3 4 7 1 0 . 2 6 7 6 4 0 . 2 1 7 5 7 5 1 3 4 1 0 . 4 2 4 0 9 0 . 3 4 9 5 8

. 6 2 8 8 3 6 9 6 1 6

0 . 5 2 9 4 6 0 . 5 9 2 3 2

0 . 4 4 0 0 0 0 . 4 9 4 4 0

1 8 1 4 9 0 . 1 5 5 0 7 0 . 1 4 4 4 3 0 . 2 3 1 7 4 0 . 1 9 4 5 6 2 9 3 7 2 0 . 2 5 2 0 3 0 . 2 3 5 1 1 0 . 3 6 6 8 3 0 . 3 1 2 7 0

. 3 7 1 2 3 0 . 3 1 9 3 3 0 . 2 9 8 1 5 0 . 4 5 7 8 0 0 . 3 9 3 6 6 , 4 1 8 0 7 0 . 3 6 0 1 0 0 . 3 3 6 3 8 0 . 5 1 2 0 7 0 . 4 4 2 4 0

0 . 6 4 8 9 0 0 . 5 4 3 6 5 0 . 4 6 0 5 8 0 . 3 9 7 1 6 0 . 3 7 1 1 5 0 . 5 6 0 9 4 0 . 4 8 6 5 2 0 . 6 6 4 0 6 0 . 5 5 6 8 8 0 . 4 7 2 0 1 0 . 4 0 7 1 4 0 . 3 8 0 5 1 0 . 5 7 4 0 3 0 . 4 9 8 3 8

0 . 7

3 1 2 5 0 0 . 2 4 7 8 6 0 . 2 0 1 5 8 0 . 1 6 8 3 2 0 . 1 4 3 9 2 0 . 1 3 4 0 9 0 . 2 1 5 5 3 0 . 1 8 0 4 7 1 3 0 7 6 0 . 1 2 5 0 3 1 0 8 6 1

0 6 3 2 1 0 9 3 6 1 0 . 0 8 1 5 2 0 . 0 7 6 4 5 0 . 1 0 7 2 9 0 . 0 9 7 1 8 0 5 5 3 1 0 . 0 4 8 5 9 0 . 0 4 5 7 1 0 . 0 6 0 7 6 0 . 0 5 6 6 6

0 . 0 6 8 3 4 0 . 0 5 5 2 1 0 . 1 5 1 5 8 0 . 1 2 5 0 8 0 . 2 5 7 1 4 Ο

0 . 0 4 5 2 3 0 . 0 3 7 9 1 0 . 0 3 2 4 9 0 . 0 3 0 2 9 0 . 0 4 7 9 5 0 . 0 4 0 5 0 0 , 1 0 3 3 4 0 . 0 8 6 9 8 0 . 0 7 4 7 3 0 . 0 6 9 7 4 0 . 1 0 8 4 9 0 . 0 9 2 5 2

2 1 7 4 9 0 . 1 8 1 4 9 0 . 1 5 3 5 2 0 . 1 3 2 2 7 0 . 1 2 3 5 8 0 . 1 8 8 4 5 0 . 1 6 2 5 0 3 4 7 8 5 0 . 2 9 3 7 2 0 . 2 4 9 9 0 0 . 2 1 6 0 6 0 . 2 0 2 1 0 0 . 3 0 1 1 7 0 . 2 6 3 0 5

0 . 4 9 2 7 2 0 . 4 3 6 6 7 Ο 3 7 1 2 3 0 . 3 1 6 9 1 0 . 2 7 4 5 3 0 . 2 5 6 9 7 0 . 3 7 7 9 6 0 . 3 3 2 5 2 4 8 9 9 7 0 . 4 1 8 0 7 0 . 3 5 7 5 4 0 . 3 1 0 0 5 0 . 2 9 0 3 3 0 . 4 2 4 0 6 0 . 3 7 4 5 1 5 3 8 1 3 0 . 4 6 0 5 8 0 . 3 9 4 4 8 0 . 3 4 2 3 9 0 . 3 2 0 7 2 0 . 4 6 5 7 2 0 . 4 1 2 6 3 5 5 1 0 6 0 . 4 7 2 0 1 0 . 4 0 4 4 4 0 . 3 5 1 1 1 0 . 3 2 8 9 1 0 . 4 7 6 9 1 0 . 4 2 2 8 9



VECTOR / x = 0 . 0 / x = 0 . i μ-0 .3 /χ = 0 . 5 AVERAGE

Ν FLUX

U


Mo = 0 . 1 0 . 0 PEAK 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 0 2 3 0 . 0 0 0 0 5 0 . 0 0 0 1 1 0 . 0 0 1 1 3 0 . 0 4 4 5 4 0 . 0 8 4 5 6 0 . 0 9 9 8 4 0 . 1 0 2 8 6 0 . 1 0 2 1 8 0 . 0 6 6 6 0 0 . 0 8 7 5 5 C . 0 3 3 3 0 0 . 0 4 1 1 1 0 . 0 6 6 2 9 0 . 0 7 6 1 6 0 . 0 7 5 3 3 0 . 0 7 1 0 3 0 . 0 6 8 5 1 0 . 0 6 5 9 0 0 . 0 7 1 5 3 C . 0 3 2 9 5 0 . 0 4 2 0 1 0 . 0 5 4 9 0 0 . 0 5 5 7 5 0 . 0 5 1 9 7 0 . 0 4 7 3 8 0 . 0 4 5 1 6 0 . 0 5 0 2 4 0 . 0 5 1 0 0

SUMS α α - 0 . 2 C α - 0 . 4 0 α 0 . 6 0 α = 0 . 8 0

α 0 . 9 0 α - 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

0 . 0 0 1 6 6 0 . 0 0 2 2 7 0 . 0 1 2 0 7 0 . 0 2 0 4 7 0 . 0 2 3 4 6 0 . 0 2 3 8 5 0 . 0 2 3 5 9 0 . 0 1 6 6 5 0 . 0 2 0 8 9 0 . 0 0 8 3 6 0 . 0 1 0 8 5 0 . 0 3 3 2 8 0 . 0 5 0 8 7 0 . 0 5 6 4 8 0 . 0 5 6 5 9 0 . 0 5 5 7 2 0 . 0 4 1 8 2 0 . 0 5 0 9 4 C . 0 2 4 6 1 0 . 0 3 1 5 4 0 . 0 7 0 6 9 0 . 0 9 8 0 7 0 . 1 0 5 3 6 0 . 1 0 3 8 9 0 . 1 0 1 7 6 0 . 0 8 2 0 5 0 . 0 9 6 4 6 0 . 0 6 0 1 6 0 . 0 7 6 8 4 0 . 1 3 9 0 3 0 . 1 7 6 2 0 0 . 1 8 2 8 8 0 . 1 7 7 2 1 0 . 1 7 2 5 7 0 . 1 5 0 3 9 0 . 1 7 0 2 2

0 . 0 9 1 5 4 0 . 1 1 6 9 0 0 . 1 9 4 2 5 0 . 2 3 5 7 9 0 . 2 4 0 4 0 0 . 2 3 0 7 8 0 . 2 2 4 0 3 0 . 2 0 3 4 2 0 . 2 2 5 7 5 0 . 1 1 2 7 4 0 . 1 4 4 0 0 0 . 2 3 0 2 2 0 . 2 7 3 6 1 0 . 2 7 6 4 4 0 . 2 6 4 1 0 0 . 2 5 5 9 6 0 . 2 3 7 3 5 0 . 2 6 0 7 9 C . 1 3 3 2 6 0 . 1 7 0 2 3 0 . 2 6 4 3 2 0 . 3 0 8 9 6 0 . 3 0 9 8 8 0 . 2 9 4 8 8 0 . 2 8 5 4 1 0 . 2 6 9 2 2 0 . 2 9 3 4 3 0 . 1 3 8 9 8 0 . 1 7 7 5 4 0 . 2 7 3 7 1 0 . 3 1 8 6 3 0 . 3 1 8 9 8 0 . 3 0 3 2 4 0 . 2 9 3 4 0 0 . 2 7 7 9 5 0 . 3 0 2 3 4


Mo = 0 . 3 C O C O PEAK 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 5 9 4 6 0 . 0 3 5 6 7 0 . 0 2 9 7 3 0 . 0 4 4 5 4 0 . 0 9 9 0 9 0 . 1 2 4 5 8 0 . 1 2 7 4 9 0 . 1 2 2 3 0 0 . 1 1 8 6 4 0 . 1 0 3 4 8 0 . 1 1 8 5 0 0 . 0 5 1 7 4 0 . 0 6 6 2 9 0 . 0 9 2 1 7 0 . 0 9 6 1 1 0 . 0 9 0 6 7 0 . 0 8 3 2 1 0 . 0 7 9 5 0 0 . 0 8 5 4 6 0 . 0 8 8 2 6 0 . 0 4 2 7 3 0 . G 5 4 9 C 0 . 0 6 7 5 0 0 . 0 6 5 9 2 0 . 0 6 0 2 2 0 . 0 5 4 2 5 0 . 0 5 1 4 9 0 . 0 6 0 3 1 0 . 0 5 9 8 1

SUMS α α = C . 2 0 α - 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α 0 . 9 0 α - 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

C . 0 0 8 4 1 0 . 0 1 2 0 7 0 . 0 2 4 1 3 0 . 0 2 9 3 7 0 . 0 2 9 6 8 0 . 0 2 8 2 9 0 . 0 2 7 3 8 0 . 0 8 4 1 3 0 . 0 6 3 4 5 0 . 0 2 3 9 5 0 . 0 3 3 2 8 0 . 0 6 0 2 6 0 . 0 7 0 9 0 0 . 0 7 0 6 8 0 . 0 6 6 8 8 0 . 0 6 4 5 9 0 . 1 1 9 7 5 0 . 1 0 2 3 5 C . 0 5 2 1 0 0 . C 7 C 6 9 0 . 1 1 6 7 2 0 . 1 3 2 5 4 0 . 1 3 0 1 7 0 . 1 2 2 2 2 0 . 1 1 7 7 0 0 . 1 7 3 6 7 0 . 1 5 9 5 4 C . 1 0 4 3 4 0 . 1 3 9 0 3 0 . 2 1 0 5 0 0 . 2 3 0 3 4 0 . 2 2 2 5 9 0 . 2 0 7 1 4 0 . 1 9 8 8 9 0 . 2 6 0 8 4 0 . 2 4 9 5 2

C . 1 4 6 8 4 0 . 1 9 4 2 5 0 . 2 8 2 1 5 0 . 3 0 2 8 8 0 . 2 9 0 1 4 0 . 2 6 8 7 2 0 . 2 5 7 5 8 0 . 3 2 6 3 1 0 . 3 1 5 8 8 0 . 1 7 4 5 9 0 . 2 3 0 2 2 0 . 3 2 7 6 5 0 . 3 4 8 3 1 0 . 3 3 2 1 6 0 . 3 0 6 8 6 0 . 2 9 3 8 9 0 . 3 6 7 5 5 0 . 3 5 7 3 2 C . 2 0 0 9 4 0 . 2 6 4 3 2 0 . 3 7 0 1 8 0 . 3 9 0 4 3 0 . 3 7 0 9 6 0 . 3 4 2 0 0 0 . 3 2 7 3 0 0 . 4 0 5 9 3 0 . 3 9 5 6 9 C . 2 0 8 2 0 0 . 2 7 3 7 1 0 . 3 8 1 8 0 0 . 4 0 1 8 9 0 . 3 8 1 5 0 0 . 3 5 1 5 3 0 . 3 3 6 3 6 0 . 4 1 6 4 0 0 . 4 0 6 1 3


SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

Μο = 0 . 5 0 . 0 0 . 0 0 . 0 PEAK 0 . 0 0 . 0 0 . 0 0 . 1 3 5 3 4 0 . 1 3 5 3 4 0 . 0 6 7 6 7 0 . 0 8 4 5 6 0 . 1 2 4 5 8 0 . 1 3 5 3 4 0 . 1 3 0 3 9 0 . 1 2 1 1 6 0 . 1 1 6 2 7 0 . 1 1 8 9 7 0 . 1 2 5 5 5 0 . C 5 9 4 8 0 . 0 7 6 1 6 0 . 0 9 6 1 1 0 . 0 9 5 4 9 0 . 0 8 8 0 3 0 . 0 7 9 7 3 0 . 0 7 5 8 3 0 . 0 8 6 7 8 0 . 0 8 6 9 6 0 . 0 4 3 3 9 0 . 0 5 5 7 5 0 . 0 6 5 9 2 0 . 0 6 3 0 6 0 . 0 5 7 0 2 0 . 0 5 1 0 6 0 . 0 4 8 3 6 0 . 0 5 8 3 0 0 . 0 5 7 0 2

0 . 0 1 6 3 1 0 . 0 2 0 4 7 0 . 0 2 9 3 7 0 . 0 3 1 4 6 0 . 0 3 0 1 2 0 . 0 2 7 8 9 0 . 0 2 6 7 3 0 . 1 6 3 1 3 0 . 1 6 4 4 5 0 . 0 4 0 3 7 0 . 0 5 0 8 7 0 . 0 7 0 9 0 0 . 0 7 4 8 3 0 . 0 7 1 1 2 0 . 0 6 5 5 9 0 . 0 6 2 7 7 0 . 2 0 1 8 3 0 . 2 0 4 3 6 C . 0 7 7 5 1 0 . 0 9 8 0 7 0 . 1 3 2 5 4 0 . 1 3 7 5 8 0 . 1 2 9 7 3 0 . 1 1 9 0 9 0 . 1 1 3 8 0 0 . 2 5 8 3 8 0 . 2 6 1 8 3 0 . 1 3 8 7 4 0 . 1 7 6 2 G 0 . 2 3 0 3 4 0 . 2 3 4 7 6 0 . 2 1 9 4 0 0 . 2 0 0 3 7 0 . 1 9 1 1 3 0 . 3 4 6 8 6 0 . 3 5 0 4 2

C . 1 8 5 3 4 0 . 2 3 5 7 9 0 . 3 0 2 8 8 0 . 3 0 5 6 5 0 . 2 8 4 2 5 0 . 2 5 8 8 7 0 . 2 4 6 6 8 0 . 4 1 1 8 6 0 . 4 1 4 8 4 0 . 2 1 4 8 8 0 . 2 7 3 6 1 0 . 3 4 8 3 1 0 . 3 4 9 6 8 0 . 3 2 4 3 7 0 . 2 9 4 9 7 0 . 2 8 0 9 3 0 . 4 5 2 3 9 0 . 4 5 4 7 9 C . 2 4 2 4 8 0 . 3 0 8 9 6 0 . 3 9 0 4 3 0 . 3 9 0 3 2 0 . 3 6 1 3 0 0 . 3 2 8 1 5 0 . 3 1 2 3 9 0 . 4 8 9 8 6 0 . 4 9 1 6 3 0 . 2 5 0 0 3 0 . 3 1 8 6 3 0 . 4 0 1 8 9 0 . 4 0 1 3 5 0 . 3 7 1 3 1 0 . 3 3 7 1 4 0 . 3 2 0 9 1 0 . 5 0 0 0 5 0 . 5 0 1 6 2


SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

Μο = 0 . 7 0 . 0 0 . 0 0 . 0 0 . 0 PEAK 0 . 0 0 . 0 0 . 1 7 1 1 8 0 . 2 3 9 6 5 0 . 0 8 5 5 9 0 . 0 9 9 8 4 0 . 1 2 7 4 9 0 . 1 3 0 3 9 0 . 1 2 2 2 7 0 . 1 1 1 9 3 0 . 1 0 6 8 6 0 . 1 1 8 1 4 0 . 1 1 9 8 2 C . 0 5 9 0 7 0 . 0 7 5 3 3 0 . 0 9 0 6 7 0 . 0 8 8 0 3 0 . 0 8 0 2 4 0 . 0 7 2 2 1 0 . 0 6 8 5 2 0 . 0 8 0 9 6 0 . 0 7 9 8 7 0 . 0 4 0 4 8 0 . 0 5 1 9 7 0 . 0 6 0 2 2 0 . 0 5 7 0 2 0 . 0 5 1 2 8 0 . 0 4 5 7 9 0 . 0 4 3 3 2 0 . 0 5 3 0 0 0 . 0 5 1 4 8

0 . 0 1 9 8 5 0 . 0 2 3 4 6 0 . 0 2 9 6 8 0 . 0 3 0 1 2 0 . 0 2 8 1 3 0 . 0 2 5 6 9 0 . 0 2 4 5 1 0 . 1 9 8 5 3 0 . 2 6 7 2 8 C . 0 4 7 1 9 0 . 0 5 6 4 8 0 . 0 7 0 6 8 0 . 0 7 1 1 2 0 . 0 6 6 1 3 0 . 0 6 0 2 3 0 . 0 5 7 4 0 0 . 2 3 5 9 4 0 . 3 0 4 7 6 0 . 0 8 6 9 0 0 . 1 0 5 3 6 0 . 1 3 0 1 7 0 . 1 2 9 7 3 0 . 1 2 0 0 1 0 . 1 0 8 9 8 0 . 1 0 3 7 5 0 . 2 8 9 6 7 0 . 3 5 8 1 6 C . 1 4 8 9 4 0 . 1 8 2 8 8 0 . 2 2 2 5 9 0 . 2 1 9 4 0 0 . 2 0 1 7 8 0 . 1 8 2 6 1 0 . 1 7 3 6 2 0 . 3 7 2 3 4 0 . 4 3 9 5 8

C 1 9 4 5 7 0 . 2 4 0 4 0 0 . 2 9 0 1 4 0 . 2 8 4 2 5 0 . 2 6 0 5 8 0 . 2 3 5 3 7 0 . 2 2 3 6 3 0 . 4 3 2 3 9 0 . 4 9 8 3 4 0 . 2 2 3 0 6 0 . 2 7 6 4 4 0 . 3 3 2 1 6 0 . 3 2 4 3 7 0 . 2 9 6 8 5 0 . 2 6 7 8 6 0 . 2 5 4 4 1 0 . 4 6 9 6 1 0 . 5 3 4 6 5 G . 2 4 9 4 4 0 . 3 0 9 8 8 0 . 3 7 0 9 6 0 . 3 6 1 3 0 0 . 3 3 0 1 8 0 . 2 9 7 6 8 0 . 2 8 2 6 5 0 . 5 0 3 9 2 0 . 5 6 8 0 5 0 . 2 5 6 6 1 0 . 3 1 8 9 8 0 . 3 8 1 5 0 0 . 3 7 1 3 1 0 . 3 3 9 2 1 0 . 3 0 5 7 6 0 . 2 9 0 3 0 0 . 5 1 3 2 3 0 . 5 7 7 1 1



VECTOR μ-0.0

b = 1 . 0 0 0 0 0 Η-ο


C . 2 7 7 7 8 C . 0 8 9 6 2 C . 0 4 6 0 8

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

0 . 0 5 9 5 6 0 . 1 2 9 2 9 0 . 2 1 4 2 7 0 . 3 2 4 2 5

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

0 . 3 9 4 5 3 0 . 4 3 5 3 7 0 . 4 7 1 5 5 0 . 4 8 1 1 7

b = i . O O O O C μ0 F I R S T ORDER SECOND ORDER THIRD ORDER

0 . 2 5 0 0 0 C . 0 8 2 6 7 C . 0 4 3 0 1

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

C . 0 5 3 7 0 C . 1 1 6 8 2 0 . 1 9 4 1 1 0 . 2 9 4 7 4

α = 0 . 9 0 α = C . 9 5 α = 0 . 9 9 α = 1 . 0 0

C . 3 5 9 3 7 C . 3 9 7 0 3 0 . 4 3 0 4 6 0 . 4 3 9 3 4

M = 0 . i μ =0.3 M = 0 . 5 μ~0,1 M

s0 . 9 μ*1.0 AVERAGE

N FLUX

U

0 . 1 0 9 5 3 0 . 1 0 7 1 2 0 . 0 9 4 0 6 0 . 0 8 1 5 2 0 . 0 7 1 2 3 0 . 0 6 6 8 7 0 . 0 5 8 5 3 0 . 0 6 1 4 2 0 . 0 5 5 3 8 0 . 0 4 8 5 9 0 . 0 4 2 7 6 0 . 0 4 0 2 5

C . 1 2 2 4 1 0 . 1 0 4 6 7 0 . 0 8 7 9 8 0 . 0 7 4 7 3 0 . 0 6 4 5 6 0 . 0 6 0 3 7 0 . 2 0 8 9 8 0 . 1 8 2 8 7 0 . 1 5 5 0 7 0 . 1 3 2 2 7 0 . 1 1 4 5 5 0 . 1 0 7 2 2 0 . 3 2 7 0 8 0 . 2 9 4 1 7 0 . 2 5 2 0 3 0 . 2 1 6 0 6 0 . 1 8 7 6 5 0 . 1 7 5 8 2

0 . 0 9 2 1 7 0 . 0 8 4 2 5 0 . 0 5 3 0 2 0 . 0 4 9 6 7

0 . 0 9 0 9 1 0 . 0 7 8 8 5 0 . 1 5 8 6 9 0 . 1 3 8 9 9 0 . 2 5 5 0 7 0 . 2 2 5 9 4

4 0 5 5 5 0 . 3 7 0 5 2 0 . 3 1 9 3 3 0 . 2 7 4 5 3 0 . 2 3 8 8 2 0 . 2 2 3 8 8 4 5 2 0 4 0 . 4 1 6 5 1 0 . 3 6 0 1 0 0 . 3 1 0 0 5 0 . 2 6 9 9 5 0 . 2 5 3 1 5 4 9 3 7 2 0 . 4 5 8 1 6 0 . 3 9 7 1 6 0 . 3 4 2 3 9 0 . 2 9 8 3 3 0 . 2 7 9 8 3 5 0 4 8 6 0 . 4 6 9 3 5 0 . 4 0 7 1 4 0 . 3 5 1 1 1 0 . 3 0 5 9 8 0 . 2 8 7 0 3

. 2 2 7 2 7 0 . 1 8 9 7 8 0 . 1 5 8 3 7 0 . 1 3 4 0 9 0 . 1 1 5 6 4 0 . 1 0 8 0 8

. 1 0 1 2 9 0 . 0 9 9 8 6 0 . 0 8 8 0 3 0 . 0 7 6 4 5 0 . 0 6 6 8 7 0 . 0 6 2 8 1 , 0 5 4 6 6 0 . 0 5 7 6 0 0 . 0 5 2 0 4 0 . 0 4 5 7 1 0 . 0 4 0 2 5 0 . 0 3 7 8 9

3 2 1 1 8 0 . 2 8 6 3 0 3 6 1 0 1 0 . 3 2 2 8 8 3 9 7 0 8 0 . 3 5 6 1 3 4 0 6 7 8 0 . 3 6 5 0 9

1 6 5 3 4 0 . 1 4 2 0 2 0 8 6 0 1 0 . 0 7 8 8 8 0 4 9 7 6 0 . 0 4 6 6 9

, 0 5 0 0 0 0 . 0 4 2 4 7 0 . 0 3 5 6 7 0 . 0 3 0 2 9 0 . 0 2 6 1 7 0 . 0 2 4 4 7 0 . 0 3 6 9 6 0 . 0 3 1 9 8 1 1 1 6 9 0 . 0 9 6 7 5 0 . 0 8 1 8 3 0 . 0 6 9 7 4 0 . 0 6 0 3 7 0 . 0 5 6 5 0 0 . 0 8 4 0 9 0 . 0 7 3 3 8 1 9 1 1 2 0 . 1 6 9 3 3 0 . 1 4 4 4 3 0 . 1 2 3 5 8 0 . 1 0 7 2 2 0 . 1 0 0 4 2 0 . 1 4 7 0 4 0 . 1 2 9 5 1

, 3 0 0 0 3 0 . 2 7 2 9 7 0 . 2 3 5 1 1 0 . 2 0 2 1 0 0 . 1 7 5 8 2 0 . 1 6 4 8 2 0 . 2 3 6 8 5 0 . 2 1 0 8 5

0 . 3 7 2 6 7 0 . 3 4 4 2 4 0 . 2 9 8 1 5 0 . 2 5 6 9 7 0 . 2 2 3 8 8 0 . 2 1 0 0 0 0 . 2 9 8 6 0 0 . 2 6 7 4 1 0 . 4 1 5 8 0 0 . 3 8 7 2 2 0 . 3 3 6 3 8 0 . 2 9 0 3 3 0 . 2 5 3 1 5 0 . 2 3 7 5 2 0 . 3 3 5 8 5 0 . 3 0 1 7 2 0 . 4 5 4 5 2 0 . 4 2 6 1 8 0 . 3 7 1 1 5 0 . 3 2 0 7 2 0 . 2 7 9 8 3 0 . 2 6 2 6 2 0 . 3 6 9 6 1 0 . 3 3 2 9 2 0 . 4 6 4 8 8 0 . 4 3 6 6 6 0 . 3 8 0 5 1 0 . 3 2 8 9 1 0 . 2 8 7 0 3 0 . 2 6 9 3 9 0 . 3 7 8 6 9 0 . 3 4 1 3 3

b = 1 . 0 0 0 0 0 NARROW SURFACE LAYER AT TOP

F I R S T ORDER I N F I N I T E SECOND ORDER 0 . 1 7 1 3 5 THIRD ORDER 0 . 0 6 6 1 9

SUMS α = 0 . 2 0 INF I N I TE α 0 . 4 0 I N F I N I T E α = 0 . 6 0 I N F I N I T E α = 0 . 8 0 I N F I N I T E

α 0 . 9 0 INF I N I TE α = 0 . 9 5 I N F I N I T E α = 0 . 9 9 I N F I N I T E α = 1 . 0 0 I N F I N I T E

b = 1 . 0 0 0 0 0


SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

0 . 5 9 9 4 7 0 . 3 6 5 2 6 0 . 2 7 0 4 9 0 . 2 1 5 5 3 0 . 1 7 9 2 4 0 . 1 6 5 3 4 0 . 3 4 2 7 0 0 . 2 4 4 4 9 0 . 1 8 2 7 3 0 . 1 5 3 8 6 0 . 1 2 7 4 2 0 . 1 0 7 2 9 0 . 0 9 2 1 7 0 . 0 8 6 0 1 0 . 1 3 2 3 7 0 . 1 1 3 9 1 0 . 0 8 1 9 2 0 . 0 8 0 2 0 0 . 0 7 0 2 3 0 . 0 6 0 7 6 0 . 0 5 3 0 2 0 . 0 4 9 7 6 0 . 0 6 8 8 0 0 . 0 6 2 8 5

. 1 2 7 9 4 0 . 0 7 9 9 3 0 . 0 5 9 8 3 0 . 0 4 7 9 5 0 . 0 4 0 0 2

. 2 7 5 7 3 0 . 1 7 7 4 0 0 . 1 3 4 4 7 0 . 1 0 8 4 9 0 . 0 9 0 9 1 0 . 0 3 6 9 6 0 . 0 . 0 8 4 0 9 Ο,

1 5 8 6 9 0 . 1 4 7 0 4 0 . 2 5 5 0 7 0 . 2 3 6 8 5 0 .

0 7 4 4 6 0 . 0 5 4 0 2 1 6 4 0 0 0 . 1 2 1 2 9 2 7 6 1 6 0 . 2 0 8 8 2 4 2 5 4 7 0 . 3 3 0 2 1

0 . 5 7 7 7 9 0 . 4 5 7 8 0 0 . 3 7 7 9 6 0 . 3 2 1 1 8 0 . 2 9 8 6 0 0 . 5 2 2 8 6 0 . 4 1 1 9 2

0 . 9 7 2 4 4 0 . 7 1 5 6 6 0 . 5 7 4 0 3 0 . 4 7 6 9 1 3 9 7 0 8 0 . 3 6 9 6 1 0 . 6 3 0 9 7 0 . 5 0 4 5 4 4 0 6 7 8 0 . 3 7 8 6 9 0 . 6 4 4 5 4 0 . 5 1 6 2 9


0 . 5 0 0 0 0 0 . 3 8 0 1 0 0 . 2 7 8 2 0 0 . 2 1 9 4 6 0 . 1 8 0 4 7 0 . 1 5 2 9 2 0 . 1 4 2 0 2 0 . 2 4 4 4 9 0 . 1 9 6 7 0 0 . 1 2 2 2 4 0 . 1 4 4 6 5 0 . 1 3 2 8 9 0 . 1 1 3 6 4 0 . 0 9 7 1 8 0 . 0 8 4 2 5 0 . 0 7 8 8 8 0 . 1 1 3 9 1 0 . 1 0 1 6 1 C . 0 5 6 9 6 0 . 0 7 1 7 7 0 . 0 7 3 0 8 0 . 0 6 5 0 0 0 . 0 5 6 6 6 0 . 0 4 9 6 7 0 . 0 4 6 6 9 0 . 0 6 2 8 5 0 . 0 5 8 2 3

C . 1 0 5 4 0 0 . 0 8 2 4 5 0 . 0 6 1 6 2 0 . 0 4 9 0 3 0 . 0 4 0 5 0 0 . 0 3 4 4 1 0 . 0 3 1 9 8 0 . 0 5 4 0 2 0 . 0 4 3 9 3 C . 2 2 4 2 6 0 . 1 8 1 1 3 0 . 1 3 8 6 6 0 . 1 1 1 4 4 0 . 0 9 2 5 2 0 . 0 7 8 8 5 0 . 0 7 3 3 8 0 . 1 2 1 2 9 0 . 0 9 9 8 4 0 . 3 6 2 6 5 0 . 3 0 3 7 4 0 : 2 3 9 2 6 0 . 1 9 4 5 6 0 . 1 6 2 5 0 0 . 1 3 8 9 9 0 . 1 2 9 5 1 0 . 2 0 8 8 2 0 . 1 7 4 3 0 0 . 5 3 2 0 9 0 . 4 6 5 0 1 0 . 3 7 9 1 5 0 . 3 1 2 7 0 0 . 2 6 3 0 5 0 . 2 2 5 9 4 0 . 2 1 0 8 5 0 . 3 3 0 2 1 0 . 2 8 0 1 5

a = 0 . 9 0 0 . 6 3 5 3 6 0 . 5 6 8 9 9 0 . 4 7 3 4 1 0 . 3 9 3 6 6 0 . 3 3 2 5 2 0 . 2 8 6 3 0 0 . 2 6 7 4 1 0 . 4 1 1 9 2 0 . 3 5 2 7 1 a = 0 . 9 5 0 . 6 9 3 8 1 0 . 6 2 9 5 9 0 . 5 2 9 6 5 0 . 4 4 2 4 0 0 . 3 7 4 5 1 0 . 3 2 2 8 8 0 . 3 0 1 7 2 0 . 4 6 0 6 6 0 . 3 9 6 4 0 a = 0 . 9 9 C . 7 4 4 7 5 0 . 6 8 3 3 7 0 . 5 8 0 3 0 0 . 4 8 6 5 2 0 . 4 1 2 6 3 0 . 3 5 6 1 3 0 . 3 3 2 9 2 0 . 5 0 4 5 4 0 . 4 3 5 9 6 a = 1 . 0 0 0 . 7 5 8 1 5 0 . 6 9 7 6 6 0 . 5 9 3 8 7 0 . 4 9 8 3 8 0 . 4 2 2 6 9 0 . 3 6 5 0 9 0 . 3 4 1 3 3 0 . 5 1 6 2 9 0 . 4 4 6 5 9




VECTOR μ=0,0 μ = 0. / i - 0 . 3 p.=0.b μ = 0*7 μ = 0·9 = 1 . 0 AVERAGE

Ν FLUX

U

BE 1 . 0 0 0 0 0 ZERO ORDER F I R S T ORDER SECOND JRDER THIRD ORDER

μο = 0 . 9 0 . 0 O . C 0 . 0 0 . C . 0 9 1 4 4 0 . 1 0 2 8 6 0 . 1 2 2 3 0 0 . C . 0 5 5 8 8 0 . 0 7 1 0 3 0 . 0 8 3 2 1 0 . 0 . 0 3 6 9 3 0 . 0 4 7 3 8 0 . 0 5 4 2 5 0 .

0 0 . 0 PEAK 0 . 0 0 . 1 8 2 8 8 0 . 3 2 9 1 9 1 2 1 1 6 0 . 1 1 1 9 3 0 . 1 0 1 6 0 0 . 0 9 6 7 2 0 . 1 1 1 7 5 0 . 1 1 0 8 1 0 7 9 7 3 0 . 0 7 2 2 1 0 . 0 6 4 7 5 0 . 0 6 1 3 6 0 . 0 7 3 8 6 0 . 0 7 2 2 0 0 5 1 0 6 0 . 0 4 5 7 9 0 . 0 4 0 8 1 0 . 0 3 8 5 9 0 . 0 4 7 6 2 0 . 0 4 6 0 6

SUMS α = 0 . 2 0 C . 0 2 0 8 6 0 . 0 2 3 8 5 0 . 0 2 8 2 9 0 , a = 0 . 4 0 C . C 4 8 7 C G . C 5 6 5 9 0 . 0 6 6 8 8 0 . a = 0 . 6 0 C . 0 8 7 9 2 0 . 1 0 3 8 9 G . 1 2 2 2 2 0 , a = 0 . 8 0 0 . 1 4 7 4 4 0 . 1 7 7 2 1 0 . 2 0 7 1 4 0 ,

0 2 7 8 9 0 . 0 2 5 6 9 0 . 0 2 3 2 8 0 . 0 2 2 1 5 0 . 2 0 8 6 3 0 . 3 5 4 6 6 C 6 3 5 9 0 . 0 6 0 2 3 0 . 0 5 4 4 8 0 . 0 5 1 7 9 0 . 2 4 3 4 8 0 . 3 8 9 0 1 1 1 9 0 9 0 . 1 0 8 9 8 0 . 0 9 8 3 6 0 . 0 9 3 4 3 0 . 2 9 3 0 6 0 . 4 3 7 6 2 2 0 0 3 7 0 . 1 8 2 6 1 0 . 1 6 4 3 9 0 . 1 5 6 0 1 0 . 3 6 8 6 0 0 . 5 1 1 2 8

C . 9 0 0 . 1 9 C 3 9 C . 2 3 0 7 8 0 . 2 6 8 7 2 0 , 0 . 9 5 0 . 2 1 6 9 6 0 . 2 6 4 1 C 0 . 3 0 6 8 6 0 . 0 . 9 9 0 . 2 4 1 4 2 0 . 2 9 4 8 8 0 . 3 4 2 0 0 0 , 1 . 0 0 0 . 2 4 8 C 5 0 . 3 0 3 2 4 0 . 3 5 1 5 3 0 ,

2 5 8 8 7 0 . 2 3 5 3 7 0 . 2 1 1 5 9 0 . 2 0 0 7 0 0 . 4 2 3 0 8 0 . 5 6 4 2 0 2 9 4 9 7 0 . 2 6 7 8 6 0 . 2 4 0 6 2 0 . 2 2 8 1 8 0 . 4 5 6 7 5 0 . 5 9 6 8 2 3 2 8 1 5 0 . 2 9 7 6 8 0 . 2 6 7 2 4 0 . 2 5 3 3 7 0 . 4 8 7 7 1 0 . 6 2 6 7 9 3 3 7 1 4 0 . 3 0 5 7 6 0 . 2 7 4 4 5 0 . 2 6 0 1 8 0 . 4 9 6 1 1 0 . 6 3 4 9 1

b = 1 . 0 0 0 C C ZERO ORDER F I R S T ORDER SECOND URDER THIRD ORDER

SUMS a - 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0

μο = 1 . 0 C O 0 . 0 0 . 0 0 , 0 . 0 9 1 9 7 0 . 1 0 2 1 8 0 . 1 1 8 6 4 0 , 0 . 0 5 3 9 7 0 . 0 6 8 5 1 0 . C 7 9 5 0 0 . 0 . 0 3 5 2 1 0 . 0 4 5 1 6 0 . 0 5 1 4 9 0 .

0 . 0 2 0 8 8 0 . C 2 3 5 9 0 . 0 2 7 3 8 0 . C . 0 4 8 4 5 0 . 0 5 5 7 2 0 . C 6 4 5 9 0 , C . 0 8 6 9 1 0 . 1 0 1 7 6 0 . 1 1 7 7 0 0 . 0 . 1 4 4 7 1 0 . 1 7 2 5 7 0 . 1 9 8 8 9 0 .

0 0 . 0 0 . 0 PEAK 0 . 1 8 3 9 4 0 . 3 6 7 8 8 1 1 6 2 7 0 . 1 0 6 8 6 0 . 0 9 6 7 2 0 . 0 9 1 9 7 0 . 1 0 7 9 3 0 . 1 0 6 1 7 0 7 5 8 3 0 . 0 6 8 5 2 0 . 0 6 1 3 6 0 . 0 5 8 1 2 0 . 0 7 0 4 3 0 . 0 6 8 6 2 0 4 8 3 6 0 . 0 4 3 3 2 0 . 0 3 8 5 9 0 . 0 3 6 4 8 0 . 0 4 5 1 6 0 . 0 4 3 6 1

0 2 6 7 3 0 . 0 2 4 5 1 0 . 0 2 2 1 5 0 . 0 2 1 0 5 0 . 2 0 8 7 6 0 . 3 9 2 2 6 0 6 2 7 7 0 . 0 5 7 4 0 0 . 0 5 1 7 9 0 . 0 4 9 2 0 0 . 2 4 2 2 4 0 . 4 2 5 0 5 1 1 3 8 0 0 . 1 0 3 7 5 0 . 0 9 3 4 3 0 . 0 8 8 6 8 0 . 2 8 9 7 1 0 . 4 7 1 3 7 1 9 1 1 3 0 . 1 7 3 6 2 0 . 1 5 6 0 1 0 . 1 4 7 9 6 0 . 3 6 1 7 8 0 . 5 4 1 4 0

a =

0 . 9 0 C . 1 8 6 1 4 0 . 2 2 4 0 3 0 . 2 5 7 5 8 0 , 0 . 9 5 C . 2 1 1 6 8 0 . 2 5 5 9 6 0 . 2 9 3 8 9 0 , 0 . 9 9 G . 2 3 5 1 5 0 . 2 8 5 4 1 0 . 3 2 7 3 0 0 , 1 . 0 0 0 . 2 4 1 5 1 C . 2 9 3 4 0 0 . 3 3 6 3 6 0 ,

2 4 6 6 8 0 . 2 2 3 6 3 0 . 2 0 0 7 0 0 . 1 9 0 2 6 0 . 4 1 3 6 4 0 . 5 9 1 6 3 2 8 0 9 3 0 . 2 5 4 4 1 0 . 2 2 8 1 8 0 . 2 1 6 2 6 0 . 4 4 5 6 4 0 . 6 2 2 5 7 3 1 2 3 9 0 . 2 8 2 6 5 0 . 2 5 3 3 7 0 . 2 4 0 0 8 0 . 4 7 5 0 6 0 . 6 5 0 9 8 3 2 0 9 1 0 . 2 9 0 3 0 0 . 2 6 0 1 8 0 . 2 4 6 5 3 0 . 4 8 3 0 3 0 . 6 5 8 6 7


SUMS a 0 . 2 0 0 . 4 0

a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = I . 0 0

b= I . 0 0 0 0 0 ZERO ORDER F I R S T ORDER SECOND ORDER THIRD ORDER

SUMS a 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0

0 . 9 0 0 . 9 3 0 . 9 9 1 . 0 0

NARROW SURFACE LAYER AT TOP C O 0 . C 0 0 2 3 0 . 0 5 9 4 6 0 . 0 . 0 5 4 8 5 0 . 0 6 6 6 0 0 . 1 0 3 4 8 0 . C . 0 5 1 8 1 0 . 0 6 5 9 0 0 . 0 8 5 4 6 0 . C . 0 3 9 1 7 0 . 0 5 0 2 4 0 . 0 6 0 3 1 0 .

0 . 0 1 3 4 0 0 . 0 1 6 6 5 0 . 0 8 4 1 3 0 , 0 . 0 3 3 6 6 0 . 0 4 1 8 2 0 . 1 1 9 7 5 0 . C . 0 6 5 6 9 0 . 0 8 2 0 5 0 . 1 7 3 6 7 0 , 0 . 1 1 9 6 5 0 . 1 5 0 3 9 0 . 2 6 0 8 4 0 ,

C . 1 6 1 3 2 0 . 2 0 3 4 2 0 . 3 2 6 3 1 0 , 0 . 1 8 7 9 4 0 . 2 3 7 3 5 0 . 3 6 7 5 5 0 . C . 2 1 2 9 0 0 . 2 6 9 2 2 0 . 4 0 5 9 3 0 , 0 . 2 1 9 7 4 0 . 2 7 7 9 5 0 . 4 1 6 4 0 0 ,

LAMBERT SURFACE ON TOP C O 0 . 0 0 0 0 5 0 . 0 3 5 6 7 0 . 0 . 0 7 4 2 5 0 . 0 8 7 5 5 0 . 1 1 8 5 0 0 . 0 . 0 5 6 1 2 0 . 0 7 1 5 3 0 . 0 8 8 2 6 0 . 0 . 0 3 9 7 3 0 . 0 5 1 0 0 0 . 0 5 9 8 1 0 ,

0 . 0 1 7 4 6 0 . 0 2 0 8 9 0 . 0 6 3 4 5 0 . G . 0 4 2 1 3 0 . 0 5 0 9 4 0 . 1 0 2 3 5 0 . 0 . 0 7 8 8 7 0 . 0 9 6 4 6 0 . 1 5 9 5 4 0 , 0 . 1 3 7 6 4 0 . 1 7 0 2 2 0 . 2 4 9 5 2 0 ,

C . 1 8 1 5 5 0 . 2 2 5 7 5 0 . 3 1 5 8 8 0 . 0 . 2 0 9 1 7 * 0 . 2 6 0 7 9 0 . 3 5 7 3 2 0 , C . 2 3 4 8 5 0 . 2 9 3 4 3 0 . 3 9 5 6 9 0 , 0 . 2 4 1 8 5 0 . 3 0 2 3 4 0 . 4 0 6 1 3 0 .

1 3 5 3 4 0 . 1 7 1 1 8 0 . 1 8 2 8 8 0 . 1 8 3 9 4 0 . 1 0 9 6 9 0 . 1 4 8 5 0 1 1 8 9 7 0 . 1 1 8 1 4 0 . 1 1 1 7 5 0 . 1 0 7 9 3 0 . 1 0 3 6 3 0 . 1 1 2 2 4 0 8 6 7 8 0 . 0 8 0 9 6 0 . 0 7 3 8 6 0 . 0 7 0 4 3 0 . 0 7 8 3 3 0 . 0 7 9 4 6 0 5 8 3 0 0 . 0 5 3 0 0 0 . 0 4 7 6 2 0 . 0 4 5 1 6 0 . 0 5 3 6 6 0 . 0 5 2 8 3

1 6 3 1 3 0 . 1 9 8 5 3 0 . 2 0 8 6 3 0 . 2 0 8 7 6 0 . 1 3 4 0 4 0 . 1 7 4 6 1 2 0 1 3 3 0 . 2 3 5 9 4 0 . 2 4 3 4 8 0 . 2 4 2 2 4 0 . 1 6 8 3 1 0 . 2 1 0 6 5 2 5 8 3 8 0 . 2 8 9 6 7 0 . 2 9 3 0 6 0 . 2 8 9 7 1 0 . 2 1 8 9 7 0 . 2 6 2 9 2 3 4 6 8 6 0 . 3 7 2 3 4 0 . 3 6 8 6 0 0 . 3 6 1 7 8 0 . 2 9 9 1 2 0 . 3 4 4 0 9

4 1 1 8 6 0 . 4 3 2 3 9 0 . 4 2 3 0 8 0 . 4 1 3 6 4 0 . 3 5 8 4 9 0 . 4 0 3 4 4 4 5 2 3 9 0 . 4 6 9 6 1 0 . 4 5 6 7 5 0 . 4 4 5 6 4 0 . 3 9 5 6 6 0 . 4 4 0 3 6 4 8 9 8 6 0 . 5 0 3 9 2 0 . 4 8 7 7 1 0 . 4 7 5 0 6 0 . 4 3 0 1 1 0 . 4 7 4 4 5 5 0 0 0 5 0 . 5 1 3 2 3 0 . 4 9 6 1 1 0 . 4 8 3 0 3 0 . 4 3 9 4 9 0 . 4 8 3 7 1

. 1 3 5 3 4 0 . 2 3 9 6 5 0 . 3 2 9 1 9 0 . 3 6 7 8 8 0 . 1 4 8 5 0 0 . 2 1 9 3 8 , 1 2 5 5 5 0 . 1 1 9 8 2 0 . 1 1 0 8 1 0 . 1 0 6 1 7 0 . 1 1 2 2 4 0 . 1 1 6 2 4 , 0 8 6 9 6 0 . 0 7 9 8 7 0 . 0 7 2 2 0 0 . 0 6 8 6 2 0 . 0 7 9 4 6 0 . 0 7 9 1 3 , 0 5 7 0 2 0 . 0 5 1 4 8 0 . 0 4 6 0 6 0 . 0 4 3 6 1 0 . 0 5 2 8 3 0 . 0 5 1 5 4

, 1 6 4 4 5 0 . 2 6 7 2 8 0 . 3 5 4 6 6 0 . 3 9 2 2 6 0 . 1 7 4 6 1 0 . 2 4 6 2 7 , 2 0 4 3 6 0 . 3 0 4 7 6 0 . 3 8 9 0 1 0 . 4 2 5 0 5 0 . 2 1 0 6 5 0 . 2 8 2 9 6 . 2 6 1 8 3 0 . 3 5 8 1 6 0 . 4 3 7 6 2 0 . 4 7 1 3 7 0 . 2 6 2 9 2 0 . 3 3 5 5 5 , 3 5 0 4 2 0 . 4 3 9 5 8 0 . 5 1 1 2 8 0 . 5 4 1 4 0 0 . 3 4 4 0 9 0 . 4 1 6 2 4

, 4 1 4 8 4 0 . 4 9 8 3 4 0 . 5 6 4 2 0 0 . 5 9 1 6 3 0 . 4 0 3 4 4 0 . 4 7 4 7 5 , 4 5 4 7 9 0 . 5 3 4 6 5 0 . 5 9 6 8 2 0 . 6 2 2 5 7 0 . 4 4 0 3 6 0 . 5 1 0 9 8 . 4 9 1 6 3 0 . 5 6 8 0 5 0 . 6 2 6 7 9 0 . 6 5 0 9 8 0 . 4 7 4 4 5 0 . 5 4 4 3 5 , 5 0 1 6 2 0 . 5 7 7 1 1 0 . 6 3 4 9 1 0 . 6 5 8 6 7 0 . 4 8 3 7 1 0 . 5 5 3 4 1




μ=0.0 μ = 0.1 / χ = 0 . 3 μ = 0. b /χ = 0 . 7 μ=0.9 μ = 1 . 0 AVERAGE Ν

FLUX U

b = 2 . 0 0 0 0 0


SUMS α =

Mo = 0 . 1

α - 0 . 2 0 α - 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α - 0 . 9 0 α - 0 . 9 5 α = 0 . 9 9 α - 1 . 0 0

2 . 5 0 0 0 C 1 . 2 5 0 0 0 0 . 6 2 5 0 0 0 . 4 1 6 6 7 0 . 3 1 2 5 0 0 . 2 5 0 0 0 0 . 2 2 7 2 7 0 . 5 9 9 4 7 0 . 3 8 0 1 1 0 . 2 9 9 7 4 0 . 2 9 9 7 4 0 . 2 1 2 4 0 0 . 1 6 4 3 2 0 . 1 3 4 3 1 0 . 1 1 3 6 9 0 . 1 0 5 6 1 0 . 1 8 4 9 1 0 . 1 4 7 7 4 0 . 0 9 2 4 6 0 . 1 1 0 4 2 0 . 0 9 9 0 7 0 . 0 8 5 8 0 0 . 0 7 4 9 2 0 . 0 6 6 2 1 0 . 0 6 2 5 2 0 . 0 8 6 7 5 0 . 0 7 7 9 8

0 . 5 1 2 8 1 0 . 2 6 2 9 7 0 . 1 3 4 3 9 0 . 0 9 0 6 9 0 . 0 6 8 5 6 0 . 0 5 5 1 6 0 . 0 5 0 2 6 0 . 1 2 8 0 8 0 . 0 8 2 6 4 1 . 0 5 5 3 4 0 . 5 5 6 8 7 0 . 2 9 2 2 7 0 . 2 0 0 3 0 0 . 1 5 3 0 0 0 . 1 2 4 0 0 0 . 1 1 3 3 1 0 . 2 7 6 7 C 0 . 1 8 2 3 9 1 . 6 3 6 8 4 0 . 8 9 3 0 6 0 . 4 8 5 0 1 0 . 3 3 9 3 7 0 . 2 6 2 9 1 0 . 2 1 5 2 6 0 . 1 9 7 5 1 0 . 4 5 6 1 4 C . 3 0 8 9 8 2 . 2 7 5 8 1 1 . 2 9 4 6 6 0 . 7 3 7 4 9 0 . 5 3 2 1 4 0 . 4 2 1 1 1 0 . 3 5 0 1 0 0 . 3 2 3 2 0 0 . 6 8 9 5 2 0 . 4 8 5 1 4

2 . 6 3 0 0 8 1 . 5 3 6 6 8 0 . 9 0 5 2 4 0 . 6 6 8 0 3 0 . 5 3 6 9 5 0 . 4 5 1 4 3 0 . 4 1 8 6 0 0 . 8 4 4 6 2 0 . 6 1 0 1 8 2 . 8 2 1 3 6 1 . 6 7 4 9 0 1 . 0 0 7 6 8 0 . 7 5 4 3 4 0 . 6 1 2 3 6 0 . 5 1 8 4 8 0 . 4 8 2 1 3 0 . 9 3 9 7 1 0 . 6 9 0 0 7 2 . 9 8 4 3 0 1 . 7 9 7 6 9 1 . 1 0 3 3 3 0 . 8 3 7 2 6 0 . 6 8 6 0 7 0 . 5 8 4 7 5 0 . 5 4 5 1 6 1 . 0 2 8 8 8 0 . 7 6 7 1 6 3 . 0 2 6 7 9 1 . 8 3 0 5 7 1 . 1 2 9 7 4 0 . 8 6 0 5 4 0 . 7 0 6 9 7 0 . 6 0 3 6 6 0 . 5 6 3 1 9 1 . 0 5 3 5 7 0 . 7 8 8 8 7

b = 2 . 0 0 0 0 0


SUMS α =

Mo 0 . 3

α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

C . 8 3 3 3 3 0 . 6 2 5 0 0 0 . 4 1 6 6 7 0 . 3 1 2 4 9 0 . 2 4 9 9 8 0 . 2 0 8 3 0 0 . 1 9 2 2 7 0 . 3 6 6 5 7 0 . 2 8 0 C 3 0 . 1 8 3 2 9 0 . 2 1 2 4 0 0 . 1 8 3 2 7 0 . 1 5 4 4 3 0 . 1 3 2 2 9 0 . 1 1 5 3 2 0 . 1 0 8 3 0 0 . 1 5 8 6 8 0 . 1 3 9 3 1 0 . 0 7 9 3 4 0 . 0 9 9 0 7 0 . 0 9 9 4 0 0 . 0 9 0 9 2 0 . 0 8 1 8 6 0 . 0 7 3 7 5 0 . 0 7 0 1 3 0 . 0 8 8 4 6 0 . 0 8 3 3 5

0 . 1 7 4 7 1 0 . 1 3 4 3 9 0 . 0 9 1 5 7 0 . 0 6 9 5 1 0 . 0 5 6 0 5 0 . 0 4 6 9 6 0 . 0 4 3 4 4 0 . 0 8 0 4 7 0 . 0 6 2 3 5 0 . 3 6 9 2 7 0 . 2 9 2 2 7 0 . 2 0 4 5 2 0 . 1 5 7 6 5 0 . 1 2 8 3 9 0 . 1 0 8 3 4 0 . 1 0 0 5 1 0 . 1 7 9 6 9 0 . 1 4 1 6 2 0 . 5 9 2 6 7 . 0 . 4 8 5 0 1 0 . 3 5 1 1 1 0 . 2 7 6 2 4 0 . 2 2 8 0 7 0 . 1 9 4 3 2 0 . 1 8 0 9 5 0 . 3 0 8 9 1 0 . 2 4 8 8 5 0 . 8 6 4 5 6 0 . 7 3 7 4 9 0 . 5 5 9 6 0 0 . 4 5 3 5 3 0 . 3 8 2 0 3 0 . 3 3 C 1 3 0 . 3 0 9 1 4 0 . 4 9 4 7 4 0 . 4 1 0 6 8

1 . 0 3 3 7 3 0 . 9 0 5 2 4 0 . 7 0 9 5 8 0 . 5 8 7 3 8 0 . 5 0 1 8 9 0 . 4 3 8 0 7 0 . 4 1 1 8 2 0 . 6 3 0 5 2 0 . 5 3 4 1 5 1 . 1 3 3 0 7 1 . 0 0 7 6 8 0 . 8 0 5 8 7 0 . 6 7 5 9 6 0 . 5 8 2 7 0 0 . 5 1 1 7 5 0 . 4 8 2 2 4 0 . 7 1 8 7 5 0 . 6 1 6 4 5 1 . 2 2 3 3 9 1 . 1 0 3 3 3 0 . 8 9 8 9 9 0 . 7 6 3 4 0 0 . 6 6 3 5 0 0 . 5 8 6 0 2 0 . 5 5 3 4 4 0 . 8 0 4 8 3 0 . 6 9 8 1 2 1 . 2 4 7 9 4 1 . 1 2 9 7 4 0 . 9 2 5 2 2 0 . 7 8 8 3 3 0 . 6 8 6 7 0 0 . 6 0 7 4 4 0 . 5 7 4 0 2 0 . 8 2 9 2 1 0 . 7 2 1 4 7

b = 2 . 0 0 0 0 0


SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

Mo 0 . 5

0 . 5 0 0 0 0 0 . 4 1 6 6 7 0 . 3 1 2 4 9 0 . 2 4 9 9 2 0 . 2 0 8 1 1 0 . 1 7 8 2 2 0 . 1 6 6 2 5 0 . 2 7 4 5 2 0 . 2 2 5 1 4 0 . 1 3 7 2 6 0 . 1 6 4 3 2 0 . 1 5 4 4 3 0 . 1 3 6 9 4 0 . 1 2 1 1 2 0 . 1 0 7 8 8 0 . 1 0 2 1 5 0 . 1 3 6 1 9 0 . 1 2 4 8 5 0 . 0 6 8 0 9 0 . 0 8 5 8 0 0 . 0 9 0 9 2 0 . 0 8 6 2 7 0 . 0 7 9 5 0 0 . 0 7 2 7 5 0 . 0 6 9 5 9 0 . 0 8 2 5 9 0 . 0 7 9 9 1

0 . 1 0 6 1 1 0 . 2 2 7 7 8 0 . 3 7 3 3 1 0 . 5 6 1 7 8

C . 6 8 7 3 6 0 . 7 6 4 6 7 0 . 8 3 7 4 7 C . 8 5 7 6 7

. C 9 C 6 9 , 2 0 0 3 0 , 3 3 9 3 7 , 5 3 2 1 4

, 6 6 8 0 3 , 7 5 4 3 4 • 8 3 7 2 6

b = 2 . 0 0 0 0 0 Mo

F I R S T ORDER C . 3 5 7 1 4 SECOND ORDER 0 . 1 1 0 7 3 THIRD ORDER 0 . 0 5 9 2 9

SUMS a = 0 . 2 0 0 . 0 7 6 4 0 a = 0 . 4 0 0 . 1 6 5 7 1 a = 0 . 6 0 0 . 2 7 5 5 1 a = 0 . 8 0 0 . 4 2 3 7 1

a = 0 . 9 0 0 . 5 2 6 8 9 a = 0 . 9 5 0 . 5 9 2 2 9 a = 0 . 9 9 0 . 6 5 5 1 5 a = 1 . 0 0 0 . 6 7 2 8 1

= 0 . 7

0 . 3 1 2 5 C 0 . 1 3 4 3 1 0 . 0 7 4 9 2

0 . 0 6 8 5 6 0 . 1 5 3 0 0 0 . 2 6 2 9 1 0 . 4 2 1 1 1

0 . 5 3 6 9 5 0 . 6 1 2 3 6 0 . 6 8 6 0 7 0 . 7 0 6 9 7

0 . 0 6 9 5 1 0 . 0 5 6 2 6 0 . 0 4 7 2 1 0 . 0 4 0 6 4 0 . 0 3 7 9 9 0 . 0 6 1 1 2 0 . 0 5 0 7 6 0 . 1 5 7 6 5 0 . 1 2 9 5 3 0 . 1 0 9 7 4 0 . 0 9 5 1 0 0 . 0 8 9 1 3 0 . 1 3 8 8 8 0 . 1 1 7 1 5 0 . 2 7 6 2 4 0 . 2 3 1 5 1 0 . 1 9 8 7 1 0 . 1 7 3 7 4 0 . 1 6 3 4 0 0 . 2 4 4 3 5 0 . 2 1 0 1 7 0 . 4 5 3 5 3 0 . 3 9 1 0 2 0 . 3 4 1 8 8 0 . 3 0 2 7 7 0 . 2 8 6 1 6 0 . 4 0 4 4 6 0 . 3 5 7 1 4

0 . 5 8 7 3 8 0 . 5 1 6 4 8 0 . 4 5 7 4 2 0 . 4 0 8 6 6 0 . 3 8 7 5 5 0 . 5 2 7 4 7 0 . 4 7 3 9 4 0 . 6 7 5 9 6 0 . 6 0 1 5 6 0 . 5 3 6 9 4 0 . 4 8 2 2 5 0 . 4 5 8 2 7 0 . 6 0 9 8 4 0 . 5 5 3 6 6 0 . 7 6 3 4 0 0 . 6 8 6 9 3 0 . 6 1 7 5 3 0 . 5 5 7 2 9 0 . 5 3 0 5 6 0 . 6 9 1 8 5 0 . 6 3 4 0 2 0 . 7 8 8 3 3 0 . 7 1 1 4 9 0 . 6 4 0 8 4 0 . 5 7 9 0 8 0 . 5 5 1 5 7 0 . 7 1 5 3 5 0 . 6 5 7 2 0

0 . 2 4 9 9 8 0 . 2 0 8 1 1 0 . 1 7 7 9 8 0 . 1 5 5 2 8 0 . 1 4 5 9 2 0 . 2 2 1 4 6 0 . 1 8 8 8 7 0 . 1 3 2 2 9 0 . 1 2 1 1 2 0 . 1 0 9 4 1 0 . 0 9 8 8 7 0 . 0 9 4 1 4 0 . 1 1 8 5 7 0 . 1 1 1 4 2 0 . 0 8 1 8 6 0 . 0 7 9 5 0 0 . 0 7 4 4 0 0 . 0 6 8 7 8 0 . 0 6 6 0 5 0 . 0 7 5 5 0 0 . 0 7 4 2 2

0 . 0 5 6 0 5 0 . 0 4 7 2 1 0 . 0 4 0 6 7 0 . 0 3 5 6 5 0 . 0 3 3 5 7 0 . 0 4 9 7 4 0 . 0 4 2 9 2 0 . 1 2 8 3 9 0 . 1 0 9 7 4 0 . 0 9 5 4 1 0 . 0 8 4 1 7 0 . 0 7 9 4 3 0 . 1 1 4 2 9 0 . 1 0 0 0 5 0 . 2 2 8 0 7 0 . 1 9 8 7 1 0 . 1 7 4 8 7 0 . 1 5 5 5 3 0 . 1 4 7 2 4 0 . 2 0 4 0 7 0 . 1 8 1 8 7 0 . 3 8 2 0 3 0 . 3 4 1 8 8 0 . 3 0 6 0 4 0 . 2 7 5 3 3 0 . 2 6 1 8 0 0 . 3 4 5 0 0 0 . ^ 1 4 8 5

0 . 5 0 1 8 9 0 . 4 5 7 4 2 0 . 4 1 4 2 2 0 . 3 7 5 5 4 0 . 3 5 8 1 3 0 . 4 5 6 5 9 0 . 4 2 3 1 6 0 . 5 8 2 7 0 0 . 5 3 6 9 4 0 . 4 8 9 6 1 0 . 4 4 5 9 3 0 . 4 2 6 0 1 0 . 5 3 2 6 4 0 . 4 9 8 1 4 0 . 6 6 3 5 0 0 . 6 1 7 5 3 0 . 5 6 6 6 2 0 . 5 1 8 2 0 0 . 4 9 5 8 1 0 . 6 0 9 2 5 0 . 5 7 4 4 1 0 . 6 8 6 7 0 0 . 6 4 0 8 4 0 . 5 8 9 0 0 0 . 5 3 9 2 6 0 . 5 1 6 1 8 0 . 6 3 1 3 4 0 . 5 9 6 5 2




VECTOR μ = 0 . 0 f t ' O . l / x = 0 . 3 μ=0.5 / Α * 0 · 7 / x = 0 . 9 • 1 . 0 AVERAGE

Ν FLUX

U

b = 2 . 0 0 0 0 0 ZERO OROER F I R S T ORDER SECOND ORDER T H I R D ORDER

SUMS a : 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

Mo = 0 · 1 0 . 0 PEAK 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 1 5 9 0 . 0 1 1 4 5 0 . 0 2 3 9 3 0 . 0 3 3 8 7 0 . 0 3 7 5 9 0 . 0 1 4 2 2 0 . 0 2 1 6 1 0 . 0 0 7 1 1 0 . 0 0 8 3 0 0 . 0 1 3 1 2 0 . 0 2 1 4 1 0 . 0 2 8 5 2 0 . 0 3 2 9 9 0 . 0 3 4 3 9 0 . 0 2 0 8 7 0 . 0 2 6 1 8 0 . 0 1 0 4 3 0 . 0 1 2 7 8 0 . 0 1 8 4 6 0 . 0 2 4 6 4 0 . 0 2 8 6 8 0 . 0 3 0 6 0 0 . 0 3 1 0 0 0 . 0 2 3 0 0 0 . 0 2 6 7 8

0 . 0 0 0 3 9 0 . 0 0 0 4 6 0 . 0 0 1 0 3 0 . 0 0 3 3 9 0 . 0 0 6 2 1 0 . 0 0 8 3 9 0 . 0 0 9 1 9 0 . 0 0 3 9 1 0 . 0 0 5 6 3 0 . 0 0 2 2 7 0 . 0 0 2 7 3 0 . 0 0 4 7 1 0 . 0 1 0 5 3 0 . 0 1 6 9 9 0 . 0 2 1 8 2 0 . 0 2 3 5 5 0 . 0 1 1 3 7 0 . 0 1 5 5 1 0 . 0 0 8 0 9 0 . 0 0 9 8 6 0 . 0 1 5 1 5 0 . 0 2 6 3 6 0 . 0 3 7 7 0 0 . 0 4 5 7 4 0 . 0 4 8 5 1 0 . 0 2 6 9 7 0 . 0 3 4 7 2 0 . 0 2 5 8 7 0 . 0 3 1 9 4 0 . 0 4 5 5 5 0 . 0 6 6 0 1 0 . 0 8 4 1 8 0 . 0 9 5 9 2 0 . 0 9 9 6 4 0 . 0 6 4 6 7 0 . 0 7 8 4 0

C . 0 4 7 4 4 0 . 0 5 8 9 7 0 . 0 8 1 4 9 0 . 1 0 9 5 4 0 . 1 3 2 2 0 0 . 1 4 5 7 1 0 . 1 4 9 6 0 0 . 1 0 5 4 3 0 . 1 2 3 9 6 0 . 0 6 5 5 6 0 . 0 8 1 7 4 0 . 1 1 1 3 0 0 . 1 4 4 5 1 0 . 1 6 9 7 9 0 . 1 8 3 9 7 0 . 1 8 7 7 3 0 . 1 3 8 0 3 0 . 1 5 9 8 0 0 . 0 8 6 2 4 0 . 1 0 7 7 8 0 . 1 4 5 0 7 0 . 1 8 3 4 4 0 . 2 1 1 0 2 0 . 2 2 5 4 9 0 . 2 2 8 9 3 0 . 1 7 4 2 2 0 . 1 9 9 2 1 0 . 0 9 2 6 1 0 . 1 1 5 8 1 0 . 1 5 5 4 3 0 . 1 9 5 2 8 0 . 2 2 3 4 6 0 . 2 3 7 9 5 0 . 2 4 1 2 7 0 . 1 8 5 2 1 0 . 2 1 1 1 3


SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a * 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

Mo - 0 ·

3

C O 0 . 0 PEAK 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 2 1 2 0 . 0 0 1 2 7 0 . 0 0 1 0 6 0 . 0 0 1 5 9 0 . 0 0 7 0 7 0 . 0 2 1 3 0 0 . 0 3 5 1 0 0 . 0 4 4 6 2 0 . 0 4 7 8 8 0 . 0 2 1 9 8 0 . 0 3 1 3 2 0 . 0 1 0 9 9 0 . 0 1 3 1 2 0 . 0 2 0 9 9 0 . 0 3 1 7 5 0 . 0 3 9 6 6 0 . 0 4 4 0 2 0 . 0 4 5 2 0 0 . 0 2 9 8 9 0 . 0 3 6 4 9 0 . 0 1 4 9 5 0 . 0 1 8 4 6 0 . 0 2 6 4 3 0 . 0 3 4 0 1 0 . 0 3 8 4 1 0 . 0 4 0 1 3 0 . 0 4 0 3 3 0 . 0 3 1 4 3 0 . 0 3 6 0 0

0 . 0 0 0 8 0 0 . 0 0 1 0 3 0 . 0 0 2 5 2 0 . 0 0 5 8 6 0 . 0 0 8 9 8 0 . 0 1 1 0 7 0 . 0 1 1 7 7 0 . 0 0 8 0 2 0 . 0 0 9 3 5 0 . 0 0 3 7 7 0 . 0 0 4 7 1 0 . 0 0 8 9 4 0 . 0 1 7 0 2 0 . 0 2 4 1 7 0 . 0 2 8 7 9 0 . 0 3 0 2 8 0 . 0 1 8 8 5 0 . 0 2 3 2 0 0 . 0 1 2 1 9 0 . 0 1 5 1 5 0 . 0 2 4 7 8 0 . 0 3 9 9 9 0 . 0 5 2 5 1 0 . 0 6 0 1 5 0 . 0 6 2 4 8 0 . 0 4 0 6 2 0 . 0 4 9 4 4 0 . 0 3 6 5 1 0 . 0 4 5 5 5 0 . 0 6 7 1 9 0 . 0 9 4 2 5 0 . 1 1 4 2 2 0 . 1 2 5 1 8 0 . 1 2 8 0 8 0 . 0 9 1 2 8 0 . 1 0 7 4 3

0 . 0 6 5 1 2 0 . 0 8 1 4 9 0 . 1 1 5 2 6 0 . 1 5 1 8 3 C . 1 7 6 6 9 0 . 1 8 9 0 0 0 . 1 9 1 7 4 0 . 1 4 4 7 2 0 . 1 6 6 7 2 0 . 0 8 8 7 9 0 . 1 1 1 3 0 0 . 1 5 4 4 0 0 . 1 9 7 3 9 0 . 2 2 5 0 7 0 . 2 3 7 7 3 0 . 2 4 0 0 8 0 . 1 8 6 9 3 0 . 2 1 2 8 7 0 . 1 1 5 5 6 0 . 1 4 5 0 7 0 . 1 9 8 2 7 0 . 2 4 7 6 6 0 . 2 7 7 8 1 0 . 2 9 0 4 0 0 . 2 9 2 1 6 0 . 2 3 3 4 5 0 . 2 6 3 3 2 0 . 1 2 3 7 7 0 . 1 5 5 4 3 0 . 2 1 1 6 6 0 . 2 6 2 8 8 0 . 2 9 3 6 7 0 . 3 0 6 1 8 0 . 3 0 7 7 4 0 . 2 4 7 5 3 0 . 2 7 8 5 3

b = 2 . 0 0 0 0 0 ZERO ORDER F I R S T OROER SECOND ORDER T H I R D ORDER

SUMS a ; 0 . 2 0 0 . 4 0

a = 0 . 6 0 a = 0 . 8 0

a » 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a » I . 0 0


SUMS a = 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0

Mo ~ 0 . 5 0 . 0 0 . 0 C O PEAK 0 . 0 0 . 0 0 . 0 0 . 0 1 8 3 2 0 . 0 1 8 3 2 0 . 0 0 9 1 6 0 . 0 1 1 4 5 0 . 0 2 1 3 0 0 . 0 3 6 6 3 0 . 0 4 8 9 0 0 . 0 5 6 2 8 0 . 0 5 8 5 1 0 . 0 3 4 9 1 0 . 0 4 4 5 2 0 . 0 1 7 4 6 0 . 0 2 1 4 1 0 . 0 3 1 7 5 0 . 0 4 2 6 4 0 . 0 4 9 4 8 0 . 0 5 2 5 8 0 . 0 5 3 1 7 0 . 0 3 9 5 2 0 . 0 4 6 1 1 0 . 0 1 9 7 6 0 . 0 2 4 6 4 0 . 0 3 4 0 1 0 . 0 4 1 5 1 0 . 0 4 5 2 2 0 . 0 4 6 1 8 0 . 0 4 6 0 3 0 . 0 3 8 2 2 0 . 0 4 2 7 3

0 . 0 0 2 7 2 0 . 0 0 3 3 9 0 . 0 0 5 8 6 0 . 0 0 9 4 4 0 . 0 1 2 2 0 0 . 0 1 3 8 0 0 . 0 1 4 2 7 0 . 0 2 7 2 5 0 . 0 2 9 4 8 C 0 0 8 4 7 0 . 0 1 0 5 3 0 . 0 1 7 0 2 0 . 0 2 5 5 6 0 . 0 3 1 8 6 0 . 0 3 5 3 6 0 . 0 3 6 3 2 0 . 0 4 2 3 6 0 . 0 4 7 6 6 0 . 0 2 1 1 7 C . 0 2 6 3 6 0 . 0 3 9 9 9 0 . 0 5 5 8 8 0 . 0 6 6 8 7 0 . 0 7 2 5 2 0 . 0 7 3 8 9 0 . 0 7 0 5 7 0 . 0 8 0 3 8 C . 0 5 2 8 0 0 . 0 6 6 0 1 0 . 0 9 4 2 5 0 . 1 2 2 2 2 0 . 1 3 9 6 3 0 . 1 4 7 3 1 0 . 1 4 8 6 4 0 . 1 3 1 9 9 0 . 1 4 9 1 8

C . 0 8 7 3 6 0 . 1 0 9 5 4 0 . 1 5 1 8 3 0 . 1 8 9 4 8 0 . 2 1 1 0 5 0 . 2 1 9 1 8 0 . 2 1 9 9 0 0 . 1 9 4 1 4 0 . 2 1 7 2 4 0 . 1 1 5 0 9 0 . 1 4 4 5 1 0 . 1 9 7 3 9 0 . 2 4 1 6 0 0 . 2 6 5 5 6 0 . 2 7 3 4 6 0 . 2 7 3 5 3 0 . 2 4 2 2 9 0 . 2 6 9 3 9 C . 1 4 5 9 1 0 . 1 8 3 4 4 0 . 2 4 7 6 6 0 . 2 9 8 4 1 0 . 3 2 4 4 4 0 . 3 3 1 7 6 0 . 3 3 0 9 8 0 . 2 9 4 7 7 0 . 3 2 5 8 7 0 . 1 5 5 2 8 0 . 1 9 5 2 8 0 . 2 6 2 8 8 0 . 3 1 5 5 0 0 . 3 4 2 0 8 0 . 3 4 9 1 6 0 . 3 4 8 1 1 0 . 3 1 0 5 6 0 . 3 4 2 8 0

^ o = 0 . 7 C O 0 . 0 0 . 0 0 . 0 PEAK 0 . 0 0 . 0 0 . 0 4 1 0 2 0 . 0 5 7 4 3 0 . 0 2 0 5 1 0 . 0 2 3 9 3 0 . 0 3 5 1 0 0 . 0 4 8 9 0 0 . 0 5 8 6 0 0 . 0 6 3 6 7 0 . 0 6 4 9 2 0 . 0 4 6 0 C 0 . 0 5 4 4 6 0 . 0 2 3 0 0 0 . 0 2 8 5 2 0 . 0 3 9 6 6 0 . 0 4 9 4 8 0 . 0 5 4 8 6 0 . 0 5 6 7 1 0 . 0 5 6 7 9 C . 0 4 5 7 5 0 . 0 5 1 6 8 0 . 0 2 2 8 8 0 . 0 2 8 6 8 0 . 0 3 8 4 1 0 . 0 4 5 2 2 0 . 0 4 8 1 0 0 . 0 4 8 3 9 0 . 0 4 7 9 7 0 . 0 4 1 6 5 0 . 0 4 5 7 5

0 . 0 0 5 2 5 0 . 0 0 6 2 1 0 . 0 0 8 9 8 0 . 0 1 2 2 0 0 . 0 1 4 3 8 0 . 0 1 5 4 7 0 . 0 1 5 7 1 0 . 0 5 2 4 6 0 . 0 7 0 8 3 0 . 0 1 4 1 6 0 . 0 1 6 9 9 0 . 0 2 4 1 7 0 . 0 3 1 8 6 0 . 0 3 6 8 4 0 . 0 3 9 1 6 0 . 0 3 9 6 2 0 . 0 7 0 7 8 0 . 0 9 1 8 9 0 . 0 3 0 9 8 0 . 0 3 7 7 0 0 . 0 5 2 5 1 0 . 0 6 6 8 7 0 . 0 7 5 5 3 0 . 0 7 9 1 3 0 . 0 7 9 6 4 0 . 1 0 3 2 6 0 . 1 2 8 4 1 0 . 0 6 8 2 3 0 . 0 8 4 1 8 0 . 1 1 4 2 2 0 . 1 3 9 6 3 0 . 1 5 3 3 2 0 . 1 5 7 7 6 0 . 1 5 7 7 5 0 . 1 7 0 5 6 0 . 2 0 2 4 3

0 . 9 0 0 . 1 0 6 4 5 0 . 1 3 2 2 0 0 . 1 7 6 6 9 0 . 2 1 1 0 5 0 . 2 2 7 9 8 0 . 2 3 2 0 9 0 . 2 3 1 1 5 0 . 2 3 6 5 5 0 . 2 7 3 8 7 0 . 9 5 0 . 1 3 6 2 8 0 . 1 6 9 7 9 0 . 2 2 5 0 7 0 . 2 6 5 5 6 0 . 2 8 4 3 4 0 . 2 8 7 7 7 0 . 2 8 5 9 9 0 . 2 8 6 9 0 0 . 3 2 7 9 5 0 . 9 9 0 . 1 6 8 9 4 0 . 2 1 1 0 2 0 . 2 7 7 8 1 0 . 3 2 4 4 4 0 . 3 4 4 8 2 0 . 3 4 7 2 7 0 . 3 4 4 4 9 0 . 3 4 1 2 9 0 . 3 8 6 1 0 1 . 0 0 C . 1 7 8 7 9 0 . 2 2 3 4 6 0 . 2 9 3 6 7 0 . 3 4 2 0 8 0 . 3 6 2 8 7 0 . 3 6 4 9 9 0 . 3 6 1 8 9 0 . 3 5 7 5 7 0 . 4 0 3 4 8




VECTOR

b = 2 . 0 0 0 0 0


SUMS α = 0 . 2 0 α = 0 . 4 0 α Β 0 . 6 0 α c 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

b = 2 . 0 0 0 C 0


/ x - 0 . 0

Mo

μ.<=0 .5 μ = 0.7 /x = 0 . 9 μ~1.0 AVERAGE Ν

FLUX U

0 . 9

C . 2 7 7 7 8 0 . 2 5 0 0 0 0 . 2 0 8 3 0 0 . 1 7 8 2 2 0 . 1 5 5 2 8 0 . 1 3 7 2 6 0 . 1 2 9 6 5 0 . 1 8 6 2 0 0 . 1 6 2 8 0 C . C 9 3 1 0 0 . 1 1 3 6 9 0 . 1 1 5 3 2 0 . 1 0 7 8 8 0 . 0 9 8 8 7 0 . 0 9 0 2 3 0 . 0 8 6 2 4 0 . 1 0 4 6 6 0 . 0 9 9 9 2 C . 0 5 2 3 3 0 . 0 6 6 2 1 0 . 0 7 3 7 5 0 . 0 7 2 7 5 0 . 0 6 8 7 8 0 . 0 6 4 0 3 0 . 0 6 1 6 6 0 . 0 6 8 7 6 0 . 0 6 8 2 9

0 . 0 5 9 7 6 0 . 0 5 5 1 6 0 . 0 4 6 9 6 0 . 0 4 0 6 4 0 . 0 3 5 6 5 0 . 0 3 1 6 6 0 . 0 2 9 9 6 0 . 0 4 2 0 7 0 . 0 3 7 2 0 C . 1 3 0 5 9 0 . 1 2 4 0 0 0 . 1 0 8 3 4 0 . 0 9 5 1 0 0 . 0 8 4 1 7 0 . 0 7 5 1 8 0 . 0 7 1 2 9 0 . 0 9 7 4 0 0 . 0 8 7 2 8 0 . 2 1 9 3 7 0 . 2 1 5 2 6 0 . 1 9 4 3 2 0 . 1 7 3 7 4 0 . 1 5 5 5 3 0 . 1 3 9 9 6 0 . 1 3 3 1 0 0 . 1 7 5 6 8 0 . 1 6 0 0 9 0 . 3 4 2 7 5 0 . 3 5 0 1 C 0 . 3 3 0 1 3 0 . 3 0 2 7 7 0 . 2 7 5 3 3 0 . 2 5 0 3 4 0 . 2 3 9 0 1 0 . 3 0 1 3 2 0 . 2 8 0 6 3

0 . 4 3 1 2 4 0 . 4 5 1 4 3 0 . 4 3 8 0 7 0 . 4 0 8 6 6 0 . 3 7 5 5 4 0 . 3 4 3 8 2 0 . 3 2 9 1 0 0 . 4 0 2 7 6 0 . 3 8 0 4 0 0 . 4 8 8 4 1 0 . 5 1 8 4 8 0 . 5 1 1 7 5 0 . 4 8 2 2 5 0 . 4 4 5 9 3 0 . 4 0 9 9 2 0 . 3 9 2 9 6 0 . 4 7 2 6 8 0 . 4 5 0 0 8 C . 5 4 4 1 0 0 . 5 8 4 7 5 0 . 5 8 6 0 2 0 . 5 5 7 2 9 0 . 5 1 8 2 0 0 . 4 7 8 0 7 0 . 4 5 8 9 1 0 . 5 4 3 6 3 0 . 5 2 1 3 7 C . 5 5 9 8 7 0 . 6 0 3 6 6 0 . 6 0 7 4 4 0 . 5 7 9 0 8 0 . 5 3 9 2 6 0 . 4 9 7 9 7 0 . 4 7 8 1 9 0 . 5 6 4 1 8 0 . 5 4 2 1 0

Mo 1 . 0

0 . 2 5 0 0 0 0 . 2 2 7 2 7 0 . 1 9 2 2 7 0 . 1 6 6 2 5 0 . 1 4 5 9 2 0 . 1 2 9 6 5 0 . 1 2 2 7 1 0 . 1 7 2 5 8 0 . 1 5 2 3 1 0 . 0 8 6 2 9 0 . 1 0 5 6 1 0 . 1 0 8 3 0 0 . 1 0 2 1 5 0 . 0 9 4 1 4 0 . 0 8 6 2 4 0 . 0 8 2 5 5 0 . 0 9 8 7 9 0 . 0 9 4 8 8 C . 0 4 9 4 0 0 . 0 6 2 5 2 0 . 0 7 0 1 3 0 . 0 6 9 5 9 0 . 0 6 6 0 5 0 . 0 6 1 6 6 0 . 0 5 9 4 3 0 . 0 6 5 6 7 0 . 0 6 5 4 7

SUMS α = α = C . 2 0 α - 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

b = 2 . 0 0 0 0 0


SUMS α α - 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

b = 2 . 0 0 0 0 0


SUMS α α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

0 . 0 5 3 9 1 0 . 0 5 0 2 6 0 . 0 4 3 4 4 0 . 0 . 1 1 8 1 5 C 1 1 3 3 1 C 1 0 0 5 1 0 . 0 . 1 9 9 3 0 0 . 1 9 7 5 1 0 . 1 8 0 9 5 0 . C . 3 1 3 3 7 0 . 3 2 3 2 0 0 . 3 0 9 1 4 0 .

0 . 3 9 6 1 3 0 . 4 1 8 6 C 0 . 4 1 1 8 2 0 . C . 4 4 9 9 8 0 . 4 8 2 1 3 0 . 4 8 2 2 4 0 . 0 . 5 0 2 6 9 0 . 5 4 5 1 6 0 . 5 5 3 4 4 0 . C . 5 1 7 6 6 0 . 5 6 3 1 9 0 . 5 7 4 0 2 0 .


I N F I N I T E 0 . 5 9 9 4 7 0 . 3 6 6 5 7 0 , C . 1 7 3 1 7 0 . 1 8 4 9 1 0 . 1 5 8 6 8 0 . C . 0 7 0 1 4 0 . 0 8 6 7 5 0 . 0 8 8 4 6 0 ,

I N F I N I T E 0 . 1 2 8 0 8 0 . 0 8 0 4 7 0 . I N F I N I T E 0 . 2 7 6 7 0 0 . 1 7 9 6 9 0 , I N F I N I T E 0 . 4 5 6 1 4 0 . 3 0 8 9 1 0 , I N F I N I T E 0 . 6 8 9 5 2 0 . 4 9 4 7 4 0 ,

I N F I N I T E 0 . 8 4 4 6 2 0 . 6 3 0 5 2 0 , I N F I N I T E 0 . 9 3 9 7 1 0 . 7 1 8 7 5 0 . I N F I N I T E 1 . 0 2 8 8 8 0 . 8 0 4 8 3 0 , I N F I N I T E 1 . 0 5 3 5 7 0 . 8 2 9 2 1 0 ,


0 . 5 0 0 0 0 0 . 3 8 0 1 1 0 . 2 8 0 0 3 0 . 0 . 1 2 4 8 2 0 . 1 4 7 7 4 0 . 1 3 9 3 1 0 . 0 . 0 6 2 0 1 0 . 0 7 7 9 8 0 . 0 8 3 3 5 0 .

0 . 1 0 5 5 6 0 . 0 8 2 6 4 0 . 0 6 2 3 5 0 . C . 2 2 5 2 9 0 . 1 8 2 3 9 0 . 1 4 1 6 2 0 . 0 . 3 6 6 8 9 0 . 3 0 8 9 8 0 . 2 4 8 8 5 0 . 0 . 5 4 8 2 6 0 . 4 8 5 1 4 0 . 4 1 0 6 8 0 .

0 . 6 6 8 3 2 0 . 6 1 0 1 8 0 . 5 3 4 1 5 0 . C . 7 4 2 0 9 0 . 6 9 0 0 7 0 . 6 1 6 4 5 0 , C . 8 1 1 5 1 0 . 7 6 7 1 6 0 . 6 9 8 1 2 0 , C . 8 3 0 7 9 0 . 7 8 8 8 7 0 . 7 2 1 4 7 0 ,

0 3 7 9 9 0 . 0 3 3 5 7 0 . 0 2 9 9 6 0 . 0 2 8 4 0 0 . 0 3 9 0 8 0 . 0 3 4 8 7 0 8 9 1 3 0 . 0 7 9 4 3 0 . 0 7 1 2 9 0 . 0 6 7 7 3 0 . 0 9 0 7 5 0 . 0 8 2 0 4 1 6 3 4 0 0 . 1 4 7 2 4 0 . 1 3 3 1 0 0 . 1 2 6 8 0 0 . 1 6 4 3 3 0 . 1 5 0 9 8 2 8 6 1 6 0 . 2 6 1 8 0 0 . 2 3 9 0 1 0 . 2 2 8 5 4 0 . 2 8 3 4 2 0 . 2 6 5 9 4

3 8 7 5 5 0 . 3 5 8 1 3 0 . 3 2 9 1 0 0 . 3 1 5 4 5 0 . 3 8 0 2 8 0 . 3 6 1 6 5 4 5 8 2 7 0 . 4 2 6 0 1 0 . 3 9 2 9 6 0 . 3 7 7 2 1 0 . 4 4 7 3 3 0 . 4 2 8 7 2 5 3 0 5 6 0 . 4 9 5 8 1 0 . 4 5 8 9 1 0 . 4 4 1 0 7 0 . 5 1 5 5 4 0 . 4 9 7 4 9 5 5 1 5 7 0 . 5 1 6 1 8 0 . 4 7 8 1 9 0 . 4 5 9 7 5 0 . 5 3 5 3 3 0 . 5 1 7 5 1

2 7 4 5 2 0 . 2 2 1 4 6 0 . 1 8 6 2 0 0 . 1 7 2 5 8 0 . 3 4 6 3 5 0 . 2 4 9 6 5 1 3 6 1 9 0 . 1 1 8 5 7 0 . 1 0 4 6 6 0 . 0 9 8 7 9 0 . 1 4 0 2 8 0 . 1 2 4 0 3 0 8 2 5 9 0 . 0 7 5 5 0 0 . 0 6 8 7 6 0 . 0 6 5 6 7 0 . 0 7 9 9 8 0 . 0 7 6 3 4

0 6 1 1 2 0 . 0 4 9 7 4 0 . 0 4 2 0 7 0 . 0 3 9 0 8 0 . 0 7 5 6 2 0 . 0 5 5 6 0 1 3 8 8 8 0 . 1 1 4 2 9 0 . 0 9 7 4 0 0 . 0 9 0 7 5 0 . 1 6 7 9 8 0 . 1 2 6 4 7 2 4 4 3 5 0 . 2 0 4 0 7 0 . 1 7 5 6 8 0 . 1 6 4 3 3 0 . 2 8 7 5 9 0 . 2 2 2 9 8 4 0 4 4 6 0 . 3 4 5 0 0 0 . 3 0 1 3 2 0 . 2 8 3 4 2 0 . 4 5 9 7 0 0 . 3 7 0 6 6

5 2 7 4 7 0 . 4 5 6 5 9 0 . 4 0 2 7 6 0 . 3 8 0 2 8 0 . 5 8 6 3 9 0 . 4 8 5 1 5 6 0 9 8 4 0 . 5 3 2 6 4 0 . 4 7 2 6 8 3 . 4 4 7 3 3 0 . 6 6 9 2 9 0 . 5 6 2 2 9 6 9 1 8 5 0 . 6 0 9 2 5 0 . 5 4 3 6 3 0 . 5 1 5 5 4 0 . 7 5 0 6 5 0 . 6 3 9 4 2 7 1 5 3 5 0 . 6 3 1 3 4 0 . 5 6 4 1 8 0 . 5 3 5 3 3 0 . 7 7 3 7 8 0 . 6 6 1 5 8

, 2 2 5 1 4 0 . 1 8 8 8 7 0 . 1 6 2 8 0 0 . 1 5 2 3 1 0 . 2 4 9 6 5 0 . 2 0 4 0 1 1 2 4 8 5 0 . 1 1 1 4 2 0 . 0 9 9 9 2 0 . 0 9 4 8 8 0 . 1 2 4 0 3 0 . 1 1 4 4 3

, 0 7 9 9 1 0 . 0 7 4 2 2 0 . 0 6 8 2 9 0 . 0 6 5 4 7 0 . 0 7 6 3 4 0 . 0 7 4 3 6

0 5 0 7 6 0 . 0 4 2 9 2 0 . 0 3 7 2 0 0 . 0 3 4 8 7 0 . 0 5 5 6 0 0 . 0 4 6 0 7 1 1 7 1 5 0 . 1 0 0 0 5 0 . 0 8 7 2 8 0 . 0 8 2 0 4 0 . 1 2 6 4 7 0 . 1 0 6 5 7

, 2 1 0 1 7 0 . 1 8 1 8 7 0 . 1 6 0 0 9 0 . 1 5 0 9 8 0 . 2 2 2 9 8 0 . 1 9 1 8 7 , 3 5 7 1 4 0 . 3 1 4 8 5 0 . 2 8 0 6 3 0 . 2 6 5 9 % 0 . 3 7 0 6 6 0 . 3 2 7 9 5

, 4 7 3 9 4 0 . 4 2 3 1 6 0 . 3 8 0 4 0 0 . 3 6 1 6 5 0 . 4 8 5 1 5 0 . 4 3 7 1 5 5 5 3 6 6 0 . 4 9 8 1 4 0 . 4 5 0 0 8 0 . 4 2 8 7 2 0 . 5 6 2 2 9 0 . 5 1 2 1 3

, 6 3 4 0 2 0 . 5 7 4 4 1 0 . 5 2 1 3 7 0 . 4 9 7 4 9 0 . 6 3 9 4 2 0 . 5 8 8 0 0 , 6 5 7 2 0 0 . 5 9 6 5 2 0 . 5 4 2 1 0 0 . 5 1 7 5 1 0 . 6 6 1 5 8 0 . 6 0 9 9 4



VECTOR

b « 2 , 0 0 0 0 0 ZERO OROER F I R S T ORDER SECOND ORDER T H I R D ORDER

SUMS α * 0 . 2 0 α = 0 . 4 0 α « 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0


SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

b = 2 . 0 0 0 0 0 ZERO ORDER F I R S T ORDER SECOND ORDER T H I R D OROER

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 , 9 5 α = 0 . 9 9 α = 1 . 0 0

b » 2 . 0 0 0 0 0 ZERO OROER F I R S T ORDER SECOND ORDER THIRD ORDER

Μ = 0 . 0 μ=0. 1 μ =0.3 μ*0.5 μ*0·7 / * » 0 . 9 μ = 1.0 AVERAGE Ν

FLUX

υ

SUMS 0 . 2 0 α = 0 . 4 0 α - 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = I . 0 0

Mo

= 0 . 9

G . 0 0 . 0 0 . 0 0 . 0 0 . 0 PEAK 0 . 0 0 . 0 6 0 2 0 0 . 1 0 8 3 7 0 . 0 3 0 1 0 0 . 0 3 3 8 7 0 . 0 4 4 6 2 0 . 0 5 6 2 8 0 . 0 6 3 6 7 0 . 0 6 6 8 9 0 . 0 6 7 4 2 0 . 0 5 3 0 1 0 . 0 6 0 0 1 0 . 0 2 6 5 1 0 . 0 3 2 9 9 0 . 0 4 4 0 2 0 . 0 5 2 5 8 0 . 0 5 6 7 1 0 . 0 5 7 6 3 0 . 0 5 7 3 6 0 . 0 4 8 6 7 0 . 0 5 3 8 3 0 . 0 2 4 3 3 0 . 0 3 0 6 C 0 . 0 4 0 1 3 0 . 0 4 6 1 8 0 . 0 4 8 3 9 0 . 0 4 8 2 2 0 . 0 4 7 6 4 0 . 0 4 2 5 8 0 . 0 4 6 2 4

0 . 0 0 7 3 2 0 . 0 0 8 3 9 0 . 0 1 1 0 7 0 . 0 1 3 8 0 0 . 0 1 5 4 7 0 . 0 1 6 1 5 0 . 0 1 6 2 3 0 . 0 7 3 1 6 0 . 1 2 2 9 7 0 . 0 1 8 6 6 0 . 0 2 1 8 2 0 . 0 2 8 7 9 0 . 0 3 5 3 6 0 . 0 3 9 1 6 0 . 0 4 0 5 5 0 . 0 4 0 6 6 0 . 0 9 3 2 9 0 . 1 4 5 4 0 C . 0 3 8 3 4 0 . 0 4 5 7 4 0 . 0 6 0 1 5 0 . 0 7 2 5 2 0 . 0 7 9 1 3 0 . 0 8 1 1 5 0 . 0 8 1 0 5 0 . 1 2 7 8 1 0 . 1 8 3 4 2 C . 0 7 8 8 3 0 . 0 9 5 9 2 0 . 1 2 5 1 8 0 . 1 4 7 3 1 0 . 1 5 7 7 6 0 . 1 5 9 7 7 0 . 1 5 8 8 3 0 . 1 9 7 0 7 0 . 2 5 8 5 9

0 . 1 1 8 5 8 0 . 1 4 5 7 1 0 . 1 8 9 0 0 0 . 2 1 9 1 8 0 . 2 3 2 0 9 0 . 2 3 3 2 4 0 . 2 3 1 2 0 0 . 2 6 3 5 2 0 . 3 2 9 9 3 0 . 1 4 9 0 1 0 . 1 8 3 9 7 0 . 2 3 7 7 3 0 . 2 7 3 4 6 0 . 2 8 7 7 7 0 . 2 8 7 9 7 0 . 2 8 4 9 9 0 . 3 1 3 7 0 0 . 3 8 3 5 0 0 . 1 8 1 9 5 0 . 2 2 5 4 9 0 . 2 9 0 4 0 0 . 3 3 1 7 6 0 . 3 4 7 2 7 0 . 3 4 6 2 5 0 . 3 4 2 1 9 0 . 3 6 7 5 7 0 . 4 4 0 8 2 0 . 1 9 1 8 3 0 . 2 3 7 9 5 0 . 3 0 6 1 8 0 . 3 4 9 1 6 0 . 3 6 4 9 9 0 . 3 6 3 5 7 0 . 3 5 9 1 8 0 . 3 8 3 6 5 0 . 4 5 7 9 0

Mo = 1 · 0 C O 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 PEAK 0 . 0 6 7 6 7 0 . 1 3 5 3 4 C . 0 3 3 8 3 0 . 0 3 7 5 9 0 . 0 4 7 8 8 0 . 0 5 8 5 1 0 . 0 6 4 9 2 0 . 0 6 7 4 2 0 . 0 6 7 6 7 0 . 0 5 5 2 1 0 . 0 6 1 5 1 0 . 0 2 7 6 0 0 . 0 3 4 3 9 0 . 0 4 5 2 0 0 . 0 5 3 1 7 0 . 0 5 6 7 9 0 . 0 5 7 3 6 0 . 0 5 6 9 6 0 . 0 4 9 2 5 0 . 0 5 4 0 6 0 . 0 2 4 6 2 0 . 0 3 1 0 0 0 . 0 4 0 3 3 0 . 0 4 6 0 3 0 . 0 4 7 9 7 0 . 0 4 7 6 4 0 . 0 4 6 9 9 0 . 0 4 2 4 6 0 . 0 4 5 9 1

0 . 0 0 8 1 1 0 . 0 0 9 1 9 0 . 0 1 1 7 7 0 . 0 1 4 2 7 0 . 0 1 5 7 1 0 . 0 1 6 2 3 0 . 0 1 6 2 6 0 . 0 8 1 0 9 0 . 1 5 0 2 4 C . 0 2 0 3 4 0 . 0 2 3 5 5 0 . 0 3 0 2 8 0 . 0 3 6 3 2 0 . 0 3 9 6 2 0 . 0 4 0 6 6 0 . 0 4 0 6 2 0 . 1 0 1 6 9 0 . 1 7 2 9 6 C . 0 4 0 9 9 0 . 0 4 8 5 1 0 . 0 6 2 4 8 0 . 0 7 3 8 9 0 . 0 7 9 6 4 0 . 0 8 1 0 5 0 . 0 8 0 7 3 0 . 1 3 6 6 4 0 . 2 1 1 1 4 0 . 0 8 2 3 7 0 . 0 9 9 6 4 0 . 1 2 8 0 8 0 . 1 4 8 6 4 0 . 1 5 7 7 5 0 . 1 5 8 8 3 0 . 1 5 7 5 5 0 . 2 0 5 9 2 0 . 2 8 5 9 5

0 . 1 2 2 3 3 0 . 1 4 9 6 0 0 . 1 9 1 7 4 0 . 2 1 9 9 0 0 . 2 3 1 1 5 0 . 2 3 1 2 0 0 . 2 2 8 7 7 0 . 2 7 1 8 5 0 . 3 5 6 5 0 0 . 1 5 2 6 8 0 . 1 8 7 7 3 0 . 2 4 0 0 8 0 . 2 7 3 5 3 0 . 2 8 5 9 9 0 . 2 8 4 9 9 0 . 2 8 1 6 0 0 . 3 2 1 4 3 0 . 4 0 9 3 1 0 . 1 8 5 4 0 0 . 2 2 8 9 3 0 . 2 9 2 1 6 0 . 3 3 0 9 8 0 . 3 4 4 4 9 0 . 3 4 2 1 9 0 . 3 3 7 7 1 0 . 3 7 4 5 4 0 . 4 6 5 7 0 0 . 1 9 5 1 9 0 . 2 4 1 2 7 0 . 3 0 7 7 4 0 . 3 4 8 1 1 0 . 3 6 1 8 9 0 . 3 5 9 1 8 0 . 3 5 4 3 6 0 . 3 9 0 3 8 0 . 4 8 2 4 9

NARROW SURFACE LAYER AT TOP C O 0 . 0 0 0 0 0 0 . 0 0 2 1 2 0 . 0 . 0 1 2 2 3 0 . 0 1 4 2 2 0 . 0 2 1 9 8 0 , C . 0 1 7 0 1 0 . 0 2 0 8 7 0 . 0 2 9 8 9 0 . 0 . 0 1 8 4 4 0 . 0 2 3 0 0 0 . 0 3 1 4 3 0 ,

0 . 0 0 3 3 1 0 . 0 0 3 9 1 0 . 0 0 8 0 2 0 , 0 . 0 0 9 4 9 0 . 0 1 1 3 7 0 . 0 1 8 8 5 0 . 0 . 0 2 2 1 7 0 . 0 2 6 9 7 0 . 0 4 0 6 2 0 , 0 . 0 5 2 4 1 0 . 0 6 4 6 7 0 . 0 9 1 2 8 0 .

0 . 0 8 4 8 8 0 . 1 0 5 4 3 0 . 1 4 4 7 2 0 . 0 . 1 1 0 7 7 0 . 1 3 8 0 3 0 . 1 8 6 9 3 0 . 0 . 1 3 9 4 6 0 . 1 7 4 2 2 0 . 2 3 3 4 5 0 . 0 . 1 4 8 1 7 0 . 1 8 5 2 1 0 . 2 4 7 5 3 0 .

LAMBERT SURFACE ON TOP C O 0 . 0 0 0 0 0 0 . 0 0 1 2 7 0 . C . 0 1 8 7 7 0 . 0 2 1 6 1 0 . 0 3 1 3 2 0 . 0 . 0 2 1 1 8 0 . 0 2 6 1 8 0 . 0 3 6 4 9 0 . 0 . 0 2 1 3 9 0 . 0 2 6 7 8 0 . 0 3 6 0 0 0 .

0 . 0 0 4 8 1 0 . 0 0 5 6 3 0 . 0 0 9 3 5 0 . 0 . 0 1 3 0 3 0 . 0 1 5 5 1 0 . 0 2 3 2 0 0 . 0 . 0 2 8 6 9 0 . 0 3 4 7 2 0 . 0 4 9 4 4 0 . 0 . 0 6 3 7 6 0 . 0 7 8 4 0 . 0 . 1 0 7 4 3 0 ,

C . 1 0 0 0 6 0 . 1 2 3 9 6 0 . 1 6 6 7 2 0< C 1 2 8 5 2 0 . 1 5 9 8 0 0 . 2 1 2 8 7 0 . 0 . 1 5 9 7 7 0 . 1 9 9 2 1 0 . 2 6 3 3 2 0 , 0 . 1 6 9 2 1 0 . 2 1 1 1 3 0 . 2 7 8 5 3 0 ,

0 1 8 3 2 0 . 0 4 1 0 2 0 . 0 6 0 2 0 0 . 0 6 7 6 7 0 . 0 2 4 4 5 0 . 0 3 7 5 3 0 3 4 9 1 0 . 0 4 6 0 0 0 . 0 5 3 0 1 0 . 0 5 5 2 1 0 . 0 3 4 0 3 0 . 0 4 2 3 6 0 3 9 5 2 0 . 0 4 5 7 5 0 . 0 4 8 6 7 0 . 0 4 9 2 5 0 . 0 3 6 8 8 0 . 0 4 2 7 7 0 3 8 2 2 0 . 0 4 1 6 5 0 . 0 4 2 5 8 0 . 0 4 2 4 6 0 . 0 3 5 2 9 0 . 0 3 9 4 0

0 2 7 2 5 0 . 0 5 2 4 6 0 . 0 7 3 1 6 0 . 0 8 1 0 9 0 . 0 3 3 0 7 0 . 0 4 8 1 0 0 4 2 3 6 0 . 0 7 0 7 8 0 . 0 9 3 2 9 0 . 1 0 1 6 9 0 . 0 4 7 4 3 0 . 0 6 5 1 6 0 7 0 5 7 0 . 1 0 3 2 6 0 . 1 2 7 8 1 0 . 1 3 6 6 4 0 . 0 7 3 9 0 0 . 0 9 5 6 4 1 3 1 9 9 0 . 1 7 0 5 6 0 . 1 9 7 0 7 0 . 2 0 5 9 2 0 . 1 3 1 0 3 0 . 1 5 9 4 1

1 9 4 1 4 0 . 2 3 6 5 5 0 . 2 6 3 5 2 0 . 2 7 1 8 5 0 . 1 8 8 6 3 0 . 2 2 2 3 6 2 4 2 2 9 0 . 2 8 6 9 0 0 . 3 1 3 7 0 0 . 3 2 1 4 3 0 . 2 3 3 1 9 0 . 2 7 0 5 7 2 9 4 7 7 0 . 3 4 1 2 9 0 . 3 6 7 5 7 0 . 3 7 4 5 4 0 . 2 8 1 7 3 0 . 3 2 2 7 7 3 1 0 5 6 0 . 3 5 7 5 7 0 . 3 8 3 6 5 0 . 3 9 0 3 8 0 . 2 9 6 3 3 0 . 3 3 8 4 2

. 0 1 8 3 2 0 . 0 5 7 4 3 0 . 1 0 8 3 7 0 . 1 3 5 3 4 0 . 0 3 7 5 3 0 . 0 6 0 2 7

. 0 4 4 5 2 0 . 0 5 4 4 6 0 . 0 6 0 0 1 0 . 0 6 1 5 1 0 . 0 4 2 3 6 0 . 0 5 0 5 6

. 0 4 6 1 1 0 . 0 5 1 6 8 0 . 0 5 3 8 3 0 . 0 5 4 0 6 0 . 0 4 2 7 7 0 . 0 4 8 6 0

. 0 4 2 7 3 0 . 0 4 5 7 5 0 . 0 4 6 2 4 0 . 0 4 5 9 1 0 . 0 3 9 4 0 0 . 0 4 3 4 6

. 0 2 9 4 8 0 . 0 7 0 8 3 0 . 1 2 2 9 7 0 . 1 5 0 2 4 0 . 0 4 8 1 0 0 . 0 7 2 7 4 , 0 4 7 6 6 0 . 0 9 1 8 9 0 . 1 4 5 4 0 0 . 1 7 2 9 6 0 . 0 6 5 1 6 0 . 0 9 2 4 6 . 0 8 0 3 8 0 . 1 2 8 4 1 0 . 1 8 3 4 2 0 . 2 1 1 1 4 0 . 0 9 5 6 4 0 . 1 2 6 9 1 , 1 4 9 1 8 0 . 2 0 2 4 3 0 . 2 5 8 5 9 0 . 2 8 5 9 5 0 . 1 5 9 4 1 0 . 1 9 7 2 7

. 2 1 7 2 4 0 . 2 7 3 8 7 0 . 3 2 9 9 3 0 . 3 5 6 5 0 0 . 2 2 2 3 6 0 . 2 6 5 5 8 , 2 6 9 3 9 0 . 3 2 7 9 5 0 . 3 8 3 5 0 0 . 4 0 9 3 1 0 . 2 7 0 5 7 0 . 3 1 7 4 6 . 3 2 5 8 7 0 . 3 8 6 1 0 0 . 4 4 0 8 2 0 . 4 6 5 7 0 0 . 3 2 2 7 7 0 . 3 7 3 3 5 , 3 4 2 8 0 0 . 4 0 3 4 8 0 . 4 5 7 9 0 0 . 4 8 2 4 9 0 . 3 3 8 4 2 0 . 3 9 0 0 6




VECTOR

b = 4 . 0 0 0 0 0

μ . = 0 . 0 f i = 0 . 1 Μ * 0 · 5 μ=0.7 μ. « 0 . 9 μ'1,0 AVERAGE Ν

F L U X U

F I R S T ORDER SECOND ORDER

2 . 5 0 0 0 0 1 . 2 5 0 0 0 0 . 6 2 5 0 0 0 . 4 1 6 6 7 0 . 3 1 2 5 0 0 . 2 9 9 7 4 0 . 2 9 9 7 4 0 . 2 1 2 4 0 0 . 1 6 4 3 9 0 . 1 3 4 5 1

THIRD ORDER 0 . 0 9 2 5 2

SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

0 . 5 1 2 8 1 1 . 0 5 5 3 6 1 . 6 3 7 0 2 2 . 2 7 7 5 2

a = 0 . 9 0 a = 0 . 9 5 Q = 0 . 9 9 a * 1 . 0 0

2 . 6 3 6 4 8 2 . 8 3 5 1 5 3 . 0 1 2 2 8 3 . 0 6 0 8 3

b = 4 . 0 0 0 0 0


0 . 8 3 3 3 3 0 . 1 8 3 2 9 0 . 0 7 9 4 4

SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

0 . 1 7 4 7 2 0 . 3 6 9 3 0 0 . 5 9 2 9 3 0 . 8 6 6 9 3

Q= 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a * 1 . 0 0

1 . 0 4 2 3 9 1 . 1 5 1 5 2 1 . 2 6 0 5 0 1 . 2 9 2 9 9

b = 4 . 0 0 0 0 0 Mo


0 . 5 0 0 0 0 0 . 1 3 7 3 3 0 . 0 6 8 3 5

SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

0 . 1 0 6 1 2 0 . 2 2 7 8 4 0 . 3 7 3 7 3 0 . 5 6 5 1 3

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = I . 0 0

0 . 6 9 8 7 7 0 . 7 8 8 2 4 C . 8 8 3 7 3 0 . 9 1 3 5 1

0 . 2 5 0 0 0 0 . 2 2 7 2 7 0 . 5 9 9 4 7 0 . 1 1 4 0 3 0 . 1 0 6 0 1 0 . 1 8 5 0 4

3 8 0 1 1 1 4 7 9 4

0 . 1 1 0 4 9 0 . 0 9 9 2 0 0 . 0 8 6 1 0 0 . 0 7 5 5 1 0 . 0 6 7 0 9 0 . 0 6 3 5 3 0 . 0 8 7 1 5 0 . 0 7 8 5 5

0 . 2 6 2 9 7 0 . 1 3 4 3 9 0 . 0 9 0 7 0 0 . 0 6 8 5 8 0 . 0 5 5 1 8 0 . 0 5 0 2 8 0 . 1 2 8 0 9 0 . 0 8 2 6 6 0 . 5 5 6 8 9 0 . 2 9 2 3 1 0 . 2 0 0 3 8 0 . 1 5 3 1 3 0 . 1 2 4 1 9 0 . 1 1 3 5 3 0 . 2 7 6 8 0 0 . 1 8 2 5 2 0 . 8 9 3 2 8 0 . 4 8 5 3 2 0 . 3 3 9 8 9 0 . 2 6 3 7 2 0 . 2 1 6 3 4 0 . 1 9 8 7 0 0 . 4 5 6 7 3 0 . 3 0 9 7 5 1 . 2 9 6 7 6 0 . 7 4 0 4 0 0 . 5 3 6 2 6 0 . 4 2 6 6 2 0 . 3 5 6 8 4 0 . 3 3 0 4 4 0 . 6 9 3 7 9 0 . 4 9 0 4 0

0 . 4 4 0 4 2 0 . 8 5 8 8 5 0 . 6 2 7 0 9 0 . 5 2 4 4 4 0 . 9 6 8 7 5 0 . 7 2 4 0 1

1 . 8 3 2 6 1 1 . 1 4 9 6 3 0 . 8 9 4 9 8 0 . 7 5 4 1 6 0 . 6 6 0 7 9 0 . 6 2 4 1 7 1 . 0 8 5 4 1 0 . 8 3 2 3 5 1 . 8 7 3 0 8 1 . 1 8 5 9 8 0 . 9 3 0 2 5 0 . 7 8 8 6 9 0 . 6 9 4 4 7 0 . 6 5 7 3 7 1 . 1 2 1 6 7 0 . 8 6 7 1 3

1 . 0 3 0 6 3 0 . 7 8 3 6 6 0 . 6 4 7 8 6 0 . 5 5 8 9 0

0 . 3

0 . 2 1 2 4 0 , 4 1 6 6 7 0 . 3 1 2 5 0 , 1 8 3 2 9 0 . 1 5 4 5 6

2 5 0 0 0 0 . 2 0 8 3 3 0 . 1 9 2 3 1 0 . 3 6 6 5 8 1 3 2 6 3 0 . 1 1 5 8 7 0 . 1 0 8 9 4 0 . 1 5 8 8 9

2 8 0 0 5 1 3 9 6 4

0 . 0 4 3 4 9 0 . 0 8 0 4 9 0 . 0 6 2 3 8

0 . 4 8 5 3 2 0 . 3 5 1 5 6 0 . 2 7 6 9 9 0 . 2 2 9 2 4 0 . 1 9 5 8 8 0 . 1 8 2 6 8 0 . 7 4 0 4 0 0 . 5 6 3 6 2 0 . 4 5 9 2 2 0 . 3 8 9 6 5 0 . 3 3 9 4 4 0 . 3 1 9 1 4

0 . 3 0 9 7 6 0 . 2 4 9 9 7 0 . 5 0 0 6 5 0 . 4 1 7 9 5

0 . 7 2 4 0 9 0 . 6 0 6 5 2 0 . 5 2 5 8 2 0 . 4 6 5 9 8

. 1 4 9 6 3

. 1 8 5 9 8 9 6 0 3 8 0 . 9 9 9 6 4 0 ,

8 3 9 9 4 0 . 7 5 3 8 0 8 8 0 5 7 0 . 7 9 4 8 4

4 1 6 6 7 0 . 3 1 2 5 0 0 . 2 5 0 0 0 0 . 2 0 8 3 3 1 6 4 3 9 0 . 1 5 4 5 6 0 . 1 3 7 3 3 0 . 1 2 1 9 2

6 8 6 8 4 7 2 7 5 9

4 4 1 3 1 5 3 8 8 2 6 5 8 2 1

6 4 9 7 5 0 . 5 5 7 0 2 7 5 7 5 9 0 . 6 6 1 8 5 8 7 9 7 9 0 . 7 8 4 5 6

6 9 8 6 2 0 . 9 1 9 3 2 0 . 8 2 5 0 2

1 7 8 5 7 0 . 1 6 6 6 7 0 . 2 7 4 6 5 0 . 2 2 5 3 5 1 0 9 0 8 0 . 1 0 3 5 3 0 . 1 3 6 7 1 0 . 1 2 5 6 1

0 8 6 1 0 0 . 0 9 1 3 9 0 . 0 8 7 2 1 0 . 0 8 1 1 2 0 . 0 7 5 0 0 0 . 0 7 2 1 2 0 . 0 8 3 7 1 0 . 0 8 1 4 5

0 . 0 9 0 7 0 0 . 0 6 9 5 3 0 . 0 5 6 3 0 0 . 0 4 7 3 0 0 . 0 4 0 7 8 0 . 0 3 8 1 5 0 . 0 6 1 1 8 0 . 0 5 0 8 5 0 . 2 0 0 3 8 0 . 1 5 7 7 6 0 . 1 2 9 7 7 0 . 1 1 0 1 9 0 . 0 9 5 7 5 0 . 0 8 9 8 6 0 . 1 3 9 1 9 0 . 1 1 7 5 8 0 . 3 3 9 8 9 0 . 2 7 6 9 9 0 . 2 3 2 7 9 0 . 2 0 0 6 8 0 . 1 7 6 3 6 0 . 1 6 6 3 0 0 . 2 4 5 7 8 0 . 2 1 2 0 5 0 . 5 3 6 2 6 0 . 4 5 9 2 2 0 . 3 9 9 0 7 0 . 3 5 2 6 6 0 . 3 1 5 9 2 0 . 3 0 0 2 8 0 . 4 1 2 8 2 0 . 3 6 7 4 1

0 . 6 8 2 1 8 0 . 6 0 6 5 2 0 . 5 4 1 7 2 0 . 4 8 8 9 9 0 . 4 4 5 4 5 0 . 4 2 6 4 1 0 . 5 5 2 8 3 0 . 5 0 4 0 9 0 . 7 8 3 6 6 0 . 7 1 5 1 8 0 . 6 5 1 6 6 0 . 5 9 7 5 9 0 . 5 5 1 2 7 0 . 5 3 0 5 3 0 . 6 5 9 4 5 0 . 6 1 1 6 5 0 . 8 9 4 9 8 0 . 8 3 9 9 4 0 . 7 8 2 3 6 0 . 7 3 0 1 1 0 . 6 8 2 9 8 0 . 6 6 1 1 4 0 . 7 8 5 3 1 0 . 7 4 1 7 9 0 . 9 3 0 2 5 0 . 8 8 0 5 7 0 . 8 2 5 8 2 0 . 7 7 4 8 7 0 . 7 2 7 9 7 0 . 7 0 5 9 8 0 . 8 2 7 0 2 0 . 7 8 5 5 3

b«= 4 . 0 0 0 0 0 Mo

F I R S T ORDER 0 . 3 5 7 1 4 SECONO ORDER 0 . 1 1 0 9 1 T H I R D ORDER 0 . 0 5 9 7 9

SUMS Q = 0 . 2 0 0 . 0 7 6 4 2 a = 0 . 4 0 0 . 1 6 5 8 3 a = 0 . 6 0 0 . 2 7 6 1 7 a = 0 . 8 0 0 . 4 2 8 1 9

a = 0 . 9 0 0 . 5 4 1 1 7 o » 0 . 9 5 0 . 6 2 0 8 2 a = 0 . 9 9 0 . 7 0 9 7 3 a = 1 . 0 0 0 . 7 3 8 2 8

0 . 7

0 . 1 3 4 5 1 0 . 1 3 2 6 3 2 0 8 3 3 1 2 1 9 2

1 7 8 5 7 0 . 1 5 6 2 4 0 . 1 4 7 0 5 0 . 2 2 1 8 2 0 . 1 8 9 4 4 1 1 0 9 0 0 . 1 0 1 0 4 0 . 0 9 6 6 1 0 . 1 1 9 5 8 0 . 1 1 2 8 5

0 . 0 7 5 5 1 0 . 0 8 2 7 6 0 . 0 8 1 1 2 0 . 0 7 6 9 9 0 . 0 7 2 2 8 0 . 0 6 9 9 4 0 . 0 7 7 3 5 0 . 0 7 6 6 9

0 . 0 6 8 5 8 0 . 0 5 6 0 7 0 . 0 4 7 3 0 0 . 0 4 0 8 7 0 . 0 3 5 9 7 0 . 0 3 3 9 4 0 4 9 8 7 0 . 0 4 3 1 2 1 1 4 8 5 0 . 1 0 0 8 4

0 . 4 2 6 6 2 0 . 3 8 9 6 5 0 . 3 5 2 6 6 0 . 3 2 0 4 7 0 . 2 9 2 9 6 0 . 2 8 0 7 4 0 . 3 5 6 2 0 0 . 3 2 8 6 2

0 . 5 5 4 6 5 0 . 5 2 5 8 2 0 . 4 8 8 9 9 0 . 4 5 3 7 1 0 . 4 2 1 5 9 0 . 4 0 6 7 8 0 . 4 8 8 3 2 0 . 4 6 0 8 9 0 . 5 2 9 5 1 0 . 5 1 3 5 0 0 . 5 9 2 7 0 0 . 5 6 8 3 4 0 . 6 6 6 5 0 0 . 6 4 9 9 1 0 . 7 1 9 5 2 0 . 7 0 1 5 6 . 7 5 4 1 6 0 . 7 5 3 8 0 0 . 7 3 0 1 1 0 . 6 9 9 4 4

. 7 8 8 6 9 0 . 7 9 4 8 4 0 . 7 7 4 8 7 0 . 7 4 6 1 2 0 . 7 1 3 8 4 0 . 6 9 7 2 3 0 . 7 6 2 2 7 0 . 7 4 6 9 8




VECTOR μ=0.0 μ-O.l = 1.0 AVERAGE

Ν FLUX

U

b = 4 . 0 0 0 0 0 ZERO OROER F I R S T ORDER SECOND ORDER T H I R D ORDER

SUMS Q = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

μο ~ 0 . 1 0 . 0 PEAK 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 2 1 0 . 0 0 1 3 7 0 . 0 0 3 6 7 0 . 0 0 5 0 9 0 . 0 0 1 0 8 0 . 0 0 1 8 1 0 . 0 0 0 5 4 0 . 0 0 0 6 1 0 . 0 0 0 8 6 0 . 0 0 1 5 1 0 . 0 0 2 9 4 0 . 0 0 4 9 7 0 . 0 0 6 0 7 0 . 0 0 2 2 0 0 . 0 0 3 1 0 0 . 0 0 1 1 0 0 . 0 0 1 3 1 0 . 0 0 1 8 4 0 . 0 0 2 8 2 0 . 0 0 4 3 9 0 . 0 0 6 2 5 0 . 0 0 7 1 7 0 . 0 0 3 3 3 0 . 0 0 4 3 7

0 . 0 0 0 0 3 0 . 0 0 0 0 4 0 . 0 0 0 0 6 0 . 0 0 0 1 3 0 . 0 0 0 4 4 0 . 0 0 1 0 0 0 . 0 0 1 3 3 0 . 0 0 0 3 4 0 . 0 0 0 5 3 G . 0 0 0 2 4 0 . 0 C 0 2 8 0 . 0 0 0 4 0 0 . 0 0 0 7 0 0 . 0 0 1 5 6 0 . 0 0 2 9 9 0 . 0 0 3 8 2 0 . 0 0 1 2 0 0 . 0 0 1 7 5 C . 0 0 1 1 4 0 . 0 0 1 3 8 0 . 0 0 1 9 0 0 . 0 0 2 8 8 0 . 0 0 4 9 1 0 . 0 0 7 8 7 0 . 0 0 9 5 1 0 . 0 0 3 8 1 0 . 0 0 5 1 5 0 . 0 0 5 8 8 0 . 0 0 7 2 1 0 . 0 0 9 7 8 0 . 0 1 3 2 4 0 . 0 1 8 3 7 0 . 0 2 4 6 8 0 . 0 2 7 9 4 0 . 0 1 4 6 9 0 . 0 1 8 3 2

C . 0 1 5 7 5 0 . 0 1 9 4 9 0 . 0 2 6 0 8 0 . 0 3 3 6 2 0 . 0 4 2 8 4 0 . 0 3 2 8 7 0 . 0 5 7 7 4 0 . 0 3 4 9 9 0 . 0 4 1 9 6 0 . 0 2 8 3 5 0 . 0 3 5 2 3 0 . 0 4 6 8 0 0 . 0 5 8 8 7 0 . 0 7 2 1 5 0 . 0 8 5 4 7 0 . 0 9 1 6 7 0 . 0 5 9 6 8 0 . 0 7 0 1 8 0 . 0 4 9 3 0 0 . 0 6 1 4 5 0 . 0 8 1 1 1 0 . 1 0 0 0 9 0 . 1 1 9 0 7 0 . 1 3 6 6 7 0 . 1 4 4 4 8 0 . 0 9 9 5 9 0 . 1 1 5 3 1 0 . 0 5 7 6 0 0 . 0 7 1 8 6 0 . 0 9 4 6 8 0 . 1 1 6 2 9 0 . 1 3 7 3 3 0 . 1 5 6 4 2 0 . 1 6 4 7 6 0 . 1 1 5 1 9 0 . 1 3 2 8 7


SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

μο = 0 . 3 0 . 0 0 . 0 PEAK 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 C . 0 0 0 0 0 0 . 0 0 0 0 C 0 . 0 0 0 0 2 0 . 0 0 0 4 2 0 . 0 0 2 0 6 0 . 0 0 4 8 9 0 . 0 0 6 5 4 0 . 0 0 1 5 1 0 . 0 0 2 5 0 0 . 0 0 0 7 5 0 . 0 0 0 8 6 0 . 0 0 1 2 3 0 . 0 0 2 2 0 0 . 0 0 4 1 9 0 . 0 0 6 8 3 0 . 0 0 8 2 1 0 . 0 0 3 0 8 0 . 0 0 4 3 2 0 . 0 0 1 5 4 0 . 0 0 1 8 4 0 . 0 0 2 6 0 0 . 0 0 3 9 9 0 . 0 0 6 1 2 0 . 0 0 8 5 4 0 . 0 0 9 7 2 0 . 0 0 4 6 3 0 . 0 0 6 0 4

0 . 0 0 0 0 5 0 . 0 0 0 0 6 0 . 0 0 0 0 8 0 . 0 0 0 2 2 0 . 0 0 0 6 4 0 . 0 0 1 3 4 0 . 0 0 1 7 4 0 . 0 0 0 4 8 0 . 0 0 0 7 4 0 . 0 0 0 3 3 0 . 0 0 0 4 0 0 . 0 0 0 5 6 0 . 0 0 1 0 4 0 . 0 0 2 2 4 0 . 0 0 4 0 4 0 . 0 0 5 0 3 0 . 0 0 1 6 7 0 . 0 0 2 4 2 C . 0 0 1 5 8 0 . 0 0 1 9 0 0 . 0 0 2 6 5 0 . 0 0 4 0 9 0 . 0 0 6 8 7 0 . 0 1 0 6 3 0 . 0 1 2 6 2 0 . 0 0 5 2 6 0 . 0 0 7 0 7 0 . 0 0 7 9 6 0 . 0 0 9 7 8 0 . 0 1 3 2 8 0 . 0 1 8 1 1 0 . 0 2 5 0 4 0 . 0 3 3 1 0 0 . 0 3 7 1 0 0 . 0 1 9 9 0 0 . 0 2 4 7 6

C . 0 2 1 0 7 0 . 0 2 6 0 8 0 . 0 3 4 9 4 0 . 0 4 5 1 8 0 . 0 5 7 4 9 0 . 0 7 0 3 1 0 . 0 7 6 3 5 0 . 0 4 6 8 1 0 . 0 5 6 0 6 C . C 3 7 6 5 0 . 0 4 6 8 C 0 . 0 6 2 2 1 0 . 0 7 8 4 1 0 . 0 9 6 0 0 0 . 1 1 3 0 7 0 . 1 2 0 7 8 0 . 0 7 9 2 6 0 . 0 9 3 1 3 0 . 0 6 5 0 5 0 . C 8 1 U 0 . 1 0 7 0 9 0 . 1 3 2 3 4 0 . 1 5 7 3 3 0 . 1 7 9 8 9 0 . 1 8 9 6 5 0 . 1 3 1 4 2 0 . 1 5 2 0 9 0 . 0 7 5 8 8 0 . 0 9 4 6 8 0 . 1 2 4 8 0 0 . 1 5 3 4 6 0 . 1 8 1 1 4 0 . 2 0 5 6 0 0 . 2 1 6 0 5 0 . 1 5 1 7 7 0 . 1 7 4 9 8


SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 ο = 1 . 0 0


SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α » 0 . 9 9 α » 1 . 0 0

μ0 = 0 . 5 0 . 0 0 . 0 0 . 0 PEAK 0 . 0 0 . 0 0 . 0 0 . 0 0 0 3 4 0 . 0 0 0 3 4 0 . 0 0 0 1 7 0 . 0 C 0 2 1 0 . 0 0 0 4 2 0 . 0 0 1 3 4 0 . 0 0 3 7 0 0 . 0 0 7 1 3 0 . 0 0 8 9 9 0 . 0 0 2 5 9 0 . 0 0 4 0 2 0 . 0 0 1 3 0 0 . 0 0 1 5 1 0 . 0 0 2 2 0 0 . 0 0 3 7 4 0 . 0 0 6 3 6 0 . 0 0 9 5 2 0 . 0 1 1 1 0 0 . 0 0 4 6 9 0 . 0 0 6 3 7 0 . 0 0 2 3 4 0 . 0 0 2 8 2 0 . 0 0 3 9 9 0 . 0 0 5 9 3 0 . 0 0 8 6 1 0 . 0 1 1 4 5 0 . 0 1 2 7 8 0 . 0 0 6 5 8 0 . 0 0 8 3 9

0 . 0 0 0 1 1 0 . 0 0 0 1 3 0 . 0 0 0 2 2 0 . 0 0 0 4 8 0 . 0 0 1 0 9 0 . 0 0 1 9 2 0 . 0 0 2 3 7 0 . 0 0 1 1 1 0 . 0 0 1 4 8 0 . 0 0 0 5 8 0 . 0 0 0 7 0 0 . 0 0 1 0 4 0 . 0 0 1 8 7 0 . 0 0 3 5 2 0 . 0 0 5 6 7 0 . 0 0 6 7 9 0 . 0 0 2 9 1 0 . 0 0 3 9 5 0 . 0 0 2 3 7 0 . 0 0 2 8 8 0 . 0 0 4 0 9 0 . 0 0 6 3 1 0 . 0 0 9 9 9 0 . 0 1 4 4 7 0 . 0 1 6 7 2 0 . 0 0 7 9 1 0 . 0 1 0 3 3 0 . 0 1 0 7 6 0 . 0 1 3 2 4 0 . 0 1 8 1 1 0 . 0 2 4 7 5 0 . 0 3 3 5 4 0 . 0 4 3 0 3 0 . 0 4 7 5 4 0 . 0 2 6 9 0 0 . 0 3 3 1 4

0 . 0 2 7 1 2 0 . 0 3 3 6 2 0 . 0 4 5 1 8 0 . 0 5 8 4 9 0 . 0 7 3 7 2 0 . 0 8 8 7 2 0 . 0 9 5 5 2 0 . 0 6 0 2 8 0 . 0 7 1 8 2 0 . 0 4 7 3 4 0 . 0 5 8 8 7 0 . 0 7 8 4 1 0 . 0 9 8 9 3 0 . 1 2 0 4 1 0 . 1 4 0 2 9 0 . 1 4 8 9 7 0 . 0 9 9 6 5 0 . 1 1 6 7 1 0 . 0 8 0 2 6 0 . 1 0 C 0 9 0 . 1 3 2 3 4 0 . 1 6 3 6 3 0 . 1 9 3 8 1 0 . 2 2 0 0 1 0 . 2 3 1 0 1 0 . 1 6 2 1 3 0 . 1 8 7 2 5 0 . 0 9 3 1 7 0 . 1 1 6 2 9 0 . 1 5 3 4 6 0 . 1 8 8 8 0 0 . 2 2 2 1 3 0 . 2 5 0 5 3 0 . 2 6 2 3 0 0 . 1 8 6 3 5 0 . 2 1 4 4 7

μ0 = 0 . 7 0 . 0 0 . 0 0 . 0 0 . 0 PEAK 0 . 0 0 . 0 0 . 0 0 2 3 6 0 . 0 0 3 3 0 0 . 0 0 1 1 8 0 . 0 0 1 3 7 0 . 0 0 2 0 6 0 . 0 0 3 7 0 0 . 0 0 6 7 3 0 . 0 1 0 5 6 0 . 0 1 2 5 1 0 . 0 0 4 9 1 0 . 0 0 6 8 3 C . 0 0 2 4 6 0 . 0 0 2 9 4 0 . 0 0 4 1 9 0 . 0 0 6 3 6 0 . 0 0 9 5 1 0 . 0 1 2 9 6 0 . 0 1 4 6 1 0 . 0 0 7 2 1 0 . 0 0 9 3 2 0 . 0 0 3 6 1 0 . 0 0 4 3 9 0 . 0 0 6 1 2 0 . 0 0 8 6 1 0 . 0 1 1 6 9 0 . 0 1 4 7 4 0 . 0 1 6 1 2 0 . 0 0 9 1 2 0 . 0 1 1 3 0

0 . 0 0 0 3 7 0 . 0 0 0 4 4 0 . 0 0 0 6 4 0 . 0 0 1 0 9 0 . 0 0 1 8 5 0 . 0 0 2 7 8 0 . 0 0 3 2 5 0 . 0 0 3 7 2 0 . 0 0 5 1 5 0 . 0 0 1 3 1 0 . 0 0 1 5 6 0 . 0 0 2 2 4 0 . 0 0 3 5 2 0 . 0 0 5 5 4 0 . 0 0 7 9 2 0 . 0 0 9 1 0 0 . 0 0 6 5 3 0 . 0 0 8 8 0 0 . 0 0 4 0 5 0 . 0 0 4 9 1 0 . 0 0 6 8 7 0 . 0 0 9 9 9 0 . 0 1 4 4 3 0 . 0 1 9 3 3 0 . 0 2 1 6 9 0 . 0 1 3 4 9 0 . 0 1 7 4 5 0 . 0 1 4 9 4 0 . 0 1 8 3 7 0 . 0 2 5 0 4 0 . 0 3 3 5 4 0 . 0 4 3 7 9 0 . 0 5 4 0 9 0 . 0 5 8 7 9 0 . 0 3 7 3 4 0 . 0 4 5 7 7

C . 0 3 4 5 7 0 . 0 4 2 8 4 0 . 0 5 7 4 9 0 . 0 7 3 7 2 0 . 0 9 1 1 3 0 . 1 0 7 3 3 0 . 1 1 4 4 1 0 . 0 7 6 8 3 0 . 0 9 1 3 0 0 . 0 5 8 0 1 0 . 0 7 2 1 5 0 . 0 9 6 0 0 0 . 1 2 0 4 1 0 . 1 4 4 7 2 0 . 1 6 6 1 4 0 . 1 7 5 1 9 0 . 1 2 2 1 3 0 . 1 4 2 7 9 0 . 0 9 5 4 7 0 . 1 1 9 0 7 0 . 1 5 7 3 3 0 . 1 9 3 8 1 0 . 2 2 7 6 8 0 . 2 5 5 8 9 0 . 2 6 7 3 8 0 . 1 9 2 8 7 0 . 2 2 2 4 9 C . 1 1 0 0 4 0 . 1 3 7 3 3 0 . 1 8 1 1 4 0 . 2 2 2 1 3 0 . 2 5 9 4 5 0 . 2 9 0 0 3 0 . 3 0 2 3 4 0 . 2 2 0 0 8 0 . 2 5 3 0 2




VECTOR

b = 4 . 0 0 0 0 0


SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . C 0

μ =0.1 μ =0.3 / I = 0 . 5 μ =0.1 μ. = 0 . 9 μ=1.0 AVERAGE Ν

FLUX U

Mo = 0 . 9

0 . 2 7 7 7 8 0 . 2 5 0 0 0 0 . 2 0 8 3 3 0 . 1 7 8 5 7 0 . 1 5 6 2 4 0 . 1 3 8 8 7 0 . 1 3 1 5 5 0 . 1 8 6 8 0 0 . 1 6 3 7 4 0 . 0 9 3 4 C 0 . 1 1 4 0 3 0 . 1 1 5 8 7 0 . 1 0 9 0 8 0 . 1 0 1 0 4 0 . 0 9 3 3 5 0 . 0 8 9 7 8 0 . 1 0 6 1 5 0 . 1 0 2 0 0 C . 0 5 3 0 7 0 . 0 6 7 0 9 0 . 0 7 5 0 7 0 . C 7 5 0 0 0 . 0 7 2 2 8 0 . 0 6 8 6 9 0 . 0 6 6 8 0 0 . 0 7 1 2 9 0 . 0 7 1 6 2

C . 0 5 9 7 8 0 . 0 5 5 1 8 0 . 0 4 7 0 0 0 . 0 4 0 7 8 0 . 0 3 5 9 7 0 . 0 3 2 1 6 0 . 0 3 0 5 4 0 . 0 4 2 2 7 0 . 0 3 7 5 0 0 . 1 3 0 7 5 0 . 1 2 4 1 9 0 . 1 0 8 6 4 0 . 0 9 5 7 5 0 . 0 8 5 3 6 0 . 0 7 6 9 1 0 . 0 7 3 2 5 0 . 0 9 8 2 2 0 . 0 8 8 4 3 G . 2 2 0 2 6 0 . 2 1 6 3 4 0 . 1 9 5 8 8 0 . 1 7 6 3 6 0 . 1 5 9 6 1 0 . 1 4 5 4 2 0 . 1 3 9 1 4 0 . 1 7 8 6 5 0 . 1 6 3 9 8 C . 3 4 8 2 3 0 . 3 5 6 8 4 0 . 3 3 9 4 4 0 . 3 1 5 9 2 0 . 2 9 2 9 6 0 . 2 7 1 9 2 0 . 2 6 2 2 1 0 . 3 1 5 0 1 0 . 2 9 7 4 6

0 . 4 4 7 9 0 0 . 4 7 2 0 8 0 . 4 6 5 9 8 0 . 4 4 5 4 5 0 . 4 2 1 5 9 0 . 3 9 7 5 6 0 . 3 8 5 9 0 0 . 4 3 9 7 7 0 . 4 2 4 4 0 C . 5 2 0 8 9 0 . 5 5 8 9 0 0 . 5 6 5 8 0 0 . 5 5 1 2 7 0 . 5 2 9 5 1 0 . 5 0 5 1 2 0 . 4 9 2 6 6 0 . 5 4 1 0 7 0 . 5 3 0 0 2 0 . 6 0 5 C 5 0 . 6 6 0 7 9 0 . 6 8 6 8 4 0 . 6 8 2 9 8 0 . 6 6 6 5 0 0 . 6 4 3 7 1 0 . 6 3 1 0 7 0 . 6 6 6 7 6 0 . 6 6 3 3 6 0 . 6 3 2 6 0 0 . 6 9 4 4 7 0 . 7 2 7 5 9 0 . 7 2 7 9 7 0 . 7 1 3 8 4 0 . 6 9 2 0 0 0 . 6 7 9 4 6 0 . 7 0 9 6 5 0 . 7 0 9 3 0

b = 4 . 0 0 0 0 0 Mo

F I R S T ORDER C . 2 5 0 C 0 SECOND ORDER 0 . 0 8 6 6 4 THIRD ORDER C . 0 5 0 2 4

SUMS a <= 0 . 2 0 C . 0 5 3 9 3 a = 0 . 4 0 C . 1 1 8 3 3 a = 0 . 6 0 C . 2 0 0 2 8 α Β 0 . 8 0 0 . 3 1 9 2 5

a «= 0 . 9 0 0 . 4 1 3 7 2 a «= 0 . 9 5 G . 4 8 3 9 9 a = 0 . 9 9 0 . 5 6 ' 6 0 3 a = 1 . 0 0 C . 5 9 3 1 0

1 . 0

. 1 9 2 3 1 0 . 1 6 6 6 7 0 . 1 4 7 0 5 0 . 1 3 1 5 5 0 . 1 2 4 9 6 0 , , 1 0 8 9 4 0 . 1 0 3 5 3 0 . 0 9 6 6 1 0 . 0 8 9 7 8 0 . 0 8 6 5 5 0 . . 0 7 1 6 3 0 . 0 7 2 1 2 0 . 0 6 9 9 4 0 . 0 6 6 8 0 0 . 0 6 5 1 0 0 ,

1 7 3 2 8 1 0 0 4 8

, 1 5 3 4 1 , 0 9 7 2 4

0 6 8 4 9 0 . 0 6 9 1 6

. 0 5 0 2 8 0 . 0 4 3 4 9 0

. 1 1 3 5 3 0 . 1 0 0 8 5 0

. 1 9 8 7 0 0 . 1 8 2 6 8 0

. 3 3 0 4 4 0 . 3 1 9 1 4 0

0 . 4 4 0 4 2 0 C . 5 2 4 4 4 0

4 4 1 3 1 0 5 3 8 8 2 0

0 . 6 5 8 2 1 0 0 . 6 9 8 6 2 0

, 0 3 8 1 5 0 . 0 3 3 9 4 0 . 0 3 0 5 4 0 . 0 2 9 0 7 0 . 0 3 9 3 2 0 . 0 3 5 2 3 , 0 8 9 8 6 0 . 0 8 0 7 8 0 . 0 7 3 2 5 0 . 0 6 9 9 6 0 . 0 9 1 6 7 0 . 0 8 3 3 3 , 1 6 6 3 0 0 . 1 5 1 7 5 0 . 1 3 9 1 4 0 . 1 3 3 5 0 0 . 1 6 7 6 1 0 . 1 5 5 2 9 , 3 0 0 2 8 0 . 2 8 0 7 4 0 . 2 6 2 2 1 0 . 2 5 3 4 9 0 . 2 9 8 1 3 0 . 2 8 4 0 3

, 4 2 6 4 1 0 . 4 0 6 7 8 0 . 3 8 5 9 0 0 . 3 7 5 5 0 0 . 4 1 9 3 8 0 . 4 0 8 1 4 , 5 3 0 5 3 0 . 5 1 3 5 0 0 . 4 9 2 6 6 0 . 4 8 1 6 4 0 . 5 1 8 9 3 0 . 5 1 2 4 2 , 6 6 1 1 4 0 . 6 4 9 9 1 0 . 6 3 1 0 7 0 . 6 2 0 0 2 0 . 6 4 3 4 9 0 . 6 4 5 0 5 , 7 0 5 9 8 0 . 6 9 7 2 3 0 . 6 7 9 4 6 0 . 6 6 8 5 5 0 . 6 8 6 2 1 0 . 6 9 0 9 3

b = 4 . 0 0 0 0 C

F I R S T URDER SECOND ORDER T H I R D ORDER

SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

b = 4 . 0 0 0 0 0


SUMS a = 0 . 2 0 a = C . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0


I N F I N I T E 0 . 5 9 9 4 7 0 . 3 6 6 5 8 0 0 . 1 7 3 2 9 0 . 1 8 5 0 4 0 . 1 5 8 8 9 0 C . 0 7 0 4 8 0 . 0 8 7 1 5 0 . 0 8 9 0 7 0

I N F I N I T E 0 . 1 2 8 0 9 0 . 0 8 0 4 9 0 I N F I N I T E 0 . 2 7 6 8 C 0 . 1 7 9 8 3 0 I N F I N I T E 0 . 4 5 6 7 3 0 . 3 0 9 7 6 0 I N F I N I T E C . 6 9 3 7 9 0 . 5 0 0 6 5 0

I N F I N I T E 0 . 8 5 8 8 5 0 . 6 4 9 7 5 0 I N F I N I T E 0 . 9 6 8 7 5 0 . 7 5 7 5 9 0 I N F I N I T E 1 . 0 8 5 4 1 0 . 8 7 9 7 9 0 I N F I N I T E 1 . 1 2 1 6 7 0 . 9 1 9 3 2 0


C . 5 0 0 0 0 0 . 3 8 0 1 1 0 . 2 8 0 0 5 0 0 . 1 2 5 0 0 0 . 1 4 7 9 4 0 . 1 3 9 6 4 0 C . 0 6 2 5 0 0 . 0 7 8 5 5 0 . 0 8 4 2 1 0

0 . 1 0 5 5 7 C . C 8 2 6 6 0 . 0 6 2 3 8 0 0 . 2 2 5 4 0 0 . 1 8 2 5 2 0 . 1 4 1 8 2 0 0 . 3 6 7 5 3 0 . 3 0 9 7 5 0 . 2 4 9 9 7 0 C . 5 5 2 5 4 0 . 4 9 C 4 Q 0 . 4 1 7 9 5 0

C . 6 8 1 9 6 0 . 6 2 7 0 9 0 . 5 5 7 0 2 0 C . 7 6 9 3 7 0 . 7 2 4 0 1 0 . 6 6 1 8 5 0 0 . 8 6 3 7 7 0 . 8 3 2 3 5 0 . 7 8 4 5 6 0 C . 8 9 3 4 8 0 . 8 6 7 1 3 0 . 8 2 5 0 2 0

. 2 7 4 6 5 0 . 2 2 1 8 2 0 . 1 8 6 8 0 0 . 1 7 3 2 8 0 . 3 4 6 5 7 0 . 2 5 0 0 0 , 1 3 6 7 1 0 . 1 1 9 5 8 0 . 1 0 6 1 5 0 . 1 0 0 4 8 0 . 1 4 0 9 5 0 . 1 2 4 9 9 , 0 8 3 7 1 0 . 0 7 7 3 5 0 . 0 7 1 2 9 0 . 0 6 8 4 9 0 . 0 8 1 2 8 0 . 0 7 8 1 0

, 0 6 1 1 8 0 . 0 4 9 8 7 0 . 0 4 2 2 7 0 . 0 3 9 3 2 0 . 0 7 5 7 0 0 . 0 5 5 7 3 , 1 3 9 1 9 0 . 1 1 4 8 5 0 . 0 9 8 2 2 0 . 0 9 1 6 7 0 . 1 6 8 3 7 0 . 1 2 7 0 1 , 2 4 5 7 8 0 . 2 0 6 2 9 0 . 1 7 8 6 5 0 . 1 6 7 6 1 0 . 2 8 9 2 0 0 . 2 2 5 1 0 , 4 1 2 8 2 0 . 3 5 6 2 0 0 . 3 1 5 0 1 0 . 2 9 8 1 3 0 . 4 6 8 3 9 0 . 3 8 1 3 4

, 5 5 2 8 3 0 . 4 8 8 3 2 0 . 4 3 9 7 7 0 . 4 1 9 3 8 0 . 6 1 1 8 9 0 . 5 1 5 4 7 , 6 5 9 4 5 0 . 5 9 2 7 0 0 . 5 4 1 0 7 0 . 5 1 8 9 3 0 . 7 1 8 4 3 0 . 6 1 9 7 3 , 7 8 5 3 1 0 . 7 1 9 5 2 0 . 6 6 6 7 6 0 . 6 4 3 4 9 0 . 8 4 2 2 0 0 . 7 4 4 9 9 , 8 2 7 0 2 0 . 7 6 2 2 7 0 . 7 0 9 6 5 0 . 6 8 6 2 1 0 . 8 8 2 8 8 0 . 7 8 6 9 6

, 2 2 5 3 5 0 . 1 8 9 4 4 0 . 1 6 3 7 4 0 . 1 5 3 4 1 0 . 2 5 0 0 0 0 . 2 0 4 5 6 , 1 2 5 6 1 0 . 1 1 2 8 5 0 . 1 0 2 0 0 0 . 0 9 7 2 4 0 . 1 2 4 9 9 0 . 1 1 5 7 9 , 0 8 1 4 5 0 . 0 7 6 6 9 0 . 0 7 1 6 2 0 . 0 6 9 1 6 0 . 0 7 8 1 0 0 . 0 7 6 7 0

, 0 5 0 8 5 0 . 0 4 3 1 2 0 . 0 3 7 5 0 0 . 0 3 5 2 3 0 . 0 5 5 7 3 0 . 0 4 6 2 6 , 1 1 7 5 8 0 . 1 0 0 8 4 0 . 0 8 8 4 3 0 . 0 8 3 3 3 0 . 1 2 7 0 1 0 . 1 0 7 3 2 , 2 1 2 0 5 0 . 1 8 4 7 9 0 . 1 6 3 9 8 0 . 1 5 5 2 9 0 . 2 2 5 1 0 0 . 1 9 4 6 5 , 3 6 7 4 1 0 . 3 2 8 6 2 0 . 2 9 7 4 6 0 . 2 8 4 0 3 0 . 3 8 1 3 4 0 . 3 4 1 0 9

, 5 0 4 0 9 0 . 4 6 0 8 9 0 . 4 2 4 4 0 0 . 4 0 8 1 4 0 . 5 1 5 4 7 0 . 4 7 3 1 9 , 6 1 1 6 5 0 . 5 6 8 3 4 0 . 5 3 0 0 2 0 . 5 1 2 4 2 0 . 6 1 9 7 3 0 . 5 7 9 2 6 . 7 4 1 7 9 0 . 7 0 1 5 6 0 . 6 6 3 3 6 0 . 6 4 5 0 5 0 . 7 4 4 9 9 0 . 7 0 9 7 3 , 7 8 5 5 3 0 . 7 4 6 9 8 0 . 7 0 9 3 0 0 . 6 9 0 9 3 0 . 7 8 6 9 6 0 . 7 5 4 0 3




VECTOR /x = 0 . 0 μ = 0.1 μ=0.3 μ = 0.5 μ = 0.7 AVERAGE

Ν FLUX

U


α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α - 1 . 0 0

μο = 0 . 9 C O 0 . 0 0 . 0 0 . 0 0 . 0 PEAK 0 . 0 0 . 0 0 6 5 2 0 . 0 1 1 7 4 0 . 0 0 3 2 6 0 . 0 0 3 6 7 0 . 0 0 4 8 9 0 . 0 0 7 1 3 0 . 0 1 0 5 6 0 . 0 1 4 5 0 0 . 0 1 6 4 3 0 . 0 0 8 1 7 0 . 0 1 0 4 4 C 0 0 4 0 9 0 . 0 0 4 9 7 0 . 0 0 6 8 3 0 . 0 0 9 5 2 0 . 0 1 2 9 6 0 . 0 1 6 4 9 0 . 0 1 8 1 1 0 . 0 1 0 1 7 0 . 0 1 2 5 9 0 . C 0 5 0 8 0 . 0 0 6 2 5 0 . 0 0 8 5 4 0 . 0 1 1 4 5 0 . 0 1 4 7 4 0 . 0 1 7 8 3 0 . 0 1 9 1 9 0 . 0 1 1 7 7 0 . 0 1 4 2 0

0 . 0 0 0 8 7 0 . 0 0 1 0 0 0 . 0 0 1 3 4 0 . 0 0 1 9 2 0 . 0 0 2 7 8 0 . 0 0 3 7 4 0 . 0 0 4 2 0 0 . 0 0 8 6 9 0 . 0 1 4 4 8 C . 0 0 2 5 5 0 . 0 0 2 9 9 0 . 0 0 4 0 4 0 . 0 0 5 6 7 0 . 0 0 7 9 2 0 . 0 1 0 3 7 0 . 0 1 1 5 3 0 . 0 1 2 7 3 0 . 0 1 9 5 0 C 0 0 6 5 6 0 . 0 0 7 8 7 0 . 0 1 0 6 3 0 . 0 1 4 4 7 0 . 0 1 9 3 3 0 . 0 2 4 3 5 0 . 0 2 6 6 7 0 . 0 2 1 8 7 0 . 0 3 0 5 5 0 . 0 2 0 1 7 0 . 0 2 4 6 8 0 . 0 3 3 1 0 0 . 0 4 3 0 3 0 . 0 5 4 0 9 0 . 0 6 4 5 8 0 . 0 6 9 2 1 0 . 0 5 0 4 1 0 . 0 6 4 0 1

C . 0 4 2 7 8 0 . 0 5 2 8 7 0 . 0 7 0 3 1 0 . 0 8 8 7 2 0 . 1 0 7 3 3 0 . 1 2 3 8 1 0 . 1 3 0 8 0 0 . 0 9 5 0 6 0 . 1 1 5 1 7 0 . 0 6 8 8 5 0 . 0 8 5 4 7 0 . 1 1 3 0 7 0 . 1 4 0 2 9 0 . 1 6 6 1 4 0 . 1 8 7 9 6 0 . 1 9 6 9 2 0 . 1 4 4 9 5 0 . 1 7 1 6 7 0 . 1 0 9 7 2 0 . 1 3 6 6 7 0 . 1 7 9 8 9 0 . 2 2 0 0 1 0 . 2 5 5 8 9 0 . 2 8 4 7 0 0 . 2 9 6 1 0 0 . 2 2 1 6 5 0 . 2 5 7 9 0 0 . 1 2 5 4 7 0 . 1 5 6 4 2 0 . 2 0 5 6 0 0 . 2 5 0 5 3 0 . 2 9 0 0 3 0 . 3 2 1 2 9 0 . 3 3 3 5 2 0 . 2 5 0 9 3 0 . 2 9 0 7 0


α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α s 0 . 9 0 α » 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

μο = 1 · 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 PEAK 0 . 0 0 9 1 6 0 . 0 1 8 3 2 C . 0 0 4 5 8 0 . 0 0 5 0 9 0 . 0 0 6 5 4 0 . 0 0 8 9 9 0 . 0 1 2 5 1 0 . 0 1 6 4 3 0 . 0 1 8 3 2 0 . 0 0 9 9 4 0 . 0 1 2 3 1 0 . 0 0 4 9 7 0 . C 0 6 0 7 0 . 0 0 8 2 1 0 . 0 1 1 1 0 0 . 0 1 4 6 1 0 . 0 1 8 1 1 0 . 0 1 9 7 0 0 . 0 1 1 6 3 0 . 0 1 4 1 6 0 . 0 0 5 8 1 0 . 0 0 7 1 7 0 . 0 0 9 7 2 0 . 0 1 2 7 8 0 . 0 1 6 1 2 0 . 0 1 9 1 9 0 . 0 2 0 5 1 0 . 0 1 2 9 9 0 . 0 1 5 5 2

0 . 0 0 1 1 7 0 . 0 0 1 3 3 0 . 0 0 1 7 4 0 . 0 0 2 3 7 0 . 0 0 3 2 5 0 . 0 0 4 2 0 0 . 0 0 4 6 6 0 . 0 1 1 7 4 0 . 0 2 1 5 0 0 . 0 0 3 2 8 0 . 0 0 3 8 2 0 . 0 0 5 0 3 0 . 0 0 6 7 9 0 . 0 0 9 1 0 0 . 0 1 1 5 3 0 . 0 1 2 6 6 0 . 0 1 6 4 2 0 . 0 2 7 2 0 0 . 0 0 7 9 8 0 . 0 0 9 5 1 0 . 0 1 2 6 2 0 . 0 1 6 7 2 0 . 0 2 1 6 9 0 . 0 2 6 6 7 0 . 0 2 8 9 4 0 . 0 2 6 6 1 0 . 0 3 9 3 7 0 . 0 2 2 9 0 0 . 0 2 7 9 4 0 . 0 3 7 1 0 0 . 0 4 7 5 4 0 . 0 5 8 7 9 0 . 0 6 9 2 1 0 . 0 7 3 7 5 0 . 0 5 7 2 6 0 . 0 7 5 0 8

C C 4 6 8 2 0 . 0 5 7 7 4 0 . 0 7 6 3 5 0 . 0 9 5 5 2 0 . 1 1 4 4 1 0 . 1 3 0 8 0 0 . 1 3 7 6 5 0 . 1 0 4 0 4 0 . 1 2 8 5 2 0 . 0 7 3 9 5 0 . 0 9 1 6 7 0 . 1 2 0 7 8 0 . 1 4 8 9 7 0 . 1 7 5 1 9 0 . 1 9 6 9 2 0 . 2 0 5 7 1 0 . 1 5 5 6 9 0 . 1 8 6 9 2 0 . 1 1 6 1 0 0 . 1 4 4 4 8 0 . 1 8 9 6 5 0 . 2 3 1 0 1 0 . 2 6 7 3 8 0 . 2 9 6 1 0 0 . 3 0 7 3 3 0 . 2 3 4 5 5 0 . 2 7 5 4 8 0 . 1 3 2 2 8 0 . 1 6 4 7 6 0 . 2 1 6 0 5 0 . 2 6 2 3 0 0 . 3 0 2 3 4 0 . 3 3 3 5 2 0 . 3 4 5 5 8 0 . 2 6 4 5 6 0 . 3 0 9 0 7

b = 4 . 0 0 0 0 0 ZERO OROER F I R S T ORDER SECOND ORDER THIRD ORDER

NARROW SURFACE LAYER AT TOP C O 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 3 4 0 . 0 0 2 3 6 0 . 0 0 6 5 2 0 . 0 0 9 1 6 0 . 0 0 1 8 9 0 . 0 0 3 2 0 0 . 0 0 0 9 4 0 . 0 0 1 0 8 0 . 0 0 1 5 1 0 . 0 0 2 5 9 0 . 0 0 4 9 1 0 . 0 0 8 1 7 0 . 0 0 9 9 4 0 . 0 0 3 6 8 0 . 0 0 5 1 5 0 . 0 0 1 8 4 0 . 0 0 2 2 0 0 . 0 0 3 0 8 0 . 0 0 4 6 9 0 . 0 0 7 2 1 0 . 0 1 0 1 7 0 . 0 1 1 6 3 0 . 0 0 5 4 9 0 . 0 0 7 1 6 C . 0 0 2 7 4 0 . 0 0 3 3 3 0 . 0 0 4 6 3 0 . 0 0 6 5 8 0 . 0 0 9 1 2 0 . 0 1 1 7 7 0 . 0 1 2 9 9 0 . 0 0 7 1 0 0 . 0 0 8 8 7

SUMS a = a - 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = I . 0 0

0 . 0 0 0 2 9 0 . 0 0 0 3 4 0 . 0 0 0 4 8 0 . 0 0 1 1 1 0 . 0 0 3 7 2 0 . 0 0 8 6 9 0 . 0 1 1 7 4 0 . 0 0 2 9 2 0 . 0 0 4 6 1 C . 0 0 1 0 1 0 . 0 0 1 2 0 0 . 0 0 1 6 7 0 . 0 0 2 9 1 0 . 0 0 6 5 3 0 . 0 1 2 7 3 0 . 0 1 6 4 2 0 . 0 0 5 0 7 0 . 0 0 7 4 2 0 . 0 0 3 1 6 0 . 0 0 3 8 1 0 . 0 0 5 2 6 0 . 0 0 7 9 1 0 . 0 1 3 4 9 0 . 0 2 1 8 7 0 . 0 2 6 6 1 0 . 0 1 0 5 4 0 . 0 1 4 2 5 0 . 0 1 1 9 7 0 . 0 1 4 6 9 0 . 0 1 9 9 0 0 . 0 2 6 9 0 0 . 0 3 7 3 4 0 . 0 5 0 4 1 0 . 0 5 7 2 6 0 . 0 2 9 9 2 0 . 0 3 7 3 4

0 . 0 2 8 2 7 0 . 0 3 4 9 9 0 . 0 4 6 8 1 0 . 0 6 0 2 8 0 . 0 7 6 8 3 0 . 0 9 5 0 6 0 . 1 0 4 0 4 0 . 0 6 2 8 3 0 . 0 7 5 3 5 0 . 0 4 8 0 3 0 . 0 5 9 6 8 0 . 0 7 9 2 6 0 . 0 9 9 6 5 0 . 1 2 2 1 3 0 . 1 4 4 9 5 0 . 1 5 5 6 9 C l O l l l 0 . 1 1 8 9 1 0 . 0 7 9 8 9 0 . 0 9 9 5 9 0 . 1 3 1 4 2 0 . 1 6 2 1 3 0 . 1 9 2 8 7 0 . 2 2 1 6 5 0 . 2 3 4 5 5 0 . 1 6 1 3 9 0 . 1 8 6 8 9 C 0 9 2 3 4 0 . 1 1 5 1 9 0 . 1 5 1 7 7 0 . 1 8 6 3 5 0 . 2 2 0 0 8 0 . 2 5 0 9 3 0 . 2 6 4 5 6 0 . 1 8 4 6 7 0 . 2 1 3 0 4


LAMBERT SURFACE ON TOP 0 . 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 3 4 0 . 0 0 3 3 0 0 . 0 1 1 7 4 0 . 0 1 8 3 2 0 . 0 0 3 2 0 0 . 0 0 5 5 2 0 . 0 0 1 6 0 0 . 0 0 1 8 1 0 . 0 0 2 5 0 0 . 0 0 4 0 2 0 . 0 0 6 8 3 0 . 0 1 0 4 4 0 . 0 1 2 3 1 0 . 0 0 5 1 5 0 . 0 0 6 9 5 0 . 0 0 2 5 8 0 . 0 0 3 1 0 0 . 0 0 4 3 2 0 . 0 0 6 3 7 0 . 0 0 9 3 2 0 . 0 1 2 5 9 0 . 0 1 4 1 6 0 . 0 0 7 1 6 0 . 0 0 9 1 5 0 . 0 0 3 5 8 0 . 0 0 4 3 7 0 . 0 0 6 0 4 0 . 0 0 8 3 9 0 . 0 1 1 3 0 0 . 0 1 4 2 0 0 . 0 1 5 5 2 0 . 0 0 8 8 7 0 . 0 1 0 9 4

SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

0 . 0 0 0 4 6 0 . 0 0 0 5 3 0 . 0 0 0 7 4 0 . 0 0 1 4 8 0 . 0 0 5 1 5 0 . 0 1 4 4 8 0 . 0 2 1 5 0 0 . 0 0 4 6 1 0 . 0 0 7 3 9 0 . 0 0 1 4 8 0 . 0 0 1 7 5 0 . 0 0 2 4 2 0 . 0 0 3 9 5 0 . 0 0 8 8 0 0 . 0 1 9 5 0 0 . 0 2 7 2 0 0 . 0 0 7 4 2 0 . 0 1 1 0 0 0 . 0 0 4 2 8 0 . 0 0 5 1 5 0 . 0 0 7 0 7 0 . 0 1 0 3 3 0 . 0 1 7 4 5 0 . 0 3 0 5 5 0 . 0 3 9 3 7 0 . 0 1 4 2 5 0 . 0 1 9 4 4 0 . 0 1 4 9 3 0 . 0 1 8 3 2 0 . 0 2 4 7 6 0 . 0 3 3 1 4 0 . 0 4 5 7 7 0 . 0 6 4 0 1 0 . 0 7 5 0 8 0 . 0 3 7 3 4 0 . 0 4 6 7 8

C . 0 3 3 9 1 0 . 0 4 1 9 6 0 . 0 5 6 0 6 0 . 0 7 1 8 2 0 . 0 9 1 3 0 0 . 1 1 5 1 7 0 . 1 2 8 5 2 0 . 0 7 5 3 5 0 . 0 9 0 5 6 0 . 0 5 6 4 8 0 . 0 7 0 1 8 0 . 0 9 3 1 3 0 . 1 1 6 7 1 0 . 1 4 2 7 9 0 . 1 7 1 6 7 0 . 1 8 6 9 2 0 . 1 1 8 9 1 0 . 1 4 0 0 5 0 . 0 9 2 5 1 0 . 1 1 5 3 1 0 . 1 5 2 0 9 0 . 1 8 7 2 5 0 . 2 2 2 4 9 0 . 2 5 7 9 0 0 . 2 7 5 4 8 0 . 1 8 6 8 9 0 . 2 1 6 6 3 0 . 1 0 6 5 2 0 . 1 3 2 8 7 0 . 1 7 4 9 8 0 . 2 1 4 4 7 0 . 2 5 3 0 2 0 . 2 9 0 7 0 0 . 3 0 9 0 7 0 . 2 1 3 0 4 0 . 2 4 5 9 7




VECTOR

b= β·ooooo


SUMS α

μ=0.0 μ=0.1 μ=0.3 μ = 0·7 μ=0·9

α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

b= 8 . C 0 0 0 0


SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

b = 8 . 0 0 0 0 0


AVERAGE Ν

FLUX U

Mo 0 . I

2 . 5 0 0 0 0 1 . 2 5 0 0 G 0 . 6 2 5 0 0 0 . 4 1 6 6 7 0 . 3 1 2 5 0 0 . 2 5 0 0 0 0 . 2 2 7 2 7 0 . 5 9 9 4 7 0 . 3 8 0 1 1 0 . 2 9 9 7 4 0 . 2 9 9 7 4 0 . 2 1 2 4 0 0 . 1 6 4 3 9 0 . 1 3 4 5 2 0 . 1 1 4 0 4 0 . 1 0 6 0 2 0 . 1 8 5 0 4 0 . 1 4 7 9 4 0 . 0 9 2 5 2 0 . U C 4 9 0 . 0 9 9 2 0 0 . 0 8 6 1 0 0 . 0 7 5 5 2 0 . 0 6 7 1 0 0 . 0 6 3 5 4 0 . 0 8 7 1 6 0 . 0 7 8 5 6

0 . 5 1 2 8 1 0 . 2 6 2 9 7 0 . 1 3 4 3 9 0 . 0 9 0 7 0 0 . 0 6 8 5 8 0 . 0 5 5 1 8 0 . 0 5 0 2 9 0 . 1 2 8 0 9 0 . 0 8 2 6 6 1 . 0 5 5 3 6 0 . 5 5 6 8 9 0 . 2 9 2 3 1 0 . 2 0 0 3 8 0 . 1 5 3 1 3 0 . 1 2 4 2 0 0 . 1 1 3 5 4 0 . 2 7 6 8 0 0 . 1 8 2 5 2 1 . 6 3 7 0 2 C . 8 9 3 2 8 0 . 4 8 5 3 2 0 . 3 3 9 9 0 0 . 2 6 3 7 4 0 . 2 1 6 3 7 0 . 1 9 8 7 4 0 . 4 5 6 7 4 0 . 3 0 9 7 7 2 . 2 7 7 6 2 1 . 2 9 6 8 8 0 . 7 4 0 5 6 0 . 5 3 6 4 8 0 . 4 2 6 9 2 0 . 3 5 7 2 5 0 . 3 3 0 9 2 0 . 6 9 4 0 4 0 . 4 9 0 7 1

2 . 6 3 7 3 1 1 . 5 4 5 6 4 C . 9 1 7 3 4 0 . 6 8 3 9 3 0 . 5 5 6 8 9 0 . 4 7 4 8 8 0 . 4 4 3 5 2 0 . 8 6 0 6 9 0 . 6 2 9 3 0 2 . 8 3 8 5 1 1 . 6 9 6 2 3 1 . 0 3 6 1 6 0 . 7 9 0 6 0 0 . 6 5 6 3 5 0 . 5 6 9 0 7 0 . 5 3 5 4 7 0 . 9 7 5 8 1 0 . 7 3 2 3 3 3 . 0 2 7 3 0 1 . 8 5 1 3 2 1 . 1 7 4 3 2 0 . 9 2 5 3 9 0 . 7 9 0 3 1 0 . 7 0 2 6 1 0 . 6 6 8 7 4 1 . 1 1 5 7 5 0 . 8 6 7 5 1 3 . 0 8 5 2 8 1 . 9 0 3 5 7 1 . 2 2 6 1 3 0 . 9 7 9 4 8 0 . 8 4 6 7 9 0 . 7 6 1 1 2 0 . 7 2 8 0 8 1 . 1 7 0 5 5 0 . 9 2 3 5 6

Mo 0 . 3

0 . 8 3 3 3 3 0 . 6 2 5 0 C 0 . 4 1 6 6 7 0 . 3 1 2 5 0 0 . 2 5 0 0 0 0 . 2 0 8 3 3 0 . 1 9 2 3 1 0 . 3 6 6 5 8 0 . 2 8 0 0 5 0 . 1 8 3 2 9 0 . 2 1 2 4 C 0 . 1 8 3 2 9 0 . 1 5 4 5 6 0 . 1 3 2 6 3 0 . 1 1 5 8 7 0 . 1 0 8 9 5 0 . 1 5 8 8 9 0 . 1 3 9 6 4 C . 0 7 9 4 5 0 . 0 9 9 2 C 0 . 0 9 9 6 0 0 . 0 9 1 3 9 0 . 0 8 2 7 6 0 . 0 7 5 0 8 0 . 0 7 1 6 5 0 . 0 8 9 0 7 0 . 0 8 4 2 2

0 . 1 7 4 7 2 0 . 1 3 4 3 9 0 . 0 9 1 5 8 0 . 0 6 9 5 3 0 . 0 5 6 0 7 0 . 0 4 7 0 0 0 . 0 4 3 4 9 0 . 0 8 0 4 9 0 . 0 6 2 3 8 0 . 3 6 9 3 0 0 . 2 9 2 3 1 0 . 2 0 4 5 7 0 . 1 5 7 7 6 0 . 1 2 8 6 0 0 . 1 0 8 6 5 0 . 1 0 0 8 5 0 . 1 7 9 8 3 0 . 1 4 1 8 3 0 . 5 9 2 9 3 0 . 4 8 5 3 2 0 . 3 5 1 5 7 0 . 2 7 7 0 1 0 . 2 2 9 2 6 0 . 1 9 5 9 2 0 . 1 8 2 7 2 0 . 3 0 9 7 8 0 . 2 4 9 9 9 C . 8 6 7 0 6 0 . 7 4 0 5 6 0 . 5 6 3 8 4 0 . 4 5 9 5 2 0 . 3 9 0 0 6 0 . 3 4 0 0 0 0 . 3 1 9 7 9 0 . 5 0 0 9 8 0 . 4 1 8 3 6

1 . 0 4 3 5 0 0 . 9 1 7 3 4 0 . 7 2 5 9 2 0 . 6 0 8 8 7 0 . 5 2 8 8 1 0 . 4 6 9 7 3 0 . 4 4 5 4 6 0 . 6 5 2 2 1 0 . 5 5 9 9 7 1 . 1 5 5 9 7 1 . 0 3 6 1 6 0 . 8 4 3 9 2 0 . 7 2 4 3 9 0 . 6 4 1 4 5 0 . 5 7 9 3 0 0 . 5 5 3 4 7 0 . 7 6 6 9 6 0 . 6 7 2 8 9 1 . 2 8 0 3 1 1 . 1 7 4 3 2 0 . 9 9 2 9 5 0 . 8 8 0 0 5 0 . 8 0 1 4 8 0 . 7 4 2 0 1 0 . 7 1 6 9 9 0 . 9 1 9 8 1 0 . 8 3 0 9 4 1 . 3 2 5 1 8 1 . 2 2 6 1 3 1 . 0 5 2 5 2 0 . 9 4 5 4 0 0 . 8 7 1 3 5 0 . 8 1 5 3 7 0 . 7 9 1 7 5 0 . 9 8 3 7 0 0 . 8 9 9 3 4

0 . 5

0 . 5 0 0 0 0 0 . 4 1 6 6 7 0 . 3 1 2 5 0 0 . 2 5 0 0 0 0 . 2 0 8 3 3 0 . 1 7 8 5 7 0 . 1 6 6 6 7 0 . 2 7 4 6 5 0 . 2 2 5 3 5 0 . 1 3 7 3 3 0 . 1 6 4 3 9 0 . 1 5 4 5 6 0 . 1 3 7 3 3 0 . 1 2 1 9 2 0 . 1 0 9 0 9 0 . 1 0 3 5 4 0 . 1 3 6 7 1 0 . 1 2 5 6 2 0 . 0 6 8 3 6 0 . C 8 6 1 0 0 . 0 9 1 3 9 0 . 0 8 7 2 1 0 . 0 8 1 1 3 0 . 0 7 5 0 3 0 . 0 7 2 1 5 0 . 0 8 3 7 2 0 . 0 8 1 4 7

a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

0 . 1 0 6 1 2 0 . 0 9 0 7 0 0 . 0 6 9 5 3 0 . 0 5 6 3 0 0 . 0 4 7 3 0 0 . 0 4 0 7 8 0 . 0 3 8 1 5 0 . 0 6 1 1 8 0 . 0 5 0 8 5 0 . 2 2 7 8 4 0 . 2 0 0 3 8 0 . 1 5 7 7 6 0 . 1 2 9 7 8 0 . 1 1 0 2 0 0 . 0 9 5 7 6 0 . 0 8 9 8 7 0 . 1 3 9 1 9 0 . 1 1 7 5 9 0 . 3 7 3 7 4 0 . 3 3 9 9 C 0 . 2 7 7 0 1 0 . 2 3 2 8 1 0 . 2 0 0 7 1 0 . 1 7 6 4 2 0 . 1 6 6 3 7 0 . 2 4 5 8 1 0 . 2 1 2 0 8 0 . 5 6 5 3 0 0 . 5 3 6 4 8 0 . 4 5 9 5 2 0 . 3 9 9 4 6 0 . 3 5 3 2 1 0 . 3 1 6 6 8 0 . 3 0 1 1 6 0 . 4 1 3 2 6 0 . 3 6 7 9 7

C . 7 0 0 1 9 0 . 6 8 3 9 3 0 . 6 0 8 8 7 0 . 5 4 4 7 3 0 . 4 9 2 8 2 0 . 4 5 0 2 6 0 . 4 3 1 7 4 0 . 5 5 5 9 9 0 . 5 0 7 8 8 0 . 7 9 3 8 2 0 . 7 9 0 6 0 0 . 7 2 4 3 9 0 . 6 6 3 2 1 0 . 6 1 1 7 3 0 . 5 6 8 2 2 0 . 5 4 8 8 9 0 . 6 7 1 2 1 0 . 6 2 5 4 9 0 . 9 0 8 1 3 0 . 9 2 5 3 9 0 . 8 8 0 0 5 0 . 8 3 1 7 7 0 . 7 8 8 8 4 0 . 7 5 0 9 3 0 . 7 3 3 5 4 0 . 8 3 4 6 0 0 . 7 9 8 9 1 0 . 9 5 2 9 8 0 . 9 7 9 4 8 0 . 9 4 5 4 0 0 . 9 0 5 3 1 0 . 8 6 8 6 8 0 . 8 3 5 5 9 0 . 8 2 0 1 5 0 . 9 0 5 9 5 0 . 8 7 6 6 5

b = 8 . 0 0 0 0 0


Mo 0 . 7

C . 3 5 7 1 4 C . 3 1 2 5 0 0 . 2 5 0 0 0 0 . 2 0 8 3 3 0 . 1 7 8 5 7 0 . 1 5 6 2 5 0 . 1 4 7 0 6 0 . 2 2 1 8 3 0 . 1 8 9 4 4 C . 1 1 0 9 1 0 . 1 3 4 5 2 0 . 1 3 2 6 3 0 . 1 2 1 9 2 0 . 1 1 0 9 1 0 . 1 0 1 0 6 0 . 0 9 6 6 4 0 . 1 1 9 5 8 0 . 1 1 2 8 6 0 . 0 5 9 7 9 0 . 0 7 5 5 2 0 . 0 8 2 7 6 0 . 0 8 1 1 3 0 . 0 7 7 0 1 0 . 0 7 2 3 3 0 . 0 7 0 0 1 0 . 0 7 7 3 7 0 . 0 7 6 7 1

a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = I . 0 0

0 . 0 7 6 4 2 0 . 0 6 8 5 8 0 . 0 5 6 0 7 0 . 0 4 7 3 0 0 . 0 4 0 8 8 0 . 0 3 5 9 8 0 . 0 3 3 9 4 0 . 0 4 9 8 7 0 . 0 4 3 1 2 0 . 1 6 5 8 3 0 . 1 5 3 1 3 0 . 1 2 8 6 0 0 . 1 1 0 2 0 0 . 0 9 6 2 5 0 . 0 8 5 3 8 0 . 0 8 0 8 0 0 . 1 1 4 8 6 0 . 1 0 0 8 5 0 . 2 7 6 1 9 0 . 2 6 3 7 4 0 . 2 2 9 2 6 0 . 2 0 0 7 1 0 . 1 7 7 9 8 0 . 1 5 9 7 0 0 . 1 5 1 8 7 0 . 2 0 6 3 4 0 . 1 8 4 8 5 0 . 4 2 8 4 4 0 . 4 2 6 9 2 0 . 3 9 0 0 6 0 . 3 5 3 2 1 0 . 3 2 1 2 3 0 . 2 9 4 0 1 0 . 2 8 1 9 5 0 . 3 5 6 8 1 0 . 3 2 9 3 8

0 . 5 4 2 9 8 0 . 5 5 6 8 9 0 . 5 2 8 8 1 0 . 4 9 2 8 2 0 . 4 5 8 5 9 0 . 4 2 7 7 1 0 . 4 1 3 5 6 0 . 4 9 2 3 3 0 . 4 6 5 7 1 0 . 6 2 7 6 5 0 . 6 5 6 3 5 0 . 6 4 1 4 5 0 . 6 1 1 7 3 0 . 5 8 0 3 3 0 . 5 5 0 2 5 0 . 5 3 5 9 8 0 . 6 0 7 0 9 0 . 5 8 5 2 9 0 . 7 3 8 7 3 0 . 7 9 0 3 1 0 . 8 0 1 4 8 0 . 7 8 8 8 4 0 . 7 6 9 2 5 0 . 7 4 7 2 8 0 . 7 3 5 9 7 0 . 7 7 8 1 0 0 . 7 6 9 4 6 0 . 7 8 4 8 5 0 . 8 4 6 7 9 0 . 8 7 1 3 5 0 . 8 6 8 6 8 0 . 8 5 6 8 3 0 . 8 4 0 8 5 0 . 8 3 1 9 8 0 . 8 5 5 4 1 0 . 8 5 4 5 1

9 isotropic Scattering, Finite Slabs 269



VECTOR μ-0.0 μ =0.1 μ-0·3 μ<*0·9 μ « 1 · 0 AVERAGE

Ν FLUX

U


SUMS α s 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

μ0 = 0 . 1 C O PEAK 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 4 0 . 0 0 0 0 9 0 . 0 0 0 0 1 0 . 0 0 0 0 2 0 . 0 0 0 0 1 0 . 0 0 0 0 1 O.OOOOl 0 . 0 0 0 0 1 0 . 0 0 0 0 3 0 . 0 0 0 0 8 0 . 0 0 0 1 4 0 . 0 0 0 0 3 0 . 0 0 0 0 4 0 . 0 0 0 0 1 0 . 0 0 0 0 2 0 . 0 0 0 0 2 0 . 0 0 0 0 3 0 . 0 0 0 0 6 0 . 0 0 0 1 4 0 . 0 0 0 2 1 0 . 0 0 0 0 6 0 . 0 0 0 0 8

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 1 0 . 0 0 0 0 3 0 . 0 0 0 0 0 0 . 0 0 0 0 1 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 0 0 0 2 0 . 0 0 0 0 5 0 . 0 0 0 0 9 0 . 0 0 0 0 2 0 . 0 0 0 0 3 C . 0 0 0 0 3 0 . C 0 0 0 3 0 . 0 0 0 0 5 0 . 0 0 0 0 6 0 . 0 0 0 1 0 0 . 0 0 0 2 1 0 . 0 0 0 3 0 0 . 0 0 0 0 9 0 . 0 0 0 1 3 0 . C 0 0 3 4 0 . 0 0 0 4 1 0 . 0 0 0 5 6 0 . 0 0 0 7 4 0 . 0 0 1 0 1 0 . 0 0 1 4 6 0 . 0 0 1 7 9 0 . 0 0 0 8 4 0 . 0 0 1 0 6

0 . 0 0 1 9 1 0 . 0 0 2 3 6 0 . 0 0 3 1 5 0 . 0 0 4 0 3 0 . 0 0 5 1 0 0 . 0 0 6 5 2 0 . 0 0 7 3 8 0 . 0 0 4 2 4 0 . 0 0 5 1 0 0 . 0 0 6 1 0 0 . 0 0 7 5 8 0 . 0 1 0 0 6 0 . 0 1 2 5 9 0 . 0 1 5 3 9 0 . 0 1 8 5 9 0 . 0 2 0 3 8 0 . 0 1 2 8 5 0 . 0 1 5 1 4 C . 0 2 1 7 4 C . 0 2 7 1 C 0 . 0 3 5 7 4 0 . 0 4 3 9 8 0 . 0 5 2 2 4 0 . 0 6 0 6 6 0 . 0 6 4 9 3 0 . 0 4 3 9 2 0 . 0 5 0 9 2 C . 0 3 3 1 0 0 . 0 4 1 2 9 0 . 0 5 4 3 7 0 . 0 6 6 6 3 0 . 0 7 8 5 9 0 . 0 9 0 3 8 0 . 0 9 6 2 1 0 . 0 6 6 2 0 0 . 0 7 6 4 4

D * 8 . 0 0 0 0 0 ZERO ORDER F I R S T ORDER SECOND ORDER THIRD ORDER

SUMS a * 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

μο = 0 . 3 0 . 0 C O PEAK C O 0 . 0 0 . 0 0 . 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 C . 0 0 0 0 0 O.OOOOC 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 1 0 . 0 0 0 0 6 0 . 0 0 0 1 2 0 . 0 0 0 0 1 0 . 0 0 0 0 3 C . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 0 0 0 2 0 . 0 0 0 0 4 0 . 0 0 0 1 1 0 . 0 0 0 1 9 0 . 0 0 0 0 4 0 . 0 0 0 0 6 0 . 0 0 0 0 2 0 . 0 0 0 0 2 0 . 0 0 0 0 3 0 . 0 0 0 0 5 0 . 0 0 0 0 9 0 . 0 0 0 1 9 0 . 0 0 0 2 8 0 . 0 0 0 0 8 0 . 0 0 0 1 1

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 2 0 . 0 0 0 0 3 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 0 0 0 0 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 0 0 0 2 0 . 0 0 0 0 7 0 . 0 0 0 1 2 0 . 0 0 0 0 2 0 . 0 0 0 0 4 0 . 0 0 0 0 4 0 . 0 0 0 0 5 0 . 0 0 0 0 6 0 . 0 0 0 0 9 0 . 0 0 0 1 4 0 . 0 0 0 2 8 0 . 0 0 0 4 1 0 . 0 0 0 1 3 0 . 0 0 0 1 7 0 . 0 0 0 4 5 0 . 0 0 0 5 6 0 . 0 0 0 7 5 0 . 0 0 1 0 0 0 . 0 0 1 3 6 0 . 0 0 1 9 7 0 . 0 0 2 4 1 0 . 0 0 1 1 3 0 . 0 0 1 4 3

C . 0 0 2 5 5 0 . 0 0 3 1 5 0 . 0 0 4 2 1 0 . 0 0 5 3 8 0 . 0 0 6 8 2 0 . 0 0 8 7 0 0 . 0 0 9 8 5 0 . 0 0 5 6 6 0 . 0 0 6 8 1 0 . 0 0 8 1 0 0 . 0 1 0 0 6 0 . 0 1 3 3 4 0 . 0 1 6 7 1 0 . 0 2 0 4 2 0 . 0 2 4 6 7 0 . 0 2 7 0 3 0 . 0 1 7 0 4 0 . 0 2 0 0 8 0 . 0 2 8 6 7 0 . 0 3 5 7 4 0 . 0 4 7 1 3 0 . 0 5 8 0 1 0 . 0 6 8 9 0 0 . 0 8 0 0 0 0 . 0 8 5 6 2 0 . 0 5 7 9 2 0 . 0 6 7 1 5 0 . 0 4 3 5 9 0 . 0 5 4 3 7 0 . 0 7 1 6 0 0 . 0 8 7 7 4 0 . 1 0 3 4 9 0 . 1 1 9 0 2 0 . 1 2 6 6 8 0 . 0 8 7 1 8 0 . 1 0 0 6 6

b « 8 . 0 0 0 0 0 ZERO ORDER F I R S T ORDER SECOND ORDER T H I R D ORDER

SUMS a = 0 . 2 0 Q= 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0


SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a * 1 . 0 0

H-o C O 0 . 0 0 0 0 0 0 . 0 0 0 0 1 0 . 0 0 0 0 3

0 . 0 0 0 0 0 0 . 0 0 0 0 1 0 . 0 0 0 0 5 0 . 0 0 0 6 0

0 . 0 0 3 2 6 0 . 0 1 0 1 4 0 . 0 3 5 2 9 0 . 0 5 3 4 1

μο C O 0 . 0 0 0 0 0 0 . 0 0 0 0 2 0 . 0 0 0 0 5

= 0 . 5 0 . 0 0 . 0 O.OOOOC 0 . 0 0 0 0 0 0 . 0 0 0 0 1 0 . 0 0 0 0 2 0 . 0 0 0 0 3 0 . 0 0 0 0 5

PEAK 0 . 0 0 . 0 0 . 0 . 0 0 0 0 0 0 . 0 0 0 0 1 0 . 0 0 0 0 9 0 . 0 . 0 0 0 0 3 0 . 0 0 0 0 6 0 . 0 0 0 1 7 0 . 0 . 0 0 0 0 7 0 . 0 0 0 1 3 0 . 0 0 0 2 8 0 .

0 0 0 0 0 0 0 0 0 1 0 0 0 0 9 0 0 0 8 2

0 . 0 0 4 1 2 0 . 0 1 2 3 9 0 . 0 4 1 9 2 0 . 0 6 3 0 0

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 1 0 . 0 0 0 0 3 0 . 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 0 0 0 2 0 . 0 0 0 0 4 0 . 0 0 0 1 0 0 . 0 . 0 0 0 0 6 0 . 0 0 0 0 9 0 . 0 0 0 1 3 0 . 0 0 0 2 0 0 . 0 0 0 4 0 0 . 0 . 0 0 0 7 4 0 . 0 0 1 0 0 0 . 0 0 1 3 2 0 . 0 0 1 8 1 0 . 0 0 2 6 1 0 .

0 . 0 0 4 0 3 0 . 0 0 5 3 8 0 . 0 0 6 8 8 0 . 0 0 8 7 2 0 . 0 1 1 1 1 0 . C . 0 1 2 5 9 0 . 0 1 6 7 1 0 . 0 2 0 9 2 0 . 0 2 5 5 7 0 . 0 3 0 8 8 0 . 0 . 0 4 3 9 8 0 . 0 5 8 0 1 0 . 0 7 1 4 0 0 . 0 8 4 8 1 0 . 0 9 8 4 5 0 . 0 . 0 6 6 6 3 0 . 0 8 7 7 4 0 . 1 0 7 5 2 0 . 1 2 6 8 2 0 . 1 4 5 8 4 0 .

= 0 . 7 0 . 0 0 . 0 0 . 0 PEAK 0 . 0 C 0 . 0 0 0 0 0 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 0 0 0 4 0 . 0 0 0 1 6 0 . 0 . 0 0 0 0 3 0 . 0 0 0 0 4 0 . 0 0 0 0 6 0 . 0 0 0 1 2 0 . 0 0 0 2 8 0 . 0 . 0 0 0 0 6 0 . 0 0 0 0 9 0 . 0 0 0 1 3 0 . 0 0 0 2 3 0 . 0 0 0 4 4 0 .

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 1 0 . 0 0 0 0 2 0 . 0 0 0 0 5 0 . 0 . 0 0 0 0 2 0 . 0 0 0 0 2 0 . 0 0 0 0 4 . 0 . 0 0 0 0 7 0 . 0 0 C 1 7 0 . 0 . 0 0 0 1 0 0 . 0 0 0 1 4 0 . 0 0 0 2 0 0 . 0 0 0 3 3 0 . 0 0 0 6 0 0 . 0 . 0 0 I C 1 - 0 . 0 0 1 3 6 0 . 0 Q 1 8 1 0 . 0 0 2 4 8 0 . 0 0 3 5 3 0 .

0 . 0 0 5 1 0 0 . 0 0 6 8 2 0 . 0 0 8 7 2 0 . 0 1 1 0 5 0 . 0 1 4 0 4 0 . 0 . C 1 5 3 9 0 . 0 2 0 4 2 0 . 0 2 5 5 7 0 . 0 3 1 2 4 0 . 0 3 7 6 8 0 . 0 . 0 5 2 2 4 0 . 0 6 8 9 0 0 . 0 8 4 8 1 0 . 1 0 0 7 4 0 . 1 1 6 8 9 0 , 0 . 0 7 8 5 9 0 . 1 0 3 4 9 0 . 1 2 6 8 2 0 . 1 4 9 5 9 0 . 1 7 1 9 7 0 .

0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 1 7 0 . 0 0 0 0 2 0 . 0 0 0 0 4 0 0 0 2 7 0 . 0 0 0 0 6 0 . 0 0 0 0 9 0 0 0 4 0 0 . 0 0 0 1 1 0 . 0 0 0 1 6

0 0 0 0 5 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 0 0 1 6 0 . 0 0 0 0 4 0 . 0 0 0 0 6 0 0 0 5 6 0 . 0 0 0 1 8 0 . 0 0 0 2 5 0 0 3 1 7 0 . 0 0 1 5 1 0 . 0 0 1 8 9

0 1 2 5 6 0 . 0 0 7 2 3 0 . 0 0 8 7 0 0 3 3 8 1 0 . 0 2 1 3 4 0 . 0 2 5 1 4 1 0 5 3 4 0 . 0 7 1 2 9 0 . 0 8 2 6 5 1 5 5 2 1 0 . 1 0 6 8 3 0 . 1 2 3 3 5

0 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 0 0 2 7 0 . 0 0 0 0 5 0 . 0 0 0 0 8 0 0 0 4 1 0 . 0 0 0 1 1 0 . 0 0 0 1 6 0 0 0 5 9 0 . 0 0 0 1 9 0 . 0 0 0 2 7

0 0 0 0 8 0 . 0 0 0 0 2 0 . 0 0 0 0 4 0 0 0 2 5 0 . 0 0 0 0 7 0 . 0 0 0 1 1 0 0 0 8 2 0 . 0 0 0 2 9 0 . 0 0 0 3 9 0 0 4 2 4 0 . 0 0 2 0 6 0 . 0 0 2 5 8

0 1 5 8 1 0 . 0 0 9 1 7 0 . 0 1 1 0 2 0 4 1 2 1 0 . 0 2 6 0 7 0 . 0 3 0 7 2 1 2 5 0 2 0 . 0 8 4 6 8 0 . 0 9 8 1 6 1 8 2 9 7 0 . 1 2 6 0 0 0 . 1 4 5 4 9



VECTOR μ=0.0

b = 8 . 0 0 0 0 0 μο


C . 2 7 7 7 8 0 . 0 9 3 4 0 0 . 0 5 3 0 8

SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

0 . 0 5 9 7 8 C . 1 3 0 7 6 0 . 2 2 0 2 8 0 . 3 4 8 5 6

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

0 . 4 5 0 1 6 0 . 5 2 9 0 9 0 . 6 3 8 6 0 C . 6 8 6 0 4

b= 8 . 0 0 0 0 0 Mo


C . 2 5 0 0 0 0 . 0 8 6 6 4 0 . 0 5 0 2 5

SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

C . 0 5 3 9 3 0 . 1 1 8 3 4 0 . 2 0 0 3 1 0 . 3 1 9 6 4

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

0 . 4 1 6 2 3 0 . 4 9 2 8 7 0 . 6 0 1 7 8 0 . 6 4 9 7 9

μ = 0.1 « . = 0 . 3

= C . 9

/x =C . 9 ^ = 1 . 0 AVERAGE N

FLUX U

2 5 0 0 0 0 . 2 0 8 3 3 0 1 1 4 0 4 0 . 1 1 5 8 7 0 0 6 7 1 C 0 . 0 7 5 0 8 0 . 0 7 5 0 3 0

1 7 8 5 7 0 . 1 5 6 2 5 0 . 1 3 8 8 9 0 . 1 0 9 0 9 0 . 1 0 1 0 6 0 . 0 9 3 4 0 0 .

0 7 2 3 3 C . 0 6 8 7 9 0 .

1 3 1 5 8 0 . 1 8 6 8 0 0 . 1 6 3 7 5 0 8 9 8 4 0 . 1 0 6 1 6 0 . 1 0 2 0 3 C 6 6 9 3 0 . 0 7 1 3 3 0 . 0 7 1 6 8

0 4 7 0 0 1 0 8 6 5 1 9 5 9 2

0 4 0 7 8 0 . 0 3 5 9 8 0 . 0 3 2 1 7 0 . 0 3 0 5 5 0 . 0 4 2 2 8 0 . 0 3 7 5 1 0 9 5 7 6 0 . 0 8 5 3 8 0 . 0 7 6 9 4 0 . C 7 3 3 0 0 . 0 9 8 2 3 0 . 0 8 8 4 5 1 7 6 4 2 1 5 9 7 0 0 . 1 4 5 5 7 0 . 1 3 9 3 4 0 . 1 7 8 7 2 0 . 1 6 4 0 8

. 2 9 4 0 1 0 . 2 7 3 3 6 0 . 2 6 3 8 8 0 . 3 1 5 8 5 0 . 2 9 8 5 2

0 . 4 6 9 7 3 0 . 4 5 0 2 6 0 . 4 2 7 7 1 0 . 4 0 5 2 4 0 . 3 9 4 4 0 0 . 4 4 4 8 1 0 . 4 3 0 4 5 0 . 5 7 9 3 0 0 . 5 6 8 2 2 5 5 0 2 5

7 4 7 2 8 5 2 9 9 7 0 . 5 1 9 6 1 0 . 5 5 8 3 1 0 . 5 5 0 3 2 7 3 7 1 7 0 . 7 3 0 6 5 0 . 7 3 4 5 5 C . 7 4 1 9 2 8 3 7 7 1 0 . 8 3 4 0 4 0 . 8 1 6 5 2 0 . 8 3 2 6 6

1 4 7 0 6 0 . 1 3 1 5 8 0 . 1 2 5 0 0 0 0 9 6 6 4 0 . 0 8 9 8 4 0 . 0 8 6 6 4 0 . 0 7 0 0 1 0 . 0 6 6 9 3 0 . 0 6 5 2 7 0 ,

1 7 3 2 9 0 . 1 5 3 4 3 1 0 0 5 1 C . 0 9 7 2 7 0 6 8 5 4 0 . 0 6 9 2 4

, 0 3 3 9 4 0 . 0 3 0 5 5 0 . 0 2 9 0 9 0 . 0 3 9 3 2 0 . 0 3 5 2 3 , 0 8 0 8 0 0 . 0 7 3 3 0 0 . 0 7 0 0 2 0 . 0 9 1 6 9 0 . 0 8 3 3 6 . 1 5 1 8 7 0 . 1 3 9 3 4 0 . 1 3 3 7 5

0 . 4 4 3 5 2 0 . 4 4 5 4 6 0 . 4 3 1 7 4 0 . 4 1 3 5 6 0 . 3 9 4 4 0 0 . 5 3 5 4 7 0 . 5 5 3 4 7 0 . 5 4 8 8 9 0 . 5 3 5 9 8 0 . 5 1 9 6 1

0 . 7 3 5 9 7 0 . 7 3 0 6 5

3 8 4 9 2 5 1 0 8 4 7 2 6 1 2

1 6 7 7 0 0 . 1 5 5 4 1 2 9 9 1 1 0 . 2 8 5 2 5

4 2 4 9 6 0 . 4 1 4 8 4 5 3 7 6 2 0 . 5 3 4 4 3 7 1 5 7 2 0 . 7 2 8 7 5

b = 8 . 0 0 0 0 0

6 4 9 7 9 0 . 7 2 8 0 8 0 . 7 9 1 7 5 0 . 8 2 0 1 5 0 . 8 3 1 9 8 0 . 8 3 4 0 4 0 . 8 3 2 5 6 0 . 7 9 9 5 8 0 . 8 2 1 8 0



SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = I . C O

b= 8 . 0 0 0 0 0

I N F I N I T E 0 . 5 9 9 4 7 0 . 3 6 6 5 8 0 . 0 . 1 7 3 2 9 0 . 1 8 5 0 4 0 . 1 5 8 8 9 0 . 0 . 0 7 0 4 8 0 . 0 8 7 1 6 0 . 0 8 9 0 7 0 .

I N F I N I T E 0 . 1 2 8 0 9 0 . 0 8 0 4 9 0 , I N F I N I T E 0 . 2 7 6 8 C 0 . 1 7 9 8 3 0 . I N F I N I T E 0 . 4 5 6 7 4 0 . 3 0 9 7 8 0 . I N F I N I T E 0 . 6 9 4 0 4 0 . 5 0 0 9 8 0 ,

I N F I N I T E 0 . 8 6 0 6 9 0 . 6 5 2 2 1 0 . I N F I N I T E 0 . 9 7 5 8 1 0 . 7 6 6 9 6 0 . I N F I N I T E 1 . 1 1 5 7 5 0 . 9 1 9 8 1 0 . I N F I N I T E 1 . 1 7 0 5 5 0 . 9 8 3 7 0 0 .


2 7 4 6 5 0 . 2 2 1 8 3 C . 1 8 6 8 0 0 . 1 7 3 2 9 0 . 3 4 6 5 7 0 . 2 5 0 0 0 1 3 6 7 1 0 . 1 1 9 5 8 0 . 1 0 6 1 6 0 . 1 0 0 5 1 0 . 1 4 0 9 6 0 . 1 2 5 0 0 0 8 3 7 2 0 . 0 7 7 3 7 0 . 0 7 1 3 3 0 . 0 6 8 5 4 0 . 0 8 1 3 0 0 . 0 7 8 1 2

0 6 1 1 8 0 . 0 4 9 8 7 0 . 0 4 2 2 8 0 . 0 3 9 3 2 0 . 0 7 5 7 1 0 . 0 5 5 7 3 1 3 9 1 9 0 . 1 1 4 8 6 0 . 0 9 8 2 3 0 . 0 9 1 6 9 0 . 1 6 8 3 8 0 . 1 2 7 0 2 2 4 5 8 1 0 . 2 0 6 3 4 0 . 1 7 8 7 2 0 . 1 6 7 7 0 0 . 2 8 9 2 4 0 . 2 2 5 1 5 4 1 3 2 6 0 . 3 5 6 8 1 0 . 3 1 5 8 5 0 . 2 9 9 1 1 0 . 4 6 8 8 9 0 . 3 8 1 9 6

5 5 5 9 9 0 . 4 9 2 3 3 0 . 4 4 4 8 1 0 . 4 2 4 9 6 0 . 6 1 5 1 9 0 . 5 1 9 4 3 6 7 1 2 1 0 . 6 0 7 0 9 0 . 5 5 8 3 1 0 . 5 3 7 6 2 0 . 7 3 0 3 9 0 . 6 3 3 8 1 8 3 4 6 0 0 . 7 7 8 1 0 0 . 7 3 4 5 5 0 . 7 1 5 7 2 0 . 8 9 1 3 7 0 . 8 0 1 9 7 9 0 5 9 5 0 . 8 5 5 4 1 0 . 8 1 6 5 2 0 . 7 9 9 5 8 0 . 9 6 1 2 5 0 . 8 7 7 4 3


SUMS a = 0 . 2 C a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

0 . 5 0 0 0 0 0 . 3 8 0 1 1 0 . 2 8 0 0 5 0 . 0 . 1 2 5 0 0 C . 1 4 7 9 4 0 . 1 3 9 6 4 0 . 0 . 0 6 2 5 0 0 . 0 7 8 5 6 0 . 0 8 4 2 2 0 .

C . 1 0 5 5 7 0 . C 8 2 6 6 0 . 0 6 2 3 8 0 . 0 . 2 2 5 4 0 0 . 1 8 2 5 2 0 . 1 4 1 8 3 0 . 0 . 3 6 7 5 4 0 . 3 0 9 7 7 0 . 2 4 9 9 9 0 . 0 . 5 5 2 7 9 0 . 4 9 0 7 1 0 . 4 1 8 3 6 0 ,

0 . 6 8 3 7 5 0 . 6 2 9 3 0 0 . 5 5 9 9 7 0 . 0 . 7 7 6 0 6 0 . 7 3 2 3 3 0 . 6 7 2 8 9 0 . 0 . 8 9 1 9 8 0 . 8 6 7 5 1 0 . 8 3 0 9 4 0 . 0 . 9 3 8 7 2 0 . 9 2 3 5 6 0 . 8 9 9 3 4 0 .

2 2 5 3 5 0 . 1 8 9 4 4 0 . 1 6 3 7 5 0 . 1 5 3 4 3 0 . 2 5 0 0 0 0 . 2 0 4 5 7 1 2 5 6 2 0 . 1 1 2 8 6 0 . 1 0 2 0 3 0 . 0 9 7 2 7 0 . 1 2 5 0 0 0 . 1 1 5 8 0 0 8 1 4 7 0 . 0 7 6 7 1 0 . 0 7 1 6 8 0 . 0 6 9 2 4 0 . 0 7 8 1 2 0 . 0 7 6 7 4

0 5 0 8 5 0 . 0 4 3 1 2 0 . 0 3 7 5 1 0 . 0 3 5 2 3 0 . 0 5 5 7 3 0 . 0 4 6 2 6 1 1 7 5 9 0 . 1 0 0 8 5 0 . 0 8 8 4 5 0 . 0 8 3 3 6 0 . 1 2 7 0 2 0 . 1 0 7 3 3 2 1 2 0 8 0 . 1 8 4 8 5 0 . 1 6 4 0 8 0 . 1 5 5 4 1 0 . 2 2 5 1 5 0 . 1 9 4 7 2 3 6 7 9 7 0 . 3 2 9 3 8 0 . 2 9 8 5 2 0 . 2 8 5 2 5 0 . 3 8 1 9 6 0 . 3 4 1 8 6

5 0 7 8 8 0 . 4 6 5 7 1 0 . 4 3 0 4 5 0 . 4 1 4 8 4 0 . 5 1 9 4 3 0 . 4 7 7 9 5 6 2 5 4 9 0 . 5 8 5 2 9 0 . 5 5 G 3 2 0 . 5 3 4 4 3 0 . 6 3 3 8 1 0 . 5 9 5 8 5 7 9 8 9 1 0 . 7 6 9 4 6 0 . 7 4 1 9 2 0 . 7 2 8 7 5 0 . 8 0 1 9 7 0 . 7 7 5 7 7 8 7 6 6 5 0 . 8 5 4 5 1 0 . 8 3 2 6 6 0 . 8 2 1 8 0 0 . 8 7 7 4 3 0 . 8 5 8 4 7




VECTOR

b = 8 . 0 0 0 0 0 ZERO ORDfck F I R S T ORDER SECOND ORDER THIRD ORDER

SUMS

μ<=0.0 μ =0.1 μ=0.3 μ =0.5 μ = 0.1 μ =0.9

α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0


AVERAGE Ν

FLUX U

μ0 = 0 . 9 0 . 0 C O 0 . 0 0 . 0 0 . 0 PEAK 0 . 0 0 . 0 0 0 0 8 0 . 0 0 0 1 4 C C 0 0 0 4 C . 0 0 0 0 4 0 . 0 0 0 0 6 0 . 0 0 0 0 9 0 . 0 0 0 1 6 0 . 0 0 0 3 4 0 . 0 0 0 4 9 0 . 0 0 0 1 4 0 . 0 0 0 2 0 C . C 0 C 0 7 0 . 0 0 0 0 8 0 . 0 0 0 1 1 0 . 0 0 0 1 7 0 . 0 0 0 2 8 0 . 0 0 0 5 1 0 . 0 0 0 6 9 0 . 0 0 0 2 4 0 . 0 0 0 3 2 0 . 0 0 0 1 2 0 . 0 0 0 1 4 0 . C 0 0 1 9 0 . 0 0 0 2 8 0 . 0 0 0 4 4 0 . 0 0 0 7 2 0 . 0 0 0 9 3 0 . 0 0 0 3 6 0 . 0 0 0 4 8

C C O O O l 0 . 0 0 0 0 1 0 . 0 0 0 0 2 0 . 0 0 0 0 3 0 . 0 0 0 0 5 0 . 0 0 0 1 0 0 . 0 0 0 1 4 0 . 0 0 0 1 2 0 . 0 0 0 2 0 C . C 0 0 C 4 C C 0 0 0 5 0 . 0 0 0 0 7 0 . 0 0 0 1 0 0 . 0 0 0 1 7 0 . 0 0 0 3 1 0 . 0 0 0 4 3 0 . 0 0 0 2 2 0 . 0 0 0 3 3 C 0 0 0 1 7 0 . 0 0 0 2 1 0 . 0 0 0 2 8 0 . 0 0 0 4 0 0 . 0 0 0 6 0 0 . 0 0 0 9 9 0 . 0 0 1 2 7 0 . 0 0 0 5 8 0 . 0 0 0 8 0 0 . 0 0 1 1 9 0 . 0 0 1 4 6 0 . 0 0 1 9 7 0 . 0 0 2 6 1 0 . 0 0 3 5 3 0 . 0 0 4 8 9 0 . 0 0 5 7 7 0 . 0 0 2 9 8 0 . 0 0 3 7 5

C . 0 0 5 2 7 C 0 0 6 5 2 0 . 0 0 8 7 0 O . O l l l l 0 . 0 1 4 0 4 0 . 0 1 7 7 0 0 . 0 1 9 8 2 0 . 0 1 1 7 0 0 . 0 1 4 0 9 0 . C 1 4 9 7 0 . 0 1 8 5 9 0 . 0 2 4 6 7 0 . 0 3 0 8 8 0 . 0 3 7 6 8 0 . 0 4 5 3 1 0 . 0 4 9 4 3 0 . 0 3 1 5 1 0 . 0 3 7 1 3 C . 0 4 8 6 7 0 . C 6 0 6 6 0 . 0 8 0 0 0 0 . 0 9 8 4 5 0 . 1 1 6 8 9 0 . 1 3 5 4 9 0 . 1 4 4 7 9 0 . 0 9 8 3 2 0 . 1 1 3 9 8 C . C 7 2 4 6 0 . 0 9 0 3 8 0 . 1 1 9 0 2 0 . 1 4 5 8 4 0 . 1 7 1 9 7 0 . 1 9 7 5 7 0 . 2 1 0 0 7 0 . 1 4 4 9 2 0 . 1 6 7 3 4

μ0 = 1 . 0 C O 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 PEAK 0 . 0 0 0 1 7 0 . 0 0 0 3 4 C . 0 0 0 0 8 C . 0 0 0 0 9 0 . 0 0 0 1 2 0 . 0 0 0 1 7 0 . 0 0 0 2 7 0 . 0 0 C 4 9 0 . 0 0 0 6 7 0 . 0 0 0 2 3 0 . 0 0 0 3 1 0 . 0 0 0 1 2 0 . 0 0 0 1 4 0 . 0 0 0 1 9 0 . 0 0 0 2 7 0 . 0 0 0 4 1 0 . 0 0 0 6 9 0 . 0 0 0 9 0 0 . 0 0 0 3 4 0 . 0 0 0 4 6 C . 0 0 0 1 7 0 . 0 0 0 2 1 0 . 0 0 0 2 8 0 . 0 0 0 4 0 0 . 0 0 0 5 9 0 . 0 0 0 9 3 0 . 0 0 1 1 5 0 . 0 0 0 4 9 0 . 0 0 0 6 3

SUMS a = a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = I . 0 0

0 . 0 0 0 0 2 C C 0 0 0 3 0 . 0 0 0 0 3 0 . 0 0 0 0 5 0 . 0 0 0 0 8 0 . 0 0 0 1 4 0 . 0 0 0 1 8 0 . 0 0 0 2 3 0 . 0 0 0 4 2 C . 0 0 0 0 8 0 . 0 0 0 0 9 0 . 0 0 0 1 2 0 . 0 0 0 1 6 0 . 0 0 0 2 5 0 . 0 0 0 4 3 0 . 0 0 0 5 6 0 . 0 0 0 3 8 0 . 0 0 0 6 2 C 0 0 0 2 5 0 . 0 0 0 3 0 0 . 0 0 0 4 1 0 . 0 0 0 5 6 0 . 0 0 0 8 2 0 . 0 0 1 2 7 0 . 0 0 1 5 8 0 . 0 0 0 8 4 0 . 0 0 1 2 1 0 . 0 0 1 4 6 0 . 0 0 1 7 9 0 . 0 0 2 4 1 0 . 0 0 3 1 7 0 . 0 0 4 2 4 0 . 0 0 5 7 7 0 . 0 0 6 7 4 0 . 0 0 3 6 6 0 . 0 0 4 6 5

0 . 0 0 5 9 7 0 . 0 0 7 3 8 0 . 0 0 9 8 5 0 . 0 1 2 5 6 0 . 0 1 5 8 1 0 . 0 1 9 8 2 0 . 0 2 2 1 2 0 . 0 1 3 2 7 0 . 0 1 6 0 2 0 . 0 1 6 4 1 0 . 0 2 0 3 8 0 . 0 2 7 0 3 0 . 0 3 3 8 1 0 . 0 4 1 2 1 0 . 0 4 9 4 3 0 . 0 5 3 8 4 0 . 0 3 4 5 4 0 . 0 4 0 7 6 0 . 0 5 2 1 0 C . 0 6 4 9 3 0 . C 8 5 6 2 0 . 1 0 5 3 4 0 . 1 2 5 0 2 0 . 1 4 4 7 9 0 . 1 5 4 6 4 0 . 1 0 5 2 4 0 . 1 2 2 0 7 C C 7 7 1 4 0 . 0 9 6 2 1 0 . 1 2 6 6 8 0 . 1 5 5 2 1 0 . 1 8 2 9 7 0 . 2 1 0 0 7 0 . 2 2 3 2 8 0 . 1 5 4 2 8 0 . 1 7 8 2 0


NARROW SURFACE LAYER AT TOP C O 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 , 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 . 0 0 0 0 2 0 . 0 0 0 0 3 0 . 0 0 0 0 4 0 . 0 . 0 0 0 0 5 0 . 0 0 0 0 6 0 . 0 0 0 0 8 0 .

0 0 0 0 0 0 . 0 0 0 0 1 0 . 0 0 0 0 8 0 . 0 0 0 1 7 0 . 0 0 0 0 2 0 . 0 0 0 0 3 0 0 0 0 2 0 . 0 0 0 0 5 0 . 0 0 C 1 4 0 . 0 0 0 2 3 0 . 0 0 0 0 5 0 . 0 0 0 0 7 0 0 C 0 6 0 . 0 0 0 1 1 0 . 0 0 0 2 4 0 . 0 0 0 3 4 0 . 0 0 0 1 0 0 . 0 0 0 1 4 0 0 0 1 1 0 . 0 0 0 1 9 0 . 0 0 C 3 6 0 . 0 0 0 4 9 0 . 0 0 0 1 6 0 . 0 0 0 2 2

SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0


SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

0 . 0 0 0 0 0 C O O C O O 0 . 0 0 0 0 1 0 , 0 . 0 0 0 0 2 0 . 0 0 0 0 2 0 . 0 0 0 0 2 0 . 0 . 0 0 0 0 8 0 . 0 0 0 0 9 0 . 0 0 0 1 3 0 . 0 . 0 0 0 6 9 0 . 0 0 0 8 4 0 . 0 0 1 1 3 0 .

0 . 0 0 3 4 3 0 . 0 0 4 2 4 0 . 0 0 5 6 6 0 . 0 . 0 1 0 3 4 C . 0 1 2 8 5 0 . 0 1 7 0 4 0 . 0 . 0 3 5 2 4 0 . 0 4 3 9 2 0 . 0 5 7 9 2 0 , 0 . 0 5 3 0 7 0 . C 6 6 2 0 0 . 0 8 7 1 8 0 ,

LAMBERT SURFACE ON TOP C O 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 . 0 C 0 0 2 0 . 0 0 0 0 2 0 . 0 0 0 0 3 0 . 0 . 0 0 0 0 4 0 . 0 0 0 0 4 0 . 0 0 0 0 6 0 · 0 . 0 0 0 0 7 0 . 0 0 0 0 8 O . O O O l l 0 .

0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 . 0 0 0 0 2 0 . 0 0 0 0 3 0 . Q 0 0 0 4 0 . 0 . 0 0 0 1 1 0 . 0 0 0 1 3 0 . 0 0 0 1 7 0 . C . 0 0 0 8 6 0 . 0 0 1 0 6 0 . 0 0 1 4 3 0 .

0 . 0 0 4 1 2 0 . 0 0 5 1 C 0 . 0 0 6 8 1 0 . C . 0 1 2 1 9 0 . 0 1 5 1 4 0 . 0 2 0 0 8 0 . 0 . 0 4 0 8 5 0 . 0 5 0 9 2 0 . 0 6 7 1 5 0 , 0 . 0 6 1 2 8 0 . 0 7 6 4 4 0 . 1 0 0 6 6 0 .

0 0 0 0 1 0 . 0 0 0 0 2 0 . 0 0 0 1 2 0 . 0 0 0 2 3 0 . 0 0 0 0 3 0 . 0 0 0 0 6 0 0 0 0 4 0 . 0 0 0 0 7 0 . 0 0 0 2 2 0 . 0 0 0 3 8 0 . 0 0 0 0 8 0 . 0 0 0 1 2 0 0 0 1 8 0 . 0 0 0 2 9 0 . 0 0 0 5 8 0 . 0 0 0 8 4 0 . 0 0 0 2 6 0 . 0 0 0 3 6 0 0 1 5 1 0 . 0 0 2 0 6 0 . 0 0 2 9 8 0 . 0 0 3 6 6 0 . 0 0 1 7 1 0 . 0 0 2 1 6

0 0 7 2 3 0 . 0 0 9 1 7 0 . 0 1 1 7 0 0 . 0 1 3 2 7 0 . 0 0 7 6 1 0 . 0 0 9 1 6 0 2 1 3 4 0 . 0 2 6 0 7 0 . 0 3 1 5 1 0 . 0 3 4 5 4 0 . 0 2 1 7 7 0 . 0 2 5 6 6 0 7 1 2 9 0 . 0 8 4 6 8 0 . 0 9 8 3 2 0 . 1 0 5 2 4 0 . 0 7 1 1 9 0 . 0 8 2 5 3 1 0 6 8 3 0 . 1 2 6 0 0 0 . 1 4 4 9 2 0 . 1 5 4 2 8 0 . 1 0 6 1 5 0 . 1 2 2 5 7

0 0 0 0 0 0 . 0 0 0 0 1 0 . 0 0 0 1 4 0 . 0 0 0 3 4 0 . 0 0 0 0 3 0 . 0 0 0 0 6 0 0 0 0 4 0 . 0 0 0 0 8 0 . 0 0 0 2 0 0 . 0 0 0 3 1 0 . 0 0 0 0 7 0 . 0 0 0 1 1 0 0 0 0 9 0 . 0 0 0 1 6 0 . 0 0 0 3 2 0 . 0 0 0 4 6 0 . 0 0 0 1 4 0 . 0 0 0 1 9 0 0 0 1 6 0 . 0 0 0 2 7 0 . 0 0 0 4 8 0 . 0 0 0 6 3 0 . 0 0 0 2 2 0 . 0 0 0 3 0

0 0 0 0 1 0 . 0 0 0 0 4 0 . 0 0 0 2 0 0 . 0 0 0 4 2 0 . 0 0 0 0 6 0 . 0 0 0 1 0 0 0 0 0 6 0 . 0 0 0 1 1 0 . 0 0 0 3 3 0 . 0 0 0 6 2 0 . 0 0 0 1 2 0 . 0 0 0 1 8 0 0 0 2 5 0 . 0 0 0 3 9 0 . 0 0 0 8 0 0 . 0 0 1 2 1 0 . 0 0 0 3 6 0 . 0 0 0 4 9 0 0 1 8 9 0 . 0 0 2 5 8 0 . 0 0 3 7 5 0 . 0 0 4 6 5 0 . 0 0 2 1 6 0 . 0 0 2 7 1

0 0 8 7 0 0 . 0 1 1 0 2 0 . 0 1 4 0 9 0 . 0 1 6 0 2 0 . 0 0 9 1 6 0 . 0 1 1 0 3 0 2 5 1 4 0 . 0 3 0 7 2 0 . 0 3 7 1 3 0 . 0 4 0 7 6 0 . 0 2 5 6 6 0 . 0 3 0 2 3 0 8 2 6 5 0 . 0 9 8 1 6 0 . 1 1 3 9 8 0 . 1 2 2 0 7 0 . 0 8 2 5 3 0 . 0 9 5 6 8 1 2 3 3 5 0 . 1 4 5 4 9 0 . 1 6 7 3 4 0 . 1 7 8 2 0 0 . 1 2 2 5 7 0 . 1 4 1 5 3




VECTOR

b = 1 6 . 0 0 0 0 0

F I R S T OROER SECOND ORDER THIRD ORDER

SUMS a = C . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a - 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

b = 1 6 . 0 0 0 0 0


SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

b = 1 6 . 0 0 0 0 0


SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

b = 1 6 . 0 0 0 0 0

F I R S T ORDER SECONO ORDER THIRD ORDER

f t = 0 . 0 f t » 0 . 1 M = 0 . 3 /χ = 0 . 5 / x = 0 . 7 μ-0·9

SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a » 0 . 9 9 a « 1 . 0 0

AVERAGE Ν

FLUX U

Mo 0 . 1

2 . 5 0 0 0 0 1 . 2 5 0 0 0 0 . 6 2 5 0 0 0 . 4 1 6 6 7 0 . 3 1 2 5 0 0 . 2 5 0 0 0 0 . 2 2 7 2 7 0 . 5 9 9 4 7 0 . 3 8 0 1 1 0 . 2 9 9 7 4 0 . 2 9 9 7 4 0 . 2 1 2 4 0 0 . 1 6 4 3 9 0 . 1 3 4 5 2 0 . 1 1 4 0 4 0 . 1 0 6 0 2 0 . 1 8 5 0 4 0 . 1 4 7 9 4 0 . 0 9 2 5 2 0 . 1 1 C 4 9 0 . 0 9 9 2 0 0 . 0 8 6 1 0 0 . 0 7 5 5 2 0 . 0 6 7 1 0 0 . 0 6 3 5 4 0 . 0 8 7 1 6 0 . 0 7 8 5 6

0 . 5 1 2 8 1 0 . 2 6 2 9 7 0 . 1 3 4 3 9 0 . 0 9 0 7 0 0 . 0 6 8 5 8 0 . 0 5 5 1 8 0 . 0 5 0 2 9 0 . 1 2 8 0 9 0 . 0 8 2 6 6 1 . 0 5 5 3 6 0 . 5 5 6 8 9 0 . 2 9 2 3 1 0 . 2 0 0 3 8 0 . 1 5 3 1 3 0 . 1 2 4 2 0 0 . 1 1 3 5 4 0 . 2 7 6 8 0 0 . 1 8 2 5 2 1 . 6 3 7 0 2 0 . 8 9 3 2 8 0 . 4 8 5 3 2 0 . 3 3 9 9 0 0 . 2 6 3 7 4 0 . 2 1 6 3 7 0 . 1 9 8 7 4 0 . 4 5 6 7 4 0 . 3 0 9 7 7 2 . 2 7 7 6 2 1 . 2 9 6 8 8 0 . 7 4 0 5 6 0 . 5 3 6 4 8 0 . 4 2 6 9 2 0 . 3 5 7 2 5 0 . 3 3 0 9 2 0 . 6 9 4 0 4 0 . 4 9 0 7 1

2 . 6 3 7 3 2 1 . 5 4 5 6 6 0 . 9 1 7 3 6 0 . 6 8 3 9 6 0 . 5 5 6 9 2 0 . 4 7 4 9 3 0 . 4 4 3 5 7 0 . 8 6 0 7 2 0 . 6 2 9 3 4 2 . 8 3 8 6 8 1 . 6 9 6 4 4 1 . 0 3 6 4 4 0 . 7 9 0 9 4 0 . 6 5 6 7 7 0 . 5 6 9 5 8 0 . 5 3 6 0 2 0 . 9 7 6 1 6 0 . 7 3 2 7 4 3 . 0 3 1 3 1 1 . 8 5 6 3 2 1 . 1 8 0 9 1 0 . 9 3 3 5 0 0 . 7 9 9 9 4 0 . 7 1 3 8 0 0 . 6 8 0 7 2 1 . 1 2 3 8 5 0 . 8 7 6 9 0 3 . 1 0 0 4 8 1 . 9 2 2 5 3 1 . 2 5 1 1 0 1 . 0 1 0 0 7 0 . 8 8 2 8 7 0 . 8 0 2 6 2 0 . 7 7 2 2 8 1 . 2 0 0 9 5 0 . 9 5 8 6 6

Mo 0 . 3

0 . 8 3 3 3 3 0 . 6 2 5 0 0 0 . 4 1 6 6 7 0 , 0 . 1 8 3 2 9 0 . 2 1 2 4 0 0 . 1 8 3 2 9 0 . 0 . 0 7 9 4 5 0 . 0 9 9 2 0 0 . 0 9 9 6 0 0«

3 1 2 5 0 0 . 2 5 0 0 0 0 . 2 0 8 3 3 0 . 1 9 2 3 1 0 . 3 6 6 5 8 0 . 2 8 0 0 5 1 5 4 5 6 0 . 1 3 2 6 3 0 . 1 1 5 8 7 0 . 1 0 8 9 5 0 . 1 5 8 8 9 0 . 1 3 9 6 4 0 9 1 3 9 0 . 0 8 2 7 6 0 . 0 7 5 0 8 0 . 0 7 1 6 5 0 . 0 8 9 0 7 0 . 0 8 4 2 2

C . 1 7 4 7 2 0 . 1 3 4 3 9 0 . 0 9 1 5 8 0 . 0 6 9 5 3 0 . 0 5 6 0 7 0 . 0 4 7 0 0 0 . 0 4 3 4 9 0 . 0 8 0 4 9 0 . 0 6 2 3 8 C . 3 6 9 3 0 0 . 2 9 2 3 1 0 . 2 0 4 5 7 0 . 1 5 7 7 6 0 . 1 2 8 6 0 0 . 1 0 8 6 5 0 . 1 0 0 8 5 0 . 1 7 9 8 3 0 . 1 4 1 8 3 C . 5 9 2 9 3 C . 4 8 5 3 2 0 . 3 5 1 5 7 0 . 2 7 7 0 1 0 . 2 2 9 2 6 0 . 1 9 5 9 2 0 . 1 8 2 7 2 0 . 3 0 9 7 8 0 . 2 4 9 9 9 0 . 8 6 7 0 6 0 . 7 4 0 5 6 0 . 5 6 3 8 4 0 . 4 5 9 5 2 0 . 3 9 0 0 6 0 . 3 4 0 0 0 0 . 3 1 9 7 9 0 . 5 0 0 9 8 0 . 4 1 8 3 6

1 . 0 4 3 5 1 0 . 9 1 7 3 6 0 . 7 2 5 9 5 0 . 6 0 8 9 0 0 . 5 2 8 8 6 0 . 4 6 9 7 9 0 . 4 4 5 5 2 0 . 6 5 2 2 5 0 . 5 6 0 0 2 1 . 1 5 6 1 9 1 . 0 3 6 4 4 0 . 8 4 4 2 8 0 . 7 2 4 8 4 0 . 6 4 2 0 0 0 . 5 7 9 9 7 0 . 5 5 4 2 0 0 . 7 6 7 4 2 0 . 6 7 3 4 3 1 . 2 8 5 5 9 1 . 1 8 0 9 1 1 . 0 0 1 6 4 0 . 8 9 0 7 5 0 . 8 1 4 1 9 0 . 7 5 6 7 6 0 . 7 3 2 7 9 0 . 9 3 0 4 9 0 . 8 4 3 3 2 1 . 3 4 5 2 0 1 . 2 5 1 1 0 1 . 0 8 5 4 0 0 . 9 8 5 6 9 0 . 9 1 8 8 7 0 . 8 7 0 0 2 0 . 8 4 9 9 5 1 . 0 2 3 7 3 0 . 9 4 5 5 6

Mo = 0 . 5

C . 5 0 0 0 0 0 . 4 1 6 6 7 0 . 3 1 2 5 0 0 . 2 5 0 0 0 0 . 2 0 8 3 3 0 . 1 7 8 5 7 0 . 1 6 6 6 7 0 . 2 7 4 6 5 0 . 2 2 5 3 5 0 . 1 3 7 3 3 0 . 1 6 4 3 9 0 . 1 5 4 5 6 0 . 1 3 7 3 3 0 . 1 2 1 9 2 0 . 1 0 9 0 9 0 . 1 0 3 5 4 0 . 1 3 6 7 1 0 . 1 2 5 6 2 0 . 0 6 8 3 6 C . 0 8 6 1 0 0 . 0 9 1 3 9 0 . 0 8 7 2 1 0 . 0 8 1 1 3 0 . 0 7 5 0 3 0 . 0 7 2 1 5 0 . 0 8 3 7 2 0 . 0 8 1 4 7

0 . 1 0 6 1 2 0 . 0 9 0 7 0 0 . 0 6 9 5 3 0 . 0 5 6 3 0 0 . 0 4 7 3 0 0 . 0 4 0 7 8 0 . 0 3 8 1 5 0 . 0 6 1 1 8 0 . 0 5 0 8 5 0 . 2 2 7 8 4 0 . 2 0 0 3 8 0 . 1 5 7 7 6 0 . 1 2 9 7 8 0 . 1 1 0 2 0 0 . 0 9 5 7 6 0 . 0 8 9 8 7 0 . 1 3 9 1 9 0 . 1 1 7 5 9 0 . 3 7 3 7 4 0 . 3 3 9 9 0 0 . 2 7 7 0 1 0 . 2 3 2 8 1 0 . 2 0 0 7 1 0 . 1 7 6 4 2 0 . 1 6 6 3 7 0 . 2 4 5 8 1 C . 2 1 2 0 8 0 . 5 6 5 3 1 0 . 5 3 6 4 8 0 . 4 5 9 5 2 0 . 3 9 9 4 6 0 . 3 5 3 2 1 0 . 3 1 6 6 8 0 . 3 0 1 1 6 0 . 4 1 3 2 6 0 . 3 6 7 9 7

0 . 7 0 0 2 2 0 . 6 8 3 9 6 0 . 6 0 8 9 0 0 . 5 4 4 7 8 0 . 4 9 2 8 8 0 . 4 5 0 3 4 0 . 4 3 1 8 2 0 . 5 5 6 0 3 0 . 5 0 7 9 4 C . 7 9 4 1 0 0 . 7 9 0 9 4 0 . 7 2 4 8 4 0 . 6 6 3 7 8 0 . 6 1 2 4 2 0 . 5 6 9 0 6 0 . 5 4 9 8 1 0 . 6 7 1 7 9 0 . 6 2 6 1 8 0 . 9 1 4 6 3 0 . 9 3 3 5 C 0 . 8 9 0 7 5 0 . 8 4 4 9 3 0 . 8 0 4 4 7 0 . 7 6 9 0 8 0 . 7 5 2 9 9 0 . 8 4 7 7 5 0 . 8 1 4 1 5 0 . 9 7 7 5 0 1 . 0 1 0 0 7 0 . 9 8 5 6 9 0 . 9 5 4 6 8 0 . 9 2 6 9 1 0 . 9 0 2 5 7 0 . 8 9 1 4 7 0 . 9 5 5 0 1 0 . 9 3 3 2 9

MO = 0 . 7

0 . 3 5 7 1 4 0 . 3 1 2 5 C 0 . 2 5 0 0 0 0 . 2 0 8 3 3 0 . 1 7 8 5 7 0 . 1 5 6 2 5 0 . 1 4 7 0 6 0 . 2 2 1 8 3 0 . 1 8 9 4 4 0 . 1 1 0 9 1 0 . 1 3 4 5 2 0 . 1 3 2 6 3 0 . 1 2 1 9 2 0 . 1 1 0 9 1 0 . 1 0 1 0 6 0 . 0 9 6 6 4 0 . 1 1 9 5 8 0 . 1 1 2 8 6 0 . 0 5 9 7 9 0 . 0 7 5 5 2 0 . 0 8 2 7 6 0 . 0 8 1 1 3 0 . 0 7 7 0 1 0 . 0 7 2 3 3 0 . 0 7 0 0 1 0 . 0 7 7 3 7 0 . 0 7 6 7 1

0 . 0 7 6 4 2 0 . 0 6 8 5 8 0 . 0 5 6 0 7 0 . 0 4 7 3 0 0 . 0 4 0 8 8 0 . 0 3 5 9 8 0 . 0 3 3 9 4 0 . 0 4 9 8 7 0 . 0 4 3 1 2 0 . 1 6 5 8 3 0 . 1 5 3 1 3 0 . 1 2 8 6 0 0 . 1 1 0 2 0 0 . 0 9 6 2 5 0 . 0 8 5 3 8 0 . 0 8 0 8 0 0 . 1 1 4 8 6 0 . 1 0 0 8 5 0 . 2 7 6 1 9 0 . 2 6 3 7 4 0 . 2 2 9 2 6 0 . 2 0 0 7 1 0 . 1 7 7 9 8 0 . 1 5 9 7 0 0 . 1 5 1 8 7 0 . 2 0 6 3 4 0 . 1 8 4 8 5 0 . 4 2 8 4 4 0 . 4 2 6 9 2 0 . 3 9 0 0 6 0 . 3 5 3 2 1 0 . 3 2 1 2 3 0 . 2 9 4 0 1 0 . 2 8 1 9 5 0 . 3 5 6 8 1 0 . 3 2 9 3 9

0 . 5 4 3 0 0 0 . 5 5 6 9 2 0 . 5 2 8 8 6 0 . 4 9 2 8 8 0 . 4 5 8 6 6 0 . 4 2 7 8 0 0 . 4 1 3 6 6 0 . 4 9 2 3 9 0 . 4 6 5 7 8 0 . 6 2 7 9 9 0 . 6 5 6 7 7 0 . 6 4 2 0 0 0 . 6 1 2 4 2 0 . 5 8 1 1 8 0 . 5 5 1 2 8 0 . 5 3 7 1 1 0 . 6 0 7 8 0 0 . 5 8 6 1 2 0 . 7 4 6 4 6 0 . 7 9 9 9 4 0 . 8 1 4 1 9 0 . 8 0 4 4 7 0 . 7 8 7 8 2 0 . 7 6 8 8 4 0 . 7 5 9 0 6 0 . 7 9 3 7 2 0 . 7 8 7 5 6 0 . 8 1 3 7 8 0 . 8 8 2 8 7 0 . 9 1 8 8 7 0 . 9 2 6 9 1 0 . 9 2 5 5 1 0 . 9 1 9 8 4 0 . 9 1 6 0 9 0 . 9 1 3 2 7 0 . 9 2 1 3 2



μ=0.0 μ = 0. 1 μ=0.1 μ = Ό.5 μ = 0.7 = 0 . 9 • 1 . 0 AVERAGE Ν

FLUX U

b = 1 6 . 0 0 0 0 0 Μο ZERO ORDER C O F I R S T ORDER 0 . 0 0 0 0 0 SECOND ORDER 0 . 0 0 0 0 0 T H I R D ORDER 0 . 0 0 0 0 0

SUMS α = 0 . 2 0 0 . 0 0 0 0 0 α = 0 . 4 0 0 . 0 0 0 0 0 α = 0 . 6 0 0 . 0 0 0 0 0 Q = 0 . 8 0 0 . 0 0 0 0 0

α = 0 . 9 0 C 0 0 0 0 3 α = 0 . 9 5 C . 0 0 0 2 9 α = 0 . 9 9 0 . 0 0 5 2 7 α = 1 . 0 0 0 . 0 1 7 9 0

b = l 6 . 0 0 0 0 0 ZERO ORDER F I R S T ORDER SECOND ORDER T H I R D ORDER

Mo C O C O C O O O 0 . 0 0 0 0 0 0 . 0 0 0 0 0

• 0 . 1 PEAK 0 . 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 . 0 0 0 0 4 0 . 0 0 0 0 5 0 . 0 0 0 3 6 0 . 0 0 0 4 8 0 . 0 0 6 5 7 0 . 0 0 8 6 6 0 . 0 2 2 3 3 0 . 0 2 9 4 0

= 0 . 3 C O PEAK O.OOOOC 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 1 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 . 0 0 0 0 6 0 . 0 0 0 0 8 0 . 0 0 0 1 0 0 . 0 0 0 1 1 0 . 0 0 0 0 6 0 . 0 0 0 0 8 0 . 0 0 0 6 0 0 . 0 0 0 7 4 0 . 0 0 0 8 9 0 . 0 0 0 9 8 0 . 0 0 0 6 2 0 . 0 0 0 7 3 0 . 0 1 0 6 6 0 . 0 1 2 6 6 0 . 0 1 4 7 1 0 . 0 1 5 7 5 0 . 0 1 0 6 5 0 . 0 1 2 3 4 0 . 0 3 6 0 3 0 . 0 4 2 4 9 0 . 0 4 8 8 8 0 . 0 5 2 0 5 0 . 0 3 5 8 0 0 . 0 4 1 3 4

0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 '

O.OOOOC 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 O.OOOOC 0 . 0 0 0 0 0 O.OOOOC 0 . 0 0 0 0 0

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 0 0 0 0 0 . 0 0 0 0 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

b = l 6 . 0 0 0 0 0 ZERO ORDER F I R S T ORDER SECOND ORDER THIRD ORDER

SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

0 . 0 0 0 0 4 0 . 0 0 0 3 9 0 . 0 0 6 9 5 0 . 0 2 3 5 7

Mo C O 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 . 0 0 0 0 0 0 . 0 0 0 0 0

cooooo O.OOOOC

0 . 0 0 0 0 5 0 . 0 0 0 4 9 0 . 0 0 8 5 6 0 . 0 2 8 8 8

b x l 6 . 0 0 0 0 0 Mo ZERO ORDER C O F I R S T ORDER 0 . 0 0 0 0 0 SECOND OROER 0 . 0 0 0 0 0 T H I R D ORDER 0 . 0 0 0 0 0

SUMS a = 0 . 2 0 0 . 0 0 0 0 0 a » 0 . 4 0 0 . 0 0 0 0 0 a = 0 . 6 0 0 . 0 0 0 0 0 a = 0 . 8 0 0 . 0 0 0 0 0

a * 0 . 9 0 0 . 0 0 0 0 6 a = 0 . 9 5 0 . 0 0 0 5 9 a = 0 . 9 9 0 . 0 1 0 1 6 a = 1 . 0 0 0 . 0 3 4 0 7

0 . 0 0 0 0 5 0 . 0 0 0 0 6 0 . 0 0 0 4 8 0 . 0 0 0 6 4 0 . 0 0 8 6 6 0 . 0 1 1 4 3 0 . 0 2 9 4 C 0 . 0 3 8 7 2

0 . 5 C O 0 . 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 . 0 0 0 0 6 0 . 0 0 0 0 8 0 . 0 0 0 6 0 0 . 0 0 0 8 0 0 . 0 1 0 6 6 0 . 0 1 4 0 6 0 . 0 3 6 0 3 0 . 0 4 7 4 4

= 0 . 7 0 . 0 0 . 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 O.OOOOC 0 . 0 0 0 0 0

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 . 0 0 0 0 8 0 . 0 0 0 1 0 0 . 0 0 0 7 4 0 . 0 0 0 9 8 0 . 0 1 2 6 6 0 . 0 1 6 7 0 0 . 0 4 2 4 9 0 . 0 5 5 9 6

0 . 0 0 0 0 8 0 . 0 0 0 1 0 0 . 0 0 0 1 3 0 . 0 0 0 1 5 0 . 0 0 0 0 8 0 . 0 0 0 1 0 0 . 0 0 0 8 0 0 . 0 0 0 9 8 0 . 0 0 1 1 8 0 . 0 0 1 3 0 0 . 0 0 0 8 2 0 . 0 0 0 9 6 0 . 0 1 4 0 6 0 . 0 1 6 7 0 0 . 0 1 9 4 0 0 . 0 2 0 7 7 0 . 0 1 4 0 4 0 . 0 1 6 2 8 0 . 0 4 7 4 4 0 . 0 5 5 9 6 0 . 0 6 4 3 6 0 . 0 6 8 5 4 0 . 0 4 7 1 4 0 . 0 5 4 4 4

PEAK 0 . 0 0 . 0 0 . 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 0 0 0 1

0 . 0 0 0 1 0 0 . 0 0 0 1 3 0 . 0 0 0 1 7 0 . 0 0 0 1 9 0 . 0 0 0 1 1 0 . 0 0 0 1 3 0 . 0 0 1 0 0 0 . 0 0 1 2 3 0 . 0 0 1 4 8 0 . 0 0 1 6 3 0 . 0 0 1 0 2 0 . 0 0 1 2 1 0 . 0 1 7 3 1 0 . 0 2 0 5 6 0 . 0 2 3 8 7 0 . 0 2 5 5 7 0 . 0 1 7 2 8 0 . 0 2 0 0 4 0 . 0 5 8 1 4 0 . 0 6 8 5 7 0 . 0 7 8 8 7 0 . 0 8 3 9 9 0 . 0 5 7 7 7 0 . 0 6 6 7 1

0 . 0 PEAK 0 . 0 0 . 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 cooooo 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 0 0 0 2 0 . 0 0 0 0 1 0 . 0 0 0 0 1

0 . 0 0 0 1 3 0 . 0 0 0 1 6 0 . 0 0 0 2 1 0 . 0 0 0 2 4 0 . 0 0 0 1 4 0 . 0 0 0 1 6 0 . 0 0 1 2 3 0 . 0 0 1 5 0 0 . 0 0 1 8 1 0 . 0 0 1 9 9 0 . 0 0 1 2 5 0 . 0 0 1 4 7 0 . 0 2 0 5 6 0 . 0 2 4 4 1 0 . 0 2 8 3 5 0 . 0 3 0 3 6 0 . 0 2 0 5 3 0 . 0 2 3 8 0 0 . 0 6 8 5 7 0 . 0 8 0 8 8 0 . 0 9 3 0 3 0 . 0 9 9 0 6 0 . 0 6 8 1 4 0 . 0 7 8 6 8



VECTOR μ--0.0

b = 1 6 . 0 0 0 0 0 Μο

F I R S T OROER SECOND OROER THIRD ORDER

C . 2 7 7 7 8 0 . 0 9 3 4 0 0 . 0 5 3 0 8

SUMS c = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

0 . 0 5 9 7 8 0 . 1 3 0 7 6 0 . 2 2 0 2 8 0 . 3 4 8 5 7

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

0 . 4 5 0 2 0 C . 5 2 9 4 9 C . 6 4 7 5 8 C . 7 1 9 3 1

b = l 6 . 0 0 0 0 0 Mo


0 . 2 5 0 0 0 C . 0 8 6 6 4 0 . 0 5 0 2 5

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

0 . 0 5 3 9 3 0 . 1 1 8 3 4 C . 2 0 0 3 1 0 . 3 1 9 6 4

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

0 . 4 1 6 2 7 0 . 4 9 3 3 2 0 . 6 1 1 3 9 C . 6 8 5 2 2

μ=0.1 μ-0.3 μ=0.5 μ=0·7 μ =C ,9 μ =1.0 AVERAGE FLUX Ν U

2 0 8 3 3 0 . 1 7 8 5 7 0 . 1 5 6 2 5 0 . 1 3 8 8 9 1 1 5 8 7 0 . 1 0 9 0 9 0 . 1 0 1 0 6 0 . 0 9 3 4 0

0 . 1 3 1 5 8 0 . 1 8 6 8 0 0 . 0 8 9 8 4 0 . 1 0 6 1 6

0 . 0 5 5 1 8

1 6 3 7 5 1 0 2 0 3

0 . 0 7 5 0 8 0 . 0 7 5 0 3 0 . 0 7 2 3 3 0 . 0 6 8 7 9 0 . 0 6 6 9 3 0 . 0 7 1 3 3 0 . 0 7 1 6 8

0 . 0 4 7 0 0 0 . 0 4 0 7 8 0 . 0 3 5 9 8 0 . 0 3 2 1 7 0 . 0 3 0 5 5 0 . 0 4 2 2 8 0 . 0 3 7 5 1 0 . 1 0 8 6 5 0 . 0 9 5 7 6 0 . 0 8 5 3 8 0 . 0 7 6 9 4 0 . 0 7 3 3 0 0 . 0 9 8 2 3 0 . 0 8 8 4 5 0 . 1 9 5 9 2 0 . 1 7 6 4 2 0 . 1 5 9 7 0 0 . 1 4 5 5 7 0 . 1 3 9 3 4 0 . 1 7 8 7 2 0 . 1 6 4 0 8 0 . 3 4 0 0 0 0 . 3 1 6 6 8 0 . 2 9 4 0 1 0 . 2 7 3 3 7 0 . 2 6 3 8 8 0 . 3 1 5 8 6 0 . 2 9 8 5 3

. 4 7 4 9 3 0 . 4 6 9 7 9 0 . 4 5 0 3 4 0 . 4 2 7 8 0 0 . 4 0 5 3 5 0 , . 5 6 9 5 8 . 7 1 3 8 0 . 8 0 2 6 2

1 . 0

2 2 7 2 7 1 0 6 0 2 0 6 3 5 4

5 7 9 9 7 0 . 5 6 9 0 6 0 . 5 5 1 2 8 0 . 5 3 1 2 1 0 . 5 2 0 9 7 7 5 6 7 6 0 . 7 6 9 0 8 0 . 7 6 8 8 4 0 . 7 6 2 2 1 0 . 7 5 7 4 6 8 7 0 0 2 0 . 9 0 2 5 7 0 . 9 1 9 8 4 0 . 9 2 8 5 7 0 . 9 3 0 8 0

3 9 4 5 4 0 . 4 4 4 8 8 0 . 4 3 0 5 4 0 . 5 5 9 1 7 0 . 5 5 1 3 3 0 . 7 5 2 6 8 0 . 7 6 2 9 4 0 . 8 8 3 0 7 0 . 9 0 9 5 0

1 9 2 3 1 0 . 1 6 6 6 7 0 . 1 4 7 0 6 0 . 1 3 1 5 8 0 . 1 2 5 0 0 0 . 1 7 3 2 9 0 . 1 5 3 4 3 1 0 8 9 5 0 . 1 0 3 5 4 0 . 0 9 6 6 4 0 . 0 8 9 8 4 0 . 0 8 6 6 4 0 . 1 0 0 5 1 0 . 0 9 7 2 7 0 7 1 6 5 0 . 0 7 2 1 5 0 . 0 7 0 0 1 0 . 0 6 6 9 3 0 . 0 6 5 2 7 0 . 0 6 8 5 4 0 . 0 6 9 2 4

0 5 0 2 9 0 . 0 4 3 4 9 0 . 0 3 8 1 5 0 . 0 3 3 9 4 0 . 0 3 0 5 5 0 . 0 2 9 0 9 0 , 1 1 3 5 4 0 . 1 0 0 8 5 0 . 0 8 9 8 7 0 . 0 8 0 8 0 0 . 0 7 3 3 0 0 . 0 7 0 0 2 0 ,

1 8 2 7 2 0 . 1 6 6 3 7 0 . 1 5 1 8 7 0 . 1 3 9 3 4 0 . 1 3 3 7 5 0 , 3 0 1 1 6 0 . 2 8 1 9 5 0 . 2 6 3 8 8 0 . 2 5 5 4 3 0 ,

1 9 8 7 4 0 . 3 3 0 9 2 0 . 3 1 9 7 9

0 3 9 3 2 0 . 0 3 5 2 3 0 9 1 6 9 0 . 0 8 3 3 6 1 6 7 7 0 0 . 1 5 5 4 1 2 9 9 U 0 . 2 8 5 2 5

0 . 4 1 6 2 7 0 . 4 4 3 5 7 0 . 4 4 5 5 2 0 . 4 3 1 8 2 0 . 4 1 3 6 6 0 . 3 9 4 5 4 0 . 3 8 5 0 7 0 . 4 2 5 0 5 0 . 4 1 4 9 5 0 . 5 3 6 0 2 5 5 4 2 0

7 3 2 7 9 8 4 9 9 5

5 4 9 8 1 0 . 5 3 7 1 1 0 . 5 2 0 9 7 0 . 5 1 2 3 4 0 . 5 3 8 5 6 0 . 5 3 5 5 4 7 5 2 9 9 0 . 7 5 9 0 6 0 . 7 5 7 4 6 0 . 7 5 4 8 3 0 . 7 3 5 1 4 0 . 7 5 1 2 6 8 9 1 4 7 0 . 9 1 6 0 9 0 . 9 3 0 8 0 0 . 9 3 5 5 8 0 . 8 7 0 4 5 0 . 9 0 3 6 3

b - = l 6 . 0 0 0 0 0 NARROW SURFACE LAYER AT TOP

F I R S T ORDER SECOND ORDER THIRO ORDER

α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

0 = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

b = 1 6 . 0 0 0 0 0

I N F I N I T E 0 . 5 9 9 4 7 0 . 3 6 6 5 8 0 . 2 7 4 6 5 0 . 2 2 1 8 3 0 . 1 8 6 8 0 0 . 1 7 3 2 9 0 . 3 4 6 5 7 0 . 2 5 0 0 0 0 . 1 7 3 2 9 0 . 1 8 5 0 4 0 . 1 5 8 8 9 0 . 1 3 6 7 1 0 . 1 1 9 5 8 0 . 1 0 6 1 6 0 . 1 0 0 5 1 0 . 1 4 0 9 6 0 . 1 2 5 0 0 0 . 0 7 0 4 8 0 . 0 8 7 1 6 0 . 0 8 9 0 7 0 . 0 8 3 7 2 0 . 0 7 7 3 7 0 . 0 7 1 3 3 0 . 0 6 8 5 4 0 . 0 8 1 3 0 0 . 0 7 8 1 3


I N F I N I T E 0 . 8 6 0 7 2 0 . 6 5 2 2 5 0 . 5 5 6 0 3 0 . 4 9 2 3 9 0 . 4 4 4 8 8 0 . 4 2 5 0 5 0 . 6 1 5 2 4 0 . 5 1 9 4 9 I N F I N I T E 0 . 9 7 6 1 6 0 . 7 6 7 4 2 0 . 6 7 1 7 9 0 . 6 0 7 8 0 0 . 5 5 9 1 7 0 . 5 3 8 5 6 0 . 7 3 0 9 8 0 . 6 3 4 5 1 I N F I N I T E 1 . 1 2 3 8 5 0 . 9 3 0 4 9 0 . 8 4 7 7 5 0 . 7 9 3 7 2 0 . 7 5 2 6 8 0 . 7 3 5 1 4 0 . 9 0 4 5 0 0 . 8 1 7 1 9 I N F I N I T E 1 . 2 0 0 9 5 1 . C 2 3 7 3 0 . 9 5 5 0 1 0 . 9 1 3 2 7 0 . 8 8 3 0 7 0 . 8 7 0 4 5 1 . 0 1 0 0 0 0 . 9 3 3 7 2



SUMS α α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α 0 . 9 0 α - 0 . 9 5 α - 0 . 9 9 α 1 . 0 0

0 . 5 0 0 0 0 0 . 3 8 0 1 1 0 . 2 8 0 0 5 0 . 2 2 5 3 5 0 . 1 8 9 4 4 0 . 1 6 3 7 5 0 . 1 5 3 4 3 0 . 2 5 0 0 0 0 . 2 0 4 5 7 0 . 1 2 5 0 0 0 . 1 4 7 9 4 0 . 1 3 9 6 4 0 . 1 2 5 6 2 0 . 1 1 2 8 6 0 . 1 0 2 0 3 0 . 0 9 7 2 7 0 . 1 2 5 0 0 0 . 1 1 5 8 0 0 . 0 6 2 5 0 0 . 0 7 8 5 6 0 . 0 8 4 2 2 0 . 0 8 1 4 7 0 . 0 7 6 7 1 0 . 0 7 1 6 8 0 . 0 6 9 2 4 0 . 0 7 8 1 3 0 . 0 7 6 7 4

0 . 1 0 5 5 7 0 . C 8 2 6 6 0 . 0 6 2 3 8 0 . 0 5 0 8 5 0 . 0 4 3 1 2 0 . 0 3 7 5 1 0 . 0 3 5 2 3 0 . 0 5 5 7 3 0 . 0 4 6 2 6 C . 2 2 5 4 0 0 . 1 8 2 5 2 0 . 1 4 1 8 3 0 . 1 1 7 5 9 0 . 1 0 0 8 5 0 . 0 8 8 4 5 0 . 0 8 3 3 6 0 . 1 2 7 0 2 0 . 1 0 7 3 3 C . 3 6 7 5 4 0 . 3 0 9 7 7 0 . 2 4 9 9 9 0 . 2 1 2 0 8 0 . 1 8 4 8 5 0 . 1 6 4 0 8 0 . 1 5 5 4 1 0 . 2 2 5 1 5 0 . 1 9 4 7 2 0 . 5 5 2 7 9 0 . 4 9 0 7 1 0 . 4 1 8 3 6 0 . 3 6 7 9 7 0 . 3 2 9 3 9 0 . 2 9 8 5 3 0 . 2 8 5 2 5 0 . 3 8 1 9 7 0 . 3 4 1 8 7

0 . 6 8 3 7 7 0 . 6 2 9 3 4 0 . 5 6 0 0 2 0 . 5 0 7 9 4 0 . 4 6 5 7 8 0 . 4 3 0 5 4 0 . 4 1 4 9 5 0 . 5 1 9 4 9 0 . 4 7 8 0 2 C . 7 7 6 3 9 0 . 7 3 2 7 4 0 . 6 7 3 4 3 0 . 6 2 6 1 8 0 . 5 8 6 1 2 0 . 5 5 1 3 3 0 . 5 3 5 5 4 0 . 6 3 4 5 1 0 . 5 9 6 6 7 0 . 8 9 9 5 1 0 . 8 7 6 9 0 0 . 8 4 3 3 2 0 . 8 1 4 1 5 0 . 7 8 7 5 6 0 . 7 6 2 9 4 0 . 7 5 1 2 6 0 . 8 1 7 1 9 0 . 7 9 3 4 2 0 . 9 6 6 8 6 0 . 9 5 8 6 6 0 . 9 4 5 5 6 0 . 9 3 3 2 9 0 . 9 2 1 3 2 0 . 9 0 9 5 0 0 . 9 0 3 6 3 0 . 9 3 3 7 2 0 . 9 2 3 4 6


VECTOR



= 0 . 0 = 0 . 1 j = 0 . 3 μ*0·5 μ = 0.7 μ=0.9 / x « 1 . 0 AVERAGE Ν

FLUX U

b = 1 6 . 0 0 0 0 0 ZERO ORDER F I R S T ORDER SECOND ORDER THIRD ORDER

SUMS a= 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a = 0 . 9 0 a = 0 . 9 5 a= 0 . 9 9 a = i . 0 0

b * 1 6 . 0 0 0 0 C ZERO ORDER F I R S T ORDER SECOND ORDER THIRD ORDER

SUMS a = 0 . 2 0 a= 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

a= 0 . 9 0 a = 0 . 9 5 a= 0 . 9 9 a = i . C O

t = 1 6 . 0 0 0 0 0 ZERO ORDER F I R S T ORDER SECOND ORDER THIRD ORDER

SUMS a= 0 . 2 0 a = 0 . 4 0 a= 0 . 6 0 a= 0 . 8 0

a= 0 . 9 0 a= 0 . 9 5 a= 0 . 9 9 a= 1 .00

b = 1 6 . 0 0 0 0 0 ZERU ORDER F I R S T ORDER SECOND ORDER THIRD ORDER

0 . 0 0 0 0 0 0 . 0 0 0 0 0

SUMS a= 0 . 2 0 a = 0 . 4 0 a= 0 . 6 0 a = 0 . 8 0

a= 0 . 9 0 a= 0 . 9 5 a= 0 . 9 9 a= 1 .00

Mo = 0 . 9 O .C 0 . 0 C O 0 . 0 0 . 0 PEAK 0 . 0 COOOOO 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 C . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 COOOOO 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

C.COOOO COOOOO 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 COOOOO COOOOO 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 COOOOO COOOOC 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 COOOOO O.OOOOC 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 0 0 0 2 0 . 0 0 0 0 2 0 . 0 0 0 0 1 0 . 0 0 0 0 1

0 . 0 0 0 0 8 0 . 0 0 0 1 0 0 . 0 0 0 1 3 0 . 0 0 0 1 7 0 . 0 0 0 2 1 0 . 0 0 0 2 7 0 . 0 0 0 3 1 0 . 0 0 0 1 7 0 . 0 0 0 2 1 0 . 0 0 0 7 2 C . 0 0 C 8 9 0 . 0 0 1 1 8 0 . 0 0 1 4 8 0 . 0 0 1 8 1 0 . 0 0 2 1 9 0 . 0 0 2 4 0 0 . 0 0 1 5 1 0 . 0 0 1 7 8 0 . 0 1 1 8 0 0 . 0 1 4 7 1 0 . 0 1 9 4 0 0 . 0 2 3 8 7 0 . 0 2 8 3 5 0 . 0 3 2 9 2 0 . 0 3 5 2 6 0 . 0 2 3 8 4 0 . 0 2 7 6 3 0 . 0 3 9 1 9 0 . 0 4 8 8 8 0 . 0 6 4 3 6 0 . 0 7 8 8 7 0 . 0 9 3 0 3 0 . 1 0 7 0 0 0 . 1 1 3 9 4 0 . 0 7 8 3 7 0 . 0 9 0 5 0

0 . 0 0 0 0 0 0 . 0 0 0 0 0 Mo * I · 0 C O 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 PEAK COOOOO 0 . 0 0 0 0 0 COOOOO 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 O.OOOOC 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 O.OOOOC 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 . 0 0 0 0 0 O.OOOOC 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 COOOOO O.OOOOC 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 O.OOOOC 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 O.OOOOC 0 . 0 0 0 0 0 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 0 0 0 1 O.OOOCl 0 . 0 0 0 0 2 0 . 0 0 0 0 2 0 . 0 0 0 0 3 0 . 0 0 0 0 1 0 . 0 0 0 0 2

C 0 0 0 C 9 0 . 0 0 0 1 1 0 . 0 0 0 1 5 0 . 0 0 0 1 9 0 . 0 0 0 2 4 0 . 0 0 0 3 1 0 . 0 0 0 3 5 0 . 0 0 0 2 0 0 . 0 0 0 2 4 0 . 0 0 0 7 9 0 . 0 0 0 9 8 0 . 0 0 1 3 0 0 . 0 0 1 6 3 0 . 0 0 1 9 9 0 . 0 0 2 4 0 0 . 0 0 2 6 4 0 . 0 0 1 6 6 0 . 0 0 1 9 6 0 . 0 1 2 6 4 0 . 0 1 5 7 5 0 . 0 2 0 7 7 0 . 0 2 5 5 7 0 . 0 3 0 3 6 0 . 0 3 5 2 6 0 . 0 3 7 7 6 0 . 0 2 5 5 3 0 . 0 2 9 6 0 0 . 0 4 1 7 3 C . 0 5 2 0 5 0 . 0 6 8 5 4 0 . 0 8 3 9 9 0 . 0 9 9 0 6 0 . 1 1 3 9 4 0 . 1 2 1 3 4 0 . 0 8 3 4 6 0 . 0 9 6 3 7

NARROW SURFACE LAYER AT TOP C O 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 COOOOO 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 COOOOO 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 COOOOO COOOOO 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

O.OOOOC 0 . 0 0 0 0 0 0 . 0 0 0 0 0 O.OOOOC COOOOO O.OOOOC COOOOO 0 . 0 0 0 0 0

0 . 0 0 0 0 5 0 . 0 0 0 0 6 0 . 0 0 0 5 0 0 . 0 0 0 6 2 0 . 0 0 8 5 4 0 . C 1 0 6 5 0 . 0 2 8 7 0 0 . 0 3 5 8 0

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 0 0 0 1

0 . 0 0 0 0 8 0 . 0 0 0 1 1 0 . 0 0 0 1 4 0 . 0 0 0 1 7 0 . 0 0 0 2 0 0 . 0 0 0 1 1 0 . 0 0 0 1 4 0 . 0 0 0 8 2 0 . 0 0 1 0 2 0 . 0 0 1 2 5 0 . 0 0 1 5 1 0 . 0 0 1 6 6 0 . 0 0 1 0 4 0 . 0 0 1 2 3 0 . 0 1 4 0 4 0 . 0 1 7 2 8 0 . 0 2 0 5 3 0 . 0 2 3 8 4 0 . 0 2 5 5 3 0 . 0 1 7 2 6 0 . 0 2 0 0 1 0 . 0 4 7 1 4 0 . 0 5 7 7 7 0 . 0 6 8 1 4 0 . 0 7 8 3 7 0 . 0 8 3 4 6 0 . 0 5 7 4 0 0 . 0 6 6 2 8


C O 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

COOOOO COOOOO COOOOO COÔÔÔÔ Ô'.ÔOÔÔÔ 0 . 0 0 0 0 0 o . o o o o o o . o o o o o o . o o o o o ο 00000 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 COOOOO 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 COOOOO 0 . 0 0 0 0 0

0 . 0 0 0 0 6 0 . 0 0 0 0 8 C . 0 0 0 5 8 0 . 0 0 0 7 3 C . 0 0 9 9 0 0 . 0 1 2 3 4 0 . 0 3 3 1 4 0 . 0 4 1 3 4

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 Ο.σΟΟΟΟ 0 . 0 0 0 0 0 ' 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 0 0 0 2 0 . 0 0 0 0 1 0 . 0 0 0 0 1

0 . 0 0 0 1 0 0 . 0 0 0 1 3 0 . 0 0 0 1 6 0 . 0 0 0 2 1 0 . 0 0 0 2 4 0 . 0 0 0 1 4 0 . 0 0 0 1 6 0 . 0 0 0 9 6 0 . 0 0 1 2 1 0 . 0 0 1 4 7 0 . 0 0 1 7 8 0 . 0 0 1 9 6 0 . 0 0 1 2 3 0 . 0 0 1 4 5 0 . 0 1 6 2 8 0 . 0 2 0 0 4 0 . 0 2 3 8 0 0 . 0 2 7 6 3 0 . 0 2 9 6 0 0 . 0 2 0 0 1 0 . 0 2 3 2 0 0 . 0 5 4 4 4 0 . 0 6 6 7 1 0 . 0 7 8 6 8 0 . 0 9 0 5 0 0 . 0 9 6 3 7 0 . 0 6 6 2 8 0 . 0 7 6 5 4



VECTOR μ=0.0

b = 3 2 . 0 0 0 0 0 μ0 F I R S T ORDER SECOND ORDER THIRD ORDER

2 . 5 0 0 0 0 0 . 2 9 9 7 4 0 . 0 9 2 5 2

SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

0 . 5 1 2 8 1 1 . 0 5 5 3 6 1 . 6 3 7 0 2 2 . 2 7 7 6 2

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

2 . 6 3 7 3 2 2 . 8 3 8 6 8 3 . 0 3 1 5 7 3 . 1 0 9 0 5

b = 3 2 . 0 0 0 0 0 μο


0 . 8 3 3 3 3 0 . 1 8 3 2 9 0 . 0 7 9 4 5

SUMS a « 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

C . 1 7 4 7 2 0 . 3 6 9 3 0 0 . 5 9 2 9 3 0 . 8 6 7 0 6

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = 1 . 0 0

1 . 0 4 3 5 1 1 . 1 5 6 1 9 1 . 2 8 5 9 4 1 . 3 5 6 4 8

b * 3 2 . 0 0 0 0 0 μ0 F I R S T ORDER SECOND ORDER THIRD ORDER

0 . 5 0 0 0 0 0 . 1 3 7 3 3 0 . 0 6 8 3 6

SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

0 . 1 0 6 1 2 0 . 2 2 7 8 4 0 . 3 7 3 7 4 0 . 5 6 5 3 1

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a * 1 . 0 0

0 . 7 0 0 2 2 0 . 7 9 4 1 0 0 . 9 1 5 0 6 0 . 9 9 1 3 3

b = 3 2 . 0 0 0 0 0 μο


0 . 3 5 7 1 4 0 . 1 1 0 9 1 0 . 0 5 9 7 9

SUMS a = 0 . 2 0 « = 0 . 4 0 a = 0 . 6 0 a * 0 . 8 0

0 . 0 7 6 4 2 0 . 1 6 5 8 3 0 . 2 7 6 1 9 0 . 4 2 8 4 4

a = 0 . 9 0 a = 0 . 9 5 a = 0 . 9 9 a = I . 0 0

0 . 5 4 3 0 0 0 . 6 2 7 9 9 0 . 7 4 6 9 6 0 . 8 3 0 0 9

μ=0. 1 = 0 . 3 μ=0.5 μ=0.7 μ=0.9 L t = 1 . 0 AVERAGE Ν

0 . 1

2 5 0 0 0 2 9 9 7 4

0 . 6 2 5 0 0 0 . 4 1 6 6 7 0 . 3 1 2 5 0 0 . 2 5 0 0 0 0 . 2 2 7 2 7 0 . 5 9 9 4 7 0 . 0 . 2 1 2 4 0 0 . 1 6 4 3 9 0 . 1 3 4 5 2 0 . 1 1 4 0 4 0 . 1 0 6 0 2 0 . 1 8 5 0 4 0 . 0 . 0 9 9 2 0 0 . 0 8 6 1 0 0 . 0 7 5 5 2 0 . 0 6 7 1 0 0 . 0 6 3 5 4 0 . 0 8 7 1 6 0 ,

FLUX U

3 8 0 1 1 1 4 7 9 4 0 7 8 5 6

0 . 5 5 6 8 9 0 . 2 9 2 3 1 0 . 2 0 0 3 8 0 . 1 5 3 . 1 3 0 . 1 2 4 2 0 0 . 1 1 3 5 4 0 . 2 7 6 8 0 0 . 1 8 2 5 2 0 . 8 9 3 2 8 0 . 4 8 5 3 2 0 . 3 3 9 9 0 0 . 2 6 3 7 4 0 . 2 1 6 3 7 0 . 1 9 8 7 4 0 . 4 5 6 7 4 0 . 3 0 9 7 7 1 . 2 9 6 8 8 0 . 7 4 0 5 6 0 . 5 3 6 4 8 0 . 4 2 6 9 2 0 . 3 5 7 2 5 0 . 3 3 0 9 2 0 . 6 9 4 0 4 0 . 4 9 0 7 1

1 . 0 3 6 4 4 0 . 7 9 0 9 4

1 . 2 6 5 1 7 1 . 0 2 7 3 2

0 . 5 5 6 9 2 0 . 4 7 4 9 3 0 . 4 4 3 5 7 0 . 8 6 0 7 2 0 . 6 2 9 3 4 0 . 6 5 6 7 7 0 . 5 6 9 5 8 0 . 5 3 6 0 3 0 . 9 7 6 1 6 0 . 7 3 2 7 4 0 . 8 0 0 5 7 0 . 7 1 4 5 2 0 . 6 8 1 4 9 1 . 1 2 4 3 7 0 . 8 7 7 5 1 0 . 9 0 3 2 2 0 . 8 2 6 0 2 0 . 7 9 7 2 0 1 . 2 1 8 0 9 0 . 9 7 8 4 5

4 1 6 6 7 0 . 3 1 2 5 0 0 . 2 5 0 0 0 0 . 2 0 8 3 3 0 . 1 9 2 3 1 0 . 3 6 6 5 8 0 . 2 8 0 0 5 , 1 8 3 2 9 0 . 1 5 4 5 6 0 . 1 3 2 6 3 0 . 1 1 5 8 7 0 . 1 0 8 9 5 0 . 1 5 8 8 9 0 . 1 3 9 6 4

0 . 0 9 9 2 0 0 . 0 9 9 6 0 0 . 0 9 1 3 9 0 . 0 8 2 7 6 0 . 0 7 5 0 8 0 . 0 7 1 6 5 0 . 0 8 9 0 7 0 . 0 8 4 2 2

0 . 1 3 4 3 9 0 . 0 9 1 5 8 0 . 0 6 9 5 3 0 . 0 5 6 0 7 0 . 0 4 7 0 0 0 . 2 9 2 3 1 0 . 2 0 4 5 7 0 . 1 5 7 7 6 0 . 1 2 8 6 0 0 . 1 0 8 6 5 0 . 4 8 5 3 2 0 . 3 5 1 5 7 0 . 2 7 7 0 1 0 . 2 2 9 2 6 0 . 1 9 5 9 2 0 . 7 4 0 5 6 0 . 5 6 3 8 4 0 . 4 5 9 5 2 0 . 3 9 0 0 6 0 . 3 4 0 0 0

. 9 1 7 3 6 0 . 7 2 5 9 5 0 . 6 0 8 9 0

. 0 3 6 4 4 0 . 8 4 4 2 8 0 . 7 2 4 8 4 • 1 8 1 3 3 1 . 0 0 2 2 0 0 . 8 9 1 4 4

5 2 8 8 6 6 4 2 0 1

, 8 1 5 0 1 9 4 5 6 6

0 . 4 6 9 7 9 0 . 5 7 9 9 8 0 . 7 5 7 7 2 0 . 9 0 0 8 4

0 4 3 4 9 1 0 0 8 5 1 8 2 7 2 3 1 9 7 9

0 8 0 4 9 0 . 0 6 2 3 8 1 7 9 8 3 0 . 1 4 1 8 3 3 0 9 7 8 0 . 2 4 9 9 9 5 0 0 9 8 0 . 4 1 8 3 6

4 4 5 5 2 0 . 6 5 2 2 5 0 . 5 6 0 0 2 5 5 4 2 1 0 . 7 6 7 4 2 0 . 6 7 3 4 3 7 3 3 8 1 0 . 9 3 1 1 9 0 . 8 4 4 1 3

0 . 8 8 2 7 6 1 . 0 4 6 3 0 0 . 9 7 1 6 3

, 4 1 6 6 7 0 . 3 1 2 5 0 0 . 2 5 0 0 0 0 . 2 0 8 3 3 0 . 1 7 8 5 7 0 . 1 6 6 6 7 0 . 2 7 4 6 5 0 . 2 2 5 3 5 1 6 4 3 9 0 . 1 5 4 5 6 , 1 3 7 3 3

, 0 8 7 2 1 0 . 1 2 1 9 2 0 . 1 0 9 0 9 0 . 1 0 3 5 4 0 . 1 3 6 7 1 0 . 1 2 5 6 2 0 . 0 8 1 1 3 0 . 0 7 5 0 3 0 . 0 7 2 1 5 0 . 0 8 3 7 2 0 . 0 8 1 4 7

0 . 0 4 7 3 0 0 . 1 1 0 2 0

2 3 2 8 1 0 . 2 0 0 7 1

0 5 6 3 0 1 2 9 7 8

0 . 0 4 0 7 8 0 . 0 3 8 1 5 0 . 0 6 1 1 8 0 . 0 5 0 8 5 0 . 0 9 5 7 6 0 . 0 8 9 8 7 0 . 1 3 9 1 9 0 . 1 1 7 5 9 0 . 1 7 6 4 2 0 . 1 6 6 3 7 0 . 2 4 5 8 1 0 . 2 1 2 0 8

3 9 9 4 6 0 . 3 5 3 2 1 0 . 3 1 6 6 8 0 . 3 0 1 1 6 0 . 4 1 3 2 6 0 . 3 6 7 9 7

6 8 3 9 6 0 . 6 0 8 9 0 0 . 5 4 4 7 8 0 . 4 9 2 8 8 0 . 4 5 0 3 4 0 . 4 3 1 8 2 0 . 5 5 6 0 3 0 . 5 0 7 9 4 7 9 0 9 4 0 . 7 2 4 8 4 0 . 6 6 3 7 8 0 . 6 1 2 4 2 0 . 5 6 9 0 6 0 . 5 4 9 8 1 0 . 6 7 1 7 9 0 . 6 2 6 1 8 9 3 4 0 2 0 . 8 9 1 4 4 0 . 8 4 5 7 8 0 . 8 0 5 4 8 0 . 7 7 0 2 6 0 . 7 5 4 2 4 0 . 8 4 8 6 0 0 . 8 1 5 1 4 0 2 7 3 2 1 . 0 0 8 4 1 0 . 9 8 2 5 1 0 . 9 5 9 7 3 0 . 9 4 0 3 3 0 . 9 3 1 6 8 0 . 9 8 2 6 7 0 . 9 6 5 2 3

3 1 2 5 0 0 . 2 5 0 0 0 0 . 2 0 8 3 3 0 . 1 7 8 5 7 0 . 1 5 6 2 5 0 . 1 4 7 0 6 0 . 2 2 1 8 3 0 . 1 8 9 4 4 1 3 4 5 2 0 . 1 3 2 6 3 0 . 1 2 1 9 2 0 . 1 1 0 9 1 0 . 1 0 1 0 6 0 . 0 9 6 6 4 0 . 1 1 9 5 8 0 . 1 1 2 8 6 0 7 5 5 2 0 . 0 8 2 7 6 0 . 0 8 1 1 3 0 . 0 7 7 0 1 0 . 0 7 2 3 3 0 . 0 7 0 0 1 0 . 0 7 7 3 7 0 . 0 7 6 7 1

0 6 8 5 8 0 . 0 5 6 0 7 0 . 0 4 7 3 0 0 . 0 4 0 8 8 0 . 0 3 5 9 8 0 . 0 3 3 9 4 0 . 0 4 9 8 7 0 . 0 4 3 1 2 1 5 3 1 3 0 . 1 2 8 6 0 0 . 1 1 0 2 0 0 . 0 9 6 2 5 0 . 0 8 5 3 8 0 . 0 8 0 8 0 0 . 1 1 4 8 6 0 . 1 0 0 8 5 2 6 3 7 4 0 . 2 2 9 2 6 0 . 2 0 0 7 1 0 . 1 7 7 9 8 0 . 1 5 9 7 0 0 . 1 5 1 8 7 0 . 2 0 6 3 4 0 . 1 8 4 8 5 4 2 6 9 2 0 . 3 9 0 0 6 0 . 3 5 3 2 1 0 . 3 2 1 2 3 0 . 2 9 4 0 1 0 . 2 8 1 9 5 0 . 3 5 6 8 1 0 . 3 2 9 3 9

0 . 5 5 6 9 2 0 . 5 2 8 8 6 0 . 4 9 2 8 8 0 . 4 5 8 6 6 0 . 6 5 6 7 7 0 . 6 4 2 0 1 0 . 6 1 2 4 2 0 . 5 8 1 1 8

0 . 4 2 7 8 0 0 . 4 1 3 6 6 0 . 4 9 2 3 9 0 . 4 6 5 7 8 0 . 5 5 1 2 8 0 . 5 3 7 1 1 0 . 6 0 7 8 0 0 . 5 8 6 1 2

0 . 8 0 0 5 7 0 . 9 0 3 2 2

8 1 5 0 1 0 . 8 0 5 4 8 0 . 7 8 9 0 2 0 . 7 7 0 2 3 0 . 7 6 0 5 6 0 . 7 9 4 7 3 0 . 7 8 8 7 3 9 4 5 6 6 0 . 9 5 9 7 3 0 . 9 6 4 2 3 0 . 9 6 4 3 8 0 . 9 6 3 5 2 0 . 9 4 5 8 9 0 . 9 5 8 9 9


T A B L E 12 {continued) I n t e n s i t i e s o u t a t B o t t o m

VECTOR « . = 0 . 0 « . = 0 . 1 « . = 0 . 3 « . = 0 . 5 « . = 0 . 7 « , = 0 . 9 «.= 1 . 0 AVERAGE

Ν FLUX

U

b = 3 2 . 0 0 0 0 0 ZERO OROER F I R S T OROER SECONO OROER T H I R D ORDER

SUMS α : 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0

u-o = 0 . 1 C O PEAK 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 C O 0 . 0 0 . 0 0 . 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 C O O O O O 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 . 0 0 0 0 0 O.OOOOC 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 O.OOOOO 0 . 0 0 0 0 0 C O O O O O 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 O.OOGOO 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 . 9 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 9 5 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 9 9 0 . 0 0 0 3 3 0 . 0 0 0 4 1 0 . 0 0 0 5 5 0 . 0 0 0 6 7 0 . 0 0 0 8 0 0 . 0 0 0 9 3 0 . 0 0 0 9 9 0 . 0 0 0 6 7 0 . 0 0 0 7 8 1 . 0 0 0 . 0 0 9 3 3 0 . 0 1 1 6 4 0 . 0 1 5 3 3 0 . 0 1 8 7 8 0 . 0 2 2 1 5 0 . 0 2 5 4 8 0 . 0 2 7 1 3 0 . 0 1 8 6 6 0 . 0 2 1 5 5

b = 3 2 . 0 0 0 0 0 ZERO OROER F I R S T ORDER SECOND ORDER T H I R D ORDER

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

μο = 0 . 3 0 . 0 0 . 0 PEAK 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 O.OOOOC 0 . 0 0 0 0 0

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 O.OOOOO 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 C O O O O O 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 C O O O O O 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

C O O O O O 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 O.OOOOC 0 . 0 0 0 0 0 0 . 0 0 0 4 4 0 . 0 0 0 5 5 0 . 0 0 0 7 2 0 . 0 0 0 8 9 0 . 0 0 1 0 5 0 . 0 0 1 2 2 0 . 0 0 1 3 1 0 . 0 0 0 8 9 0 . 0 0 1 0 3 0 . 0 1 2 2 9 0 . 0 1 5 3 3 0 . 0 2 0 1 8 0 . 0 2 4 7 3 0 . 0 2 9 1 7 0 . 0 3 3 5 5 0 . 0 3 5 7 3 0 . 0 2 4 5 7 0 . 0 2 8 3 7

b = 3 2 . 0 0 0 0 0 ZERO ORDER F I R S T OROER SECOND OROER T H I R D ORDER

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

μο = 0 . 5 0 . 0 0 . 0 0 . 0 PEAK C O O O O O 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 C O O O O O 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 C O O O O O 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 5 4 0 . 0 0 0 6 7 0 . 0 0 0 8 9 0 . 0 0 1 0 9 0 . 0 1 5 0 6 0 . 0 1 8 7 8 0 . 0 2 4 7 3 0 . 0 3 0 3 1

0 . 0 0 . 0 0 . 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 1 3 0 0 . 0 0 1 5 1 0 . 0 0 1 6 1 0 . 0 0 1 0 9 0 . 0 0 1 2 6 0 . 0 3 5 7 4 0 . 0 4 1 1 1 0 . 0 4 3 7 8 0 . 0 3 0 1 1 0 . 0 3 4 7 7

b = 3 2 . 0 0 0 0 0 ZERO ORDER F I R S T ORDER SECONO ORDER THIRD ORDER

μ0 = 0 . 7 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

PEAK 0 . 0 0 . 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 O.OOOOO 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

SUMS a = 0 . 2 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 a = 0 . 4 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 a » 0 . 6 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 a * 0 . 8 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

a = 0 . 9 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 a = 0 . 9 5 C O O O O O 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 a = 0 . 9 9 0 . 0 0 0 6 4 0 . 0 0 0 8 0 0 . 0 0 1 0 5 0 . 0 0 1 3 0 a = 1 . 0 0 0 . 0 1 7 7 6 0 . 0 2 2 1 5 0 . 0 2 9 1 7 0 . 0 3 5 7 4

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 1 5 4 0 . 0 0 1 7 9 0 . 0 0 1 9 2 0 . 0 0 1 3 0 0 . 0 0 1 5 0 0 . 0 4 2 1 6 0 . 0 4 8 4 9 0 . 0 5 1 6 4 0 . 0 3 5 5 2 0 . 0 4 1 0 1




VECTOR μ=0.0

b = 3 2 . 0 0 0 0 0 Μο


0 . 2 7 7 7 8 0 . 0 9 3 4 0 0 . 0 5 3 0 8

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

0 . 0 5 9 7 8 C . 1 3 0 7 6 0 . 2 2 0 2 8 0 . 3 4 8 5 7

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

0 . 4 5 0 2 0 0 . 5 2 9 4 9 C . 6 4 8 1 6 0 . 7 3 8 0 7

b = 3 2 . 0 0 0 0 0 Μο


0 . 2 5 0 0 0 0 . 0 8 6 6 4 C . 0 5 0 2 5

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

0 . 0 5 3 9 3 0 . 1 1 8 3 4 0 . 2 0 0 3 1 0 . 3 1 9 6 4

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

0 . 4 1 6 2 7 C . 4 9 3 3 2 0 . 6 1 2 0 1 0 . 7 0 5 2 0

μ =0.1 μ =0.3 μ=0.5 μ = 0.7 /χ = 0 . 9 AVERAGE Ν

FLUX U

0 . 9

0 . 1 7 8 5 7 0 . 1 0 9 0 9

0 6 7 1 0 0 . 0 7 5 0 8 0 . 0 7 5 0 3 1 1 4 0 4 0 . 1 1 5 8 7

1 5 6 2 5 1 0 1 0 6 0 7 2 3 3

0 5 5 1 8 1 2 4 2 0 2 1 6 3 7 3 5 7 2 5

4 7 4 9 3 5 6 9 5 8

, 0 4 7 0 0 , 1 0 8 6 5 , 1 9 5 9 2 , 3 4 0 0 0

, 4 6 9 7 9 5 7 9 9 8

0 4 0 7 8 0 9 5 7 6

0 . 0 3 5 9 8 0 . 0 8 5 3 8

1 3 8 8 9 0 . 1 3 1 5 8 0 9 3 4 0 0 . 0 8 9 8 4 0 6 8 7 9 0 . 0 6 6 9 3

0 . 1 8 6 8 0 0 . 1 0 6 1 6 0 . 0 7 1 3 3

0 . 1 6 3 7 5 0 . 1 0 2 0 3 0 . 0 7 1 6 8

0 3 2 1 7 0 . 0 3 0 5 5 0 . 0 4 2 2 8 0 . 0 3 7 5 1 0 7 6 9 4 0 . 0 7 3 3 0 0 . 0 9 8 2 3 0 . 0 8 8 4 5

1 7 6 4 2 0 . 1 5 9 7 0 0 . 1 4 5 5 7 0 . 1 3 9 3 4 0 . 1 7 8 7 2 0 , 0 . 3 1 6 6 8 0 . 2 9 4 0 1 0 . 2 7 3 3 7 0 . 2 6 3 8 8 0 . 3 1 5 8 6 0

1 6 4 0 8 2 9 8 5 3

0 . 4 5 0 3 4 0 . 4 2 7 8 0 0 . 5 6 9 0 6 0 . 5 5 1 2 8

0 . 4 0 5 3 5 0 . 3 9 4 5 4 0 . 4 4 4 8 8 0 . 4 3 0 5 4 0 . 5 3 1 2 2 0 . 5 2 0 9 7 0 . 5 5 9 1 7 0 . 5 5 1 3 3 0 . 7 6 3 8 3 0 . 7 5 9 2 0 0 . 7 5 3 8 5 0 . 7 6 4 3 0 0 . 9 7 9 8 0 0 . 9 8 5 3 5 0 . 9 2 0 5 9 0 . 9 5 2 8 3

1 4 7 0 6 0 . 1 3 1 5 8 0 9 6 6 4 0 . 0 8 9 8 4

0 . 1 2 5 0 0 0 . 1 7 3 2 9 0 . 1 5 3 4 3 0 . 0 8 6 6 4 0 . 1 0 0 5 1 0 . 0 9 7 2 7 0 . 0 6 5 2 7 0 . 0 6 8 5 4 0 . 0 6 9 2 4

0 5 0 2 9 0 . 0 4 3 4 9 0 . 0 3 8 1 5 0 . 0 3 3 9 4 0 . 0 3 0 5 5 0 . 0 2 9 0 9 1 1 3 5 4 0 . 1 0 0 8 5 0 . 0 8 9 8 7 0 . 0 8 0 8 0 0 . 0 7 3 3 0 0 . 0 7 0 0 2 1 9 8 7 4 0 . 1 8 2 7 2 0 . 1 6 6 3 7 0 . 1 5 1 8 7 0 . 1 3 9 3 4 0 . 1 3 3 7 5 3 3 0 9 2 0 . 3 1 9 7 9 0 . 3 0 1 1 6 0 . 2 8 1 9 5 0 . 2 6 3 8 8 0 . 2 5 5 4 3

4 4 3 5 7 0 . 4 4 5 5 2 0 . 4 3 1 8 2 0 . 4 1 3 6 6 0 . 3 9 4 5 4 0 . 3 8 5 0 7 5 3 6 0 3 0 . 5 5 4 2 1 0 . 5 4 9 8 1 0 . 5 3 7 1 1 0 . 5 2 0 9 7 0 . 5 1 2 3 4

0 . 7 9 7 2 0 0 . 8 8 2 7 6 0 . 9 3 1 6 8 0 . 9 6 3 5 2 7 5 9 2 0 0 . 7 5 6 6 9 9 8 5 3 5 0 . 9 9 3 6 7

0 . 0 3 9 3 2 0 . 0 3 5 2 3 0 . 0 9 1 6 9 0 . 0 8 3 3 6 0 . 1 6 7 7 0 0 . 1 5 5 4 1 0 . 2 9 9 1 1 0 . 2 8 5 2 5

0 . 4 2 5 0 5 0 . 4 1 4 9 5 0 . 5 3 8 5 6 0 . 5 3 5 5 4 0 . 7 3 6 3 9 0 . 7 5 2 7 1 0 . 9 1 0 4 0 0 . 9 4 9 7 7

b = 3 2 . 0 0 0 0 0 NARROW SURFACE LAYER AT TOP


I N F I N I T E 0 . 5 9 9 4 7 0 . 3 6 6 5 8 0 . 2 7 4 6 5 0 . 2 2 1 8 3 0 . 1 8 6 8 0 0 . 1 7 3 2 9 0 . 3 4 6 5 7 0 . 2 5 0 0 0 0 . 1 7 3 2 9 0 . 1 8 5 0 4 0 . 1 5 8 8 9 0 . 1 3 6 7 1 0 . 1 1 9 5 8 0 . 1 0 6 1 6 0 . 1 0 0 5 1 0 . 1 4 0 9 6 0 . 1 2 5 0 0 0 . 0 7 0 4 8 0 . 0 8 7 1 6 0 . 0 8 9 0 7 0 . 0 8 3 7 2 0 . 0 7 7 3 7 0 . 0 7 1 3 3 0 . 0 6 8 5 4 0 . 0 8 1 3 0 0 . 0 7 8 1 3

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0


a = 0 . 9 0 I N F I N I T E 0 . 8 6 0 7 2 0 . 6 5 2 2 5 0 . 5 5 6 0 3 0 . 4 9 2 3 9 0 . 4 4 4 8 8 0 . 4 2 5 0 5 0 . 6 1 5 2 4 0 . 5 1 9 4 9 a = 0 . 9 5 I N F I N I T E 0 . 9 7 6 1 6 0 . 7 6 7 4 2 0 . 6 7 1 7 9 0 . 6 0 7 8 0 0 . 5 5 9 1 7 0 . 5 3 8 5 6 0 . 7 3 0 9 8 0 . 6 3 4 5 1 a = 0 . 9 9 I N F I N I T E 1 . 1 2 4 3 7 0 . 9 3 1 1 9 0 . 8 4 8 6 0 0 . 7 9 4 7 3 0 . 7 5 3 8 5 0 . 7 3 6 3 9 0 . 9 0 5 3 5 0 . 8 1 8 1 8 a = 1 . 0 0 I N F I N I T E 1 . 2 1 8 0 9 1 . 0 4 6 3 0 0 . 9 8 2 6 7 0 . 9 4 5 8 9 0 . 9 2 0 5 9 0 . 9 1 0 4 0 1 . 0 3 7 4 8 0 . 9 6 5 4 5

b = 3 2 . 0 0 0 0 0 LAMBERT SURFACE ON TOP


0 . 5 0 0 0 0 0 . 3 8 0 1 1 0 . 2 8 0 0 5 0 . 2 2 5 3 5 0 . 1 8 9 4 4 0 . 1 6 3 7 5 0 . 1 5 3 4 3 0 . 2 5 0 0 0 0 . 2 0 4 5 7 0 . 1 2 5 0 0 0 . 1 4 7 9 4 0 . 1 3 9 6 4 0 . 1 2 5 6 2 0 . 1 1 2 8 6 0 . 1 0 2 0 3 0 . 0 9 7 2 7 0 . 1 2 5 0 0 0 . 1 1 5 8 0 0 . 0 6 2 5 0 0 . 0 7 8 5 6 0 . 0 8 4 2 2 0 . 0 8 1 4 7 0 . 0 7 6 7 1 0 . 0 7 1 6 8 0 . 0 6 9 2 4 0 . 0 7 8 1 3 0 . 0 7 6 7 4

SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

C . 1 0 5 5 7 C . 0 8 2 6 6 0 . 0 6 2 3 8 0 . 0 5 0 8 5 0 . 0 4 3 1 2 0 . 0 3 7 5 1 0 . 0 3 5 2 3 0 . 0 5 5 7 3 0 . 0 4 6 2 6 0 . 2 2 5 4 0 0 . 1 8 2 5 2 0 . 1 4 1 8 3 0 . 1 1 7 5 9 0 . 1 0 0 8 5 0 . 0 8 8 4 5 0 . 0 8 3 3 6 0 . 1 2 7 0 2 0 . 1 0 7 3 3 0 . 3 6 7 5 4 0 . 3 0 9 7 7 0 . 2 4 9 9 9 0 . 2 1 2 0 8 0 . 1 8 4 & 5 0 . 1 6 4 0 8 0 . 1 5 5 4 1 0 . 2 2 5 1 5 0 . 1 9 4 7 2 0 . 5 5 2 7 9 0 . 4 9 0 7 1 0 . 4 1 8 3 6 0 . 3 6 , 7 9 7 0 . 3 2 9 3 9 0 . 2 9 8 5 3 0 . 2 8 5 2 5 0 . 3 8 1 9 7 0 . 3 4 1 8 7

a = 0 . 9 0 0 . 6 8 3 7 7 0 . 6 2 9 3 4 0 . 5 6 0 0 2 0 . 5 0 7 9 4 0 . 4 6 5 7 8 0 . 4 3 0 5 4 0 . 4 1 4 9 5 0 . 5 1 9 4 9 0 . 4 7 8 0 2 a = 0 . 9 5 C . 7 7 6 3 9 0 . 7 3 2 7 4 0 . 6 7 3 4 3 0 . 6 2 6 1 8 0 . 5 8 6 1 2 0 . 5 5 1 3 3 0 . 5 3 5 5 4 0 . 6 3 4 5 1 0 . 5 9 6 6 7 a = 0 . 9 9 0 . 9 0 0 0 0 0 . 8 7 7 5 1 0 . 8 4 4 1 3 0 . 8 1 5 1 4 0 . 7 8 8 7 3 0 . 7 6 4 3 0 0 . 7 5 2 7 1 0 . 8 1 8 1 8 0 . 7 9 4 5 6 a = I . 0 0 0 . 9 8 2 7 2 0 . 9 7 8 4 5 0 . 9 7 1 6 3 0 . 9 6 5 2 3 0 . 9 5 8 9 9 0 . 9 5 2 8 3 0 . 9 4 9 7 7 0 . 9 6 5 4 5 0 . 9 6 0 1 0



VECTOR M - 0 . 5 u - * 0 . 7 U . - 0 . 9 u - . i . O AVERAGE

Ν FLUX

U

b = 3 2 . 0 0 0 0 0 ZERO OROER F I R S T OROER SECONO OROER T H I R D OROER

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

μ0

3 0 . 9

0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 PEAK 0 . 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

α = 0 . 9 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 α = 0 . 9 5 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 L 0 . 0 0 0 0 1 0 . 0 0 0 0 0 0 . 0 0 0 0 0 α * 0 . 9 9 0 . 0 0 0 7 4 0 . 0 0 0 9 3 0 . 0 0 1 2 2 0 . 0 0 1 5 1 0 . 0 0 1 7 9 0 . 0 0 2 0 8 0 . 0 0 2 2 3 0 . 0 0 1 5 0 0 . 0 0 1 7 4 α = 1 . 0 0 0 . 0 2 0 4 3 0 . 0 2 5 4 8 0 . 0 3 3 5 5 0 . 0 4 1 1 1 0 . 0 4 8 4 9 0 . 0 5 5 7 7 0 . 0 5 9 3 9 0 . 0 4 0 8 5 0 . 0 4 7 1 7

b = 3 2 . 0 0 0 0 0 ZERO ORDER F I R S T OROER SECOND OROER THIRD OROER

μ0 = 1 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 PEAK 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

C O O O O O O.OOOOC 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 C O O O O O O.OOOOC 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

a = 0 . 9 0 0 . 0 0 0 0 0 . 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 a = 0 . 9 5 0 . 0 0 0 0 0 O.OOOOC 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 1 0 . 0 0 0 0 1 0 . 0 0 0 0 0 0 . 0 0 0 0 0 a = 0 . 9 9 0 . 0 0 0 8 0 0 . 0 0 0 9 9 0 . 0 0 1 3 1 0 . 0 0 1 6 1 0 . 0 0 1 9 2 0 . 0 0 2 2 3 0 . 0 0 2 3 8 0 . 0 0 1 6 1 0 . 0 0 1 8 7 a = 1 . 0 0 0 . 0 2 1 7 5 0 . 0 2 7 1 3 0 . 0 3 5 7 3 0 . 0 4 3 7 8 0 . 0 5 1 6 4 0 . 0 5 9 3 9 0 . 0 6 3 2 5 0 . 0 4 3 5 0 0 . 0 5 0 2 3

b = 3 2 . 0 0 0 0 0 ZERO OROER F I R S T ORDER SECOND ORDER T H I R D ORDER

NARROW SURFACE LAYER AT TOP 0 . 0 0 . 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 O.OOOOC 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

0 . 0 0 0 0 0 O.OOOOC 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 O.OOGOO 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

a = 0 . 9 0 0 . 0 0 0 0 0 C O O O O O 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . a = 0 . 9 5 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . a = 0 . 9 9 0 . 0 0 0 5 4 0 . 0 0 0 6 7 0 . 0 0 0 8 9 0 . 0 0 1 0 9 0 . 0 0 1 3 0 0 . 0 0 1 5 0 0 . 0 0 1 6 1 0 . a = 1 . 0 0 0 . 0 1 4 9 6 0 . 0 1 8 6 6 0 . 0 2 4 5 7 0 . 0 3 0 1 1 0 . 0 3 5 5 2 0 . 0 4 0 8 5 0 . 0 4 3 5 0 0 .

0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 1 0 9 0 . 0 0 1 2 6 0 2 9 9 2 0 . 0 3 4 5 5

b * 3 2 . 0 0 0 0 0 ZERO ORDER F I R S T ORDER SECOND ORDER T H I R D ORDER

LAMBERT SURFACE ON TOP 0 . 0 0 . 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 C O O O O O 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0

0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0

SUMS a * 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a « 0 . 8 0

0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 .

0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0

0 . 9 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 . 9 5 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 . 9 9 0 . 0 0 0 6 3 0 . 0 0 0 7 8 0 . 0 0 1 0 3 0 . 0 0 1 2 6 0 . 0 0 1 5 0 0 . 0 0 1 7 4 0 . 0 0 1 8 7 0 , 1 . 0 0 0 . 0 1 7 2 8 0 . 0 2 1 5 5 0 . 0 2 8 3 7 0 . 0 3 4 7 7 0 . 0 4 1 0 1 0 . 0 4 7 1 7 0 . 0 5 0 2 3 0 ,

0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 1 2 6 0 . 0 0 1 4 6 0 3 4 5 5 0 . 0 3 9 9 0




VECTOR μ = 0 . 0 ζ

M = 0 . 1 μ = 0 . 3 / a = 0 . 5 μ = 0 . 7 μ . = 0 . 9 μ = 1 . 0 AVERAGE Ν

FLUX U

b = ce


Mo 0 . 1

b = oo

F I R S T ORDER SECOND DRDER THIRD ORDER

SUMS α α = ο . ? ο α = 0 . 4 0 α - 0 . 6 0 α - 0 . 8 0

α 0 . 9 0

α = 0 . 9 5 α = 0 . 9 9

α = 1 . 0 0

2 . 5 0 0 0 0 1 . 2 5 0 0 0 0 . 6 2 5 0 0 0 . 4 1 6 6 7 0 . 3 1 2 5 0 0 . 2 5 0 0 0 0 . 2 2 7 2 7 0 . 5 9 9 4 7 0 . 3 8 0 1 1 0 . 2 9 9 7 4 0 . 2 9 9 7 4 0 . 2 1 2 4 0 0 . 1 6 4 3 9 0 . 1 3 4 5 2 0 . 1 1 4 0 4 0 . 1 0 6 0 2 0 . 1 8 5 0 4 0 . 1 4 7 9 4 0 . 0 9 2 5 ? 0 . 1 1 0 4 9 0 . 0 9 9 2 0 0 . 0 8 6 1 0 0 . 0 7 5 5 2 0 . 0 6 7 1 0 0 . 0 6 3 5 4 0 . 0 8 7 1 6 0 . 0 7 8 5 6

SUMS α = 0 . 2 0 0 . 5 1 2 8 1 0 . 2 6 2 9 7 0 . 1 3 4 3 9 0 . 0 9 0 7 0 0 . 0 6 8 5 8 0 . 0 5 5 1 8 0 . 0 5 0 2 9 0 . 1 2 8 0 9 0 . 0 8 2 6 6 α = 0 . 4 0 1 . 0 5 5 3 6 0 . 5 5 6 8 9 0 . 2 9 2 3 1 0 . 2 0 0 3 8 0 . 1 5 3 1 3 0 . 1 2 4 2 3 0 . 1 1 3 5 4 0 . 2 7 6 8 0 0 . 1 8 2 5 2 0 = 0 . 6 0 1 . 6 3 7 0 2 0 . 8 9 3 2 8 0 . 4 8 5 3 2 0 . 3 3 9 9 0 0 . 2 6 3 7 4 0 . 2 1 6 3 7 0 . 1 9 8 7 4 0 . 4 5 6 7 4 3 . 3 0 9 7 7 α = 0 . 8 0 2 . 2 7 7 6 2 1 . 2 9 6 8 8 0 . 7 4 0 5 6 0 . 5 3 6 4 8 0 . 4 2 6 9 2 0 . 3 5 7 2 5 0 . 3 3 0 9 2 0 . 6 9 4 0 4 0 . 4 9 0 7 1

α = C . 9 0 2 . 6 3 7 3 2 1 . 5 4 5 6 6 0 . 9 1 7 3 6 0 . 6 8 3 9 6 0 . 5 5 6 9 2 0 . 4 7 4 9 3 0 . 4 4 3 5 7 0 . 8 6 0 7 2 0 . 6 2 9 3 4 α = 0 . 9 5 2 . 8 3 8 6 8 1 . 6 9 6 4 4 1 . 0 3 6 4 4 0 . 7 9 0 9 4 0 . 6 5 6 7 7 3 . 5 6 9 5 8 3 . 5 3 6 0 3 0 . 9 7 6 1 6 0 . 7 3 2 7 4 α = 0 . 9 9 3 . 0 3 1 5 7 1 . 8 5 6 6 5 1 . 1 8 1 3 4 0 . 9 3 4 0 3 0 . 8 0 0 5 7 0 . 7 1 4 5 2 0 . 6 8 1 4 9 1 . 1 2 4 3 8 0 . 8 7 7 5 1 α = 1 . 0 0 3 . 1 1 8 3 8 1 . 9 4 4 8 5 1 . 2 8 C 5 0 1 . 0 4 6 1 0 0 . 9 2 5 3 7 0 . 8 5 1 5 0 0 . 8 2 4 3 3 1 . 2 3 6 7 5 I . 0 0 0 0 0

Μο = 0 - ^

C . 8 3 3 3 3 0 . 6 2 5 0 C 0 . 4 1 6 6 7 0 . 3 1 2 5 0 0 . 2 5 C 0 0 0 . 2 0 8 3 3 0 . 1 9 2 3 1 0 . 3 6 6 5 8 0 . 2 8 0 0 5 0 . 1 8 3 7 9 0 . 2 1 2 4 0 0 . 1 8 3 2 9 0 . 1 5 4 5 6 0 . 1 3 2 6 3 0 . 1 1 5 8 7 0 . 1 3 8 9 5 0 . 1 5 8 8 9 0 . 1 3 9 6 4 0 . 0 7 9 4 5 0 . 0 9 9 2 0 0 . 0 9 9 6 0 0 . 0 9 1 3 9 0 . 0 8 2 7 6 0 . 0 7 5 0 8 0 . 0 7 1 6 5 0 . 0 8 9 0 7 0 . 0 8 4 2 2

C . 1 7 4 7 2 0 . 1 3 4 3 9 0 . 0 9 1 5 9 0 . 0 6 9 5 3 0 . 0 5 6 0 7 0 . 0 4 7 0 0 0 . 0 4 3 4 9 0 . 0 8 0 4 9 0 . 0 6 2 3 8 0 . 3 6 9 3 0 0 . 2 9 2 3 1 0 . 2 0 4 5 7 0 . 1 5 7 7 6 0 . 1 2 8 6 0 0 . 1 0 8 6 5 0 . 1 3 0 8 5 0 . 1 7 9 8 3 0 . 1 4 1 8 3 0 . 5 9 2 9 3 0 . 4 8 5 3 2 0 . 3 5 1 5 7 0 . 2 7 7 0 1 0 . 2 2 9 2 6 0 . 1 9 5 9 2 0 . 1 8 2 7 2 0 . 3 0 9 7 8 0 . 2 4 9 9 9 0 , 8 6 7 0 6 0 . 7 4 0 5 6 0 . 5 6 3 8 4 0 . 4 5 9 5 2 0 . 3 9 0 0 6 0 . 3 4 0 0 0 0 . 3 1 9 7 9 0 . 5 0 0 9 8 0 . 4 1 8 3 6

1 . 0 4 3 5 1 0 . 9 1 7 3 6 0 . 7 2 5 9 5 0 . 6 0 8 9 0 0 . 5 2 8 8 6 0 . 4 6 9 7 9 0 . 4 4 5 5 2 0 . 6 5 2 2 5 0 . 5 6 0 0 2 1 . 1 5 6 1 9 1 . 0 3 6 4 4 0 . 8 4 4 2 8 0 . 7 2 4 8 4 0 . 6 4 2 0 1 3 . 5 7 9 9 8 3 . 5 5 4 2 1 0 . 7 6 7 4 2 0 . 6 7 3 4 3 1 . 2 8 5 9 4 1 . 1 8 1 3 4 I . C 0 2 2 0 0 . 8 9 1 4 4 0 . 8 1 5 0 1 0 . 7 5 7 7 2 0 . 7 3 3 8 1 0 . 9 3 1 1 9 0 . 8 4 4 1 3 1 . 3 6 8 7 7 1 . 2 8 0 5 0 1 . 1 2 4 1 2 1 . 0 3 3 1 4 0 . 9 7 4 8 3 0 . 9 3 4 3 9 0 . 9 1 8 4 9 1 . 0 7 0 8 7 1 . 0 0 0 0 0

b = 00

F I R S T ORDER SECOND 3RDER THIRD ORDFR

Mo 0 . 5

0 . 5 0 0 0 0 0 . 4 1 6 6 7 0 . 3 1 2 5 0 0 . 2 5 0 0 0 0 . 2 0 8 3 3 0 . 1 7 8 5 7 0 . 1 6 6 6 7 0 . 2 7 4 6 5 0 . 2 2 5 3 5 0 . 1 3 7 3 3 3 . 1 6 4 3 9 0 . 1 5 4 5 6 0 . 1 3 7 3 3 0 . 1 2 1 9 2 0 . 1 0 9 0 9 0 . 1 0 3 5 4 0 . 1 3 6 7 1 3 . 1 2 5 6 2 0 . 0 6 8 3 6 0 . C 8 6 1 C 0 . 0 9 1 3 9 0 . 0 8 7 2 1 0 . 0 8 1 1 3 0 . 0 7 5 0 3 0 . 0 7 2 1 5 0 . 0 8 3 7 2 0 . 0 8 1 4 7

SUMS ο = 0 . 2 0 0 . 1 0 6 1 2 0 . 0 9 0 7 C 0 . 0 6 9 5 3 0 . 0 5 6 3 0 0 . 0 4 7 3 0 0 . 0 4 0 7 8 0 . 0 3 8 1 5 0 . 0 6 1 1 8 0 . 0 5 0 8 5 a = 3 . 4 0 3 . 2 2 7 8 4 0 . 2 0 0 3 8 0 . 1 5 7 7 6 0 . 1 2 9 7 8 0 . 1 1 0 2 0 0 . 0 9 5 7 6 0 . 0 8 9 8 7 0 . 1 3 9 1 9 0 . 1 1 7 5 9 a = 0 . 6 0 0 . 3 7 3 7 4 0 . 3 3 9 9 0 0 . 2 7 7 0 1 0 . 2 3 2 8 1 0 . 2 0 0 7 1 0 . 1 7 6 4 2 0 . 1 6 6 3 7 0 . 2 4 5 8 1 0 . 2 1 2 0 8 a = O.RO 0 . 5 6 5 3 1 0 . 5 3 6 4 8 0 . 4 5 9 5 2 0 . 3 9 9 4 6 0 . 3 5 3 2 1 0 . 3 1 6 6 8 0 . 3 0 1 1 6 0 . 4 1 3 2 6 0 . 3 6 7 9 7

0 = 0 . 9 0 0 . 7 0 0 2 2 0 . 6 8 3 9 6 0 . 6 0 8 9 0 0 . 5 4 4 7 8 0 . 4 9 2 8 8 0 . 4 5 0 3 4 0 . 4 3 1 8 2 0 . 5 5 6 0 3 0 . 5 0 7 9 4 a = 0 . 9 5 3 . 7 9 4 1 0 0 . 7 9 0 9 4 0 . 7 2 4 8 4 0 . 6 6 3 7 8 0 . 6 1 2 4 2 0 . 5 6 9 3 6 0 . 5 4 9 8 1 0 . 6 7 1 7 9 0 . 6 2 6 1 8 a = 0 . 9 9 0 . 9 1 5 0 6 0 . 9 3 4 0 3 0 . 8 9 1 4 4 0 . 8 4 5 7 9 0 . 8 0 5 4 9 0 . 7 7 0 2 6 0 . 7 5 4 2 5 0 . 8 4 8 6 0 0 . 8 1 5 1 4 a = 1 . 0 0 1 . 0 0 6 3 9 1 . 0 4 6 1 C 1 . 0 3 3 1 4 1 . 0 1 2 8 2 0 . 9 9 5 4 8 0 . 9 8 1 4 4 0 . 9 7 5 4 6 1 . 0 1 2 7 8 1 . 0 0 0 0 0

b= oo

F I R S T ORDER SECOND ORDER THIRD ORDFR

SUMS a = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

Mo 0 . 7

0 . 3 5 7 1 4 C . 3 1 2 5 0 0 . 2 5 0 0 0 0 . 2 0 8 3 3 0 . 1 7 8 5 7 0 . 1 5 6 2 5 0 . 1 4 7 0 6 0 . 2 2 1 8 3 0 . 1 8 9 4 4 0 . 1 1 0 9 1 0 . 1 3 4 5 2 0 . 1 3 2 6 3 0 . 1 2 1 9 2 0 . 1 1 0 9 1 0 . 1 0 1 0 6 0 . 0 9 6 6 4 0 . 1 1 9 5 8 0 . 1 1 2 8 6 0 . 0 5 9 7 9 0 . 0 7 5 5 2 0 . 0 8 2 7 6 0 . 0 8 1 1 3 0 . 0 7 7 0 1 0 . 0 7 2 3 3 0 . 0 7 0 0 1 0 . 0 7 7 3 7 0 . 0 7 6 7 1

0 . 0 7 6 4 2 0 . 0 6 8 5 8 0 . C 5 6 0 7 0 . 0 4 7 3 0 0 . 0 4 0 8 8 0 . 0 3 5 9 8 0 . 0 3 3 9 4 0 . 0 4 9 8 7 0 . 0 4 3 1 2 0 . 1 6 5 8 3 0 . 1 5 3 1 3 0 . 1 2 8 6 0 0 . 1 1 0 2 0 0 . 0 9 6 2 5 0 . 0 8 5 3 8 0 . 0 8 0 8 0 0 . 1 1 4 8 6 0 . 1 0 0 8 5 0 . 2 7 6 1 9 0 . 2 6 3 7 4 0 . 2 2 9 2 6 0 . 2 0 0 7 1 0 . 1 7 7 9 8 0 . 1 5 9 7 0 0 . 1 5 1 8 7 0 . 2 0 6 3 4 0 . 1 8 4 8 5 0 . 4 2 3 4 4 0 . 4 2 6 9 2 0 . 3 9 0 0 6 0 . 3 5 3 2 1 0 . 3 2 1 2 3 0 . 2 9 4 0 1 0 . 2 8 1 9 5 0 . 3 5 6 8 1 0 . 3 2 9 3 9

a = 0 . 9 0 0 . 5 4 3 C 0 0 . 5 5 6 9 2 0 . 5 2 8 8 6 0 . 4 9 2 8 8 0 . 4 5 8 6 6 0 . 4 2 7 8 0 0 . 4 1 3 6 6 0 . 4 9 2 3 9 0 . 4 6 5 7 8 a = 0 . 9 5 0 . 6 2 7 9 9 0 . 6 5 6 7 7 0 . 6 4 2 0 1 0 . 6 1 2 4 2 0 . 5 8 1 1 8 0 . 5 5 1 2 8 3 . 5 3 7 1 1 0 . 6 0 7 8 3 0 . 5 8 6 1 2 a = 0 . 9 9 0 . 7 4 6 9 6 0 . 8 0 0 5 7 0 . 8 1 5 0 1 0 . 8 0 5 4 9 0 . 7 8 9 0 3 0 . 7 7 0 2 4 0 . 7 6 0 5 6 0 . 7 9 4 7 3 0 . 7 8 8 7 4 a = 1 . 0 0 0 . 8 4 7 8 5 0 . 9 2 5 3 7 0 . 9 7 4 8 3 0 . 9 9 5 4 8 1 . 0 0 6 3 9 1 . 0 1 2 8 7 1 . 0 1 5 1 6 0 . 9 8 1 4 1 1 . 0 0 0 0 0




VECTOR u = 0 . Ζ

« . = 0 . 1 « = 0 . 3 « . = 0 . 5 « . = 0 . 7 « . = 0 . 9 «. = 1 . 0 AVERAGE Ν

FLUX U

b= c o

F I R S T ORDER SECOND OROER THIRD ORDER

0 . 9

0 . 2 7 7 7 8 0 . 2 5 0 0 0 0 . 2 0 8 3 3 0 . 1 7 8 5 7 0 . 1 5 6 2 5 0 . 1 3 8 8 9 0 . 1 3 1 5 8 0 . 1 8 6 8 0 0 . 1 6 3 7 5 0 . 0 9 3 4 0 0 . 1 1 4 0 4 0 . 1 1 5 8 7 0 . 1 0 9 0 9 0 . 1 0 1 0 6 0 . 0 9 3 4 0 0 . 0 8 9 8 4 0 . 1 0 6 1 6 0 . 1 0 2 0 3 0 . 0 5 3 0 8 0 . 0 6 7 1 0 0 . 0 7 5 0 8 0 . 0 7 5 0 3 0 . 0 7 2 3 3 0 . 0 6 8 7 9 0 . 0 6 6 9 3 0 . 0 7 1 3 3 0 . 0 7 1 6 8

SUMS 0 = 0 . 2 0 0 . 0 5 9 7 8 0 . 0 5 5 1 8 0 . 0 4 7 0 0 0 . 0 4 0 7 8 0 . 0 3 5 9 8 0 . 0 3 2 1 7 0 . 0 3 0 5 5 0 . 0 4 2 2 8 0 . 0 3 7 5 1 α = 0 . 4 0 0 . 1 3 0 7 6 0 . 1 2 4 2 0 0 . 1 0 8 6 5 0 . 0 9 5 7 6 0 . 0 8 5 3 8 0 . 0 7 6 9 4 0 . 0 7 3 3 0 0 . 0 9 8 2 3 0 . 0 8 8 4 5 α = 0 . 6 0 0 . 2 2 0 2 8 0 . 2 1 6 3 7 0 . 1 9 5 9 2 0 . 1 7 6 4 2 0 . 1 5 9 7 0 0 . 1 4 5 5 7 0 . 1 3 9 3 4 0 . 1 7 8 7 2 0 . 1 6 4 0 8 α = 0 . 8 0 0 . 3 4 8 5 7 0 . 3 5 7 2 5 0 . 3 4 0 0 0 0 . 3 1 6 6 8 0 . 2 9 4 0 1 0 . 2 7 3 3 7 0 . 2 6 3 8 8 0 . 3 1 5 8 6 0 . 2 9 8 5 3

α = 0 . 9 0 0 . 4 5 0 2 0 0 . 4 7 4 9 3 0 . 4 6 9 7 9 0 . 4 5 0 3 4 0 . 4 2 7 8 0 0 . 4 0 5 3 5 0 . 3 9 4 5 4 0 . 4 4 4 8 8 0 . 4 3 0 5 4 α = 0 . 9 5 0 . 5 2 9 4 9 0 . 5 6 9 5 8 0 . 5 7 9 9 8 0 . 5 6 9 0 6 0 . 5 5 1 2 8 0 . 5 3 1 2 2 0 . 5 2 0 9 7 0 . 5 5 9 1 7 0 . 5 5 1 3 3 α = 0 . 9 9 0 . 6 4 8 1 6 0 . 7 1 4 5 2 0 . 7 5 7 7 2 0 . 7 7 0 2 6 0 . 7 7 0 2 4 0 . 7 6 3 8 4 0 . 7 5 9 2 0 0 . 7 5 3 8 6 0 . 7 6 4 3 1 α = I . 0 0 0 . 7 5 8 5 0 0 . 8 5 1 5 0 0 . 9 3 4 3 9 0 . 9 8 1 4 4 1 . 0 1 2 8 7 1 . 0 3 5 5 7 1 . 0 4 4 7 4 0 . 9 6 1 4 4 1 . 0 0 0 0 0

b = c o


Mo 1 . 0

SUMS α = 0 . 2 0 a = 0 . 4 0 a = 0 . 6 0 a = 0 . 8 0

α = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

b = cO I

0 . 2 5 0 0 0 0 . 2 2 7 2 7 0 . 1 9 2 3 1 0 . 1 6 6 6 7 0 . 1 4 7 0 6 0 . 1 3 1 5 8 0 . 1 2 5 0 0 0 . 1 7 3 2 9 0 . 1 5 3 4 3 0 . 0 8 6 6 4 0 . 1 0 6 0 2 0 . 1 0 8 9 5 0 . 1 0 3 5 4 0 . 0 9 6 6 4 0 . 0 8 9 8 4 0 . 0 8 6 6 4 0 . 1 0 0 5 1 0 . 0 9 7 2 7 0 . 0 5 0 2 5 0 . 0 6 3 5 4 0 . 0 7 1 6 5 0 . 0 7 2 1 5 0 . 0 7 0 0 1 0 . 0 6 6 9 3 0 . 0 6 5 2 7 0 . 0 6 8 5 4 0 . 0 6 9 2 4

0 . 0 5 3 9 3 0 . 0 5 0 2 9 0 . 0 4 3 4 9 0 . 0 3 8 1 5 0 . 0 3 3 9 4 0 . 0 3 0 5 5 0 . 0 2 9 0 9 0 . 0 3 9 3 2 0 . 0 3 5 2 3 0 . 1 1 8 3 4 0 . 1 1 3 5 4 0 . 1 0 0 8 5 0 . 0 8 9 8 7 0 . 0 8 0 8 0 0 . 0 7 3 3 0 0 . 0 7 0 0 2 0 . 0 9 1 6 9 0 . 0 8 3 3 6 0 . 2 0 0 3 1 0 . 1 9 8 7 4 0 . 1 8 2 7 2 0 . 1 6 6 3 7 0 . 1 5 1 8 7 0 . 1 3 9 3 4 0 . 1 3 3 7 5 0 . 1 6 7 7 0 0 . 1 5 5 4 1 0 . 3 1 9 6 4 p . 3 3 0 9 2 0 . 3 1 9 7 9 0 . 3 0 1 1 6 0 . 2 8 1 9 5 0 . 2 6 3 8 8 0 . 2 5 5 4 3 0 . 2 9 9 1 1 0 . 2 8 5 2 5

0 . 4 1 6 2 7 0 . 4 4 3 5 7 0 . 4 4 5 5 2 0 . 4 3 1 8 2 0 . 4 1 3 6 6 0 . 3 9 4 5 4 0 . 3 8 5 0 7 0 . 4 2 5 0 5 0 . 4 1 4 9 5 0 . 4 9 3 3 2 0 . 5 3 6 0 3 0 . 5 5 4 2 1 0 . 5 4 9 8 1 0 . 5 3 7 1 1 0 . 5 2 0 9 7 0 . 5 1 2 3 4 0 . 5 3 8 5 6 0 . 5 3 5 5 4 0 . 6 1 2 0 2 0 . 6 8 1 4 9 0 . 7 3 3 8 1 0 . 7 5 4 2 5 0 . 7 6 0 5 6 0 . 7 5 9 2 0 0 . 7 5 6 6 9 0 . 7 3 6 4 0 0 . 7 5 2 7 2 0 . 7 2 6 9 5 0 . 8 2 4 3 3 0 . 9 1 8 4 9 0 . 9 7 5 4 6 1 . 0 1 5 1 6 1 . 0 4 4 7 4 1 . 0 5 6 9 2 0 . 9 5 3 9 1 1 . 0 0 0 0 0


F I R S T OROER SECONO ORDER THIRD ORDER

I N F I N I T E 0 . 5 9 9 4 7 0 . 3 6 6 5 8 0 . 2 7 4 6 5 0 . 2 2 1 8 3 0 . 1 8 6 8 0 0 . 1 7 3 2 9 0 . 3 4 6 5 7 0 . 2 5 0 0 0 0 . 1 7 3 2 9 0 . 1 8 5 0 4 0 . 1 5 8 8 9 0 . 1 3 6 7 1 0 . 1 1 9 5 8 0 . 1 0 5 1 6 0 . 1 0 0 5 1 0 . 1 4 0 9 6 0 . 1 2 5 0 0 0 . 0 7 0 4 8 0 . 0 8 7 1 6 0 . 0 8 9 0 7 0 . 0 8 3 7 2 0 . 0 7 7 3 7 0 . 0 7 1 3 3 0 . 0 6 8 5 4 0 . 0 8 1 3 0 0 . 0 7 8 1 2

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α 0 . 8 0

α 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

b = c O I


I N F I N I T E 0 . 8 6 0 7 2 0 . 6 5 2 2 5 0 . 5 5 6 0 3 0 . 4 9 2 3 9 0 . 4 4 4 8 8 0 . 4 2 5 0 5 I N F I N I T E 0 . 9 7 6 1 6 0 . 7 6 7 4 2 0 . 6 7 1 7 9 0 . 6 0 7 8 D 0 . 5 5 9 1 7 0 . 5 3 8 5 6 I N F I N I T E 1 . 1 2 4 3 8 0 . 9 3 1 1 9 0 . 8 4 8 6 0 0 . 7 9 4 7 3 0 . 7 5 3 8 6 0 . 7 3 6 4 0 I N F I N I T E 1 . 2 3 6 7 5 1 . 0 7 0 8 7 1 . 0 1 2 7 8 0 . 9 8 1 4 1 0 . 9 6 1 4 4 0 . 9 5 3 9 1


0 . 6 1 5 2 4 0 . 5 1 9 4 9 0 . 7 3 0 9 8 0 . 6 3 4 5 1 0 . 9 0 5 3 5 0 . 8 1 8 1 8 1 . 0 6 7 4 0 1 . 0 0 0 0 0

F I R S T ORDER SECOND ORDER THIRO ORDER

SUMS α = 0 . 2 0 α = 0 . 4 0 α = 0 . 6 0 α = 0 . 8 0

ο = 0 . 9 0 α = 0 . 9 5 α = 0 . 9 9 α = 1 . 0 0

0 . 5 0 0 0 0 0 . 3 8 0 1 1 0 . 2 8 0 0 5 0 . 2 2 5 3 5 0 . 1 8 9 4 4 0 . 1 6 3 7 5 0 . 1 5 3 4 3 0 . 2 5 0 0 0 0 . 2 0 4 5 7 0 . 1 2 5 0 0 0 . 1 4 7 9 4 0 . 1 3 9 6 4 0 . 1 2 5 6 2 0 . 1 1 2 8 6 0 . 1 0 2 0 3 0 . 0 9 7 2 7 0 . 1 2 5 0 0 0 . 1 1 5 8 0 0 . 0 6 2 5 0 0 . 0 7 8 5 6 0 . 0 8 4 2 2 0 . 0 8 1 4 7 0 . 0 7 6 7 1 0 . 0 7 1 6 8 0 . 0 6 9 2 4 0 . 0 7 8 1 2 0 . 0 7 6 7 4

0 . 1 0 5 5 7 0 . 0 8 2 6 6 0 . 0 6 2 3 8 0 . 0 5 0 8 5 0 . 0 4 3 1 2 0 . 0 3 7 5 1 0 . 0 3 5 2 3 0 . 0 5 5 7 3 0 . Ό 4 6 2 6 0 . 2 2 5 4 0 0 . 1 8 2 5 2 0 . 1 4 1 8 3 0 . 1 1 7 5 9 0 . 1 0 0 8 5 0 . 0 8 8 4 5 0 . 0 8 3 3 6 0 . 1 2 7 0 2 0 . 1 0 7 3 3 0 . 3 6 7 5 4 0 . 3 0 9 7 7 0 . 2 4 9 9 9 0 . 2 1 2 0 8 0 . 1 8 4 8 5 0 . 1 6 4 0 8 0 . 1 5 5 4 1 0 . 2 2 5 1 5 0 . 1 9 4 7 2 0 . 5 5 2 7 9 0 . 4 9 0 7 1 0 . 4 1 8 3 6 0 . 3 6 7 9 7 0 . 3 2 9 3 9 0 . 2 9 8 5 3 0 . 2 8 5 2 5 0 . 3 8 1 9 7 0 . 3 4 1 8 7

0 . 6 8 3 7 7 0 . 6 2 9 3 4 0 . 5 6 0 0 2 0 . 5 0 7 9 4 0 . 4 6 5 7 8 0 . 4 3 0 5 4 0 . 4 1 4 9 5 0 . 5 1 9 4 9 0 . 4 7 8 0 2 0 . 7 7 6 3 9 0 . 7 3 2 7 4 0 . 6 7 3 4 3 0 . 6 2 6 1 8 0 . 5 8 6 1 2 0 . 5 5 1 3 3 3 . 5 3 5 5 4 0 . 6 3 4 5 1 0 . 5 9 6 6 7 0 . 9 0 0 0 0 0 . 8 7 7 5 1 0 . 8 4 4 1 3 0 . 8 1 5 1 4 0 . 7 8 8 7 4 0 . 7 6 4 3 1 0 . 7 5 2 7 2 0 . 8 1 8 1 8 0 . 7 9 4 5 6 1 . 0 0 0 0 0 1 . 0 0 0 0 0 1 . 0 0 0 0 0 1 . 0 0 0 0 0 1 . 0 0 0 0 0 1 . 0 0 0 0 0 1 . 0 0 0 0 0 1 . 0 0 0 0 0 1 . 0 0 0 0 0


T A B L E 15

T a b l e o f F l u x e s f o r I s o t r o p i c S c a t t e r i n g ( I n c i d e n t F l u x = 10 ,000 )

FOR b = 0 . 0 5 0 0 0 0 FOR b = 0 . 0 6 2 5 0 0 FOR b « 0 . 0 7 8 1 2 5

I S O T R O P I C GRA2 THIN LAMB PERP GRAZ T H I N LAMB PERP GRAZ THIN LAMB PERP

9 = 0 . 0 0 N U 1 0 N U 1 0 N U 1

a = 0 . 2 0 REFLECTED ABSORBED TRANSMITTED

1 0 1 6 1 6 0 8 1 4 1 1 4 0 5

8 4 3 8 4 3 5

8 3 4 5 7 3 6 3 9 8

9 1 8 1 9 5 5 7

1 0 1 9 1 8 6 8 1 6 5 1 6 5 1

8 1 6 8 1 6 3

1 0 0 5 5 9 0 5 4 9 6

8 9 9 5 9 4 4 9

1 0 2 1 2 1 3 8 1 9 4 1 9 3 5

7 8 5 7 8 5 2

1 2 0 6 7 1 1 1 0 6 1 8 8 7 7 0 9 3 1 5

0 = 0 . 4 0 REFLECTED ABSORBED TRANSMITTED

2 0 6 5 3 2 7 6 2 1 6 1 0 7 6 1 7 1 9 8 5 9 7

1 7 0 9 2 5 6 3 3 0 4

9 2 6 7 9 6 0 4

2 0 7 6 3 8 0 6 2 5 4 1 2 6 9 1 6 7 0 8 3 5 1

2 0 6 1 1 2 6 9 5 3 8 2

9 0 9 9 9 5 0 6

2 0 8 8 4 3 9 6 2 9 8 1 4 9 2 1 6 1 4 8 0 6 9

2 4 8 1 3 7 8 5 6 4 7 7

8 8 9 6 9 3 8 6

a = 0 . 6 0 REFLECTED 3 1 5 0 5 0 1 2 6 0 1 4 0 3 1 7 5 5 8 4 3 1 6 1 7 3 3 2 0 3 6 7 7 3 8 3 2 1 2 ABSORBED 4 2 2 0 7 3 2 3 8 3 2 0 8 4 2 6 0 8 6 7 4 7 5 2 6 0 4 3 0 7 1 0 2 4 5 8 7 3 2 8 TRANSMITTED 2 6 3 0 8 7 6 7 9 3 5 7 9 6 5 2 2 5 6 5 8 5 4 9 9 2 0 9 9 5 6 7 2 4 9 0 8 2 9 9 9 0 3 0 9 4 6 0

0 = 0 . 8 0 REFLECTED 4 2 7 3 6 8 2 3 5 4 1 9 1 4 3 1 9 7 9 8 4 3 2 2 3 6 4 3 7 2 9 3 0 5 2 6 2 9 2 ABSORBED 2 1 4 9 3 7 4 1 9 6 1 0 6 2 1 7 8 4 4 5 2 4 4 1 3 4 2 2 1 1 5 2 7 3 0 3 1 6 9 TRANSMITTED 3 5 7 8 8 9 4 4 9 4 5 0 9 7 0 3 3 5 0 3 8 7 5 7 9 3 2 4 9 6 3 0 3 4 1 7 8 5 4 3 9 1 7 1 9 5 3 9

a = 0 . 9 0 REFLECTED 4 8 4 9 7 7 6 4 0 2 2 1 7 4 9 0 9 9 0 9 4 9 2 2 6 9 4 9 7 8 1 0 6 2 6 0 1 3 3 3 ABSORBED 1 0 8 5 1 8 9 9 9 5 4 1 1 0 1 2 2 5 1 2 4 6 8 1 1 2 0 2 6 8 1 5 3 8 6 TRANSMITTED 4 0 6 6 9 0 3 5 9 4 9 9 9 7 2 9 3 9 9 0 8 8 6 6 9 3 8 4 9 6 6 3 3 9 0 2 8 6 7 0 9 2 4 6 9 5 8 1

0 = 0 . 9 5 REFLECTED 5 1 4 1 8 2 3 4 2 7 2 3 1 5 2 0 9 9 6 6 5 2 3 2 8 6 5 2 8 6 1 1 2 9 6 3 9 3 5 5 ABSORBED 5 4 5 9 5 5 0 2 6 5 5 4 1 1 3 6 2 3 4 5 6 5 1 3 5 7 7 4 3 TRANSMITTED 4 3 1 4 9 0 8 2 9 5 2 3 9 7 4 3 4 2 3 7 8 9 2 1 9 4 1 5 9 6 8 0 4 1 4 9 8 7 3 6 9 2 8 4 9 6 0 2


5 3 7 6 8 6 2 1 1 0 1 9

4 5 1 4 9 1 1 9

4 4 7 2 4 1 1 0 6

9 5 4 3 9 7 5 3

5 4 5 1 1 0 1 2 1 1 1 2 3

4 4 3 8 8 9 6 5

5 4 8 3 0 0 1 3 7

9 4 3 9 9 6 9 3

5 5 3 6 1 1 8 4 1 1 4 2 7

4 3 5 0 8 7 8 9

6 7 0 3 7 2 1 6 9

9 3 1 4 9 6 1 9

a = l . 0 0 REFLECTED 5 4 3 6 8 7 1 4 5 2 2 4 4 5 5 1 2 1 0 2 3 5 5 4 3 0 3 5 5 9 9 1 1 9 8 6 7 8 3 7 6 ABSORBED C 0 0 0 0 0 0 0 0 0 0 0 TRANSMITTED 4 5 6 4 9 1 2 9 9 5 4 8 9 7 5 6 4 4 8 8 8 9 7 7 9 4 4 6 9 6 9 7 4 4 0 1 8 8 0 2 9 3 2 2 9 6 2 4

ANY α DIRECTLY TRANSMITTED 0 8 2 7 8 9 0 9 8 9 5 1 2 0 7 9 8 4 8 8 9 5 9 3 9 4 0 7 6 4 8 8 6 5 1 9 2 4 8



FOR b = O . l O 0 0 O 0 FOR b = 0 . 1 2 5 0 0 0 FOR b = 0 . 1 5 6 2 5 C

I S O T R O P I C GRAZ THIN LAMB PERP GRAZ THIN LAMB PERP GRAZ THIN LAMB PERP

g = C O 0 N U 1 0 N U 1 O N U l

α = 0 . 2 0 REFLECTED ABSORBED TRANSMITTED

1 0 2 5 2 4 7 8 2 2 9 2 2 9 4

7 4 6 7 4 5 9

1 4 6 8 3 1 3 8 4 7 8 6 8 4 7 0 9 1 3 1

1 0 2 8 2 7 9 8 2 6 6 2 6 6 4

7 0 6 7 0 5 7

1 7 2 9 9 1 6 8 4 9 7 7 8 1 4 4 8 9 2 4

1 0 3 1 3 1 4 8 3 0 8 3 0 7 8

6 6 1 6 6 0 8

2 0 1 1 1 9 2 0 3 8 1 2 1 0 7 7 6 1 8 6 7 1

a = C 4 0 REFLECTED ABSORBED TRANSMITTED

2 1 0 2 5 1 0 6 3 5 6 1 7 8 0 1 5 4 2 7 7 1 0

3 0 2 1 7 1 1 0 7 5 6 1 1 8 6 2 3 9 2 1 8

2 1 1 6 5 8 1 6 4 1 6 2 0 7 9 1 4 6 8 7 3 4 0

3 5 9 2 0 7 1 3 1 4 7 6 3 8 3 2 7 9 0 3 0

2 1 3 1 6 5 6 6 4 8 4 2 4 2 0 1 3 8 5 6 9 2 4

4 2 2 2 4 9 1 6 0 2 9 5 2 7 9 7 6 8 7 9 9


3 2 3 8 7 9 2 4 3 6 8 1 2 2 9 2 3 9 4 7 9 7 9

4 6 9 2 6 5 7 4 3 4 2 2

8 7 8 8 9 3 1 3

3 2 7 2 9 0 7 4 4 3 4 1 4 4 6 2 2 9 4 7 6 4 7

5 6 1 3 2 3 9 1 4 5 3 1

8 5 2 5 9 1 4 6

3 3 1 0 1 0 3 2 4 5 0 8 1 6 9 5 2 1 8 2 7 2 7 3

6 6 5 3 9 2 1 1 2 3 6 6 7 8 2 1 2 8 9 4 1

α = 0 . 8 0 REFLECTED 4 4 3 8 1 0 9 5 6 4 9 3 6 7 4 5 0 5 1 2 6 2 7 8 1 4 5 0 4 5 7 9 1 4 4 7 9 3 4 5 5 0 ABSORBED 2 2 5 5 6 3 7 3 8 5 2 1 9 2 3 0 2 7 5 5 4 7 8 2 7 8 2 3 5 7 8 9 3 5 9 1 3 5 2 TRANSMITTED 3 3 0 7 8 2 6 8 8 9 6 6 9 4 1 4 3 1 9 3 7 9 8 3 8 7 4 1 9 2 7 2 3 0 6 4 7 6 6 0 8 4 7 5 9 0 9 8


5 0 6 4 1 2 5 4 1 1 4 7 3 2 5 3 7 8 9 8 4 2 1

7 4 4 4 2 1 1 9 6 1 1 2

9 0 6 0 9 4 6 7

5 1 5 3 1 4 5 2 1 1 7 4 3 8 6 3 6 7 3 8 1 6 2

8 9 9 5 1 8 2 4 4 1 4 2

8 8 5 7 9 3 4 0

5 2 5 2 1 6 7 1 1 2 0 6 4 5 9 3 5 4 2 7 8 7 0

1 0 8 0 6 3 6 3 0 3 1 8 1

8 6 1 7 9 1 8 3

a = 0 . 9 5 REFLECTED 5 3 8 5 1 3 3 7 7 9 3 4 4 9 5 4 8 6 1 5 4 9 9 6 0 5 5 3 5 5 9 9 1 7 8 8 1 1 5 5 6 8 1 ABSORBED 5 7 8 1 6 3 9 9 5 6 5 9 3 1 9 6 1 2 3 7 2 6 1 1 2 3 2 1 5 4 9 1 TRANSMITTED 4 0 3 7 8 5 0 0 9 1 0 8 9 4 9 5 3 9 2 1 8 2 5 5 8 9 1 7 9 3 7 5 3 7 9 0 7 9 8 0 8 6 9 1 9 2 2 8


5 6 4 5 1 4 0 3 1 1 6 3 3

4 2 3 9 8 5 6 4

8 3 3 4 7 1 2 0 1 2

9 1 4 7 9 5 1 7

5 7 5 7 1 6 2 9 1 1 9 4 0

4 1 2 4 8 3 3 1

1 0 0 9 5 8 2 2 5 1 5

8 9 6 6 9 4 0 3

5 8 8 2 1 8 8 3 1 2 4 4 7

3 9 9 4 8 0 7 0

1 2 1 7 7 1 7 3 2 1 9

8 7 5 1 9 2 6 4

a = l . 0 0 REFLECTED 5 7 1 0 1 4 2 0 8 4 3 4 7 7 5 8 2 5 1 6 5 0 1 0 2 2 5 8 9 5 9 5 4 1 9 0 8 1 2 3 3 7 2 7 ABSORBED 0 0 0 0 0 0 0 0 0 0 0 0 TRANSMITTED 4 2 9 0 8 5 8 0 9 1 5 7 9 5 2 3 4 1 7 5 8 3 5 0 8 9 7 8 9 4 1 1 4 0 4 6 8 0 9 2 8 7 6 7 9 2 7 3

ANY a DIRECTLY TRANSMITTED 0 7 2 2 5 8 3 2 6 9 0 4 8 0 6 7 9 6 7 9 7 6 8 8 2 5 0 6 3 2 0 7 5 6 6 8 5 5 3



FOR b = 0 . 2 C 0 0 0 0

I S O T R O P I C GRAZ THIN LAMB PERP

0 . 0 0 Ν U 1

FOR b = 0 . 2 5 0 0 0 0

GRAZ THIN LAMB PERP

0 N U I

FOR b = 0 . 3 1 2 5 0 0

GRAZ THIN LAMB PERP

0 N U I

α = 0 . 2 0 REFLECTED ABSORBED TRANSMlTTED

1 0 3 5 3 5 2 2 3 7 1 4 3 1 0 3 9 3 8 8 2 7 0 1 6 7 1 0 4 2 4 2 2 3 0 4 1 9 3 8 3 5 9 3 5 9 1 2 4 9 8 1 5 2 9 8 4 1 0 4 0 9 9 2 9 8 3 1 8 8 2 8 4 6 5 4 6 4 8 3 5 3 3 2 3 0 5

6 0 6 6 0 5 7 7 2 6 5 8 3 2 8 5 5 1 5 5 1 3 6 7 4 7 7 9 5 1 4 9 3 4 9 3 0 6 1 6 3 7 5 0 2


2 1 4 9 6 5 6 9 1 2 8 2

7<.3 2 8 4 8 6 4 0 9

5 0 0 1 9 8 2 7 5 1 8

3 0 2 1 2 1 4 8 4 8 4

2 1 6 5 8 2 4 5 7 6 6 6 5 6 3 2 8 1 2 3 8 8 1 1 7 9 5 8 9 5 7 0 3 6

3 5 6 1 5 0 8 8 1 3 6

2 1 8 1 6 7 5 1 1 0 6 8

9 0 4 3 7 5 8 5 3 3 8

6 5 4 2 8 5 9 6 4 8 7

4 1 6 1 8 6 6 7 7 1 8


3 3 5 4 4 6 0 4 2 0 4 2

1 1 8 0 2 C 1 4 6 8 OA

7 9 5 1 4 0 3 7 8 0 2

4 8 1 8 5 9

8 6 6 C

3 3 9 6 1 3 2 0 9 2 5 4 7 0 3 2 3 4 5 1 7 0 8 1 9 0 1 6 3 3 5 7 3 6 7

5 7 3 1 0 7 9 8 3 4 8

3 4 3 9 4 8 1 5 1 7 4 6

1 4 6 3 2 7 1 8 5 8 1 9

1 0 6 3 2 0 7 0 6 8 6 7

6 7 8 1 3 5 0 7 9 7 2


4 6 6 9 1 6 7 1 1 1 2 9 6 8 3 2 4 2 9 1 0 7 3 7 4 7 4 5 7 2 9 0 2 7 2 5 6 8 1 2 4 8 8 6 0

4 7 5 6 1 8 9 0 1 3 2 7 2 5 0 5 1 2 6 3 9 2 2 2 7 3 9 6 8 4 7 7 7 5 1

8 2 4 5 8 1

8 5 9 5

4 8 4 8 2 5 9 4 2 5 5 8

2 1 2 0 1 4 8 6 6 3 9 4

1 5 4 6 1 1 3 2 7 3 2 2

9 8 7 7 3 9

8 2 7 4

a = 0 . 9 0 REFLECTED ABSORBED TRANSMlTTED

5 3 7 3 1 9 4 1 1 3 1 2 1 2 5 0 5 5 4 3'86 3 3 7 7 7 5 0 5 8 3 0 2

7 9 4 2 3 6

8 9 7 0

5 4 9 3 2 2 0 8 1 5 5 3 1 2 9 6 6 5 7 4 8 0 3 2 1 1 7 1 3 5 7 9 6 7

9 6 5 3 0 2

8 7 3 3

5 6 2 3 1 3 5 0 3 0 2 7

2 4 9 4 7 7 9

6 7 2 7

1 8 2 3 5 9 4

7 5 8 3

1 1 6 5 3 8 8

8 4 4 7


5 7 3 9 2 0 8 2 1 4 0 8 6 3 4 2 8 2 1 9 7

3 6 2 7 7 6 3 6 8 3 9 5

8 5 2 1 2 1

9 0 2 7

5 8 7 9 2 3 7 6 6 5 9 3 3 6

3 4 6 2 7 2 8 8

1 6 7 3 2 4 5

8 0 8 2

1 0 3 9 1 5 5

8 8 0 6

6 0 3 0 6 9 0

3 2 8 0

2 6 9 5 3 9 9

6 9 0 6

1 9 7 2 3 0 4

7 7 2 4

1 2 6 1 1 9 8

8 5 4 1


6 0 3 9 2 1 9 9 1 4 8 8 1 2 8 5 7 4 0

3 8 3 3 7 7 4 4 8 4 7 2

9 0 0 2 5

9 0 7 5

6 1 9 5 2 5 1 6 1 7 7 2 1 3 4 6 8 5 0

3 6 7 1 7 4 1 6 8 1 7 8

1 1 0 1 3 2

8 8 6 7

6 3 6 7 1 4 0

3 4 9 3

2 8 6 3 8 1

7 0 5 6

2 0 9 6 6 2

7 8 4 2

1 3 4 1 4 0

8 6 1 9


6 1 1 4 0

3 8 8 6

2 2 2 9 1 5 0 8 0 0

7 7 7 1 8 4 9 2

9 1 3 0

9 0 8 7

6 2 7 6 0

3 7 2 4

2 5 5 2 0

7 4 4 8

1 7 9 8 0

8 2 0 2

1 1 1 7 0

8 8 8 3

6 4 5 3 0

3 5 4 7

2 9 0 5 0

7 0 9 5

2 1 2 8 0

7 8 7 2

1 3 6 1 0

8 6 3 9




I S O T R O P I C

q = 0 . 0

GRAZ

0

FOR b = 0 . 4 0 0 0 0 0

THIN LAMB PERP GRAZ

0

FOR b = 0 . 5 0 0 0 0 0

THIN LAMB PERP

FOR b = C . 6 2 5 0 0 0

GRAZ THIN LAMB PERP

Ν U 1

α = 0 · 2 0 REFLECTED ABSORBED TRANSMITTED

1 0 4 6 4 5 6 8 5 2 9 5 2 9 4

4 2 5 4 2 5 0

3 4 1 2 2 3 4 2 1 3 2 8 6 3 5 4 4 6 6 9 1 4

1 0 4 8 4 8 4 8 5 9 0 5 8 9 8

3 6 2 3 6 1 8

3 7 2 2 5 0 4 8 8 5 3 4 5 4 4 7 4 3 6 2 9 6

1 0 5 1 5 0 7 8 6 5 0 6 5 0 7

2 9 9 2 9 8 6

3 9 9 2 7 6 5 5 9 4 4 1 2 7 4 0 0 7 5 5 9 7

a « 0 . 4 0 REFLECTED 2 1 9 8 9 8 9 7 4 2 4 8 7 2 2 1 2 1 0 5 9 8 1 8 5 5 3 2 2 2 4 1 1 2 0 8 8 7 6 1 7 ABSORBED 6 8 6 7 4 3 3 4 3 4 5 4 2 3 4 8 6 9 7 8 4 8 9 0 4 0 5 7 2 8 7 0 7 0 9 4 5 4 6 8 4 7 1 2 3 4 7 9 TRANSMITTED 9 3 5 4 6 7 7 5 8 0 4 7 1 6 5 8 1 0 4 0 5 1 5 1 2 5 6 5 7 7 6 8 2 3 4 1 2 4 4 0 1 5 9 0 4

a = 0 . 6 0 REFLECTED 3 4 8 6 1 6 2 0 1 2 2 2 8 0 5 3 5 2 7 1 7 5 6 1 3 6 5 9 2 7 3 5 6 4 1 8 8 1 1 5 0 3 1 0 5 2 ABSORBED 4 9 5 5 3 1 8 2 2 5 4 0 1 7 2 6 5 0 9 4 3 6 4 6 3 0 3 2 2 1 4 5 5 2 4 5 4 1 4 8 3 5 8 3 2 6 4 7 TRANSMITTED 1 5 5 9 5 1 9 8 6 2 3 8 7 4 6 9 1 3 7 9 4 5 9 8 5 6 0 3 6 9 2 8 1 1 9 1 3 9 7 1 4 9 1 4 6 3 0 1


4 9 5 4 2 3 8 4 2 7 0 8 1 7 7 2 2 3 3 8 5 8 4 4

1 8 0 8 1 1 9 4 1 4 1 6 9 6 3 6 7 7 6 7 8 4 3

5 0 5 0 2 6 2 5 2 8 2 8 2 0 7 0 2 1 2 2 5 3 0 5

2 0 5 6 1 4 0 1 1 7 2 4 1 2 2 1 6 2 2 0 7 3 7 8

5 1 4 4 2 8 6 0 2 9 6 4 2 4 1 0 1 8 9 2 4 7 3 0

2 3 0 7 1 6 2 4 2 0 8 7 1 5 4 4 5 6 0 6 6 8 3 2


5 7 7 4 2 8 3 1 1 4 2 3 9 3 9 2 8 0 3 6 2 3 0

2 1 5 3 1 4 2 4 7 5 1 5 1 0

7 0 9 6 8 0 6 6

5 9 1 6 3 1 4 7 1 5 0 0 1 1 1 1 2 5 8 4 5 7 4 2

2 4 7 5 1 6 9 0 9 2 6 6 5 6

6 5 9 9 7 6 5 4

6 0 6 1 3 4 6 8 1 5 9 0 1 3 1 1 2 3 4 9 5 2 2 1

2 8 1 2 1 9 8 6 1 1 3 8 8 4 2 6 0 5 0 7 1 7 2


6 2 1 0 3 0 7 4 7 3 0 4 8 4

3 0 6 0 6 4 4 2

2 3 4 1 1 5 4 9 3 8 8 2 6 4

7 2 7 1 8 1 8 7

6 3 8 2 3 4 3 6 7 7 4 5 7 7

2 8 4 4 5 9 8 7

2 7 0 8 1 8 5 1 4 8 0 3 4 1

6 8 1 2 7 8 0 8

6 5 6 0 3 8 1 1 8 2 6 6 8 6

2 6 1 4 5 5 0 3

3 0 9 9 2 1 9 2 5 9 5 4 4 1

6 3 0 6 7 3 6 7

a = 0 . 9 9 REFLECTED 6 5 7 3 3 2 7 9 2 5 0 0 1 6 5 5 6 7 7 3 3 6 8 3 2 9 0 7 1 9 8 9 6 9 8 3 4 1 0 8 3 3 4 8 2 3 7 2 ABSORBED 1 4 9 9 9 8 0 5 5 1 5 9 1 1 9 9 9 7 1 1 7 1 1 4 2 1 2 3 9 1 TRANSMITTED 3 2 7 8 6 6 2 2 7 4 2 0 8 2 9 0 3 0 6 8 6 1 9 8 6 9 9 4 7 9 4 0 2 8 4 6 5 7 5 0 6 5 2 9 7 5 3 7

a = 1 . 0 0 REFLECTED 6 6 6 6 3 3 3 2 2 5 4 1 1 6 8 3 6 8 7 3 3 7 4 7 2 9 5 8 2 0 2 5 7 0 9 3 4 1 8 5 3 4 1 3 2 4 1 9 ABSORBED 0 0 0 0 0 0 0 0 0 0 0 0 TRANSMITTED 3 3 3 4 6 6 6 8 7 4 5 9 8 3 1 7 3 1 2 7 6 2 5 3 7 0 4 2 7 9 7 5 2 9 0 7 5 8 1 5 6 5 8 7 7 5 8 1




F O R b < = 0 . 8 0 0 0 0 0

I S O T R O P I C G R A Z T H I N L A M B P E R P

g = 0 . 0 0 N U 1

F O R b = 1 . 0 0 0 C 0

G R A Z T H I N L A M B P E R P

Ο Ν U 1

F O R b = 1 . 2 5 0 0 0

G R A Z T H I N L A M B P E R P

0 N U I

o = 0 . 2 0 R E F L E C T E D A B S O R B E D T R A N S M I T T E D

1 0 5 3 5 2 7 4 2 3 3 0 1 1 0 5 4 5 4 0 4 3 9 3 2 0 1 0 5 5 5 4 9 4 5 0 3 3 4 8 7 1 6 7 1 6 2 6 3 9 4 4 9 5 6 8 7 7 1 7 7 1 4 7 0 9 8 5 7 5 7 8 8 2 0 8 2 0 4 7 7 5 0 6 5 7 6

2 3 1 2 3 1 1 3 1 8 3 4 7 4 3 1 7 5 1 7 4 6 2 4 6 3 3 9 2 3 1 2 5 1 2 4 7 1 8 0 0 3 0 9 0

α = 0 . 4 0 R E F L E C T E D A B S O R B E D T R A N S M I T T E D

2 2 3 5 1 1 7 5 9 5 2 6 8 3 2 2 4 3 1 2 1 3 9 9 8 7 3 4 2 2 4 8 7 2 2 3 6 1 1 4 5 4 7 5 4 2 4 8 7 3 3 6 6 6 8 1 6 1 7 2 5 0 1 5 7 4 4 1

5 4 2 2 7 1 1 3 5 7 3 5 0 6 9 4 2 1 2 1 0 6 2 8 3 0 4 2 5 1 3 1 1

1 2 3 9 1 0 3 2 7 7 4 7 2 0 7 6 8 4 2 5 8 2 2 1 5 5 4 2 1 2 6 3 4 0 4

α = 0 . 6 0 R E F L E C T E D A B S O R B E D T R A N S M I T T E D

3 6 0 0 5 4 2 1

9 7 9

2 0 0 1 4 7 3 6 3 2 6 3

1 6 3 9 4 2 5 6 4 1 0 5

1 1 8 5 3 3 C 7 5 5 0 8

3 6 2 6 2 0 8 8 5 5 8 5 5 2 8 3

7 8 9 2 6 2 9

1 7 4 3 1 2 9 5 4 9 0 2 3 9 9 1 3 3 5 5 4 7 1 4

3 6 4 6 5 7 4 7

6 0 7

2 1 5 4 1 8 2 4 1 3 8 7 5 8 2 1 5 5 5 6 4 7 4 3 2 0 2 5 2 6 2 0 3 8 7 0

a = 0 . 8 0 R E F L E C T E D A B S O R B E D T R A N S M I T T E D

5 2 4 2 3 1 0 5 2 5 7 7 1 8 8 0 5 3 2 1 3 3 0 2 2 8 0 2 2 1 0 8 5 3 8 8 3 4 7 0 2 9 9 7 2 3 2 0 3 1 3 3 2 8 3 3 2 5 5 5 1 9 8 8 3 3 0 3 3 2 5 7 3 0 3 6 2 4 7 8 3 4 8 5 3 7 1 1 3 5 6 4 3 0 5 3 1 6 2 5 4 0 6 2 4 8 6 8 6 1 3 2 1 3 7 6 3 4 4 1 4 1 6 2 5 4 1 4 1 1 2 7 2 8 1 9 3 4 3 9 4 6 2 7


6 2 1 8 3 8 1 8 3 1 9 2 2 3 4 1 6 3 5 4 4 1 1 9 3 5 2 7 2 6 7 4 6 4 7 8 4 3 9 5 3 8 4 1 3 0 0 5 1 7 0 8 1 5 7 2 1 4 2 0 1 1 0 5 1 8 3 1 1 8 4 7 1 7 2 6 1 4 1 0 1 9 7 1 2 1 5 9 2 0 8 0 1 7 8 5 2 0 7 4 4 6 1 0 5 3 8 8 6 5 5 4 1 8 1 5 4 0 3 4 4 7 4 7 5 9 1 6 1 5 5 1 3 4 4 6 4 0 7 9 5 2 1 0


6 7 6 0 4 2 3 1 3 5 5 1 2 6 1 1 6 9 3 8 4 6 0 7 3 9 6 4 3 0 1 7 7 1 0 9 4 9 6 6 4 3 6 8 3 4 3 8 8 9 5 8 3 2 7 5 2 5 8 6 9 7 0 9 8 9 9 2 6 7 5 7 1 0 5 9 1 1 7 6 1 1 3 5 9 7 4

2 3 4 5 4 9 3 7 5 6 9 7 6 8 0 3 2 0 9 2 4 4 0 4 5 1 1 0 6 2 2 6 1 8 3 2 3 8 5 8 4 4 9 7 5 5 8 8

0 = 0 . 9 9 R E F L E C T E D A B S O R B E D T R A N S M I T T E O

7 2 2 5 4 5 9 6 3 8 6 8 2 8 5 1 7 4 4 7 5 0 4 5 4 3 6 0 3 3 2 9 7 6 7 0 5 4 9 4 4 8 5 9 3 8 4 3 1 8 6 1 7 4 1 5 8 1 2 3 2 C 4 2 1 1 1 9 6 1 6 1 2 2 5 2 5 4 2 4 5 2 1 0

2 5 8 9 5 2 3 0 5 9 7 4 7 0 2 6 2 3 4 9 4 7 4 4 5 4 4 4 6 5 1 0 2 1 0 5 4 2 5 2 4 8 9 6 5 9 4 7

0 = 1 . 0 0 R E F L E C T E D A B S O R B E D T R A N S M I T T E D

7 3 4 6 4 6 9 2 3 9 5 3 2 9 1 4 7 5 8 1 5 1 6 3 4 4 6 6 3 4 1 3 7 8 1 9 5 6 3 8 4 9 9 3 3 9 5 4 C O O O 0 0 0 0 0 0 0 0

2 6 5 4 5 3 0 8 6 0 4 7 7 0 8 6 2 4 1 9 4 8 3 7 5 5 3 4 6 5 8 7 2 1 8 1 4 3 6 2 5 0 0 7 6 0 4 6

A N Y α D I R E C T L Y T R A N S M I T T E D 0 2 0 0 9 2 8 8 6 4 4 9 3 0 1 4 8 5 2 1 9 4 3 6 7 9 0 1 0 3 5 1 5 7 1 2 8 6 5



FOR b = 1 . 6 0 0 0 0


g = 0 . 0 0 N U i

FOR b = 2 . 0 0 0 0 0

GRAZ THIN LAMB PERP

0 N U I

FOR b = 2 . 5 0 0 0 0

GRAZ THIN LAMB PERP

0 Ν U 1


1 0 5 5 8 8 6 0

7 9

5 5 4 8 6 5 3

7 9 3

4 5 8 8 3 6 9 1 1 7 3

3 4 4 7 4 4 7 2 2 0 9

1 0 5 6 8 8 9 6

4 8

5 5 6 8 9 6 3

4 8 1

4 6 1 8 8 1 2

7 2 7

3 4 9 8 1 4 9 1 5 0 2

1 0 5 6 8 9 1 8

2 6

5 5 7 9 1 8 1

2 6 2

4 6 2 9 1 3 3

4 0 5

3 5 1 8 7 2 3

9 2 6


2 2 5 1 7 5 4 3

2 0 6

1 2 5 7 7 7 1 4 1 0 2 9

1 0 5 5 7 5 0 9 1 4 3 6

8 0 4 6 7 1 0 2 4 8 6

2 2 5 3 7 6 1 7

1 3 0

1 2 6 5 8 0 8 3

6 5 2

1 0 6 6 8 0 0 9

9 2 5

8 2 0 7 4 5 0 1 7 3 0

2 2 5 4 7 6 7 1

7 5

1 2 6 8 8 3 5 9

3 7 3

1 0 7 1 8 3 9 1

5 3 8

8 2 9 8 0 7 7 1 0 9 4


3 6 6 1 5 9 1 3

4 2 6

2 2 0 4 6 3 7 6 1 4 2 0

1 8 8 5 6 2 5 2 1 8 6 3

1 4 6 4 5 6 1 2 2 9 2 4

3 6 6 9 6 0 4 4

2 8 7

2 2 3 0 6 8 1 4

9 5 6

1 9 1 9 6 8 1 2 1 2 6 9

1 5 1 0 6 3 7 9 2 1 1 1

3 6 7 3 6 1 5 0

1 7 7

2 2 4 3 7 1 6 8

5 8 9

1 9 3 6 7 2 7 4

7 9 0

1 5 3 6 7 0 6 7 1 3 9 7


5 4 4 6 3 6 9 3

8 6 1

3 6 1 4 4 2 3 4 2 1 5 2

3 1 6 8 4 1 8 5 2 6 4 7

2 5 2 0 3 7 7 7 3 7 0 3

5 4 8 3 3 8 7 9

6 3 8

3 7 0 7 4 6 9 9 1 5 9 4

3 2 8 0 4 7 4 7 1 9 7 3

2 6 5 9 4 4 8 2 2 8 5 9

5 5 0 6 4 0 5 3

4 4 1

3 7 6 5 5 1 3 2 1 1 0 3

3 3 5 1 5 2 7 7 1 3 7 2

2 7 5 5 5 1 8 5 2 0 6 0

0 = 0 . 9 0 REFLECTED ABSORBED TRANSMlTTED

6 5 9 7 2 1 4 5 1 2 5 8

4 6 5 9 2 5 4 5 2 7 9 6

4 1 4 6 2 5 2 7 3 3 2 7

3 3 4 8 2 2 8 8 4 3 6 4

6 6 8 3 2 3 1 6 1 0 0 1

4 8 5 2 2 9 2 4 2 2 2 4

4 3 7 1 2 9 7 3 2 6 5 6

3 6 1 6 2 8 1 9 3 5 6 5

6 7 4 8 2 4 9 4

7 5 8

4 9 9 6 3 3 1 9 1 6 8 5

4 5 4 2 3 4 3 9 2 0 1 9

3 8 3 0 3 4 0 3 2 7 6 7


7 2 8 3 1 1 7 4 1 5 4 3

5 3 3 3 1 4 1 8 3 2 4 9

4 7 8 6 1 4 1 1 3 8 0 3

3 8 9 8 1 2 8 0 4 8 2 2

7 4 2 1 1 2 9 4 1 2 8 5

5 6 2 3 1 6 7 1 2 7 0 6

5 1 2 1 1 7 0 4 3 1 7 5

4 2 8 7 1 6 2 0 4 0 9 3

7 5 3 6 1 4 2 9 1 0 3 5

5 8 6 5 1 9 5 5 2 1 8 0

5 4 0 3 2 0 3 5 2 5 6 2

4 6 2 9 2 0 2 1 3 3 5 0


7 9 0 9 2 5 5

1 8 3 6

5 9 7 9 3 1 2

3 7 0 9

5 4 0 6 3 1 1

4 2 8 3

4 4 3 6 2 8 2

5 2 8 2

8 1 1 5 2 8 7

1 5 9 8

6 3 9 4 3 7 8

3 2 2 8

5 8 8 0 3 8 7

3 7 3 3

4 9 7 5 3 6 8

4 6 5 7

8 3 0 6 3 2 6

1 3 6 8

6 7 7 9 4 5 8

2 7 6 3

6 3 2 2 4 7 8

3 2 0 0

5 4 9 7 4 7 6

4 0 2 7


8 0 7 9 0

1 9 2 1

6 1 5 9 0

3 8 4 1

5 5 8 0 0

4 4 2 0

4 5 8 7 0

5 4 1 3

8 3 0 8 0

1 6 9 2

6 6 1 6 0

3 3 8 4

6 0 9 9 0

3 9 0 1

5 1 7 5 0

4 8 2 5

8 5 2 6 0

1 4 7 4

7 0 5 1 0

2 9 4 9

6 5 9 8 0

3 4 0 2

5 7 6 0 0

4 2 4 0

ANY α DIRECTLY TRANSMITTED 9 9 8 2 0 1 9 6 0 3 1 3 5 3 1 9 8 3 2 6



FOR b= 3 . 2 0 0 0 0 FOR b = 4 . 0 0 0 0 0 FOR b = 5 . 0 0 0 0 0

I S O T R O P I C GRAZ THIN LAMB PERP GRAZ THIN LAMB PERP GRAZ THIN LAMB PERP

ς = 0 . 0 0 N U 1 O N U l O N U l

α = 0 . 2 0 REFLECTED 1 0 5 6 5 5 7 4 6 3 3 5 2 1 0 5 6 5 5 7 4 6 3 3 5 2 1 0 5 6 5 5 7 4 6 3 3 5 2 ABSORBED 8 9 3 2 9 3 2 8 9 3 5 5 9 1 7 9 8 9 3 9 9 3 9 7 9 4 6 3 9 4 3 3 8 9 4 2 9 4 2 8 9 5 1 3 9 5 6 7 TRANSMITTED 1 2 1 1 5 1 8 2 4 6 9 5 4 6 7 4 2 1 5 2 1 5 2 4 8 1


2 2 5 4 1 2 7 0 7 7 1 1 8 5 5 6

3 5 1 7 4

1 0 7 3 8 3 2 8 6 7 2 8 5 9 5

2 5 5 5 7 3

2 2 5 4 1 2 7 0 7 7 3 1 8 6 5 6

1 5 7 4

1 0 7 3 8 3 3 8 8 1 7 8 8 9 5

1 1 0 2 7 2

2 2 5 4 1 2 7 0 7 7 4 1 8 7 0 4

5 2 6

1 0 7 3 8 3 4 8 8 8 8 9 0 6 0

3 9 1 0 6

α = 0 . 6 0 REFLECTED 3 6 7 5 2 2 4 9 1 9 4 4 1 5 4 9 3 6 7 5 2 2 5 1 1 9 4 7 1 5 5 3 3 6 7 5 2 2 5 1 1 9 4 7 1 5 5 4 ABSORBED 6 2 3 4 7 4 4 9 7 6 4 7 7 6 7 4 6 2 8 2 7 6 0 6 7 8 5 9 8 0 5 3 6 3 0 8 7 6 9 3 7 9 7 6 8 2 7 9 TRANSMITTED 9 1 3 0 2 4 0 9 7 7 7 4 3 1 4 3 1 9 4 3 9 4 1 7 5 6 7 7 1 6 7


5 5 2 0 3 8 0 0 4 2 1 4 5 5 3 6

2 6 6 6 6 4

3 3 9 4 2 8 1 5 5 7 7 7 5 8 9 4

8 2 9 1 2 9 1

5 5 2 5 3 8 1 3 4 3 2 6 5 8 1 4

1 4 9 3 7 3

3 4 1 1 2 8 4 0 6 1 2 1 6 4 0 9

4 6 8 7 5 1

5 5 2 7 3 8 1 8 4 4 0 0 5 9 9 9

7 3 1 8 3

3 4 1 7 2 8 5 0 6 3 5 4 6 7 7 2

2 2 9 3 7 8

α = 0 . 9 0 REFLECTED 6 7 9 5 5 1 0 1 4 6 6 7 3 9 9 4 6 8 2 0 5 1 5 5 4 7 3 2 4 0 8 1 6 8 3 1 5 1 8 1 4 7 6 3 4 1 2 5 ABSORBED 2 6 8 6 3 7 4 5 3 9 4 8 4 0 6 9 2 8 4 1 4 0 9 1 4 3 6 2 4 6 3 4 2 9 6 9 4 3 7 5 4 7 C 3 5 1 0 9 TRANSMITTED 5 1 9 1 1 5 4 1 3 8 5 1 9 3 7 3 3 9 7 5 4 9 0 6 1 2 8 5 2 0 0 4 4 4 5 3 4 7 6 6

α = 0 . 9 5 REFLECTED 7 6 3 3 6 0 7 0 5 6 4 3 4 9 3 0 7 6 9 4 6 1 9 7 5 7 9 3 5 1 2 4 7 7 3 1 6 2 7 7 5 8 8 6 5 2 4 7 ABSORBED 1 5 9 1 2 2 9 6 2 4 3 4 2 5 2 4 1 7 4 1 2 6 1 4 2 8 0 7 3 0 0 7 1 8 8 6 2 9 1 7 3 1 6 4 3 4 7 8 TRANSMITTED 7 7 6 1 6 3 4 1 9 2 3 2 5 4 6 5 6 5 1 1 8 9 1 4 0 0 1 8 6 9 3 8 3 8 0 6 9 5 0 1 2 7 5

α = 0 . 9 9 REFLECTED 8 4 9 3 7 1 5 8 6 7 5 9 6 0 2 9 8 6 3 8 7 4 5 0 7 0 9 7 6 4 5 0 8 7 5 6 7 6 9 0 7 3 7 5 6 8 0 1 ABSORBED 3 8 0 5 6 5 6 0 2 6 2 7 4 3 7 6 8 1 7 3 7 7 9 5 5 0 5 8 1 6 8 9 3 9 9 3 TRANSMITTED 1 1 2 7 2 2 7 7 2 6 3 9 3 3 4 4 9 2 5 1 8 6 9 2 1 6 6 2 7 5 5 7 3 9 1 4 9 4 1 7 3 2 2 2 0 6

σ = 1 . 0 0 REFLECTED 8 7 5 0 7 5 0 0 7 1 1 4 6 3 8 4 8 9 3 5 7 8 7 0 7 5 4 0 6 9 0 9 9 1 0 1 8 2 0 2 7 9 2 3 7 3 8 7 ABSORBED 0 0 0 0 0 0 0 0 0 0 0 0 TRANSMITTED 1 2 5 0 2 5 0 0 2 8 8 6 3 6 1 6 1 0 6 5 2 1 3 0 2 4 6 0 3 0 9 1 8 9 9 1 7 9 8 2 0 7 7 2 6 1 3

ANY α DIRECTLY TRANSMITTED 0 8 3 1 4 1 4 0 8 0 3 2 5 5 1 8 3 0 1 0 1 8 6 7


ANY α DIRECTLY TRANSMITTED


FOR b = 6 . 4 0 0 0 0


g = 0 . 0 0 N U 1

FOR b = 8 . 0 0 0 0 0

GRAZ THIN LAMB PERP

0 Ν U 1

FOR b = 1 0 . 0 0 0 0 0

GRAZ THIN LAMB PERP

0 N U I


1 0 5 6 8 9 4 4

0

5 5 7 9 4 4 0

3

4 6 3 9 5 3 2

5

3 5 2 9 6 2 8

2 0

1 0 5 6 8 9 4 4

0

5 5 7 9 4 4 2

1

4 6 3 9 5 3 6

1

3 5 2 9 6 4 4

1 0 5 6 8 9 4 4

0

5 5 7 9 4 4 3

0

4 6 3 9 5 3 7

0

3 5 2 9 6 4 7

1


2 2 5 4 7 7 4 5

I

1 2 7 0 8 7 2 4

1 0 7 3 8 9 1 8

9

8 3 4 9 1 3 8

2 8

2 2 5 4 7 7 4 6

0

1 2 7 0 8 7 2 9

1

1 0 7 3 8 9 2 5

2

8 3 4 9 1 6 0

6

2 2 5 4 7 7 4 6

0

1 2 7 0 8 7 3 0

0

1 0 7 3 8 9 2 7

0

8 3 4 9 1 6 5

1


3 6 7 5 6 3 2 0

5

2 2 5 1 7 7 3 4

15

1 9 4 7 8 0 3 2

2 1

1 5 5 4 8 3 9 7

4 9

3 6 7 5 6 3 2 4

I

2 2 5 1 7 7 4 5

4

1 9 4 7 8 0 4 8

5

1 5 5 4 8 4 3 4

12

3 6 7 5 6 3 2 5

0

2 2 5 1 7 7 4 8

I

1 9 4 7 8 0 5 2

1

1 5 5 4 8 4 4 4

2


5 5 2 8 4 4 4 5

2 7

3 8 1 9 6 1 1 4

6 7

3 4 1 8 6 4 9 7

8 5

2 8 5 2 7 0 0 5

1 4 3

5 5 2 8 4 4 6 3

9

3 8 2 0 6 1 5 8

2 2

3 4 1 9 6 5 5 4

2 7

2 8 5 3 7 1 0 1

4 6

5 5 2 8 4 4 7 0

2

3 8 2 0 6 1 7 5

5

3 4 1 9 6 5 7 4

7

2 8 5 3 7 1 3 6

1 1

0 = 0 . 9 0 REFLECTED ABSORBED TRANSMlTTED

6 8 3 6 3 0 6 8

9 6

5 1 9 2 4 5 9 5

2 1 3

4 7 7 6 4 9 6 8

2 5 6

4 1 4 4 5 4 8 6

3 7 0

6 8 3 7 3 1 2 2

4 1

5 1 9 4 4 7 1 4

9 2

4 7 8 0 5 1 1 0

1 1 0

4 1 4 8 5 6 9 2

1 6 0

6 8 3 8 3 1 4 8

1 4

5 1 9 5 4 7 7 3

3 2

4 7 8 0 5 1 8 1

3 9

4 1 4 9 5 7 9 5

5 6


7 7 5 3 2 0 2 3

2 2 4

6 3 2 2 3 2 0 6

4 7 2

5 9 3 9 3 5 0 5

5 5 6

5 3 1 8 3 9 3 4

7 4 8

7 7 6 1 2 1 1 7

1 2 2

6 3 3 8 3 4 0 5

2 5 7

5 9 5 8 3 7 4 0

3 0 2

5 3 4 4 4 2 4 8

4 0 8

7 7 6 3 2 1 8 0

5 7

6 3 4 4 3 5 3 6

1 2 0

5 9 6 5 3 8 9 4

1 4 1

5 3 5 3 4 4 5 6

1 9 1


8 8 5 6 5 8 9 5 5 5

7 8 9 2 9 8 7

1 1 2 1

7 6 0 9 1 0 9 2 1 2 9 9

7 0 9 8 1 2 4 5 1 6 5 7

8 9 2 0 6 7 1 4 0 9

8 0 2 C 1 1 5 5

8 2 5

7 7 5 8 1 2 8 5

9 5 7

7 2 8 8 1 4 9 1 1 2 2 1

8 9 6 1 7 5 5 2 8 4

8 1 0 2 1 3 2 5

5 7 3

7 8 5 3 1 4 8 3

6 6 4

7 4 0 9 1 7 4 3

8 4 8

a = 1 . 0 0 REFLECTED ABSORBED TRANSMlTTED

9 2 6 2 C

7 3 8

8 5 2 4 0

1 4 7 6

8 2 9 5 0

1 7 0 5

7 8 5 4 0

2 1 4 6

9 3 8 7 0

6 1 3

8 7 7 4 0

1 2 2 6

8 5 8 5 0

1 4 1 5

8 2 1 8 0

1 7 8 2

9 4 9 4 0

5 0 6

8 9 8 9 0

1 0 1 1

8 8 3 3 0

1 1 6 7

8 5 3 0 0

1 4 7 0



FOR b = l 2 . 8 0 0 0 0


g = 0 , 0 O N U l

FOR b = l 6 . 0 0 0 0 0

GRAZ THIN LAMB PERP

O N U l

FOR b = 0 0

GRAZ THIN LAMB PERP

O N U l

a = 0 . 2 0 REFLECTED 1 0 5 6 5 5 7 4 6 3 3 5 2 1 0 5 6 5 5 7 4 6 3 3 5 2 1 0 5 6 5 5 7 4 6 3 3 5 2 ABSORBED 8 9 4 4 9 4 4 3 9 5 3 7 9 6 4 8 8 9 4 4 9 4 4 3 9 5 3 7 9 6 4 8 8 9 4 4 9 4 4 3 9 5 3 7 9 6 4 8 TRANSMITTED 0 0 0 0 0 0 0 0 0 0 0 0

a = 0 . 4 0 REFLECTED 2 2 5 4 1 2 7 0 1 0 7 3 8 3 4 2 2 5 4 1 2 7 0 1 0 7 3 8 3 4 2 2 5 4 1 2 7 0 1 0 7 3 8 3 4 ABSORBED 7 7 4 6 8 7 3 0 8 9 2 7 9 1 6 6 7 7 4 6 8 7 3 0 8 9 2 7 9 1 6 6 7 7 4 6 8 7 3 0 8 9 2 7 9 1 6 6 TRANSMITTED 0 0 0 0 0 0 0 0 0 0 0 0

a = 0 . 6 0 REFLECTED 3 6 7 5 2 2 5 1 1 9 4 7 1 5 5 4 3 6 7 5 2 2 5 1 1 9 4 7 1 5 5 4 3 6 7 5 2 2 5 1 1 9 4 7 1 5 5 4 ABSORBED 6 3 2 5 7 7 4 9 8 0 5 3 8 4 4 6 6 3 2 5 7 7 4 9 8 0 5 3 8 4 4 6 6 3 2 5 7 7 4 9 8 C 5 3 8 4 4 6 TRANSMITTED 0 0 0 0 C O O C 0 0 0 0

a = 0 . 8 0 REFLECTED 5 5 2 8 3 8 2 0 3 4 1 9 2 8 5 3 5 5 2 8 3 8 2 0 3 4 1 9 2 8 5 3 5 5 2 8 3 8 2 0 3 4 1 9 2 8 5 3 ABSORBED 4 4 7 2 6 1 7 9 6 5 8 0 7 1 4 5 4 4 7 2 6 1 8 0 6 5 8 1 7 1 4 7 4 4 7 2 6 1 8 0 6 5 8 1 7 1 4 7 TRANSMITTED 0 1 1 2 0 0 0 0 C O O C

a = 0 . 9 0 REFLECTED 6 8 3 8 5 1 9 5 4 7 8 0 4 1 4 9 6 8 3 8 5 1 9 5 4 7 8 0 4 1 4 9 6 8 3 8 5 1 9 5 4 7 8 0 4 1 4 9 ABSORBED 3 1 5 9 4 7 9 8 5 2 1 1 5 8 3 8 3 1 6 1 4 8 0 4 5 2 1 8 5 8 4 9 3 1 6 2 4 8 C 5 5 2 2 0 5 8 5 1 TRANSMITTED 3 7 9 1 3 1 1 2 2 0 0 0 0

a = 0 . 9 5 REFLECTED 7 7 6 4 6 3 4 5 5 9 6 6 5 3 5 5 7 7 6 4 6 3 4 5 5 9 6 7 5 3 5 5 7 7 6 4 6 3 4 5 5 9 6 7 5 3 5 5 ABSORBED 2 2 1 6 3 6 1 4 3 9 8 5 4 5 7 9 2 2 3 0 3 6 4 3 4 0 1 8 4 6 2 5 2 2 3 6 3 6 5 5 4 0 3 3 4 6 4 5 TRANSMITTED 2 0 4 1 4 9 6 6 6 12 1 5 2 0 0 C 0 0

0 = 0 . 9 9 REFLECTED 8 9 8 5 8 1 5 2 7 9 1 1 7 4 8 3 8 9 9 5 8 1 7 2 7 9 3 4 7 5 1 3 9 0 0 0 8 1 8 2 7 9 4 6 7 5 2 7 ABSORBED 8 4 2 1 4 9 9 1 6 8 4 2 0 0 0 9 0 6 1 6 2 8 1 8 3 4 2 1 9 1 1 0 0 0 1 8 1 8 2 0 5 4 2 4 7 3 TRANSMITTED 1 7 3 3 4 9 4 0 5 5 1 7 9 9 2 0 0 2 3 2 2 9 6 0 0 0 0


9 5 9 4 9 1 8 8 9 0 6 2 8 8 1 9 9 6 6 9 9 3 3 7 9 2 3 5 9 0 3 6 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 C O O O

4 0 6 8 1 2 9 3 8 1 1 8 1 3 3 1 6 6 3 7 6 5 9 6 4 C O O O

ANY a DIRECTLY TRANSMITTED 0 0 0 0 O C O C C O O C


T A B L E 16

S o u r c e F u n c t i o n f o r I s o t r o p i c S c a t t e r i n g , b = \

a

Incidence from μ0 = 0.1 Incidence from μΌ = 1.0 J(T) = 2.5 a G(a, 1, τ, 0.1) J(T) = 0.25 a G(a, Ι , τ , 1)

τ . a = 0.20 0.60 0.90 0.99 1.00 a = 0.20 0.60 0.90 0.99 1.00

0 .00 5128 16356 26140 29475 29862 537 1941 3594 4305 4393 0.01 4665 15060 24332 27542 27916 534 1953 3650 4386 4478 0.02 4240 13825 22549 25612 25971 531 1957 3685 4438 4533 0.05 3181 10679 17918 20566 20880 520 1955 3749 4543 4643

0.10 1974 7000 12386 14500 14756 500 1931 3793 4636 4743 0.20 778 3213 6528 8024 8213 460 1849 3773 4674 4789 0.30 324 1681 4041 5237 5393 422 1747 3672 4595 4714 0.40 149 1027 2896 3924 4060 385 1635 3519 4440 4560

0.50 78 720 2291 3204 3327 351 1519 3327 4227 4343

0 .60 48 554 1910 2728 2838 319 1399 3106 3964 4075 0 .70 34 448 1627 2354 2454 290 1278 2858 3657 3761 0 .80 26 371 1387 2024 2111 262 1154 2582 3305 3400 0 .90 21 308 1164 1703 1777 236 1024 2270 2898 2979

0.95 19 278 1049 1535 1601 223 954 2091 2660 2734 0.98 17 260 975 1424 1485 215 907 1967 2494 2563 0.99 17 253 948 1382 1442 212 890 1920 2431 2497 1.00 17 246 915 1333 1390 209 869 1861 2352 2415

Integral 560 2249 4618 5742 5886 360 1497 3172 3985 4090

Incidence from narrow layer, Ν Uniform incidence, U J(T) = 0.25 a g -L(A,L,r) J(T) = 0 .5ago(a , Ι ,τ )

r a = 0.20 0.60 0.90 0.99 1.00 a = 0.20 0.60 0 . 9 0 0.99 1.00

0.00 1054 3626 6354 7447 7581 0.01 2099 6943 11595 13322 13529 1008 3518 6247 7357 7493 0.02 1759 5943 10136 11737 11930 973 3436 6163 7283 7421 0.05 1318 4649 8270 9727 9905 892 3234 5945 7088 7230

0.10 995 3693 6905 8270 8440 790 2965 5636 6800 6946 0.20 689 2759 5552 6828 6990 643 2547 5106 6280 6429 0.30 523 2223 4737 5945 6100 535 2215 4639 5794 5942 0.40 414 1848 4128 5268 5416 451 1938 4208 5324 5468

0.50 336 1561 3627 4691 4830 383 1698 3802 4862 5000

0 .60 276 1330 3188 4169 4298 327 1488 3413 4402 4532 0.70 230 1135 2789 3678 3795 280 1299 3035 3939 4058 0 .80 192 966 2412 3198 3302 241 1127 2660 3465 3571 0 .90 161 813 2040 2709 2798 206 963 2274 2963 3054

0.95 147 739 1845 2447 2527 190 881 2067 2688 2770 0.98 140 693 1717 2273 2347 181 829 1929 2503 2579 0.99 137 676 1670 2208 2279 178 811 1877 2433 2506 1.00 134 657 1613 2129 2198 175 789 1815 2348 2419

Integral 482 1981 4155 5202 5337 444 1838 3882 4872 5000

a T h e t a b l e g i v e s 1 0 , 0 0 0 J{x).


T A B L E 17

G a i n a n d F l u x f o r I s o t r o p i c S c a t t e r i n g

I n c i d e n c e f r o m μ0 = 0.1

α = 0 . 2 0 0 . 6 0 0 . 9 0 0 . 9 9 I . 0 0 0 . 2 0 0 . 6 0 0 . 9 0 0 . 9 9 1 . 0 0

P O I N T D I R E C T I O N GAIN X 1 0 0 0 0 NET FLUX DOWN X 1 0 0 0 0 0 . 0 3 1 2 5

0 . 0 1 0 1 0 8 1 0 3 3 4 1 0 5 1 2 1 0 5 6 7 1 0 5 7 3 9 7 4 6 9 2 1 5 8 7 9 7 8 6 6 8 8 6 5 4 0 . 0 1 5 6 2 8 6 7 8 8 9 3 9 9 1 4 4 9 2 0 8 9 2 1 5 8 5 7 3 8 6 1 3 8 6 4 4 8 6 3 3 8 6 5 4 0 . 0 3 1 2 5 7 4 2 0 7 6 3 8 7 8 0 9 7 8 6 2 7 8 6 8 7 5 6 9 8 0 9 6 8 5 1 1 8 6 4 0 8 6 5 4

0 . 0 6 2 5 0

0 . 0 1 0 1 6 1 1 0 5 0 6 1 0 7 8 7 1 0 8 7 5 1 0 8 8 5 9 5 7 5 8 6 6 1 7 9 1 5 7 6 7 9 7 6 5 3 0 . 0 1 5 6 2 8 7 4 0 9 1 3 9 9 4 6 5 9 5 6 8 9 5 7 9 8 3 9 5 8 0 4 8 7 7 5 6 7 6 6 3 7 6 5 3 0 . 0 3 1 2 5 7 5 0 1 7 9 0 0 8 2 2 6 8 3 2 9 8 3 4 1 7 3 8 2 7 5 1 6 7 6 1 8 7 6 4 9 7 6 3 3 0 . 0 4 6 8 7 6 4 3 1 6 8 0 4 7 1 1 0 7 2 0 6 7 2 1 7 6 5 1 3 7 0 5 7 7 4 9 9 7 6 3 7 7 6 5 3 0 . 0 6 2 5 0 5 4 9 9 5 8 1 5 6 0 7 4 6 1 5 6 6 1 6 5 5 7 6 9 6 6 6 3 7 3 9 6 7 6 2 7 7 6 5 3

0 . 1 2 5 Û 0

0 . 0 1 0 2 1 3 1 0 6 8 8 1 1 0 9 6 1 1 2 2 9 1 1 2 4 4 9 3 7 7 7 9 7 6 6 7 6 1 6 3 6 4 6 3 1 9 0 . 0 3 1 2 5 7 5 6 5 8 1 2 2 8 6 0 4 8 7 6 1 8 7 7 9 7 1 7 0 6 8 0 5 6 4 5 4 6 3 3 3 6 3 1 9 0 . 0 6 2 5 0 5 5 8 9 6 1 2 4 6 5 9 1 6 7 4 5 6 7 6 2 5 5 3 8 5 9 2 1 6 2 1 8 6 3 0 9 6 3 1 9 0 . 0 9 3 7 5 4 1 2 6 4 6 0 4 5 0 2 5 5 1 6 3 5 1 7 9 4 3 3 3 5 2 5 4 6 0 3 7 6 2 9 0 6 3 1 9 0 . 1 2 5 0 0 3 0 3 5 3 4 2 4 3 7 6 5 3 8 7 7 3 8 9 0 3 4 4 4 4 7 5 5 5 9 0 0 6 2 7 6 6 3 1 9

0 . 2 5 0 0 0

0 . 0 1 0 2 4 5 1 0 8 2 0 1 1 3 5 2 1 1 5 3 4 1 1 5 5 5 9 2 3 4 7 3 9 9 5 6 6 1 5 0 5 9 4 9 9 0 0 . 0 6 2 5 0 5 6 3 1 6 2 9 7 6 9 2 5 7 1 4 3 7 1 6 8 5 3 7 7 5 3 0 6 5 1 0 0 5 0 0 2 4 9 9 0 0 . 1 2 5 0 0 3 1 0 2 3 6 8 6 4 2 5 4 4 4 5 4 4 4 7 7 3 2 5 7 4 0 8 9 4 7 5 8 4 9 6 6 4 9 9 0 0 . 1 8 7 5 0 1 7 2 4 2 2 0 4 2 6 8 2 2 8 5 3 2 8 7 3 2 0 8 5 3 3 7 0 4 5 4 5 4 9 4 4 4 9 9 0 0 . 2 5 0 0 0 9 6 6 1 3 3 2 1 6 9 8 1 8 2 8 1 8 4 3 1 4 3 1 2 9 3 6 4 4 1 0 4 9 3 0 4 9 9 0

0 . 5 0 0 0 0

0 . 0 1 0 2 5 4 1 0 8 8 0 1 1 5 1 1 1 1 7 4 3 1 1 7 7 0 9 1 8 5 7 0 9 8 4 8 8 6 4 0 5 0 3 9 5 1 0 . 0 6 2 5 0 5 6 4 2 6 3 6 9 7 1 1 9 7 3 9 8 7 4 3 1 5 3 2 2 4 9 8 9 4 3 1 4 3 9 9 1 3 9 5 1 0 . 1 2 5 0 0 3 1 1 6 3 7 7 2 4 4 8 3 4 7 5 4 4 7 8 7 3 1 9 6 3 7 5 2 3 9 5 9 3 9 5 4 3 9 5 1 0 . 1 8 7 5 0 1 7 4 2 2 3 1 0 2 9 5 6 3 2 0 9 3 2 4 0 2 0 1 7 3 0 0 9 3 7 3 0 3 9 2 9 3 9 5 1 0 . 2 5 0 0 0 9 9 3 1 4 8 0 2 0 5 8 2 2 9 0 2 3 1 8 1 3 5 2 2 5 4 5 3 5 7 6 3 9 1 2 3 9 5 1 0 . 3 1 2 5 0 5 8 1 9 9 9 1 5 1 1 1 7 2 1 1 7 4 6 9 6 8 2 2 4 0 3 4 6 6 3 9 0 0 3 9 5 1 0 . 3 7 5 0 0 3 5 4 7 1 1 1 1 6 1 1 3 4 8 1 3 7 0 7 4 0 2 0 3 0 3 3 8 3 3 8 9 0 3 9 5 1 0 . 4 3 7 5 0 2 2 5 5 2 9 9 1 8 1 0 7 9 1 0 9 9 5 9 8 1 8 7 6 3 3 1 9 3 8 8 3 3 9 5 1 0 . 5 0 0 0 0 1 5 1 4 0 1 7 1 8 8 4 9 8 6 5 5 0 6 1 7 6 0 3 2 6 8 3 8 7 7 3 9 5 1

1 . 0 0 0 0 0

0 . 0 1 0 2 5 6 1 0 9 0 4 1 1 6 1 8 1 1 9 0 9 1 1 9 4 5 9 1 7 5 6 9 6 3 4 3 1 0 3 1 6 6 3 0 2 3 0 . 0 6 2 5 0 5 6 4 4 6 3 9 7 7 2 4 5 7 5 9 5 7 6 3 8 5 3 1 2 4 8 4 6 3 7 3 0 3 1 0 6 3 0 2 3 0 . 1 2 5 0 0 3 1 1 8 3 8 0 4 4 6 2 6 4 9 7 9 5 0 2 2 3 1 8 5 3 6 0 2 3 3 6 7 3 0 6 8 3 0 2 3 0 . 1 8 7 5 0 1 7 4 4 2 3 4 6 3 1 1 7 3 4 6 0 3 5 0 3 2 0 0 4 2 8 5 1 3 1 2 9 3 0 4 2 3 0 2 3 0 . 2 5 0 0 0 9 9 5 1 5 2 0 2 2 3 7 2 5 6 7 2 6 0 9 1 3 3 8 2 3 7 7 2 9 6 4 3 0 2 3 3 0 2 3 0 . 3 1 2 5 0 5 8 5 1 0 4 5 1 7 1 1 2 0 2 8 2 0 6 8 9 5 4 2 0 6 2 2 8 4 2 3 0 0 9 3 0 2 3 0 . 3 7 5 0 0 3 5 7 7 6 3 1 3 8 3 1 6 8 7 1 7 2 6 7 2 4 1 8 3 9 2 7 4 6 2 9 9 7 3 0 2 3 0 . 4 3 7 5 0 2 3 0 5 9 1 1 1 6 9 1 4 5 9 1 4 9 6 5 8 0 1 6 7 1 2 6 6 7 2 9 8 8 3 0 2 3 0 . 5 0 0 0 0 1 5 7 4 8 0 1 0 1 8 1 2 9 4 1 3 3 1 4 8 5 1 5 3 8 2 5 9 9 2 9 7 9 3 0 2 3 0 . 5 6 2 5 0 1 1 4 4 0 4 9 0 5 1 1 6 7 1 2 0 2 4 1 8 1 4 2 8 2 5 3 9 2 9 7 1 3 0 2 3 0 . 6 2 5 0 0 8 7 3 4 9 8 1 5 1 0 6 2 1 0 9 4 3 6 8 1 3 3 5 2 4 8 3 2 9 6 4 3 0 2 3 0 . 6 8 7 5 0 7 0 3 0 6 7 3 7 9 6 9 9 9 9 3 2 9 1 2 5 3 2 4 3 7 2 9 5 8 3 0 2 3 0 . 7 5 0 0 0 5 8 2 7 2 6 6 8 8 8 3 9 1 2 2 9 7 1 1 8 1 2 3 9 3 2 9 5 2 3 0 2 3 0 . 8 1 2 5 0 5 0 2 4 2 6 0 4 8 0 2 8 2 8 2 7 0 1 1 1 7 2 3 5 3 2 9 4 7 3 0 2 3 0 . 8 7 5 0 0 4 3 2 1 5 5 4 2 7 2 1 7 4 5 2 4 7 1 0 6 0 2 3 1 7 2 9 4 2 3 0 2 3 0 . 9 3 7 5 0 3 8 1 9 0 4 7 9 6 3 8 6 5 9 2 2 7 1 0 0 9 2 2 8 5 2 9 3 8 3 0 2 3 I . 0 0 0 0 0 3 3 1 6 4 4 0 7 5 3 8 5 5 6 2 0 9 9 6 5 2 2 5 8 2 9 3 4 3 0 2 3



I n c i d e n c e f r o m μ 0 = 0 .1

b =

α = 0 . 2 0 0 . 6 0 0 . 9 0 0 . 9 9 1 . 0 0 0 . 2 0 0 . 6 0 0 . 9 0 0 . 9 9 1 . 0 0

P O I N T D I R E C T I O N GAIN X 1 0 0 0 0 NET FLUX : DOWN X 1 0 0 0 0 2 . 0 0 0

0 . 0 1 0 2 5 6 1 0 9 1 2 1 1 6 8 9 1 2 0 5 8 1 2 1 0 7 9 1 7 4 6 9 1 0 3 8 9 8 2 3 2 9 2 1 1 1 0 . 1 2 5 3 1 1 8 3 8 1 5 4 7 2 0 5 1 7 5 5 2 3 7 3 1 8 3 3 5 4 5 2 9 4 4 2 2 2 8 2 1 1 1 0 . 2 5 0 9 9 6 1 5 3 3 2 3 5 0 2 8 0 4 2 8 6 8 1 3 3 6 2 3 1 4 2 5 2 9 2 1 8 0 2 1 1 1 0 . 3 7 5 3 5 8 7 7 9 1 3 1 6 1 9 6 3 2 0 2 7 7 2 1 1 7 6 8 2 2 9 6 2 1 5 1 2 1 1 1 0 . 5 0 0 1 5 8 4 9 9 1 1 7 2 1 6 0 9 1 6 7 4 4 8 1 1 4 5 9 2 1 3 0 2 1 2 9 2 l i r i 0 . 7 5 0 6 0 2 9 9 8 7 1 1 2 8 4 1 3 4 7 2 9 2 1 0 7 9 1 8 8 0 2 0 9 4 2 1 1 1 I . 0 0 0 3 5 2 1 1 6 9 6 1 0 7 5 1 1 3 4 2 0 1 8 2 8 1 6 8 5 2 0 6 4 2 1 1 1 1 . 2 5 0 2 4 1 5 5 5 5 9 8 9 4 9 4 7 1 4 2 6 4 6 1 5 2 9 2 0 4 0 2 1 1 1 1 . 5 0 0 1 6 1 1 4 4 4 0 7 2 2 7 6 7 1 0 3 5 1 3 1 4 0 4 2 0 1 9 2 1 1 1 1 . 6 2 5 1 4 9 7 3 8 5 6 3 7 6 7 7 8 8 4 6 0 1 3 5 3 2 0 1 1 2 1 1 1 1 . 7 5 0 1 1 8 2 3 3 0 5 5 0 5 8 5 7 5 4 1 6 1 3 0 8 2 0 0 4 2 1 1 1 1 . 8 7 5 1 0 6 8 2 7 5 4 5 9 4 8 9 6 5 3 7 8 1 2 7 0 1 9 9 7 2 1 1 1 2 . 0 0 0 8 5 4 2 1 1 3 4 8 3 7 0 5 6 3 4 7 1 2 4 0 1 9 9 2 2 1 1 1

4 . 0 0 0

0 . 0 1 0 2 5 6 1 0 9 1 3 1 1 7 1 7 1 2 1 7 0 1 2 2 4 3 9 1 7 3 6 9 0 3 3 7 3 0 1 6 7 7 1 3 3 0

0 . 1 2 5 3 1 1 8 3 8 1 6 4 7 5 7 5 3 2 4 5 4 1 6 3 1 8 3 3 5 3 7 2 7 7 1 1 5 7 4 1 3 2 9 0 . 2 5 0 9 9 6 1 5 3 5 2 3 9 4 2 9 8 2 3 0 8 2 1 3 3 6 2 3 0 5 2 3 5 0 1 5 2 5 1 3 2 9 0 . 3 7 5 3 5 8 7 8 1 1 5 6 8 2 1 6 8 2 2 7 4 7 2 0 1 7 5 8 2 1 1 1 1 4 9 3 1 3 2 9

0 . 5 0 0 1 5 8 5 0 1 1 2 3 1 1 8 4 1 1 9 5 3 4 8 1 1 4 4 8 1 9 3 9 1 4 6 8 1 3 2 9 0 . 7 5 0 6 0 3 0 2 9 4 4 1 5 6 9 1 6 8 9 2 9 2 1 0 6 5 1 6 7 2 1 4 2 6 1 3 2 9 I . 0 0 0 3 6 2 1 6 7 8 6 1 4 1 4 1 5 3 9 2 0 0 8 1 0 1 4 5 7 1 3 8 9 1 3 2 9 1 . 2 5 0 2 4 1 6 0 6 6 8 1 2 8 8 1 4 1 6 1 4 2 6 2 4 1 2 7 6 1 3 5 5 1 3 2 9 1 . 5 0 0 1 7 1 2 2 5 7 2 1 1 7 4 1 3 0 2 1 0 2 4 8 4 1 1 2 1 1 3 2 5 1 3 2 9 1 . 7 5 0 1 2 9 3 4 9 1 1 0 6 9 1 1 9 4 7 4 3 7 7 9 8 9 1 2 9 7 1 3 2 9 2 . 0 0 0 8 7 2 4 2 1 9 6 8 1 0 8 9 5 4 2 9 5 8 7 5 1 2 7 1 1 3 2 9

2 . 2 5 0 6 5 6 3 6 1 8 7 0 9 8 6 4 0 2 3 2 7 7 7 1 2 4 8 1 3 2 9 2 . 5 0 0 4 4 3 3 0 7 7 7 5 8 8 3 2 9 1 8 3 6 9 4 1 2 2 8 1 3 2 9 2 . 7 5 0 3 3 4 2 6 0 6 8 2 7 8 1 2 2 1 4 5 6 2 3 1 2 0 9 1 3 2 9

3 . 0 0 0 2 2 6 2 1 8 5 9 1 6 7 9 1 6 1 1 5 5 6 3 1 1 9 3 1 3 2 9

3 . 2 5 0 2 2 0 1 7 9 4 9 9 5 7 6 1 2 9 2 5 1 4 1 1 8 0 1 3 2 9

3 . 5 0 0 I 1 5 1 4 3 4 0 8 4 7 1 9 7 4 4 7 4 1 1 6 9 1 3 2 9 3 . 6 2 5 I 1 3 1 2 6 3 6 1 4 1 8 8 6 7 4 5 7 1 1 6 4 1 3 2 9 3 . 7 5 0 1 1 1 1 0 9 3 1 3 3 6 2 7 6 1 4 4 2 1 1 5 9 1 3 2 9 3 . 8 7 5 1 1 0 9 1 2 6 2 3 0 3 6 5 6 4 3 0 1 1 5 6 1 3 2 9

4 . 0 0 0 I 8 7 0 1 9 9 2 3 0 5 5 1 4 2 0 1 1 5 3 1 3 2 9

8 . 0 0 0

0 . 0 1 0 2 5 6 1 0 9 1 3 1 1 7 2 1 1 2 2 3 2 1 2 3 4 1 9 1 7 3 6 9 0 2 3 7 0 7 1 3 2 5 7 6 4 ' 4 . 0 0 0 1 1 0 1 5 0 7 9 0 1 0 8 1 5 4 4 2 8 9 6 8 1 7 6 4

8 . 0 0 0 0 0 8 8 8 1 3 2 0 1 5 1 5 0 9 7 6 4

1 6 . 0 0 0

0 . 0 1 0 2 5 6 1 0 9 1 3 1 1 7 2 1 1 2 2 4 8 1 2 4 0 2 9 1 7 3 6 9 0 2 3 7 0 7 1 2 3 1 4 1 3 8 . 0 0 0 0 0 1 8 4 5 1 1 0 8 0 0 1 3 5 2 8 9 4 1 3

1 6 . 0 0 0 0 0 0 2 1 7 2 0 0 1 1 2 3 4 1 3

3 2 . 0 0 0 -

0 . 0 1 0 2 5 6 1 0 9 1 3 1 1 7 2 1 1 2 2 4 9 1 2 4 3 6 9 1 7 3 6 9 0 2 3 7 0 7 1 2 2 5 2 1 5 1 6 . 0 0 0 0 0 0 1 1 9 1 0 8 0 0 0 1 6 9 2 1 5 3 2 . 0 0 0 0 0 0 I 3 7 0 0 0 8 2 1 5

00

0 . 0 1 0 2 5 6 1 0 9 1 3 1 1 7 2 1 1 2 2 4 9 1 2 4 7 4 9 1 7 3 6 9 0 2 3 7 0 7 1 2 2 5 0



I n c i d e n c e f r o m μ 0 = 1.0

b =

b =

α * 0 . 2 0 0 . 6 0 0 . 9 0 0 . 9 9 I . 0 0 0 . 2 0 0 . 6 0 0 . 9 0 0 . 9 9 1 . 0 0

POINT D I R E C T I O N GAIN X 1 0 0 0 0 NET FLUX DOWN X 1 0 0 0 0 0 . 0 3 1 2 5

0 . 0 1 0 1 2 2 1 0 3 7 6 1 0 5 7 7 1 0 6 3 9 1 0 6 4 6 9 9 7 1 9 9 1 0 9 8 6 3 9 8 4 8 9 8 4 6 0 . 0 1 5 6 2 9 9 8 8 1 0 2 8 7 1 0 5 2 3 1 0 5 9 5 1 0 6 0 4 9 8 4 5 9 8 4 6 9 8 4 6 9 8 4 6 9 8 4 6 0 . 0 3 1 2 5 9 8 1 4 1 0 0 6 7 1 0 2 6 7 1 0 3 2 9 1 0 3 3 6 9 7 2 1 9 7 8 2 9 8 3 0 9 8 4 4 9 8 4 6

0 . 0 6 2 5 0

0 . 0 1 0 2 0 1 1 0 6 3 3 1 0 9 8 6 1 1 0 9 7 1 1 1 1 0 9 9 4 5 9 8 2 7 9 7 3 1 9 7 0 0 9 6 9 7 0 . 0 1 5 6 2 1 0 0 8 0 1 0 5 8 6 1 0 9 9 9 1 1 1 3 0 1 1 1 4 4 9 8 1 8 9 7 6 1 9 7 1 3 9 6 9 8 9 6 9 7 0 . 0 3 1 2 5 9 9 3 4 1 0 4 5 5 1 0 8 8 1 1 1 0 1 6 1 1 0 3 1 9 6 9 3 9 6 9 5 9 6 9 6 9 6 9 7 9 6 9 7 0 . 0 4 6 8 7 9 7 7 6 1 0 2 7 8 1 0 6 8 9 1 0 8 1 8 1 0 8 3 3 9 5 7 0 9 6 3 0 9 6 7 9 9 6 9 5 9 6 9 7 0 . 0 6 2 5 0 9 5 9 3 1 0 0 2 1 1 0 3 7 2 1 0 4 8 2 1 0 4 9 5 9 4 4 9 9 5 6 7 9 6 6 3 9 6 9 3 9 6 9 7

0 . 1 2 5 0 0

0 . 0 1 0 3 1 7 1 1 0 3 0 1 1 6 5 0 1 1 8 5 2 1 1 8 7 5 9 9 0 1 9 6 7 7 9 4 8 2 9 4 1 8 9 4 1 1 0 . 0 3 1 2 5 1 0 0 7 5 1 0 9 3 8 1 1 6 8 9 1 1 9 3 5 1 1 9 6 3 9 6 4 6 9 5 3 9 9 4 4 5 9 4 1 4 9 4 1 1 0 . 0 6 2 5 0 9 7 8 8 1 0 6 8 0 1 1 4 5 7 1 1 7 1 2 1 1 7 4 1 9 3 9 7 9 4 0 4 9 4 0 9 9 4 1 0 9 4 1 1 0 . 0 9 3 7 5 9 4 8 1 1 0 3 3 0 1 1 0 7 1 1 1 3 1 4 1 1 3 4 1 9 1 5 6 9 2 7 2 9 3 7 4 9 4 0 7 9 4 1 1 0 . 1 2 5 0 0 9 1 3 4 9 8 3 3 1 0 4 4 2 1 0 6 4 1 1 0 6 6 4 8 9 2 4 9 1 4 6 9 3 4 0 9 4 0 3 9 4 1 1

0 . 2 5 0 0 0

0 . 0 1 0 4 6 7 1 1 5 9 7 1 2 6 8 0 1 3 0 5 8 1 3 1 0 1 9 8 3 3 9 4 2 7 9 0 3 5 8 8 9 9 8 8 8 3 0 . 0 6 2 5 0 9 9 8 2 1 1 4 1 3 1 2 7 9 2 1 3 2 7 4 1 3 3 3 0 9 3 2 1 9 1 3 8 8 9 5 5 8 8 9 0 8 8 8 3 0 . 1 2 5 0 0 9 4 2 8 1 0 9 0 4 1 2 3 3 3 1 2 8 3 4 1 2 8 9 2 8 8 3 6 8 8 5 8 8 8 7 7 8 8 8 2 8 8 8 3 0 . 1 8 7 5 0 8 8 5 4 1 0 2 3 4 1 1 5 7 1 1 2 0 4 0 1 2 0 9 5 8 3 7 9 8 5 9 4 8 8 0 2 8 8 7 4 8 8 8 3 0 . 2 5 0 0 0 8 2 2 9 9 3 0 4 1 0 3 4 4 1 0 7 0 8 1 0 7 5 0 7 9 5 1 8 3 4 8 8 7 3 3 8 8 6 7 8 8 8 3

0 . 5 0 0 0 0

0 . 0 1 0 6 2 5 1 2 2 9 4 1 4 1 5 7 1 4 8 8 1 1 4 9 6 8 9 7 5 0 9 0 7 3 8 3 1 0 8 0 1 1 7 9 7 5 0 . 0 6 2 5 0 1 0 1 7 0 1 2 2 5 3 1 4 5 8 6 1 5 4 9 6 1 5 6 0 5 9 2 2 9 8 7 6 5 8 2 1 9 8 0 0 1 7 9 7 5 0 . 1 2 5 0 0 9 6 5 4 1 1 9 0 2 1 4 4 4 6 1 5 4 4 3 1 5 5 6 3 8 7 3 3 8 4 6 3 8 1 2 8 7 9 9 2 7 9 7 5 0 . 1 8 7 5 0 9 1 3 4 1 1 4 4 1 1 4 0 7 5 1 5 1 1 2 1 5 2 3 7 8 2 6 4 8 1 7 1 8 0 3 9 7 9 8 2 7 9 7 5 0 . 2 5 0 0 0 8 6 2 1 1 0 9 0 9 1 3 5 4 1 1 4 5 8 1 1 4 7 0 6 7 8 2 0 7 8 9 1 7 9 5 3 7 9 7 3 7 9 7 5 0 . 3 1 2 5 0 8 1 1 6 1 0 3 2 3 1 2 8 7 1 1 3 8 8 0 1 4 0 0 2 7 4 0 1 7 6 2 6 7 8 7 0 7 9 6 4 7 9 7 5 0 . 3 7 5 0 0 7 6 2 0 9 6 8 4 1 2 0 6 9 1 3 0 1 5 1 3 1 2 8 7 0 0 8 7 3 7 5 7 7 9 2 7 9 5 5 7 9 7 5 0 . 4 3 7 5 0 7 1 2 7 8 9 7 6 1 1 1 0 9 1 1 9 5 4 1 2 0 5 5 6 6 3 9 7 1 4 2 7 7 2 0 7 9 4 8 7 9 7 5 0 . 5 0 0 0 0 6 6 0 4 8 0 8 3 9 7 8 4 1 0 4 5 7 1 0 5 3 8 6 2 9 6 6 9 2 8 7 6 5 4 7 9 4 0 7 9 7 5

1 . 0 0 0 0 0

0 . 0 1 0 7 3 9 1 2 9 4 1 1 5 9 7 2 1 7 3 9 2 1 7 5 7 4 9 6 8 0 8 7 0 5 7 3 2 6 6 6 7 1 6 5 8 7 0 . 0 6 2 5 0 1 0 3 0 1 1 3 0 0 7 1 6 7 3 4 1 8 4 8 0 1 8 7 0 3 9 1 5 3 8 3 7 9 7 2 2 3 6 6 5 9 6 5 8 7 0 . 1 2 5 0 0 9 8 0 3 1 2 7 5 8 1 6 8 8 5 1 8 8 3 0 1 9 0 7 9 8 6 5 1 8 0 5 7 7 1 1 8 6 6 4 8 6 5 8 7 0 . 1 8 7 5 0 9 3 0 4 1 2 4 0 8 1 6 8 0 6 1 8 8 9 5 1 9 1 6 3 8 1 7 3 7 7 4 2 7 0 1 2 6 6 3 6 6 5 8 7 0 . 2 5 0 0 0 8 8 1 5 1 2 0 0 0 1 6 5 8 0 1 8 7 7 1 1 9 0 5 3 7 7 2 0 7 4 3 7 6 9 0 8 6 6 2 4 6 5 8 7 0 . 3 1 2 5 0 8 3 4 1 1 1 5 5 8 1 6 2 4 5 1 8 5 0 4 1 8 7 9 5 7 2 9 1 7 1 4 2 6 8 0 5 6 6 1 3 6 5 8 7 0 . 3 7 5 0 0 7 8 8 4 1 1 0 9 3 1 5 8 2 6 1 8 1 2 2 1 8 4 1 8 6 8 8 6 6 8 5 9 6 7 0 5 6 6 0 1 6 5 8 7 0 . 4 3 7 5 0 7 4 4 6 1 0 6 1 3 1 5 3 3 7 1 7 6 4 2 1 7 9 4 0 6 5 0 3 6 5 8 8 6 6 0 8 6 5 9 0 6 5 8 7 0 . 5 0 0 0 0 7 0 2 5 1 0 1 2 4 1 4 7 8 9 1 7 0 7 7 1 7 3 7 3 6 1 4 1 6 3 2 9 6 5 1 4 6 5 7 9 6 5 8 7 0 . 5 6 2 5 0 6 6 2 2 9 6 2 7 1 4 1 8 8 1 6 4 3 5 1 6 7 2 6 5 8 0 0 6 0 8 2 6 4 2 3 6 5 6 9 6 5 8 7 0 . 6 2 5 0 0 6 2 3 7 9 1 2 6 1 3 5 4 0 1 5 7 2 2 1 6 0 0 5 5 4 7 8 5 8 4 7 6 3 3 6 6 5 5 9 6 5 8 7 0 . 6 8 7 5 0 5 8 6 8 8 6 2 0 1 2 8 4 5 1 4 9 4 0 1 5 2 1 2 5 1 7 6 5 6 2 5 6 2 5 4 6 5 4 9 6 5 8 7 0 . 7 5 0 0 0 5 5 1 3 8 1 0 8 1 2 1 0 4 1 4 0 8 9 1 4 3 4 6 4 8 9 1 5 4 1 6 6 1 7 6 6 5 4 0 6 5 8 7 0 . 8 1 2 5 0 5 1 7 2 7 5 8 7 1 1 3 1 1 1 3 1 6 2 1 3 4 0 3 4 6 2 4 5 2 2 0 6 1 0 3 6 5 3 1 6 5 8 7 0 . 8 7 5 0 0 4 8 4 1 7 0 5 1 1 0 4 5 4 1 2 1 4 6 1 2 3 6 6 4 3 7 4 5 0 3 7 6 0 3 5 6 5 2 3 6 5 8 7 0 . 9 3 7 5 0 4 5 1 6 6 4 8 2 9 5 0 2 1 1 0 0 1 1 1 1 9 6 4 1 4 0 4 8 6 8 5 9 7 2 6 5 1 6 6 5 8 7 1 . 0 0 0 0 0 4 1 7 5 5 7 9 4 8 2 7 3 9 5 0 1 9 6 6 1 3 9 2 3 4 7 1 4 5 9 1 6 6 5 1 0 6 5 8 7


T A B L E 17 {continued) I n c i d e n c e f r o m μ 0 = 1.0

b =

α = 0 . 2 0 0 . 6 0 0 . 9 0 0 . 9 9 1 . 0 0 0 . 2 0 0 . 6 0 0 . 9 0 0 . 9 9 1 . 0 0

P O I N T D I R E C T I O N GAIN X 1 0 0 0 0 NET FLUX DOWN X 1 0 0 0 0 • 2 . 0 0 0

= 0 . 0 1 0 7 8 2 1 3 2 8 7 1 7 6 0 6 2 0 3 1 1 2 0 7 0 6 9 6 5 1 8 4 9 0 6 3 8 4 5 0 2 5 4 8 2 5 0 . 1 2 5 9 8 5 6 1 3 2 0 1 1 9 0 3 0 2 2 6 9 0 2 3 2 2 5 8 6 1 7 7 8 2 2 6 1 5 2 4 9 9 8 4 8 2 5 0 . 2 5 0 8 8 7 9 1 2 5 3 9 1 9 1 6 8 2 3 4 1 5 2 4 0 4 0 7 6 8 1 7 1 7 8 5 9 1 2 4 9 6 9 4 8 2 5 0 . 3 7 5 7 9 6 3 1 1 7 3 9 1 8 8 6 2 2 3 5 2 8 2 4 2 1 8 6 8 3 9 6 5 7 1 5 6 7 4 4 9 4 0 4 8 2 5 0 . 5 0 0 7 1 2 3 1 0 8 9 5 1 8 2 9 7 2 3 2 5 2 2 3 9 9 0 6 0 8 6 6 0 0 5 5 4 4 2 4 9 1 0 4 8 2 5 0 . 7 3 0 5 6 6 9 9 2 2 4 1 6 7 1 5 2 1 9 3 6 2 2 7 2 2 4 8 1 1 4 9 9 9 5 0 0 3 4 8 5 4 4 8 2 5 i . 0 0 0 4 4 8 9 7 6 7 5 1 4 8 0 1 1 9 9 4 0 2 0 7 2 1 3 8 0 0 4 1 5 6 4 6 0 9 4 8 0 1 4 8 2 5 i . 2 5 0 3 5 4 0 6 2 8 4 1 2 7 1 9 1 7 4 8 6 1 8 2 1 6 3 0 0 0 3 4 5 9 4 2 6 4 4 7 5 4 4 8 2 5 1 . 5 0 0 2 7 7 6 5 0 4 3 1 0 5 3 5 1 4 6 8 1 1 5 3 1 9 2 3 7 1 2 8 9 4 3 9 7 4 4 7 1 4 4 8 2 5 1 . 6 2 5 2 4 5 1 4 4 6 8 9 4 0 4 1 3 1 5 3 1 3 7 3 1 2 1 1 0 2 6 5 6 3 8 4 9 4 6 9 7 4 8 2 5 1 . 7 5 0 2 1 5 7 3 9 1 4 8 2 3 3 1 1 5 2 5 1 2 0 3 3 1 8 8 0 2 4 4 7 3 7 3 9 4 6 8 1 4 8 2 5 1 . 8 7 5 1 8 8 7 3 3 6 2 6 9 8 4 9 7 4 6 1 0 1 7 3 1 6 7 8 2 2 6 5 3 6 4 3 4 6 6 8 4 8 2 5 2 . 0 0 0 1 6 2 2 2 7 3 3 5 4 3 7 7 4 9 1 7 8 0 7 1 5 0 2 2 1 1 1 3 5 6 5 4 6 5 7 4 8 2 5

» 4 . 0 0 0

' 0 . 0 1 0 7 8 6 1 3 3 5 2 1 8 3 8 7 2 2 8 6 9 2 3 7 2 3 9 6 4 8 8 4 4 7 5 9 1 9 3 5 5 0 3 0 9 1 0 . 1 2 5 9 8 6 1 1 3 2 8 4 2 0 0 4 8 2 6 0 5 5 2 7 1 9 8 8 6 1 3 7 7 7 5 5 6 7 5 3 5 1 9 3 0 9 1 0 . 2 5 0 8 8 8 6 1 2 6 3 8 2 0 3 8 7 2 7 4 3 9 2 8 7 8 9 7 6 7 6 7 1 2 6 5 4 2 2 3 4 8 5 3 0 9 1 0 . 3 7 5 7 9 7 2 1 1 8 5 5 2 0 2 7 7 2 8 1 7 5 2 9 7 0 0 6 8 3 4 6 5 1 4 5 1 6 7 3 4 5 0 3 0 9 1 0 . 5 0 0 7 1 3 2 1 1 0 3 0 1 9 9 0 9 2 8 5 0 8 3 0 1 8 3 6 0 7 9 5 9 4 2 4 9 1 6 3 4 1 5 3 0 9 1 0 . 7 5 0 5 6 8 2 9 4 0 5 1 8 7 4 2 2 8 3 9 3 3 0 3 1 0 4 8 0 3 4 9 2 1 4 4 3 2 3 3 4 3 3 0 9 1 1 . 0 0 0 4 5 0 9 7 9 1 4 1 7 2 8 5 2 7 6 0 8 2 9 6 9 9 3 7 8 8 4 0 5 6 3 9 8 1 3 2 7 3 3 0 9 1 1 . 2 5 0 3 5 6 8 6 6 0 3 1 5 7 2 2 2 6 4 0 2 2 8 6 0 6 2 9 8 4 3 3 3 2 3 5 6 9 3 2 0 6 3 0 9 1 1 . 5 0 0 2 8 1 8 5 4 7 3 1 4 1 5 1 2 4 9 1 9 2 7 1 8 0 2 3 4 8 2 7 2 9 3 1 9 5 3 1 4 2 3 0 9 1 1 . 7 5 0 2222 4 5 1 4 1 2 6 2 6 2 3 2 5 1 2 5 5 1 9 1 8 4 6 2 2 3 1 2 8 6 1 3 0 8 1 3 0 9 1 2 . 0 0 0 1 7 3 0 3 7 0 7 1 1 1 7 3 2 1 4 5 8 2 3 6 8 6 1 4 5 1 1 8 2 1 2 5 6 3 3 0 2 5 3 0 9 1 2 . 2 5 0 1 3 7 7 3 0 3 2 9 8 0 8 1 9 5 8 0 2 1 7 2 7 1 1 4 0 I 4 8 6 2 3 0 1 2 9 7 4 3 0 9 1 2 . 5 0 0 1 0 8 2 2 4 7 0 8 5 3 1 1 7 6 4 4 1 9 6 7 1 8 9 5 1 2 1 1 2 0 7 2 2 9 2 8 3 0 9 1 2 . 7 5 0 8 5 0 2 0 0 3 7 3 4 2 1 5 6 6 7 1 7 5 3 8 7 0 3 9 8 8 1 8 7 4 2 8 8 6 3 0 9 1 3 . 0 0 0 ot> 6 1 6 1 4 6 2 3 0 1 3 6 5 8 1 5 3 4 2 5 5 2 8 0 8 1 7 0 4 2 8 4 9 3 0 9 1 3 . 2 5 0 5 2 1 1 2 8 9 5 1 8 5 1 1 6 1 5 1 3 0 8 3 4 3 4 6 6 3 1 5 6 2 2 8 1 8 3 0 9 1 3 . 5 0 0 4 0 6 1 0 1 4 4 1 8 9 9 5 2 5 1 0 7 5 0 3 4 1 5 4 9 1 4 4 5 2 7 9 1 3 0 9 1 3 . 6 2 5 3 5 7 8 9 0 3 7 0 0 8 4 4 9 9 5 4 2 3 0 3 5 0 1 1 3 9 5 2 7 8 0 3 0 9 1 3 . 7 5 0 3 1 4 7 7 3 3 2 0 7 7 3 3 7 8 2 8 8 2 7 0 4 6 0 1 3 5 2 2 7 7 0 3 0 9 1 3 . 8 7 5 2 7 4 6 5 9 2 6 9 6 6 1 5 3 6 9 5 0 2 4 0 4 2 4 1 3 1 5 2 7 6 2 3 0 9 1 4 . 0 0 0 2 3 5 5 3 2 2 0 8 1 4 6 9 1 5 2 9 1 2 1 5 3 9 4 1 2 8 5 2 7 5 5 3 0 9 1

= 8 . 0 0 0

= 0 . 0 1 0 7 8 6 1 3 3 5 4 1 8 4 9 9 2 4 3 1 4 2 5 9 9 2 9 6 4 8 8 4 4 6 5 8 5 2 2 7 1 2 1 7 8 2 •• 4 . 0 0 0 2 5 3 7 2 2 4 5 1 1 1 8 7 4 5 2 5 0 0 7 2 0 8 3 3 7 8 8 9 1 6 3 1 1 7 8 2 •• 8 . 0 0 0 5 1 7 2 6 5 2 1 0 5 3 0 8 6 4 1 2 1 6 0 1 2 2 1 1 7 8 2

1 6 . 0 0 0

•• 0 . 0 1 0 7 8 6 1 3 3 5 4 1 8 5 0 1 2 4 7 0 3 2 7 4 0 9 9 6 4 8 8 4 4 6 5 8 5 1 2 4 8 7 9 6 4 = 8 . 0 0 0 5 2 3 5 7 7 1 0 8 1 7 2 5 1 8 0 4 1 0 1 1 0 6 9 2 9 6 4 = 1 6 . 0 0 0 0 0 4 5 1 1 1 6 6 9 0 0 2 2 9 6 9 6 4

3 2 . 0 0 0

0 . 0 1 0 7 8 6 1 3 3 5 4 1 8 5 0 1 2 4 7 2 8 2 8 2 0 8 9 6 4 8 8 4 4 6 5 8 5 1 2 4 7 3 5 0 2 = 1 6 . 0 0 0 0 0 9 2 8 4 7 2 5 1 8 2 0 0 2 1 6 6 5 0 2 = 3 2 . 0 0 0 0 0 0 3 2 8 7 0 0 0 0 1 9 5 0 2

: QC

0 . 0 1 0 7 8 6 1 3 3 5 4 1 8 5 0 1 2 4 7 2 8 2 9 0 7 8 9 6 4 8 8 4 4 6 5 8 5 1 2 4 7 3 0


TABLE 17 (continued)

I n c i d e n c e f r o m N a r r o w S o u r c e L a y e r (N)

a s 0 . 2 0 0 . 6 0 0 . 9 0 0 . 9 9 1 . 0 0 0 . 2 0 0 . 6 0 0 . 9 0 0 . 9 9 1 . 0 0

GAIN MOMENT g_ j X 1 0 0 0 0 NET FLUX DOWN X 1 0 0 0 0 0 . 0 3 1 2 5

0 . 0 ***** ***** ***** ***** ***** 9 8 8 4 9 6 4 2 9 4 5 1 9 3 9 2 9 3 8 6 0 . 0 1 5 6 2 3 6 5 3 2 3 7 7 0 0 3 8 6 2 2 3 8 9 0 6 3 8 9 3 8 9 3 0 3 9 3 4 4 9 3 7 5 9 3 8 5 9 3 8 6 0 . 0 3 1 2 5 2 9 6 5 8 3 0 6 2 4 3 1 3 8 7 3 1 6 2 3 3 1 6 4 9 8 8 9 4 9 1 3 3 9 3 2 1 9 3 7 9 9 3 8 5

0 . 0 6 2 5 0

0 . 0 ***** ***** ***** ***** ***** 9 8 1 4 9 4 1 6 9 0 9 1 8 9 8 8 8 9 7 7 0 . 0 1 5 6 2 3 6 7 8 0 3 8 5 1 4 3 9 9 2 9 4 0 3 7 5 4 0 4 2 5 9 2 3 1 9 1 1 4 9 0 1 3 8 9 8 0 8 9 7 7 0 . 0 3 1 2 5 2 9 9 8 7 3 1 6 9 0 3 3 0 8 6 3 3 5 2 6 3 3 5 7 6 8 8 1 8 8 8 9 6 8 9 5 6 8 9 7 4 8 9 7 6 0 . 0 4 6 0 7 2 6 0 3 1 2 7 6 2 2 2 8 9 2 7 2 9 3 4 0 2 9 3 8 6 8 4 7 0 8 7 1 2 8 9 0 8 8 9 7 0 8 9 7 7 0 . 0 6 2 5 0 2 3 1 9 7 2 4 5 4 9 2 5 6 5 9 2 6 0 0 9 2 6 0 4 9 8 1 6 3 8 5 5 0 8 8 6 6 8 9 6 6 8 9 7 7

0 . 1 2 5 0 0

0 . 0 ***** ***** ***** ***** ***** 9 7 2 1 9 0 9 3 8 5 4 8 8 3 71 8 3 5 0 0 . 0 3 1 2 5 3 0 2 8 4 3 2 7 3 2 3 4 8 5 2 3 5 5 4 5 3 5 6 2 3 8 7 1 8 8 5 6 1 8 4 0 9 8 3 5 6 8 3 5 0 0 . 0 6 2 5 0 2 3 6 1 5 2 5 9 8 5 2 8 0 5 6 2 8 7 3 5 2 8 8 1 2 8 0 5 4 8 1 9 9 8 3 1 2 8 3 4 7 8 3 5 1 0 . 0 9 3 7 5 1 9 7 6 6 2 1 9 2 9 2 3 8 2 6 2 4 4 5 1 2 4 5 2 2 7 5 1 5 7 9 0 1 8 2 3 1 8 3 3 8 8 3 5 0 0 . 1 2 5 0 0 1 7 0 1 4 1 8 7 8 5 2 0 3 3 7 2 0 8 4 8 2 0 9 0 6 7 0 5 7 7 6 4 7 8 1 6 2 8 3 3 1 8 3 5 0

0 . 2 5 0 0 0

0 . 0 ***** ***** ***** ***** ***** 9 6 1 2 8 6 8 0 7 7 9 2 7 4 8 4 7 4 4 8 0 . 0 6 2 5 0 2 3 9 2 7 2 7 2 0 4 3 0 3 2 9 3 1 4 1 6 3 1 5 4 1 7 9 3 2 7 7 6 0 7 5 4 4 7 4 5 9 7 4 4 9 0 . 1 2 5 0 0 1 7 4 9 3 2 0 5 8 6 2 3 5 9 5 2 4 6 5 2 2 4 7 7 4 6 9 1 6 7 1 7 1 7 3 7 7 7 4 4 1 7 4 4 8 0 . 1 8 7 5 0 1 3 8 5 9 1 6 5 8 7 1 9 2 6 9 2 0 2 1 6 2 0 3 2 6 6 1 3 9 6 7 1 0 7 2 4 4 7 4 2 7 7 4 4 8 0 . 2 5 0 0 0 1 1 2 9 5 1 3 4 0 5 1 5 4 7 7 1 6 2 0 8 1 6 2 9 3 5 5 1 3 6 3 3 5 7 1 3 5 7 4 1 6 7 4 4 8

0 . 5 0 0 0 0

0 . 0 ***** ***** ***** ***** ***** 9 5 1 6 8 2 4 4 6 8 5 3 6 3 1 7 6 2 5 3 0 . 0 6 2 5 0 2 4 1 4 6 2 8 2 4 4 3 2 6 6 0 3 4 3 4 5 3 4 5 4 6 7 8 2 6 7 3 0 0 6 5 9 1 6 2 9 0 6 2 5 4 0 . 1 2 5 0 0 1 7 7 5 8 2 1 8 2 4 2 6 3 4 3 2 8 0 9 6 2 8 3 0 5 6 7 9 7 6 6 8 4 6 4 0 8 6 2 7 1 6 2 5 3 0 . 1 8 7 5 0 1 4 1 8 8 1 8 0 8 9 2 2 5 3 4 2 4 2 8 1 2 4 4 9 1 6 0 0 5 6 1 8 8 6 2 5 6 6 2 5 4 6 2 5 3 0 . 2 5 0 0 0 1 1 7 6 2 1 5 4 2 7 1 9 6 8 7 2 1 3 7 9 2 1 5 8 2 5 3 6 0 5 7 7 0 6 1 2 5 6 2 4 0 6 2 5 3 0 . 3 1 2 5 0 9 9 5 7 1 3 3 3 8 1 7 3 2 6 1 8 9 2 3 1 9 1 1 6 4 8 1 9 5 4 1 2 6 0 0 9 6 2 2 8 6 2 5 3 0 . 3 7 5 0 0 8 5 3 8 1 1 5 9 3 1 5 2 3 2 1 6 6 9 6 1 6 8 7 3 4 3 5 8 5 1 0 1 5 9 0 8 6 2 1 7 6 2 5 3 0 . 4 3 7 5 0 7 3 7 4 1 0 0 5 1 1 3 2 5 3 1 4 5 4 4 1 4 7 0 0 3 9 6 1 4 8 3 0 5 8 1 9 6 2 0 7 6 2 5 3 0 . 5 0 0 0 0 6 3 6 0 8 5 2 0 1 1 0 9 0 1 2 1 2 4 1 2 2 4 9 3 6 1 8 4 5 9 8 5 7 4 2 6 1 9 8 6 2 5 3

1 . 0 0 0 0 0

0 . 0 ***** ***** ***** ***** ***** 9 4 6 0 7 9 1 2 5 8 8 1 4 9 5 5 4 8 3 7 0 . 0 6 2 5 0 2 4 2 5 3 2 8 9 2 6 3 4 7 8 7 3 7 3 8 6 3 7 7 1 3 7 7 6 4 6 9 5 2 5 6 0 6 4 9 2 6 4 8 3 7 0 . 1 2 5 0 0 1 7 8 7 9 2 2 6 0 0 2 8 7 5 9 3 1 5 4 8 3 1 9 0 1 6 7 3 1 6 3 1 7 5 4 1 0 4 9 0 4 4 8 3 7 0 . 1 8 7 5 0 1 4 3 2 6 1 8 9 6 6 2 5 2 4 1 2 8 1 3 8 2 8 5 0 7 5 9 3 2 5 8 0 1 5 2 4 2 4 8 8 6 4 8 3 7 0 . 2 5 0 0 0 1 1 9 2 1 1 6 4 1 8 2 2 7 0 1 2 5 6 5 4 2 6 0 3 2 5 2 7 9 5 3 6 0 5 0 9 2 4 8 6 9 4 8 3 7 0 . 3 1 2 5 0 1 0 1 4 1 1 4 4 6 1 2 0 6 7 7 2 3 6 4 5 2 4 0 2 6 4 7 3 0 4 9 7 5 4 9 5 7 4 8 5 4 4 8 3 7 Q . 3 7 5 0 0 8 7 5 4 1 2 8 7 7 1 8 9 6 7 2 1 9 1 8 2 2 2 9 8 4 2 5 9 4 6 3 4 4 8 3 3 4 8 3 9 4 8 3 7 0 . 4 3 7 5 0 7 6 3 6 1 1 5 4 9 1 7 4 6 8 2 0 3 7 3 2 0 7 4 9 3 8 5 0 4 3 2 9 4 7 2 0 4 8 2 6 4 8 3 7 0 . 5 0 0 0 0 6 7 1 3 1 0 4 0 7 1 6 1 1 7 1 8 9 5 1 1 9 3 1 9 3 4 9 2 4 0 5 5 4 6 1 5 4 8 1 4 4 8 3 7 0 . 5 6 2 5 0 5 9 3 6 9 4 0 8 1 4 8 7 5 1 7 6 1 5 1 7 9 7 2 3 1 7 6 3 8 0 8 4 5 1 8 4 8 0 2 4 8 3 7 0 . 6 2 5 0 0 5 2 7 3 8 5 1 9 1 3 7 1 3 1 6 3 3 9 1 6 6 8 1 2 8 9 6 3 5 8 4 4 4 2 9 4 7 9 2 4 8 3 7 0 . 6 8 7 5 0 4 7 0 1 7 7 1 8 1 2 6 1 0 1 5 1 0 2 1 5 4 2 7 2 6 4 7 3 3 8 1 4 3 4 6 4 7 8 2 4 8 3 7 0 . 7 5 0 0 0 4 2 0 2 6 9 8 6 1 1 5 5 0 1 3 8 8 7 1 4 1 9 3 2 4 2 5 3 1 9 7 4 2 7 1 4 7 7 3 4 8 3 7 0 . 8 1 2 5 0 3 7 6 2 6 3 0 9 1 0 5 1 5 1 2 6 7 8 1 2 9 6 1 2 2 2 6 3 0 3 1 4 2 0 2 4 7 6 5 4 8 3 7 0 . 8 7 5 0 0 3 3 71 5 6 7 0 9 4 8 4 1 1 4 5 0 1 1 7 0 7 2 0 4 8 2 8 8 1 4 1 3 9 4 7 5 7 4 8 3 7 0 . 9 3 7 5 0 3 0 1 7 5 0 5 1 8 4 2 2 1 0 1 6 1 1 0 3 8 8 1 8 8 9 2 7 4 7 4 0 8 3 4 7 5 0 4 8 3 7 1 . 0 0 0 0 0 2 6 8 1 4 3 7 9 7 1 7 0 8 6 0 2 8 7 9 0 1 7 4 6 2 6 2 9 4 0 3 4 4 7 4 4 4 8 3 7


T A B L E Π (continued)

I n c i d e n c e f r o m N a r r o w S o u r c e L a y e r (ΛΓ)

b =

b =

α = 0 . 2 0 0 . 6 0 0 . 9 0 0 . 9 9 1 . 0 0 0 . 2 0 0 . 6 0 0 . 9 0 0 . 9 9 1 . 0 0

GAIN MOMENT 8_ , X 1 0 0 0 0 NET FLUX DOWN X 1 0 0 0 0 • 2 . 0 0 0

: 0 . 0 ***** ***** ***** ***** ***** 9 4 4 4 7 7 7 0 5 1 4 9 3 6 0 6 3 3 8 4 0 . 1 2 5 1 7 9 0 7 2 2 8 9 3 3 0 4 2 8 3 4 7 1 3 3 5 3 2 2 6 7 1 2 6 1 6 2 4 6 5 9 3 5 5 2 3 3 8 4 0 . 2 5 0 1 1 9 5 6 1 6 7 7 4 2 4 7 1 5 2 9 4 6 4 3 0 1 4 8 5 2 5 8 5 1 8 9 4 3 1 8 3 5 1 2 3 3 8 4 0 . 3 7 5 8 7 9 8 1 3 3 0 5 2 1 3 3 1 2 6 3 5 4 2 7 0 8 6 4 2 3 3 4 4 4 3 4 0 3 2 3 4 7 7 3 3 8 4 0 . 5 0 0 6 7 6 6 1 0 9 1 9 1 8 8 4 9 2 4 0 2 0 2 4 7 8 3 3 4 6 2 3 8 4 1 3 7 8 2 3 4 4 6 3 3 8 4 0 . 7 5 0 4 2 8 8 7 7 2 9 1 5 1 4 8 2 0 3 3 8 2 1 1 1 8 2 3 8 1 2 9 2 1 3 3 5 9 3 3 9 1 3 3 8 4 I . 0 0 0 2 8 6 0 5 6 5 2 1 2 3 0 8 1 7 2 3 8 1 7 9 9 1 1 6 7 8 2 2 5 9 3 0 1 7 3 3 4 4 3 3 8 4 1 . 2 5 0 1 9 6 2 4 1 8 6 9 9 3 7 1 4 3 9 1 1 5 0 8 0 1 2 0 2 1 7 7 1 2 7 4 0 3 3 0 4 3 3 8 4 1 . 5 0 0 1 3 7 0 3 0 9 8 7 8 4 8 1 1 6 4 8 1 2 2 4 2 8 7 3 1 4 0 9 2 5 1 8 3 2 7 2 3 3 8 4 1 . 6 2 5 1 1 4 8 2 6 5 0 6 8 6 7 1 0 2 8 0 1 0 8 1 4 7 4 8 1 2 6 6 2 4 2 6 3 2 5 8 3 3 8 4 1 . 7 5 0 9 6 2 2 2 4 7 5 9 0 4 8 8 8 6 9 3 5 4 6 4 2 1 1 4 3 2 3 4 6 3 2 4 6 3 3 8 4 1 . 8 7 5 8 0 4 1 8 / 2 4 9 2 5 7 4 2 1 7 8 1 3 5 5 4 1 0 4 1 2 2 7 8 3 2 3 6 3 3 8 4 2 . 0 0 0 6 6 1 1 4 7 8 3 7 7 3 5 6 3 5 5 9 2 7 4 8 1 9 5 6 2 2 2 4 3 2 2 8 3 3 8 4

= 4 . 0 0 0

•• 0 . 0 ***** ***** ***** ***** ***** 9 4 4 3 7 7 4 9 4 8 4 6 2 5 5 1 2 1 3 1 0 . 1 2 5 1 7 9 0 9 2 2 9 3 4 3 1 0 9 2 3 7 1 2 0 3 8 1 9 3 6 7 1 1 6 1 3 9 4 3 4 3 2 4 9 3 2 1 3 0 0 . 2 5 0 1 1 9 5 9 1 6 3 2 3 2 5 5 1 0 3 2 3 4 2 3 3 5 8 1 5 2 5 6 5 1 6 4 3 9 9 9 2 4 5 0 2 1 3 0 0 . 3 7 5 8 3 0 1 1 3 3 6 2 2 2 2 5 3 2 9 6 7 8 3 1 0 4 9 4 2 3 1 4 4 1 5 3 7 0 2 2 4 1 2 2 1 3 0 0 . 5 0 0 6 7 7 0 1 0 9 8 5 1 9 9 0 1 2 7 7 8 0 2 9 2 6 1 3 4 5 9 3 8 1 0 3 4 3 9 2 3 7 6 2 1 3 0 0 . 7 5 0 4 2 9 3 7 8 1 8 1 6 4 7 0 2 4 9 5 7 2 6 6 0 5 2 3 7 8 2 8 8 3 2 9 8 7 2 3 1 0 2 1 3 0 1 . 0 0 0 2 8 6 7 5 7 7 0 1 3 9 2 8 2 2 7 2 5 2 4 4 8 4 1 6 7 4 2 2 1 0 2 6 0 8 2 2 5 1 2 1 3 0 1 . 2 5 0 1 9 7 3 4 3 4 4 1 1 8 9 7 2 0 7 7 2 2 2 5 9 4 1 1 9 6 1 7 0 8 2 2 8 6 2 1 9 6 2 1 3 0 1 . 5 0 0 1 3 8 5 3 3 1 1 1 0 2 1 0 1 8 9 7 7 2 0 8 2 2 8 6 5 1 3 2 8 2 0 1 0 2 1 4 7 2 1 3 0 1 . 7 5 0 9 8 6 2 5 4 5 8 7 7 6 1 7 2 8 5 1 9 1 1 5 6 3 0 1 0 3 7 1 7 7 3 2 1 0 1 2 1 3 0 2 . 0 0 0 7 1 0 1 9 6 7 7 5 3 9 1 5 6 6 2 1 7 4 4 4 4 6 2 8 1 3 1 5 7 0 2 0 6 0 2 1 3 0 2 . 2 5 0 5 1 5 1 5 2 7 6 4 6 0 1 4 0 9 1 1 5 7 9 4 3 4 1 6 3 9 1 3 9 5 2 0 2 3 2 1 3 0 2 . 5 0 0 3 7 6 1 1 8 7 5 5 1 0 1 2 5 5 8 1 4 1 5 6 2 5 3 5 0 4 1 2 4 6 1 9 9 0 2 1 3 0 2 . 7 5 0 2 7 6 9 2 4 4 6 6 5 1 1 0 5 4 1 2 5 2 1 1 8 8 3 9 9 1 1 1 9 I 9 6 0 2 1 3 0 3 . 0 0 0 2 0 4 7 1 7 3 9 0 6 9 5 6 9 1 0 8 8 4 1 4 0 3 1 8 1 0 1 2 1 9 3 4 2 1 3 0 3 . 2 5 0 1 5 0 5 5 4 3 2 1 4 8 0 9 1 9 2 3 4 1 0 5 2 5 4 9 2 3 1 9 1 2 2 1 3 0 3 . 5 0 0 1 1 1 4 2 3 2 5 7 1 6 6 0 4 7 5 5 5 7 9 2 0 6 8 5 1 1 8 9 4 2 1 3 0 3 . 6 2 5 9 5 3 6 7 2 2 6 1 5 8 4 6 6 6 9 4 6 9 1 8 6 8 2 0 1 8 8 6 2 1 3 0 3 . 7 5 0 8 2 3 1 4 1 9 5 2 5 0 6 7 5 8 0 5 6 0 1 6 9 7 9 4 1 8 7 9 2 1 3 0 3 . 8 7 5 7 0 2 6 4 1 6 3 5 4 2 4 2 4 8 5 9 5 2 1 5 4 7 7 2 1 8 7 4 2 1 3 0 4 . 0 0 0 5 8 2 1 1 1 2 5 6 3 2 2 8 3 6 9 3 4 6 1 4 3 7 5 3 1 8 6 9 2 1 3 0

: 8 . 0 0 0

: 0 . 0 ***** ***** ***** ***** ***** 9 4 4 3 7 7 4 9 4 8 0 6 1 9 8 0 1 2 2 6 : 4 . 0 0 0 6 3 2 8 3 2 6 9 9 1 2 8 0 1 1 7 3 2 8 4 4 1 2 1 5 1 9 1 1 0 4 1 2 2 6 : 8 . 0 0 0 1 5 1 5 2 1 4 2 4 2 1 2 3 1 4 9 2 8 2 5 1 2 2 6

= 1 6 . 0 0 0

•• 0 . 0 ***** ***** ***** ***** ***** 9 4 4 3 7 7 4 9 4 8 0 5 1 8 2 8 6 6 3 •• 8 . 0 0 0 1 7 3 3 1 7 3 1 5 1 7 3 2 1 1 3 6 3 4 6 8 6 6 3 = 1 6 . 0 0 0 0 0 2 3 4 5 1 1 4 8 0 0 1 2 0 0 6 6 3

= 3 2 . 0 0 0

= 0 . 0 ***** ***** ***** ***** ***** 9 4 4 3 7 7 4 9 4 8 0 5 1 8 1 8 3 4 6 = 1 6 . 0 0 0 0 0 5 1 9 2 5 1 7 3 2 1 0 0 1 1 1 2 3 4 6 = 3 2 . 0 0 0 0 0 0 2 2 5 9 8 0 0 0 1 3 3 4 6

= 0 . 0 ***** ***** ***** ***** ***** 9 4 4 3 7 7 4 9 4 8 0 5 1 8 1 8 0


T A B L E Π {continued) U n i f o r m I n c i d e n c e (U)

α = 0 . 2 0 0 . 6 0 0 . 9 0 0 . 9 9 1 . 0 0 0 . 2 0 0 . 6 0 0 . 9 0 0 . 9 9 1 . 0 0

GAIN MOMENT g o X 1 0 0 0 0 NET FLUX OOWN X 1 0 0 0 0 b = 0 . 0 3 1 2 5

τ - 0 . 0 1 0 1 1 6 1 0 3 5 8 1 0 5 4 9 1 0 6 0 8 1 0 6 1 4 9 9 4 5 9 8 3 0 9 7 4 0 9 7 1 2 9 7 0 9 0 . 0 1 5 6 2 9 4 1 8 9 7 0 1 9 9 2 4 9 9 9 2 1 0 0 0 0 9 7 0 2 9 7 0 5 9 7 0 8 9 7 0 9 9 7 0 9 0 . 0 3 1 2 5 8 8 9 4 9 1 3 3 9 3 2 1 9 3 7 9 9 3 8 5 9 4 7 3 9 5 8 7 9 6 7 8 9 7 0 6 9 7 0 9

b * 0 . 0 6 2 5 0

τ = 0 . 0 1 0 1 8 6 1 0 5 8 4 1 0 9 0 9 1 1 0 1 2 1 1 0 2 3 9 9 0 0 9 6 8 4 9 5 0 8 9 4 5 2 9 4 4 6 0 . 0 1 5 6 2 9 4 9 9 9 9 6 4 1 0 3 4 4 1 0 4 6 3 1 0 4 7 7 9 6 5 4 9 5 5 6 9 4 7 4 9 4 4 9 9 4 4 6 0 . 0 3 1 2 5 9 0 0 0 9 4 7 5 9 8 6 3 9 9 8 6 9 9 9 9 9 4 2 3 9 4 3 4 9 4 4 3 9 4 4 5 9 4 4 6 0 . 0 4 6 8 7 8 5 6 7 9 0 2 1 9 3 9 3 9 5 1 0 9 5 2 3 9 2 0 4 9 3 1 9 9 4 1 3 9 4 4 2 9 4 4 6 0 . 0 6 2 5 0 8 1 6 3 8 5 5 0 8 8 6 6 8 9 6 6 8 9 7 7 8 9 9 5 9 2 0 9 9 3 8 4 9 4 3 9 9 4 4 6

b = 0 . 1 2 5 0 0

τ = 0 . 0 1 0 2 7 9 1 0 9 0 7 1 1 4 5 2 1 1 6 2 9 1 1 6 5 0 9 8 2 8 9 4 3 9 9 1 0 1 8 9 9 1 8 9 7 8 0 . 0 3 1 2 5 9 1 1 4 9 d 6 8 1 0 5 2 3 1 0 7 3 7 1 0 7 6 2 9 3 4 6 9 1 8 1 9 0 3 3 8 9 8 4 8 9 7 8 0 . 0 6 2 5 0 8 3 2 2 9 0 8 8 9 7 5 6 9 9 7 5 1 0 0 0 0 8 9 1 1 8 9 4 4 8 9 6 9 8 9 7 7 8 9 7 8 0 . 0 9 3 7 5 7 6 5 9 8 3 7 9 9 0 0 8 9 2 1 4 9 2 3 8 8 5 1 2 8 7 2 6 8 9 1 1 8 9 7 1 8 9 7 8 0 . 1 2 5 0 0 7 0 5 7 7 6 4 7 8 1 6 2 8 3 3 1 8 3 5 0 8 1 4 4 8 5 2 5 8 8 5 7 8 9 6 6 8 9 7 8

b = 0 . 2 5 0 0 0

τ = 0 . 0 1 0 3 8 8 1 1 3 2 0 1 2 2 0 8 1 2 5 1 6 1 2 5 5 2 9 7 3 0 9 0 7 5 8 4 4 7 8 2 2 8 8 2 0 2 0 . 0 6 2 5 0 8 4 6 2 9 6 2 3 1 0 7 3 7 1 1 1 2 3 1 1 1 7 0 8 8 0 1 8 5 5 7 8 3 0 4 8 2 1 3 8 2 0 2 0 . 1 2 5 0 0 7 2 7 1 8 4 3 3 9 5 5 9 9 9 5 4 1 0 0 0 0 8 0 1 7 8 1 0 6 8 1 7 8 8 2 0 0 8 2 0 2 0 . 1 8 7 5 0 6 3 3 0 7 3 9 1 8 4 2 4 8 7 8 8 8 8 3 0 7 3 3 9 7 7 1 1 8 0 6 5 8 1 8 8 8 2 0 2 0 . 2 5 0 0 0 5 5 1 3 6 3 3 5 7 1 3 5 7 4 1 6 7 4 4 8 6 7 4 7 7 3 6 7 7 9 6 7 8 1 7 8 8 2 0 2

b * 0 . 5 0 0 0 0

τ * 0 . 0 1 0 4 8 4 1 1 7 5 6 1 3 1 4 7 1 3 6 8 3 1 3 7 4 7 9 6 2 8 8 6 3 5 7 5 2 5 7 0 9 3 7 0 4 2 0 . 0 6 2 5 0 8 5 7 7 1 0 1 4 8 1 1 8 7 9 1 2 5 4 8 1 2 6 2 8 8 6 8 9 8 0 9 2 7 3 6 9 7 0 7 7 7 0 4 2 0 . 1 2 5 0 0 7 4 0 9 9 0 5 7 1 0 9 0 5 1 1 6 2 5 1 1 7 1 1 7 8 9 3 7 6 1 3 7 2 2 7 7 0 6 2 7 0 4 2 0 . 1 8 7 5 0 6 5 0 2 8 1 4 8 1 0 0 2 2 1 0 7 5 8 1 0 8 4 6 7 1 9 9 7 1 8 3 7 0 9 6 7 0 4 8 7 0 4 2 0 . 2 5 0 0 0 5 7 5 5 7 3 4 8 9 1 8 6 9 9 1 3 1 0 0 0 0 6 5 8 7 6 7 9 6 6 9 7 6 7 0 3 5 7 0 4 2 0 . 3 1 2 5 0 5 1 2 1 6 6 2 4 8 3 7 4 9 0 7 0 9 1 5 4 6 0 4 4 6 4 4 7 6 8 6 7 7 0 2 3 7 0 4 2 0 . 3 7 5 0 0 4 5 6 9 5 9 5 0 7 5 6 6 8 2 1 1 8 2 8 9 5 5 6 0 6 1 3 3 6 7 6 7 7 0 1 2 7 0 4 2 0 . 4 3 7 5 0 4 0 7 8 5 3 0 0 6 7 3 2 7 3 0 4 7 3 7 3 5 1 2 8 5 8 5 2 6 6 7 8 7 0 0 3 7 0 4 2 0 . 5 0 0 0 0 3 6 1 8 4 5 9 8 5 7 4 2 6 1 9 8 6 2 5 3 4 7 4 3 5 6 0 3 6 5 9 9 6 9 9 4 7 0 4 2

b * 1 . 0 0 0 0 0

τ = 0 . 0 1 0 5 4 0 1 2 0 8 8 1 4 1 1 9 1 5 0 4 5 1 5 1 6 3 9 5 6 1 8 2 5 7 6 4 7 3 5 6 4 0 5 5 3 4 0 . 0 6 2 5 0 8 6 4 1 1 0 5 3 5 1 3 0 2 9 1 4 1 6 7 1 4 3 1 2 8 6 1 5 7 6 9 6 6 3 0 4 5 6 2 2 5 5 3 4 0 . 1 2 5 0 0 7 4 8 2 9 4 9 8 1 2 2 1 2 1 3 4 6 4 1 3 6 2 4 7 8 1 2 7 1 9 6 6 1 4 6 5 6 0 5 5 5 3 4 0 . 1 8 7 5 0 6 5 8 5 8 6 4 5 1 1 4 8 5 1 2 8 1 3 1 2 9 8 2 7 1 1 0 6 7 4 3 5 9 9 8 5 5 8 8 5 5 3 4 0 . 2 5 0 0 0 5 8 5 1 7 9 1 0 1 0 8 1 5 1 2 1 9 0 1 2 3 6 6 6 4 9 0 6 3 3 0 5 8 5 9 5 5 7 3 5 5 3 4 0 . 3 1 2 5 0 5 2 3 1 7 2 6 1 1 0 1 8 5 1 1 5 8 4 1 1 7 6 4 5 9 3 6 5 9 5 1 5 7 2 8 5 5 5 8 5 5 3 4 0 . 3 7 5 0 0 4 6 9 9 6 6 7 7 9 5 8 4 1 0 9 9 0 1 1 1 7 1 5 4 4 0 5 6 0 3 5 6 0 4 5 5 4 4 5 5 3 4 0 . 4 3 7 5 0 4 2 3 6 6 1 4 7 9 0 0 7 1 0 4 0 4 1 0 5 8 4 4 9 9 4 5 2 8 2 5 4 8 8 5 5 3 1 5 5 3 4 0 . 5 0 0 0 0 3 8 2 9 5 6 6 2 8 4 4 9 9 8 2 2 1 0 0 0 0 4 5 9 1 4 9 8 7 5 3 7 9 5 5 1 8 5 5 3 4 0 . 5 6 2 5 0 3 4 6 8 5 2 1 3 7 9 0 5 9 2 4 2 9 4 1 6 4 2 2 7 4 7 1 6 5 2 7 7 5 5 0 6 5 5 3 4 0 . 6 2 5 0 0 3 1 4 6 4 7 9 6 7 3 7 3 8 6 6 1 8 8 2 9 3 8 9 6 4 4 6 5 5 1 8 1 5 4 9 5 5 5 3 4 0 . 6 8 7 5 0 2 8 5 7 4 4 0 6 6 8 4 9 8 0 7 6 8 2 3 6 3 5 9 7 4 2 3 5 5 0 9 2 5 4 8 4 5 5 3 4 0 . 7 5 0 0 0 2 5 9 7 4 0 3 7 6 3 2 8 7 4 8 4 7 6 3 4 3 3 2 4 4 0 2 4 5 0 1 0 5 4 7 5 5 5 3 4 0 . 8 1 2 5 0 2 3 6 0 3 6 8 6 5 8 0 5 6 8 7 8 7 0 1 8 3 0 7 6 3 8 3 1 4 9 3 4 5 4 6 6 5 5 3 4 0 . 8 7 5 0 0 2 1 4 3 3 3 4 6 5 2 7 2 6 2 4 9 6 3 7 6 2 8 5 1 3 6 5 6 4 8 6 5 5 4 5 7 5 5 3 4 0 . 9 3 7 5 0 1 9 4 2 3 0 0 7 4 7 1 1 5 5 7 5 5 6 8 8 2 6 4 7 3 4 9 7 4 8 0 2 5 4 5 0 5 5 3 4 I . 0 0 0 0 0 1 7 4 6 2 6 2 9 4 0 3 4 4 7 4 4 ' 4 8 3 7 2 4 6 3 3 3 5 5 4 7 4 7 5 4 4 4 5 5 3 4


T A B L E 11 (continued) U n i f o r m I n c i d e n c e (U)

b »

b «

α * 0 . 2 0 0 . 6 0 0 . 9 0 0 . 9 9 I . 0 0 0 . 2 0 0 . 6 0 0 . 9 0 0 . 9 9 1 . 0 0

GAIN MOMENT gO X 1 0 0 0 0 NET F L U X : D O W N X 1 0 0 0 0 2 . 0 0 0

0 . 0 1 0 5 5 6 1 2 2 3 0 1 4 8 5 2 1 6 3 9 4 1 6 6 1 6 9 5 3 9 8 0 8 1 5 6 2 9 4 1 2 0 3 9 0 1 0 . 1 2 5 7 5 0 1 9 6 7 9 1 3 1 7 3 1 5 2 4 8 1 5 5 4 7 7 7 8 7 7 0 0 4 5 2 8 1 4 0 8 1 3 9 0 1 0 . 2 5 0 5 8 7 5 8 1 3 1 1 1 9 7 6 1 4 3 3 7 1 4 6 7 9 6 4 6 0 6 1 1 8 4 9 6 7 4 0 4 4 3 9 0 1 0 . 3 7 5 4 7 2 9 6 9 4 2 1 0 9 4 6 1 3 4 9 0 1 3 8 6 2 5 4 0 6 5 3 6 7 4 6 8 0 4 0 0 9 3 9 0 1 0 . 5 0 0 3 8 6 5 5 9 7 8 1 0 0 2 3 1 2 6 7 8 1 3 0 7 1 4 5 5 0 4 7 2 2 4 4 1 8 3 9 7 6 3 9 0 1 0 . 7 5 0 2 6 5 5 4 4 9 7 8 4 0 0 1 1 1 1 7 1 1 5 2 5 3 2 6 4 3 6 8 3 3 9 5 9 3 9 1 7 3 9 0 1 1 . 0 0 0 1 8 6 6 3 4 1 3 6 9 8 8 9 6 0 2 1 0 0 0 0 2 3 7 1 2 8 9 7 3 5 7 5 3 8 6 5 3 9 0 1 1 . 2 5 0 1 3 3 0 2 5 9 3 5 7 2 6 8 1 0 8 8 4 7 5 1 7 3 8 2 3 0 0 3 2 5 8 3 8 2 1 3 9 0 1 1 . 5 0 0 9 5 5 1 9 5 5 4 5 6 8 6 6 1 2 6 9 2 9 1 2 8 5 1 8 4 8 3 0 0 1 3 7 8 4 3 9 0 1 1 . 6 2 5 8 1 0 1 6 8 5 4 0 1 2 5 8 5 1 6 1 3 8 1 1 0 9 1 6 6 6 2 8 9 3 3 7 6 8 3 9 0 1 1 . 7 5 0 6 8 7 1 4 3 8 3 4 6 1 5 0 7 0 5 3 2 1 9 6 0 1 5 1 0 2 8 0 0 3 7 5 5 3 9 0 1 1 . 8 7 5 5 7 9 1 2 0 5 2 8 9 5 4 2 4 3 4 4 5 3 8 3 3 1 3 7 8 2 7 2 0 3 7 4 3 3 9 0 1 2 . 0 0 0 4 8 1 9 5 6 2 2 2 4 3 2 2 8 3 3 8 4 7 2 7 1 2 6 9 2 6 5 6 3 7 3 3 3 9 0 1

4 . 0 0 0

0 . 0 1 0 5 5 7 1 2 2 5 1 1 5 1 5 5 1 7 4 5 0 1 7 8 7 0 9 5 3 7 8 0 5 3 5 2 6 8 2 9 0 3 2 4 6 0 0 . 1 2 5 7 5 0 3 9 7 0 6 1 3 5 6 8 1 6 6 3 7 1 7 1 9 8 7 7 8 5 6 9 7 4 4 9 1 1 2 8 6 0 2 4 6 0 0 . 2 5 0 5 8 7 7 8 1 6 3 1 2 4 4 9 1 5 9 9 7 1 6 6 5 3 6 4 5 8 6 0 8 5 4 5 8 7 2 8 1 9 2 4 6 0 0 . 3 7 5 4 7 3 1 6 9 8 0 1 1 4 9 5 1 5 4 0 7 1 6 1 4 0 5 4 0 3 5 3 3 0 4 2 8 8 2 7 8 0 2 4 6 0 0 . 5 0 0 3 8 6 8 6 0 2 2 1 0 6 4 8 1 4 8 4 6 1 5 6 4 4 4 5 4 7 4 6 8 1 4 0 1 1 2 7 4 2 2 4 6 0 0 . 7 5 0 2 6 5 9 4 5 5 5 9 1 8 6 1 3 7 8 1 1 4 6 7 9 3 2 6 0 3 6 3 2 3 5 1 6 2 6 7 1 2 4 6 0 1 . 0 0 0 1 8 7 1 3 4 9 1 7 9 5 2 1 2 7 6 6 1 3 7 3 1 2 3 6 4 2 8 3 3 3 0 8 9 2 6 0 4 2 4 6 0 1 . 2 5 0 1 3 3 8 2 6 9 7 6 8 9 1 1 1 7 8 7 1 2 7 9 3 1 7 2 9 2 2 1 8 2 7 1 8 2 5 4 3 2 4 6 0 1 . 5 0 0 9 6 7 2 0 9 5 5 9 7 1 1 0 8 3 8 1 1 8 6 0 1 2 7 3 1 7 4 1 2 3 9 7 2 4 8 6 2 4 6 0 1 . 7 5 0 7 0 4 1 6 3 3 5 1 6 7 9 9 1 2 1 0 9 2 9 9 4 2 1 3 7 0 2 1 1 9 2 4 3 5 2 4 6 0 2 . 0 0 0 5 1 6 1 2 7 7 4 4 6 1 9 0 0 7 1 0 0 0 0 7 0 0 1 0 8 1 1 8 7 9 2 3 8 7 2 4 6 0 2 . 2 5 0 3 8 1 1 0 0 0 3 8 3 6 8 1 2 0 9 0 7 1 5 2 2 8 5 4 1 6 7 2 2 3 4 5 2 4 6 0 2 . 5 0 0 2 8 2 7 8 3 3 2 8 1 7 2 4 7 8 1 4 0 3 9 0 6 7 7 1 4 9 4 2 3 0 6 2 4 6 0 2 . 7 5 0 2 0 9 6 1 3 2 7 8 3 6 3 8 5 7 2 0 7 2 9 3 5 3 8 1 3 4 3 2 2 7 2 2 4 6 0 3 . 0 0 0 1 5 6 4 7 9 2 3 3 4 5 5 3 2 6 2 6 9 2 2 0 4 2 9 1 2 1 5 2 2 4 2 2 4 6 0 3 . 2 5 0 1 1 6 3 7 1 1 9 2 3 4 6 8 0 5 3 2 1 1 6 6 3 4 5 1 1 0 9 2 2 1 7 2 4 6 0 3 . 5 0 0 8 7 2 8 4 1 5 4 0 3 8 2 2 4 3 5 6 1 2 6 2 8 0 1 0 2 2 2 1 9 5 2 4 6 0 3 . 6 2 5 7 5 2 4 7 1 3 5 4 3 3 8 4 3 8 6 0 1 1 0 2 5 3 9 8 6 2 1 8 6 2 4 6 0 3 . 7 5 0 6 4 2 1 2 1 1 7 0 2 9 3 3 3 3 4 7 9 6 2 3 0 9 5 4 2 1 7 9 2 4 6 0 3 . 8 7 5 5 5 1 7 9 9 8 0 2 4 5 6 2 8 0 2 8 4 2 1 1 9 2 8 2 1 7 2 2 4 6 0 4 . 0 0 0 4 6 1 4 3 7 5 3 1 8 6 9 2 1 3 0 7 4 1 9 4 9 0 6 2 1 6 6 2 4 6 0

8 . 0 0 0

0 . 0 1 0 5 5 7 1 2 2 5 1 1 5 1 9 4 1 8 0 2 0 1 8 7 7 4 9 5 3 7 8 0 5 3 5 2 2 0 2 2 4 2 1 4 1 5 4 . 0 0 0 5 0 1 9 2 1 6 2 0 7 4 1 6 1 0 0 0 0 7 1 1 6 5 6 2 4 1 2 8 0 1 4 1 5 8 . 0 0 0 I 4 9 2 8 2 5 1 2 2 6 1 5 1 1 0 9 5 7 1 4 1 5

1 6 . 0 0 0

0 . 0 1 0 5 5 7 1 2 2 5 1 1 5 1 9 5 1 8 1 7 2 1 9 3 3 7 9 5 3 7 8 0 5 3 5 2 2 0 2 0 6 6 7 6 5 8 . 0 0 0 1 5 1 9 9 4 2 4 0 1 0 0 0 0 1 4 7 6 5 4 3 7 6 5

1 6 . 0 0 0 0 0 1 2 0 0 6 6 3 0 0 2 2 3 2 7 6 5

3 2 . 0 0 0

0 . 0 1 0 5 5 7 1 2 2 5 1 1 5 1 9 5 1 8 1 8 2 1 9 6 5 4 9 5 3 7 8 0 5 3 5 2 2 0 2 0 5 4 3 9 9 1 6 . 0 0 0 0 0 3 1 1 1 6 1 0 0 0 0 0 0 1 1 3 0 3 9 9 3 2 . 0 0 0 0 0 0 1 3 3 4 6 0 0 0 1 5 3 9 9

03

0 . 0 1 0 5 5 7 1 2 2 5 1 1 5 1 9 5 1 8 1 8 2 2 0 0 0 0 9 5 3 7 8 0 5 3 5 2 2 0 2 0 5 4 0

• Index

A

A b s o r p t i o n , see also L o s s in a t m o s p h e r e , 2 0 4 , 6 8 4 - 6 8 7

w i t h a e r o s o l , 6 8 4 d e p t h , 4 8 1

in g r o u n d , 2 0 4 , 6 8 4 - 6 8 7

b y par t i c l e , 18 , 5 7 4 - 5 7 6 a l o n g p a t h , 5 7 4 - 5 7 6 in t a u in terva l , 2 0 5 - 2 0 6

A b s o r p t i o n l ine , 3 6 , 5 7 3 in d i f fuse r e f l e c t i o n , 5 9 2 - 5 9 8

d e p t h o f f o r m a t i o n , 6 4 0 h o m o g e n e o u s s l a b , 5 9 3 - 5 9 7 i n h o m o g e n e o u s a t m o s p h e r e , 6 3 7 , 6 4 0 s q u a r e r o o t l a w , 6 3 5 w e l l - m i x e d a t m o s p h e r e , 6 3 6

b y o m n i d i r e c t i o n a l p r o b e , 6 4 0 - 6 4 2 prof i l e , 5 9 2 - 5 9 7

A b s o r p t i o n b a n d , 5 7 5 , 6 8 3 s t ruc ture , 5 9 8

A b s o r p t i o n c o e f f i c i e n t , 5 9 2 - 5 9 5 A c c u r a c y , 5 1 9 , 5 2 3

c h e c k , 2 0 3 , 3 6 2 A d d e d layer , 7 0 - 7 1 , 1 0 1 , 1 7 4 A d d i n g g r o u n d sur face , 6 2 5 - 6 2 9 A d d i n g m e t h o d , 4 3 , 5 4 , 5 6 - ^ 2 , 6 2 5 , 7 0 3

c o n v e r g e n c e , 5 7 w i t h p o l a r i z a t i o n , 6 2 , 5 0 5

A e r o s o l , 3 1 7 , 6 6 1 - 6 7 5 see also T u r b i d i t y

a l t i t u d e , 6 6 2

c a u s i n g a u r e o l e , 6 5 1 , 6 5 7 - 6 5 9 c o l l e c t i o n , 6 6 2 c o m p o s i t i o n , 6 6 2 c o n t i n e n t a l , 6 6 2 m a r i t i m e , 6 6 2 n a t u r a l , 6 6 1 - 6 7 5 o r i g i n , 6 6 2 re frac t ive i n d e x , 6 6 4 s i ze d i s t r i b u t i o n , 6 5 7 , 6 6 2

f r o m a u r e o l e , 6 6 8 - 6 7 0 f r o m e x t i n c t i o n c u r v e , 6 7 1 - 6 7 5 b y i n v e r s i o n , 6 6 8 , 6 7 0 b y m o d e l fit, 6 6 8 - 6 7 0

w i d e a n g l e s c a t t e r i n g , 6 6 4 ,

6 6 6 - 6 6 8

A i r p h a s e f u n c t i o n , 3 1 0 - 3 1 1 p h a s e m a t r i x , 5 3 2 p o l a r i z a t i o n , 3 1 1 , 5 3 2

A i r - s e a in ter face , 6 2 7 , 7 0 8 A l b e d o , see also P l a n e a l b e d o ; S p h e r i c a l

a l b e d o cr i t ica l , 2 2 5 e f fec t ive , 5 1 6 , 5 1 8 , 5 2 4 g e o m e t r i c , 6 0 0 - 6 0 1 in r e a c t o r p h y s i c s , 4 f o r s i n g l e s c a t t e r i n g , 3 , 3 5 , 3 0 3 , 3 1 2

n o t a t i o n , 3 p o w e r e x p a n s i o n , 3 8 1 , 3 8 3 t r a n s f o r m a t i o n , 4 8 1

A l t o s t r a t u s , 6 6 1

Index

Λ m a t r i x , 5 8 - 6 1 H e n y e y - G r e e n s t e i n p h a s e f u n c t i o n , 4 6 2 - 4 6 4

A m b a r t s u m i a n f u n c t i o n , 4 4 , 9 9 - 1 0 2 , 108 , 119 as m a t r i x , 102

A m b a r t s u m i a n m e t h o d , 4 3 , 5 0 - 5 4 , 174 A m p l i t u d e

c o m p l e x , 21

t r a n s f o r m a t i o n b y sca t t er ing , 1 8 - 2 0 A n g l e w i t h n o r m a l , s i g n , 4 A n g s t r o m ' s l a w , 6 7 3 A n i s o t r o p y , see A s y m m e t r y p a r a m e t e r A . N . S . , see A s t r o n o m i c a l N e t h e r l a n d s

S a t e l l i t e A n t i s y m m e t r i c s o l u t i o n , 5 0 8 A p p r o x i m a t i o n , see spec i f i c m e t h o d s A r a g o p o i n t , 5 5 5

A s t r o n o m i c a l N e t h e r l a n d s S a t e l l i t e , 7 1 5 A s y m m e t r y p a r a m e t e r , 3 5 , 6 8 , 7 0 , 7 9 ,

3 0 3 - 3 0 9 , 3 1 2 - 3 1 4 , 3 1 6 - 3 2 4 , 3 7 2 - 3 7 3 , 3 8 4 - 3 8 8

t r a n s f o r m a t i o n , 4 8 1 w a t e r d r o p , 3 1 2 - 3 1 3

A s y m p t o t i c e x p r e s s i o n , th i ck layer , 6 8 - 8 7

c o n s e r v a t i v e , 81

i s o t r o p i c s ca t t er ing , 2 0 2 A s y m p t o t i c f i t t ing , 4 5 , 8 2 , 8 5 - 8 7 , 9 3 , 2 0 2 - 2 0 3 ,

3 3 5 , 3 3 9 , 3 6 2 , 3 6 7 A s y m p t o t i c r a d i a n c e , 7 0 8 - 7 1 1

a t t e n u a t i o n , 7 0 8 - 7 0 9 pa t t ern , 7 1 0 - 7 1 1

A t m o s p h e r e , w i t h g r o u n d sur face , i n h o m o g e n e o u s , 6 4

in terna l r a d i a t i o n , 6 3 1 - 6 3 3 re f l ec t ion f u n c t i o n , 6 2 6 - 6 3 2

A t t e n u a t i o n l e n g t h , 7 0 7 A u r e o l e , 3 1 1 , 6 5 1 , 6 5 5 , 6 5 7 - 6 6 1

m e a s u r e m e n t , 6 5 7 - 6 6 0 , 6 7 0 m u l t i p l e s ca t t er ing , 6 5 9 - 6 6 1 r a d i a n c e , 6 6 9 - 6 7 0 in u l t r a v i o l e t , 6 5 7

A u x i l i a r y e q u a t i o n , 4 8 , 4 9 for R a y l e i g h sca t t er ing , 5 6 2

A z i m u t h - d e p e n d e n t t e r m s , 9 0 , 4 9 3 - 4 9 4 , 4 9 8 - 5 0 5

e f fec t ive a l b e d o , 5 1 6 - 5 1 8

Β

B a b i n e t p o i n t , 5 5 5 B a c k l a y e r , d i f fuse ly re f lec t ing , 7 0 2 B a c k s c a t t e r , 3 0 5 - 3 0 7 , 3 1 4

p r e d o m i n a n t , 4 0 3 - 4 0 4 B a c k w a r d p e a k , 3 0 7 , 3 0 9 , 3 6 8 , 4 8 9 - 4 9 2

B e a m

a t t e n u a t i o n coe f f i c i en t , 7 0 8 - 7 1 0

e x t i n c t i o n c o e f f i c i e n t , 7 0 5 s c a t t e r i n g coe f f i c i en t , 7 0 8 , 7 1 0

B e s s e l f u n c t i o n , 3 8 8

m o d i f i e d , 5 9 4 B i n o m i a l e x p a n s i o n , 109 , see also

C h a r a c t e r i s t i c b i n o m i a l B l a c k b o d y , 2 5 , 2 0 4 , 4 9 7 B l a c k b o d y r a d i a t i o n , 2 5 B l u e s k y , 3 1 0 - 3 1 1 , 5 3 1 , 5 5 5 - 5 5 7 , 6 5 1 , 661 B o n d a l b e d o , 6 0 0 , 6 0 3 , see also S p h e r i c a l

a l b e d o B o r n a p p r o x i m a t i o n , 6 7 0 B o t t o m b o u n d a r y , see G r o u n d sur face B r e w s t e r p o i n t , 5 5 5

B r i g h t n e s s , 6 5 1 , 6 5 2 , see also R a d i a n c e B u s b r i d g e p o l y n o m i a l , 4 3 , 9 1 - 9 2 , 105 ,

1 0 8 - 1 1 3 , 120

C

C a s e m e t h o d , 8 4 , 118 , 1 2 1 - 1 2 2 , 3 3 5 , 4 0 5 C h a p m a n f u n c t i o n , 6 6 7 C h a r a c t e r i s t i c b i n o m i a l , 9 4 - 9 5 , 112 C h a r a c t e r i s t i c e q u a t i o n , 5 1 6

a l t e r n a t e f o r m , 9 8 m e t h o d s o f s o l u t i o n , 9 6 - 9 9 m u l t i p l e r o o t s , 3 8 1 , 3 8 3 - 3 8 8 r o o t s , 9 6 - 9 8 , 107

C h a r a c t e r i s t i c f u n c t i o n , 4 3 , 9 1 , 9 4 , 112 , 1 1 9 - 1 2 0 , 5 1 8

d e t e r m i n a t i o n , 9 4

for f o u r - t e r m p h a s e f u n c t i o n , 3 5 6 - 3 5 7 integra l c h e c k , 9 5 for s m a l l N, 9 6

C h a r a c t e r i s t i c r o o t , 3 7 7 - 3 7 8 C i r c u l a r p o l a r i z a t i o n , s y m m e t r y r e l a t i o n s , 6 1 6 C i r r o s t r a t u s , 6 6 1 Cirrus , 6 6 1 , 6 8 0 - 6 8 2

p o l a r i z a t i o n , 6 6 8 C l e a r i n g , 7 0 6 C l i m a t e , 6 8 2 - 6 8 3 C l o s e l y p a c k e d part ic les , 6 9 9 - 7 0 3 C l o u d

a t m o s p h e r i c , 3 1 7 b r o k e n , 6 5 3 , 6 8 3 , 6 9 4 - 6 9 5 c o o l i n g , 6 8 7 c u b o i d , 6 9 4 - 6 9 8 , 7 0 5

a s y m p t o t i c c a s e , 6 9 5 - 6 9 8 c o r n e r d o m a i n , 6 9 5 - 6 9 8 H e n y e y - G r e e n s t e i n sca t ter ing , 6 9 8 i s o t r o p i c s ca t t er ing , 6 9 5 - 6 9 8

Index

h e a t i n g , 6 8 7 h e t e r o d i s p e r s e , 521 i s o l a t e d , 6 9 4 - 6 9 8 l iqu id w a t e r c o n t e n t , 6 8 7 m o d e l , 3 2 0 , 3 2 2 - 3 2 4 m o n o d i s p e r s e , 521 part ic le , r e m o t e s e n s i n g , 6 6 7 p h a s e f u n c t i o n , 3 1 5 - 3 1 7 p o l a r i z a t i o n , 6 6 7 ref lected e n e r g y , 6 8 2 terrestr ial , 3 8 8

C l o u d b o w , 6 6 8

C l o u d c o v e r , f r a c t i o n a l , 6 8 3 , 6 9 5 C l o u d d e c k , 6 5 1 - 6 5 2

o p a q u e , 6 5 3 - 6 5 4

w i t h o v e r l y i n g a t m o s p h e r e , 6 2 7 - 6 3 0 r a d i a n c e , 6 5 1 - 6 5 2 , 6 6 8

r e d u c t i o n o f i l l u m i n a t i o n , 6 5 4 - 6 5 5 C l o u d layer ,

in terna l r a d i a t i o n f ie ld , 2 0 6 re f l ec t ion f u n c t i o n , w e a k l o s s , 8 0

C o a s t a l w a t e r , 7 0 7 C o h e r e n c y m a t r i x , 2 3 , 5 3 4 C o m p u t e r t i m e , 86

C o n d e n s e d n o t a t i o n , see S h o r t h a n d n o t a t i o n C o n s e r v a t i v e a t m o s p h e r e , 188 , 190 C o n s e r v a t i v e s c a t t e r i n g , 3 8 C o n s t r a i n t , 1 0 3 - 1 0 4 , 1 1 , 6 6 3

C o n t i n u e d f r a c t i o n , 3 8 0 , 3 8 3 C o n t i n u u m

a b s o r p t i o n , 5 9 2 , 5 9 5 a l b e d o , 5 9 5 - 5 9 6 o f e i g e n v a l u e s , 159 e x t i n c t i o n , 5 9 5 o p t i c a l p a t h , 5 9 4 s c a t t e r i n g , 5 7 2

C o n t o u r in tegra l , 159 C o n t r a s t r a t i o , 6 2 9 - 6 3 2 , 7 0 3 C o n v e r g e n c e , see spec i f i c m e t h o d s C P r e p r e s e n t a t i o n o f p o l a r i z e d l ight , 4 9 7 , 5 0 8 ,

5 3 3 , 6 1 4 Cr i t i ca l a l b e d o , 140 , 2 3 3 Cr i t i ca l l imi t , 2 3 3 Cr i t i ca l s ize o f reac tor , 6 9 2 C r o s s s e c t i o n

for a b s o r p t i o n , 2 5 for e m i s s i o n , 2 5

C u b o i d , see C l o u d , c u b o i d C u m u l u s , 6 8 0 - 6 8 1 C u r v e o f g r o w t h , 5 9 3 - 5 9 8 , 6 3 4

s l a b w i t h g r o u n d sur face , 5 9 7 s q u a r e r o o t l a w , 5 9 4 - 5 9 8

C y l i n d e r , r a n d o m l y o r i e n t e d , 661

D

D a m p i n g prof i le , 5 9 3 - 5 9 6 D a y l i g h t s k y , see S k y , d a y l i g h t D e l t a - E d d i n g t o n a p p r o x i m a t i o n , 41 δ f u n c t i o n , 3 0 9 D e p o l a r i z a t i o n

o f l idar e c h o , 6 8 0 - 6 8 2

o f m o l e c u l a r s ca t t er ing , 3 1 1 , 5 3 3 D e p r e s s i o n in a b s o r p t i o n l ine , 5 9 2 - 5 9 6 D e t a i l e d b a l a n c e , 1 6 - 1 7 , 2 3 , 2 7

in a s t r o p h y s i c s , 17 inverse use , 1 7 - 1 8 , 2 0

D e t e c t o r

as f o u r - v e c t o r , 4 9 6 r e a d i n g , 1 3 3 - 1 3 5 rec iproc i ty t o s o u r c e , 6 1 5 sens i t iv i ty , 130

D i f f r a c t i o n , 6 5 7 , 6 6 0 - 6 6 1

a n o m a l o u s , 3 1 5 F r a u n h o f e r , 3 1 4

p a t t e r n , w i t h s ize d i s t r i b u t i o n , 6 6 9 - 6 7 0 p e a k , 3 1 2 , 3 1 4 , 3 2 2 - 3 2 3

o m i s s i o n , 481 D i f f u s i o n , see also specific p h a s e f u n c t i o n

a p p r o x i m a t i o n , 7 5 - 7 6 , 159

i m p r o v e d , 76 c o e f f i c i e n t , 7 5 , 159 c o n s t a n t , 144 d e p t h , 4 8 1

d o m a i n , 6 8 , 7 0 , 7 5 - 7 6 , 8 2 , 9 0 , 9 2 - 9 3 , 1 7 8 - 1 8 0 , 1 8 3 , 1 8 7 - 1 8 8 , 198 , 2 0 1 , 2 0 9 - 2 1 3 , 2 2 4 , 4 7 8 , 7 0 8 - 7 1 1

w i t h p o l a r i z a t i o n , 5 0 7 - 5 0 8 , 5 1 2 e q u a t i o n , 159

e x p o n e n t , 7 0 , 7 8 , 9 3 , 7 0 8 - 7 0 9 , 711 a s y m p t o t i c e x p r e s s i o n , 152 c h e c k , 8 7

error b y n e g l e c t o f p o l a r i z a t i o n , 511 h i g h e r r o o t , 3 8 3 w i t h p o l a r i z a t i o n , 5 0 8 - 5 1 3 as v a r i a b l e , 7 8 - 8 0 , 3 8 0 - 3 8 4 , 3 8 6

i n w a r d , 3 9 4 l e n g t h , 7 0 , 9 3 , 152 , 3 7 7 m o d e , 7 0 n o n c o n s e r v a t i v e , 9 3 o u t w a r d , 3 9 4 p a t t e r n , 109 , 1 1 1 , 4 7 8 , 7 1 0

e x p a n s i o n , 3 8 4 f o r w a r d / b a c k rat io , 511 w i t h p o l a r i z a t i o n , 5 0 7 - 5 1 2

m o m e n t s , 5 1 3 a s v e c t o r , 6 9

IV Index

D i f f u s i o n (cont.) s o l u t i o n , 121

a z i m u t h - d e p e n d e n t t e r m s , 5 1 5 - 5 1 7 c o n s e r v a t i v e , 9 2 n o n c o n s e r v a t i v e , 9 3

s t r e a m , 7 0 - 7 7 , 2 2 4 , 3 3 9 , 4 7 7

in terna l re f l ec t ion , 7 1 - 7 2 , 3 7 4 - 3 7 5 n e g a t i v e , 7 2 , 8 0 , 2 1 2 , 3 7 4 f r o m s o u r c e layer , 8 3

D i r e c t e x p e r i m e n t , 2 0 7 D i s k , see P l a n e t ; S u n D i s p e r s i o n f u n c t i o n , 9 7 , 1 0 5 - 1 0 6 D m a t r i x , 5 8 - 6 1

H e n y e y - G r e e n s t e i n p h a s e f u n c t i o n , 4 6 2 - 4 6 4 D o u b l e p e a k p h a s e f u n c t i o n

a s y m m e t r y f a c t o r , 4 8 9 flux, 4 9 0 ^ 9 1

re f l ec t ion f u n c t i o n , 4 8 3 , 4 8 6 , 4 9 0 - 4 9 1 b i m o m e n t , 4 9 1

s emi - in f in i t e m e d i u m , 4 9 0 - 4 9 1 t r a n s m i s s i o n , 4 9 0 - 4 9 1

D o u b l i n g m e t h o d , 4 3 , 4 5 , 5 4 - 5 6 , 2 0 9 , 2 2 2 ,

3 9 6 , 6 1 2 , 6 9 1 , 7 1 6 a z i m u t h - d e p e n d e n t t e r m s , 5 0 7 bas i s for a s y m p t o t i c f i t t ing, 8 5 c o n v e r g e n c e , 4 3 , 6 3

H e n y e y - G r e e n s t e i n p h a s e f u n c t i o n , 4 0 8 ^ 1 1

i m p r o v e d , 146 i s o t r o p i c s c a t t e r i n g , 2 3 2 - 2 3 4 , 4 0 8 - 4 1 0

cr i t ical l imi t , 2 3 3 - 2 3 4 e i g e n v a l u e , 2 3 2 - 2 3 4 , 4 0 8 - 4 1 1 f r o m M i l n e e q u a t i o n , 1 4 6 - 1 4 8 w i t h p o l a r i z a t i o n , 5 0 5 - 5 0 7

D u s t , 4 9 6 f r o m fores t fire, 6 6 2 i n t e r p l a n e t a r y , 6 5 9 , 7 1 3 inters te l lar , 7 1 3 s c a t t e r i n g p a t t e r n , 6 0 6 s l a b , 7 1 4

£

E a r t h a t m o s p h e r e

a l b e d o , 5 5 9 o p t i c a l d e p t h , 3 1 0 s ca t t ered l ight , 6 5 0 - 6 6 1 , 6 6 4 - 6 6 8

r a d i a t i o n b u d g e t , 6 8 2 - 6 8 3 E c h o d e l a y t i m e , 5 7 3 E d d i n g t o n a p p r o x i m a t i o n , 7 6 , 1 4 3 , 3 7 0 , 6 4 4 ,

7 1 7

E d d i n g t o n m e t h o d , m o d i f i e d , 6 9 7 Ε f u n c t i o n , see E x p o n e n t i a l in tegra l E i g e n f u n c t i o n

o f d i f f u s i o n e q u a t i o n , 7 4 H e n y e y - G r e e n s t e i n p h a s e f u n c t i o n , 3 3 5 ,

3 3 7 - 3 3 8 e x p a n s i o n , 1 2 1 , 1 3 2 , 4 0 5

o f M i l n e o p e r a t o r , 144 s i n g u l a r , 7 4 , 159

a s e x p a n s i o n t e r m , 4 3 , 1 2 1 , 1 5 9 , 2 2 2 w i t h p o l a r i z a t i o n , 122

E i g e n v a l u e , see also D o u b l i n g m e t h o d , c o n v e r g e n c e

o f a n i s o t r o p i c transfer e q u a t i o n , s p e c t r u m ,

3 7 6 - 3 8 8 o f d i f f u s i o n e q u a t i o n , 7 4 , 9 3

H e n y e y - G r e e n s t e i n p h a s e f u n c t i o n , 3 3 3 - 3 3 6

d i scre te , 4 3 , 9 3 i n s u c c e s s i v e o r d e r m e t h o d

H e n y e y - G r e e n s t e i n p h a s e f u n c t i o n , 4 0 7 ^ 1 0 8

i s o t r o p i c s ca t t er ing , 2 3 3 E l e c t r o n a s scat terer , 531 E l e c t r o n c a p t u r e , 18 E l l ip t i ca l in tegra l , 3 3 3 E l o n g a t i o n , 601

E m b e d d i n g , see I n v a r i a n t e m b e d d i n g E m e r g e n c e , see E s c a p e f u n c t i o n ;

R e f l e c t i o n ; T r a n s m i s s i o n E m i s s i o n , t h e r m a l , 18 , 2 4

d i s k , 2 6

i s o t h e r m a l b o d y , 6 4 2 n o n i s o t h e r m a l b o d y , 6 4 3 - 6 4 4 n o n t r a n s p a r e n t s p h e r e , 2 6 s l a b , 221

z e r o - o r d e r , 2 2 0 - 2 2 1 E n e r g y , see also F l u x ; L o s s

d e p o s i t i o n in c l o u d , 6 8 7 fa te

a s y m p t o t i c , 6 8 4 - 6 8 7 H e n y e y - G r e e n s t e i n p h a s e f u n c t i o n ,

4GXM02 i s o t r o p i c s ca t t er ing , 2 0 5 - 2 0 6

s i n k , 180 E q u a t o r o f p l a n e t , 6 0 2 E q u i v a l e n c e , 4 8 0

t h e o r e m , 5 7 4 - 5 7 7 E q u i v a l e n t w i d t h , 5 7 3 , 5 9 2 - 5 9 8 E r r o r

f u n c t i o n , 160 in tegra l , 3 9 3

Index V

p r o p a g a t i o n , 6 3 r o u n d - o f f , 8 6

E s c a p e f u n c t i o n , 7 2 , 1 1 1 - 1 1 4 , 1 1 6 - 1 1 7 , 151 ,

3 3 9 , 3 5 7 - 3 6 3 , 4 8 6 , 5 1 3 , 6 6 8 , see also s p e c i f i c p h a s e f u n c t i o n

m o m e n t , 1 1 6 - 1 1 7

e x p a n s i o n , 8 0 , 3 6 0 - 3 6 2 near ly c o n s e r v a t i v e ,

e x p a n s i o n , 7 9 r e d u c t i o n t o H f u n c t i o n , 91

c o n s e r v a t i v e , 117 n o n c o n s e r v a t i v e , 1 1 5 - 1 1 6

s imi lar i ty , 3 7 4 , 4 7 9 a s v e c t o r , 6 9

E s c a p e p r o b a b i l i t y , 1 8 1 - 1 8 2 , 2 0 8 , 2 1 3 - 2 1 7 i s o t r o p i c s c a t t e r i n g , 2 1 3 - 2 1 7 , 2 2 0

o n e - s i d e d , 2 1 3 - 2 1 7 t w o - s i d e d , 2 1 3 - 2 1 4

s l a b , H e n y e y - G r e e n s t e i n p h a s e f u n c t i o n , 4 0 5 ^ 0 6

E s c a p e r e g i o n , i s o t r o p i c s c a t t e r i n g , 179 ,

2 1 1 - 2 1 3 Euler 's c o n s t a n t , 9 , 6 7 2 E x p a n s i o n , see S u c c e s s i v e o r d e r E x p o n e n t i a l

in tegra l , 8 - 1 0 , 1 9 6 , 5 9 7 a s y m p t o t i c f o r m , 9 d i f f e r e n t i a t i o n , 9 e x p a n s i o n , 9 g e n e r a l i z e d , 1 0 - 1 1

s u m f i t t ing , 6 8 3 E x t i n c t i o n , see also I n t e r s t e l l a r e x t i n c t i o n

c o e f f i c i e n t , 3 1 6 c u r v e , 3 1 7

i n v e r s i o n , 6 7 0 - 6 7 5 m a x i m u m , 6 7 0 - 6 7 4 m o d e l fit , 6 7 1 - 6 7 4 p o l y d i s p e r s e m e d i u m , 6 7 2 - 6 7 5

d e p t h , 4 8 1 E x t r a p o l a t e d e n d p o i n t , 7 3 E x t r a p o l a t i o n l e n g t h , 7 2 , 1 1 7 , 1 7 4 , 2 3 4 , 4 0 9 , 5 1 3 ,

6 8 5 , see also E s c a p e f u n c t i o n f in i te layer , 2 1 8 H o p f , 2 1 8 near ly c o n s e r v a t i v e ,

e x p a n s i o n , 7 9 n o n c o n s e r v a t i v e ,

s imi lar i ty , 3 7 4 - 3 7 6 r e d u c e d , 1 1 5 , 3 5 7 - 3 6 3 s imi lar i ty , 3 5 7 - 3 6 2

E x t r a p o l a t i o n t o l a r g e t h i c k n e s s , 2 0 1 - 2 0 3

F

Far- f i e ld s c a t t e r i n g p a t t e r n , 1 8 - 2 0 , 6 9 9 F f u n c t i o n , 8 , 1 1 - 1 3 , 136 , 187 , 196

d i f f e r e n t i a t i o n , 11 e x p a n s i o n , 13 n u m e r i c a l v a l u e s , 1 2 - 1 3 recurrence , 1 1 , 13 r e d u c t i o n , 11

F l u i d m o t i o n w i t h r a d i a t i o n , 7 0 7 F l u o r e s c e n t s ca t t er ing , 231 F l u x

a b s o r b e d

H e n y e y - G r e e n s t e i n p h a s e f u n c t i o n ,

4 G X M 0 2 , 4 5 6 ^ 6 1 i s o t r o p i c s ca t t er ing , 2 0 3 - 2 0 6 , 2 1 8 , 2 2 8 ,

2 8 2 - 2 9 0 f r o m b o t t o m o f a t m o s p h e r e , 6 4 d i r e c t i o n a l , 6 9 3 d i v e r g e n c e , 6 8 4 e m e r g i n g , 8 1 - 8 2 , 188 , 190 , 4 9 7 e x t i n c t i o n c o e f f i c i e n t , 7 0 5

in c o l u m n o f f in i te w i d t h , 7 0 5 net , 5 , 7 5 , 6 9 3 , 7 1 2

in d i f f u s i o n d o m a i n , 184 in s l ab , 4 8 8

c o n s e r v a t i v e , 2 0 9 , 2 8 2 - 2 9 0 , 2 9 2 - 2 9 9 H e n y e y - G r e e n s t e i n p h a s e f u n c t i o n ,

4 0 5 , 4 6 5 ^ 7 6 i s o t r o p i c s c a t t e r i n g , 2 0 5 - 2 0 6 , 2 0 8 , 2 9 2 ,

2 9 9 o p e r a t o r , 6 9 re f lec ted , 8 1 , 4 8 6 - 4 8 7

H e n y e y - G r e e n s t e i n p h a s e f u n c t i o n , 4 0 0 - 4 0 5 , 4 5 6 - 4 6 1

i s o t r o p i c s c a t t e r i n g , 1 7 4 , 2 0 3 - 2 0 5 , 2 2 8 - 2 3 0 , 2 8 2 - 2 9 0

t r a n s m i t t e d , 4 8 6 - 4 8 7 H e n y e y - G r e e n s t e i n p h a s e f u n c t i o n ,

4 ( X M 0 1 , 4 5 6 ^ 6 1 i s o t r o p i c s c a t t e r i n g , 2 0 1 - 2 0 5 , 2 2 8 ,

2 8 2 - 2 9 0 z e r o o r d e r , 2 0 3 , 2 8 2 - 2 9 0 , 4 5 6 - 4 6 1

F m a t r i x , 21 F o g , 3 1 7 , 6 6 1 F o r w a r d a n d b a c k w a r d p e a k , 7 0 0 F o r w a r d p e a k , 3 0 7 , 3 0 9 , 3 5 8 - 3 6 0 , 4 8 9 ^ 9 2

a d d i t i o n , 3 6 2 , 3 6 8 - 3 6 9 , 4 7 9 ^ 8 1 , 4 9 1 ^ 9 2 i n f l u e n c i n g l idar e c h o , 6 7 7

F o r w a r d s c a t t e r i n g , 3 0 5 - 3 0 7 , 4 0 3 ^ 0 5 , 4 8 0

F o u r b y f o u r m a t r i x , 4 9 6 , 5 0 0 , 5 0 3

VI Index

F o u r i e r e x p a n s i o n in a z i m u t h , 4 9 3 - 4 9 4 , 4 9 8 , 5 0 1 - 5 0 5 , 5 1 4 - 5 1 5

n u m b e r o f t e r m s , 5 2 0 - 5 2 3 ra t io b e t w e e n o r d e r s , 5 2 5 R a y l e i g h sca t ter ing , 5 1 9 , 5 4 3 - 5 4 4

F o u r i e r s p e c t r o s c o p y , 6 3 4 F o u r i e r t r a n s f o r m , 158 F o u r - v e c t o r , 4 9 6 ^ 9 7 , 5 0 3 F r a u n h o f e r d i f f r a c t i o n , see D i f f r a c t i o n F r e d h o l m integra l e q u a t i o n , 110 , 122 , 162

G

G A A R S field test , 6 7 3 G a i n , see also P o i n t - d i r e c t i o n g a i n

in a n t e n n a t h e o r y , 2 6 arbi trary c o n f i g u r a t i o n , 2 6 e x t e n d e d d e f i n i t i o n , 2 7 u p o n sca t ter ing , 152

G a l a c t i c l ight d i f fuse , 7 1 4 , 7 1 6 - 7 1 7

i l l u m i n a t i n g g l o b u l e , 7 1 5 G a m m a s ize d i s t r i b u t i o n , 6 6 4 - 6 6 5 G A R P , see G l o b a l A t m o s p h e r i c R e s e a r c h

P r o g r a m

G a u s s i a n d i s t r i b u t i o n , 1 5 9 - 1 6 1 , 3 1 6 G a u s s i a n d i v i s i o n , 4 3 G a u s s i a n q u a d r a t u r e , 2 2 6 G e n e r a l i z e d s p h e r i c a l f u n c t i o n , 4 9 7 , 5 0 8 G e o m e t r i c a l b e d o

for m o d e l a t m o s p h e r e , 621

o f p lane t , 6 0 0 - 6 0 1 , 6 1 8 , 6 2 0 - 6 2 1 G e o m e t r i c fac tor , 6 0 9 , 621 G e o m e t r i c ser ies , 1 4 1 , 5 9 1 , 6 1 4 G e o m e t r y d i f ferent f r o m s l a b , 4 8 6 G e r s h u n e q u a t i o n , 7 1 2 G f u n c t i o n , 8 , 1 3 - 1 5 , 136 , 196

e x p a n s i o n , 14 , 2 2 8 n u m e r i c a l v a l u e s , 1 4 - 1 5 recurrence , 14 s y m m e t r y , 13

G l a s s b e a d , 3 1 3 G l o b a l A t m o s p h e r i c R e s e a r c h P r o g r a m , 6 8 7 G l o b u l e , 7 1 5 - 7 1 6 G l o r y , 3 1 2 , 3 1 5 - 3 1 6 , 3 2 2 - 3 2 4 G l o s s y sur face , 7 0 2 G r a f t i n g o f t h e o r i e s , 6 9 2 G r a z i n g re f l ec t ion , 5 2 5

o n s p h e r e , 3 1 3 G r e e n h o u s e effect , 6 4 4

G r e e n ' s f u n c t i o n

a s y m p t o t i c e x p a n s i o n , 3 9 3

f a c t o r i z a t i o n , 3 9 3 G r e y a b s o r p t i o n , 6 4 4

G r o u n d sur face b e l o w a t m o s p h e r e , 3 5 , 6 3 - 6 5 , 2 0 4 , 5 0 7 , 6 2 5 - 6 2 9

H

H a i l s t o n e , 3 1 4 , 3 1 6 H a l f - s p a c e , 9 0 , 691 H a l o , 3 1 4 H a z e , 6 6 6

a b s o r b i n g , 3 7 5

m o d e l , 3 1 6 - 3 1 7 , 3 2 0 - 3 2 2 p h a s e f u n c t i o n , 3 1 1 , 3 1 5 - 3 1 7 s i ze d i s t r i b u t i o n , 6 6 4 - 6 6 5

H e a t i n g rate , 6 8 4 H e a t transfer , 7 0 6 - 7 0 7

t h r o u g h i n s u l a t i n g layer , 7 0 6 b e t w e e n w a l l s , 7 0 6

H e n y e y - G r e e n s t e i n p h a s e f u n c t i o n , 3 0 7 - 3 0 9 , 3 2 0 - 3 2 2 , 3 3 1 - 3 3 2 , see also F l u x

a z i m u t h - d e p e n d e n t t e r m s , 5 2 2 - 5 2 9 c h a r a c t e r i s t i c e q u a t i o n , r o o t s , 3 8 4 - 3 8 8 in d i f fuse g a l a c t i c l ight , 7 1 6 - 7 1 7 d i f f u s i o n

e x p o n e n t , 3 3 3 - 3 3 6 , 3 6 7 , 5 1 1 , 7 0 9 - 7 1 0 e x p a n s i o n , 3 3 5

l e n g t h , 3 3 3 - 3 3 6 p a t t e r n , 3 3 5 , 3 3 7 - 3 3 8

e s c a p e f u n c t i o n , 3 3 9 - 3 4 2 , 3 6 7

c o n s e r v a t i v e , 3 5 8 - 3 6 1 e x p a n s i o n c o e f f i c i e n t s , 3 0 8 , 331 e x t r a p o l a t i o n l e n g t h ,

c o n s e r v a t i v e , 3 4 0 - 3 4 1 , 3 5 9 - 3 6 1 , 3 7 6 n o n c o n s e r v a t i v e , 3 4 0 - 3 4 1 , 3 6 7 , 3 7 6

in g l o b u l e , 7 1 5 in terna l re f l ec t ion coe f f i c i en t , 3 7 4 - 3 7 5 K u s c e r p o l y n o m i a l s , 3 7 8 - 3 7 9 L e g e n d r e coe f f i c i en t s , 3 0 8 , 331 M i n n a e r t p l o t , e x p o n e n t , 6 1 9 - 6 2 0 p l a n e a l b e d o , 3 6 7 - 3 6 9 , 4 0 0 - 4 0 3 p o i n t - d i r e c t i o n g a i n , 4 0 5 - 4 0 6 , 4 6 5 - 4 7 6

m o m e n t , 4 0 5 , 4 6 7 - 4 6 8 , 4 7 1 ^ 7 2 , 4 7 5 ^ 7 6 r a p i d - g u e s s f o r m u l a s , 3 6 3 - 3 6 4 re f l ec t ion f u n c t i o n , 3 6 7 - 3 6 8 , 3 9 6 - 3 9 9 ,

4 1 2 ^ 5 3 , 4 8 6 ^ 8 7 a z i m u t h - d e p e n d e n t t e r m s , 5 2 3 - 5 2 9

a s y m p t o t i c , 5 2 5 - 5 2 9 ra t io o f t o t a l t o first o r d e r , 5 2 4 - 5 2 5

Index VII

b i m o m e n t , 3 3 9 , 3 4 3 - 3 5 4 , 3 9 8 , 4 0 1 ,

4 0 3 ^ 0 5 , 4 1 2 ^ 5 3 , 7 0 1 - 7 0 2 first o r d e r , 3 9 9 ^ 0 0 , 4 1 2 - 4 4 3 l o w o r d e r , 4 1 2 - 4 4 3 m o m e n t , 3 3 9 , 3 4 5 - 3 5 4 , 3 9 8 , 4 1 2 - 4 5 3 near ly c o n s e r v a t i v e , 3 9 1 - 3 9 2 in o p p o s i t i o n , 6 1 7 - 6 1 8 ra t io o f t o t a l t o f irst o r d e r , 3 9 7 - 3 9 9 ,

4 5 4 ^ 5 5 , 6 7 8 s e m i - i n f i n i t e a t m o s p h e r e , 3 3 9 , 3 4 2 - 3 5 4 s l a b w i t h v a r y i n g g r o u n d , 6 2 9 - 6 3 2

s e m i - i n f i n i t e a t m o s p h e r e , 3 3 9 - 3 5 4 s imi lar i ty p a r a m e t e r , 4 7 8 s l a b , 3 9 6 ^ 7 6 s p h e r i c a l a l b e d o ,

s e m i - i n f i n i t e a t m o s p h e r e , 3 4 4 , 3 6 7 ,

3 7 0 - 3 7 1 s l a b , 4 0 3 ^ 0 5

s u c c e s s i v e o r d e r s , c o n v e r g e n c e , 5 2 2 - 5 2 3 t r a n s m i s s i o n f u n c t i o n , 3 9 6 - 3 9 9 , 4 1 2 - 4 5 3 ,

4 8 6 ^ 8 7 , 6 5 2 a z i m u t h - d e p e n d e n t t e r m s , 5 2 3 - 5 2 9

a s y m p t o t i c , 5 2 5 - 5 2 9 r a t i o o f t o t a l t o first o r d e r , 5 2 4 - 5 2 5

b i m o m e n t , 3 9 8 , 4 1 2 - 4 5 3 l o w o r d e r , 4 1 2 - 4 4 3 m o m e n t , 3 9 8 , 4 1 2 - 4 5 3 , 4 8 7

t r u n c a t i o n , 4 8 8 u n b o u n d e d m e d i u m , 3 3 1 - 3 3 8

H f u n c t i o n , 4 3 , 9 1 , 1 0 4 - 1 0 8 , see also s p e c i f i c p h a s e f u n c t i o n

e x p a n s i o n , 108 e x p l i c i t e x p r e s s i o n , 1 0 6 - 1 0 7 in tegra l r e l a t i o n , 1 0 6 - 1 0 7 i s o t r o p i c s ca t t er ing , 1 6 2 - 1 6 9 , 1 8 7 , 2 0 3 , 2 2 5 ,

5 8 4 , 5 9 7 , 621 i n c o r n e r d o m a i n s , 166 d e f i n i t i o n , 162 d e r i v a t i v e , 165 n e a r d i v e r g e n c e , 1 6 7 - 1 6 8 e x p a n s i o n , 1 6 3 - 1 6 6 in tegra l , 168 in tegra l e q u a t i o n , 162 m o m e n t s , 1 6 9 - 1 7 1

a s y m p t o t i c f o r m , 1 7 0 - 1 7 1 e x p a n s i o n , 1 6 9 - 1 7 1

n u m e r i c a l v a l u e s , 1 6 3 - 1 6 4 r e p r e s e n t a t i o n

in tegra l , 162 r a t i o n a l , 163

v ir tua l a n g l e s , 1 6 6 - 1 6 9 m o m e n t , 1 0 8 , 1 1 2

H i d i n g p o w e r , 7 0 3

H i s t o r i c a l n o t e , 5 4 , 201

H o p f f u n c t i o n , 1 6 3 , 176 , 1 8 3 , 188 , 190 for R a y l e i g h s c a t t e r i n g , 541

H o p f s o l u t i o n , 140 , 3 9 4 H y b r i d f o r m u l a t i o n , 5 7 6 H y d r o d y n a m i c s , 7 0 6 - 7 0 7 H y d r o l o g i e o p t i c s , 7 0 7 - 7 1 3

a s y m p t o t i c d o m a i n , 7 0 8 - 7 1 1 , 7 1 3 H y p e r b o l i c f u n c t i o n , 7 7 , 6 9 9 H y p e r g e o m e t r i c f u n c t i o n , 6 6 9 - 6 7 0

I

I A M A P , see I n t e r n a t i o n a l A s s o c i a t i o n o f M e t e r o l o g y a n d A t m o s p h e r i c P h y s i c s

I ce , 179

a b s o r p t i o n c o e f f i c i e n t , 7 0 5

c l o u d , 6 6 8

crys ta l , 3 1 4 , 661 I m p o r t a n c e , 2 8 In fer ior c o n j u n c t i o n , 6 0 8 I n h o m o g e n e o u s a t m o p s h e r e

b y a d d i n g m e t h o d , 5 8 - 6 1 r e f l e c t i o n f u n c t i o n , i s o t r o p i c s ca t t er ing ,

5 2 - 5 3 I n j e c t i o n f u n c t i o n , see E s c a p e f u n c t i o n I n j e c t i o n r e g i o n , i s o t r o p i c s c a t t e r i n g , 2 0 9 ,

2 1 1 - 2 1 2 I n s t a b i l i t y

in i n v e r s i o n p r o c e d u r e , 6 6 3 in recurrence , 3 7 7 - 3 8 0

I n s u l a t i o n b y f o a m , 7 0 6 I n t e g r a l c h e c k , 6 9 3 , 7 1 5 I n t e g r a l e q u a t i o n , see also M i l n e e q u a t i o n

for A m b a r t s u m i a n f u n c t i o n s l inear , 1 0 3 - 1 0 4 , 109 n o n l i n e a r , 1 0 3 - 1 0 4

for d i f f u s i u o n , 7 2 - 7 3 , 9 3 , 3 7 6

e i g e n v a l u e , 3 7 7 - 3 7 8 F r e d h o l m , 110 , 1 2 2 , 162 for H f u n c t i o n , 91

l inear , 1 0 5 - 1 0 7 , 162 n o n l i n e a r , 1 0 5 - 1 0 6 , 1 0 9 , 162

for re f l ec t i on a n d t r a n s m i s s i o n , 9 9 f o r s i ze d i s t r i b u t i o n

i n v e r s i o n , 6 6 3 m o d e l f i t t ing , 6 6 3

for S o b o l e v Φ f u n c t i o n , 118 , 120 , 1 3 8 - 1 3 9 for X a n d Y f u n c t i o n

l inear , 119 n o n l i n e a r , 5 2 , 119 , 2 2 4

VIII Index

I n t e g r a t i o n o f H f u n c t i o n , 6 0 7 o v e r τ , 135 , 2 0 9

I n t e n s i t y , 5 , 1 3 4 , see also R a d i a n c e a t arbi trary d e p t h , 6 9 , 1 1 7 - 1 2 2 a v e r a g e

h e m i s p h e r i c a l , 6 9 a t m i d l a y e r , 7 7

in d i f f u s i o n d o m a i n , 9 7 as f o u r - v e c t o r , 4 9 8 in terna l , 2 2 1 - 2 2 4

s imi lar i ty , 4 8 0 l o c a l , 4 9 4

w i t h p o l a r i z a t i o n , 2 0 , 4 9 3 - 4 9 5 spec i f i c , 5 , 2 7 a t sur face , 8 2

t r a n s f o r m a t i o n b y r o t a t i o n , 2 2 - 2 3 b y s ca t t er ing , 2 0 - 2 2

I n t e r a c t i o n pr inc ip l e , 6 2 In ter f erence , 2 0 I n t e r m e d i a t e resu l t s , u s e , 4 0 I n t e r n a t i o n a l A s s o c i a t i o n o f M e t e r o l o g y a n d

A t m o s p h e r i c P h y s i c s , 6 8 3 , 6 9 4 I n t e r p o l a t i o n b e t w e e n s m a l l b a n d ° °

i s o t r o p i c s c a t t e r i n g , 2 0 1 - 2 0 3

R a y l e i g h s c a t t e r i n g , 5 5 9 Inters te l lar d u s t , 7 1 3 Inters te l lar e x t i n c t i o n , 7 1 3 , 7 1 8 , 7 2 0 Inters te l lar g r a i n , 7 1 6 , 7 1 9 In ters te l l ar p o l a r i z a t i o n , 7 1 3 Inters te l lar r a d i o s c i n t i l l a t i o n , 7 1 7 , 7 1 9 I n v a r i a n c e pr inc ip l e , 5 3 , 7 3 , 1 0 2 , 1 6 2 , 185 I n v a r i a n t e m b e d d i n g , 4 2 , 5 3 , 9 9 , 1 0 2 , 691

R a y l e i g h s ca t t er ing , 5 3 6 I n v e r s e p r o b l e m , 3 4 , 3 6 I n v e r s i o n f r o m r a d i a n c e t o p h a s e f u n c t i o n

a s y m p t o t i c , 7 1 0 - 7 1 1 m o d e r a t e d e p t h , 7 1 2

I r r a d i a n c e , 5 d o w n w a r d , 7 1 2 sca lar , 5 , 7 1 2 u p w a r d , 7 1 2 v e c t o r , 5

I s o t r o p i c s ca t t er ing , see also spec i f i c f u n c t i o n s d i f f u s i o n , 152

c o n s t a n t , 1 5 0 , 152 e x p o n e n t , 1 5 0 , 152

e m e r g i n g r a d i a t i o n , 1 8 3 - 1 9 0 e s c a p e f u n c t i o n , 1 5 1 , 3 5 8 - 3 6 2 w i t h f o r w a r d p e a k , 3 0 7 w i t h h o m o g e n e o u s s o u r c e s , 2 2 0 - 2 2 1 in terna l in tens i ty , 2 2 1 - 2 2 4

K u s c e r p o l y n o m i a l s , 3 7 8 - 3 7 9 M i n n a e r t p l o t , e x p o n e n t , 6 1 9 p l a n e a l b e d o , 1 9 8 - 1 9 9 , 4 8 3 - 4 8 6 p h a s e m a t r i x , f a c t o r i z a t i o n , 5 3 5 p o i n t - d i r e c t i o n g a i n , 136 , 1 3 8 - 1 3 9 ,

2 0 6 - 2 1 3 , 2 1 6 - 2 1 8 , 2 9 2 - 2 9 9

d i f ferent ia l e q u a t i o n , 174 i n d i f f u s i o n d o m a i n , 175 , 2 0 9 - 2 1 3 e x p a n s i o n , 1 7 5 - 1 7 6 in tegra l , 186 , 2 1 8 - 2 2 0 a s m a t r i x , 1 3 0 , 1 3 6

m o m e n t , 1 3 6 , 1 3 8 - 1 3 9 , 1 7 4 - 1 8 2 , 2 0 6 - 2 2 1 , 2 9 2 - 2 9 9

d i f ferent ia l e q u a t i o n , 2 2 7 in tegra l , 2 2 0 z e r o o r d e r , 2 0 8

near ly c o n s e r v a t i v e , 1 7 8 - 1 8 0 n u m e r i c a l v a l u e s , 1 7 6 - 1 7 8 , 2 0 7 , 2 9 2 - 2 9 9 i n s e m i - i n f i n i t e a t m o s p h e r e , 1 7 4 - 1 8 2 a t sur face , 1 7 5 , 2 2 4 t h i c k layer , 2 0 9 - 2 1 3

r a p i d - g u e s s f o r m u l a s , 3 6 2 - 3 6 4 re f l ec t i on f u n c t i o n , 1 5 1 , 1 9 2 - 2 0 3 , 2 2 2 ,

2 3 6 - 2 8 1 , 3 9 8 , 6 1 7 - 6 1 8 b i m o m e n t , 1 3 6 , 1 7 2 - 1 7 4 , 1 9 3 - 1 9 4 , 2 0 2 ,

2 3 6 - 2 8 1 n e a r l y c o n s e r v a t i v e , 3 9 0 - 3 9 2

e i g e n f u n c t i o n e x p a n s i o n , 145 first o r d e r , 1 9 4 , 2 3 6 - 2 8 1 w i t h g r o u n d , 6 3 1 - 6 3 2 as m a t r i x , 130 , 1 3 3 - 1 3 5 m o m e n t , 1 3 6 , 1 7 2 , 1 9 3 - 1 9 4 , 2 0 0 - 2 0 2 ,

2 3 6 - 2 8 1 for μ - o o , 2 1 9 - 2 2 0 n u m e r i c a l v a l u e s , 2 3 6 - 2 8 1 , 4 8 2 - 4 8 6 ,

5 4 3 - 5 4 4

r a t i o t o t a l t o first o r d e r , 1 9 6 - 1 9 8 s e c o n d o r d e r , 2 3 6 - 2 8 1 t h i c k s l a b , 2 0 1 - 2 0 3 , 4 8 2 ^ 8 6 th i rd o r d e r , 2 3 6 - 2 8 1

s p e c i f i c a t i o n s , 1 4 9 - 1 5 1 , 1 8 5 , 2 0 1 , 2 0 9 s p h e r i c a l a l b e d o , 1 9 8 - 1 9 9 , 3 7 0 , 4 8 3 - 4 8 6 s u c c e s s i v e o r d e r s , e i g e n v a l u e , 1 4 1 - 1 4 4 , 5 8 6 t h i c k layer , 1 9 8 - 2 0 3 t r a n s m i s s i o n f u n c t i o n , 1 9 2 - 2 0 3 , 2 2 2 - 2 2 3 ,

6 5 1 - 6 5 2 b i m o m e n t , 1 3 6 , 1 9 3 , 195 , 2 3 7 - 2 7 9 d i a g o n a l v a l u e , 195 d i f fuse , 130 , 1 9 3 , 195 e i g e n f u n c t i o n e x p a n s i o n , 145 first o r d e r , 1 9 5 , 2 3 7 - 2 7 9 a s m a t r i x , 130 , 1 3 3 - 1 3 5 m o m e n t , 1 3 6 , 1 9 3 , 1 9 5 , 2 0 0 - 2 0 2 , 2 3 7 - 2 7 9

Index IX

n u m e r i c a l v a l u e s , 2 3 7 - 2 7 9

s e c o n d o r d e r , 2 3 7 - 2 7 9

t h i c k s l a b , 2 0 0 - 2 0 3

th i rd o r d e r , 2 3 7 - 2 7 9 z e r o - o r d e r , 1 9 3 - 1 9 6 , 2 3 7 - 2 7 9

a s m a t r i x , 1 3 0 I s o t r o p i c s e c t o r p h a s e f u n c t i o n , 3 0 7 , 3 0 8

e s c a p e f u n c t i o n , 3 5 8 - 3 6 1 e x t r a p o l a t i o n l e n g t h , 3 7 6 in t erna l r e f l e c t i o n c o e f f i c i e n t , 3 7 4 - 3 7 5

J

J u n g e s i ze d i s t r i b u t i o n , 6 5 5 , 6 6 4 , 6 6 9 , 6 7 2

J u p i t e r c l o u d leve l , 6 2 2 , 6 2 4 g e o m e t r i c a l b e d o , 6 2 0 - 6 2 1 , 6 2 4 in frared l i m b d a r k e n i n g , 6 4 3 M i n n a e r t p l o t , 6 1 6 , 6 1 9 p o l a r i z a t i o n , 5 5 9 , 5 6 1 , 6 2 2 - 6 2 4

Κ

K i r c h o f f s l a w , 18 for b l a c k b o d y , 2 5 for f in i te b o d y , 2 4 for p o l a r i z e d l ight , 2 5 - 2 6 f o r s u r f a c e , 3 2 f o r t h i n d i s k , 2 5

K M see K u b e l k a - M u n k f o r m u l a Κ m a t r i x , 5 7 , 5 9 K n u d s e n n u m b e r , 7 0 6 K u b e l k a - M u n k f o r m u l a , 4 9 1 , 6 9 8 - 7 0 3

a c c u r a c y t e s t , 7 0 0 - 7 0 2 c o n s e r v a t i v e s c a t t e r i n g , 7 0 0

f in i te l ayer , 6 9 9 - 7 0 2 K u s c e r p o l y n o m i a l , 9 1 , 9 4 - 9 8 , 1 2 0 , 3 7 6

a s y m p t o t i c e x p r e s s i o n , 3 7 7 c o n v e r g e n c e , 3 7 7 - 3 7 9 r a t i o o f o r d e r s , 3 7 7 - 3 8 0 r e c u r r e n c e , 9 4 - 9 5 , 3 7 7

L

L a d e n b u r g - R e i c h e f u n c t i o n , 5 9 5 , 5 9 7 Λ o p e r a t o r , 1 3 0 L a m b e r t l a w , 2 0 4 , 4 9 7 , 6 0 5 , 6 0 8 L a m b e r t s u r f a c e , 6 , 4 6 8 , 4 7 2 , 4 7 6

b e l o w a t m o s p h e r e , 6 2 3 , 6 2 6 L a p l a c e t r a n s f o r m , 5 7 4 , 5 7 7 , 5 8 5

i n v e r s e , 5 7 4 , 5 7 7 - 5 7 9 , 5 8 2 , 5 8 4 , 5 8 9 , 6 7 6 , 7 1 9

L a s e r , 3 1 4

L a s e r r a d a r , 6 7 5

L a t e x s u s p e n s i o n , 7 0 9 - 7 1 0 L e g e n d r e c o e f f i c i e n t , 3 0 8 , 3 1 7 - 3 2 4

h i g h o r d e r , 7 0 3 L e g e n d r e e x p a n s i o n , 4 8 2 , 6 1 4 , 7 1 0

e x a m p l e , 5 2 3 n u m b e r o f t e r m s , 5 1 9 - 5 2 0

L e g e n d r e f u n c t i o n , 7 9 , 5 2 8 , 6 0 5 a s s o c i a t e d , 4 7 , 9 4 - 9 5 , 3 2 4 , 4 9 8 , 5 0 9 - 5 1 0 ,

5 1 6 , 5 2 0 r e l a t e d f u n c t i o n , 3 1 1 , 3 2 5 o f s e c o n d k i n d , 9 8 , 3 6 6 , 3 7 9

L e g e n d r e p o l y n o m i a l , 4 3 , 8 9 , 9 4 - 9 5 , 109 , 3 0 4 , 3 1 7

a s s o c i a t e d , 3 2 5 - 3 2 8 , 5 9 7 L i d a r , 5 7 3 , 6 6 2 - 6 6 3 , 6 9 3 - 6 9 4 , 7 1 7

a t t e n u a t i o n f a c t o r , 6 7 7 b i s ta t i c , 6 6 3 c l o u d s , 5 7 3 , 6 7 5 - 6 8 2 d e p o l a r i z a t i o n , 6 8 0 - 6 8 2 e c h o t i m e , 6 6 3 , 6 7 6 m u l t i p l e s ca t t er ing , 6 7 6 - 6 8 0 r e c e p t i o n c o n e , 6 7 6 - 6 7 7 , 6 7 9 - 6 8 0 re turn s i g n a l

f u n c t i o n o f r e c e p t i o n c o l u m n , 6 7 8 - 6 8 0 o r d e r s , 6 7 7 r a t i o o f t o t a l t o s i n g l e s c a t t e r i n g ,

6 7 8 - 6 7 9 L i f e o n p l a n e t , 6 1 6 L i g h t h o u s e , 7 1 9 L i m b , see G l o b u l e ; P l a n e t ; S u n L i n e a b s o r p t i o n , see A b s o r p t i o n L i n e a r l y a n i s o t r o p i c p h a s e f u n c t i o n , 8 9 ,

9 6 - 9 7 , 1 2 1 , 3 0 5 - 3 0 6 B u s b r i d g e p o l y n o m i a l s , 3 6 4 c h a r a c t e r i s t i c e q u a t i o n , r o o t , 3 6 5 - 3 6 7 ,

3 8 1 - 3 8 2 i n d i f f u s e g a l a c t i c l i ght , 7 1 7 d i f f u s i o n e x p o n e n t , 3 6 5 - 3 6 7 d i s p e r s i o n f u n c t i o n , 3 6 6 e s c a p e f u n c t i o n , 3 3 9 , 3 5 8 - 3 5 9 , 3 6 7 e x t r a p o l a t i o n l e n g t h , 3 6 6 - 3 6 7 Η f u n c t i o n , 3 6 4 - 3 6 6

d e r i v a t i v e , 3 6 5 - 3 6 6 m o m e n t , 3 6 4 - 3 6 5

K u s c e r p o l y n o m i a l , 3 7 8 - 3 7 9 M i n n a e r t p l o t , e x p o n e n t , 6 1 9 - 6 2 0 p l a n e a l b e d o , 3 6 8 - 3 6 9 r e f l e c t i o n f u n c t i o n , 4 8 6 - 4 8 7 , 6 1 7 - 6 1 8

b i m o m e n t , 3 6 7 , 7 0 1 m o m e n t , 3 6 7 - 3 6 8

s p h e r i c a l a l b e d o , 3 6 7 t r a n s m i s s i o n f u n c t i o n , 4 8 6 - 4 8 7

χ Index

L o c a l s ca t t er ing , 4 9 8 - 5 0 3 , 5 0 6 - 5 1 6 L o g - n o r m a l s i ze d i s t r i b u t i o n , 6 6 4 - 6 6 5 , 6 7 5 L o m m e l - S e e l i g e r l a w , 1 9 6 , 6 0 6 - 6 1 0 L o r e n t z prof i l e , 5 9 3 - 5 9 5 L o s s , see also A b s o r p t i o n

in a t m o s p h e r e , 7 7 , 8 0 , 6 8 5 - 6 8 7 w e a k a b s o r p t i o n , 7 8 , 8 0 , 180

b y f in i te d e p t h , 8 0

i n g r o u n d , 6 8 5 - 6 8 7 at large d e p t h , 8 0 t o t a l , 6 8 5 - 6 8 7

L u m i n a n c e , 6 5 2

M

M a g n e t o - o p t i c ac t iv i ty , 6 1 6 M a g n i t u d e o f p l a n e t , 6 0 0 - 6 0 1 M a n u f a c t u r i n g c o n t r o l , 6 9 9 M a r i n e o p t i c s , see H y d r o l o g i e o p t i c s M a r s

c l ear ing , 6 3 0 h a z e , 6 3 0 i s o p h o t e s , 6 2 0 M i n n a e r t p l o t , 6 1 6 , 6 1 9 - 6 2 0

s u r f a c e c h a n g e s , 6 3 1 M a t c h i n g o f c l o u d d e c k a n d a t m o s p h e r e ,

6 2 8 - 6 3 0 , 6 4 6 M a t r i x , 128

d o t p r o d u c t , 7 6 , 145 i n v e r s i o n , 1 1 1 , 146

μ μ , 4 6 , 6 8 - 6 9 , 128 , 130 μ τ , 128 , 1 3 0 o p e r a t o r t h e o r y , 6 2 p a r t i t i o n i n g , 1 4 6 - 1 4 8

p r o d u c t , 6 8 - 6 9 , 128 , 1 3 1 , 172 s ingu lar , 6 9 , 1 2 9 - 1 3 1 τ μ , 128 , 130 rr, 128 , 1 3 0 transfer m e t h o d , 4 3 , 6 2 - 6 3 , 7 4 un i t , 6 9 , 5 3 5

M e l l i n t r a n s f o r m , 6 7 4 M e r o p e , re f l ec t ion n e b u l a , 7 1 4 , 7 1 5 M e t a l l i c o p t i c a l p r o p e r t i e s , 3 1 4 M e t h o d , see also s p e c i f i c m e t h o d s

a d v a n t a g e s , 4 2 - 4 4 o f c o m p u t a t i o n , 4 0 cr i ter ia , 3 9 - 4 4 o f d e r i v a t i o n , 3 9 d i s c r e t i z a t i o n , 4 1 , 4 3 - 4 4 d r a w b a c k s , 4 2 - 4 4 h y b r i d , 4 1 , 4 5 , 5 0 m u l t i s t r e a m , 4 4

preferred , 4 5

p r o b a b i l i s t i c , 4 1

t r a d i t i o n a l , 9 0 M i c r o w a v e , a n a l o g m e a s u r e m e n t , 3 1 4 M i d l a y e r i n t e n s i t y , 4 3

in g a l a x y , 7 1 6

H e n y e y - G r e e n s t e i n p h a s e f u n c t i o n , 2 2 3 , 4 0 5 , 4 6 2 ^ 6 4

i s o t r o p i c s ca t t er ing , 2 0 9 , 2 2 3 , 2 9 2 - 2 9 9 , 4 0 5 M i e t h e o r y , 3 0 3 , 3 0 4 , 3 1 1 - 3 1 4 , 3 1 7 - 3 2 0 ,

3 2 4 - 3 2 5 , 5 0 9 , 5 1 1 , 5 1 9 , 6 7 1 , 7 0 3

a b s o r b i n g s p h e r e , 3 1 3 - 3 1 4

n o n a b s o r b i n g s p h e r e , 3 1 2 - 3 1 3

p o l a r i z a t i o n , 3 1 2 , 3 1 5

s c a t t e r i n g p a t t e r n , 3 0 4 , 3 1 1 - 3 1 2 , 3 5 6 , 6 3 8 a m p l i t u d e , 3 2 4 a s y m m e t r y , 3 1 2 - 3 1 4 , 3 1 6 l o b e shift , 5 2 0

M i l k , 6 9 8 , 7 0 9 - 7 1 0

b l a c k e n e d , 7 0 9 - 7 1 0 M i l k y W a y , 7 1 3 M i l l i m e t e r w a v e , 3 1 7 M i l n e e q u a t i o n , 1 2 7 - 1 2 8 , 1 3 1 - 1 3 2 , 146

e i g e n f u n c t i o n s , 1 4 0 - 1 4 4 e i g e n v a l u e s , 1 4 0 - 1 4 4

g e n e r a l i z e d , 4 8 , 4 9

h o m o g e n e o u s , 150 , 1 5 2 - 1 5 3 , 185 i n h o m o g e n e o u s , 1 5 3 , 185 in m a t r i x f o r m , 131 s o l u t i o n , 1 3 1 - 1 3 2

M i l n e m a t r i x , e x p a n s i o n , 145 M i l n e o p e r a t o r , 1 2 7 - 1 3 0 , 134

e i g e n v a l u e s , 1 4 0 - 1 4 4 M i l n e p r o b l e m , 8 5 , 9 0 , 1 1 1 - 1 1 4 , 169 , 3 9 3 M i n n a e r t - B a r k s t r o m p lo t , 6 2 0 M i n n a e r t p l o t , 6 1 6 , 6 1 9 - 6 2 0 M i r r o r

a n g l e , 5 4 5 - 5 4 6 , 5 4 9 m a t r i x , 5 0 5

M i s t , 6 9 8 - 6 9 9 M i x i n g ra t io , 5 9 2 M o d e o f p r o p a g a t i o n , 1 2 1 , 3 3 3 M o d e rad ius , 3 1 6 , 3 2 2 , 6 6 4 - 6 6 5 M o d e l , u s e , 3 4 - 3 6 , 7 1 3 , 7 1 7 M o l e c u l e , 4 9 6

free p a t h , 7 0 6 M o m e n t , see s p e c i f i c f u n c t i o n M o n t e C a r l o m e t h o d , 4 4 , 5 7 5 , 6 1 0 , 6 7 6 , 691

c u b o i d c l o u d , 6 9 5 - 6 9 8 re f l ec t ion n e b u l a , 7 1 3 - 7 1 5

M u l t i p l e s ca t t er ing , 3 4 - 3 5 μ v e c t o r , 6 4 , 6 8 - 6 9 , 128 , 130 , 172

Index XI

Ν

N a s t y c o r n e r o f (a, b) d o m a i n , 8 1 , 5 9 1 , 6 8 4 - 6 8 7

N a t u r a l l ight , 2 3 , 2 8 , 4 9 6 , 5 0 8 N e a r l y c o n s e r v a t i v e s ca t t er ing , 7 8 - 8 1 ,

3 8 8 - 3 8 9 , see also specific p h a s e f u n c t i o n s N e p t u n e , g e o m e t r i c a l b e d o , 6 2 0 - 6 2 1 N e t c u r r e n t , 4 8 8 , 6 9 3 N e t flux, see F l u x N e u m a n n ser ies , 4 2 , 1 3 2 , 1 4 6 - 1 4 7 , 6 7 6 N e u t r o n s c a t t e r i n g , 7 9 N o n a s y m p t o t i c part , 159 N o n s p h e r i c a l part i c l e , 3 1 4 - 3 1 5

l idar e c h o , 6 8 0 N o n s t a t i o n a r y p r o b l e m , 5 7 4 N o n u n i q u e n e s s , 381 Ν o p e r a t o r , 6 9 , 4 9 7 N o r m a l i z a t i o n

o f e i g e n f u n c t i o n i n d i f f u s i o n , 7 4 - 7 5 o f i n c i d e n t flux, 193

o f μ v e c t o r , 172

b y o t h e r a u t h o r s , 7 8 , 8 5 o f s p e c u l a r r e f l e c t i o n f u n c t i o n , 6 2 6

N u c l e a r r e a c t o r , see R e a c t o r

Ο

O c e a n , see also H y d r o l o g i e o p t i c s u n d e r a t m o s p h e r e , 5 0 7

O m n i d i r e c t i o n a l p r o b e , 6 4 0 - 6 4 1 O n s a g e r r e l a t i o n , 17 O p a c i t y , 143 O p a l g l a s s , 6 9 8 - 6 9 9 O p a q u e a t m o s p h e r e , 6 2 5 O p e r a t o r

d e f i n i n g m o m e n t , 6 9 o n μ , 130 o n r , 128 , 130

O p p o s i t i o n , 6 0 8 , 6 1 0 ef fect , 6 0 6

in c i rcu lar p o l a r i z a t i o n , 6 1 6 l ine , 5 4 5 - 5 4 6

O p t i c a l d e p t h , 6 8 , 128

r e d u c e d , 3 9 6 t r a n s f o r m a t i o n , 4 8 1

O p t i c a l p a t h , see P h o t o n , p a t h O p t i c a l t h i c k n e s s , 7 6 , 5 7 5

r e d u c e d , 4 7 9 O r d e r , see also S u c c e s s i v e o r d e r

in F o u r i e r e x p a n s i o n , 5 0 5 , 5 1 4 - 5 1 5 , 5 2 2 - 5 2 9 o f s u c c e s s i v e s ca t t er ing , 5 0 5 , 5 1 4 - 5 1 5 ,

5 2 2 - 5 2 3

O r t h o g o n a l i t y , 1 2 1 , 3 2 6

h a l f - r a n g e , 4 3 , 121 O v e r c a s t s k y , 5 0 7 O z o n e , 5 6 7 , 6 4 1 , 6 5 7

Ρ

P a d é a p p r o x i m a t i o n , 5 8 0 , 5 8 2 P a e t z o l d p r o b e , 6 4 1 P a i n t l a y e r , 4 9 1 , 6 9 8 - 7 0 3

c o v e r e d s p o t , 7 0 3 re f l ec t ing p o w e r , 7 0 0

P a r a m e t r i z a t i o n , 6 8 3 P a r t i c u l a t e m a t e r i a l , 6 9 9 P e a k , see also D i f f r a c t i o n ; F o r w a r d peak;

T r a n s m i s s i o n f u n c t i o n , z e r o - o r d e r

d o w n w a r d , 711 P e n c i l b e a m in inverse p r o b l e m , 7 1 4 P e n e t r a t i o n d e p t h , 6 3 7 - 6 4 0

s ta t i s t i c s , 6 3 9 - 6 4 0 P h a s e a n g l e , 601

P h a s e f u n c t i o n , 5 , 3 5 , 7 0 , 3 0 3 - 3 1 7 , 4 9 6 - 4 9 7 , see also spec i f i c p h a s e f u n c t i o n s

a s y m m e t r y , 7 0 , 3 0 4 c h a n g e o f s i g n , 3 0 5

a s y m p t o t i c r a d i a n c e , 7 1 0 - 7 1 1

a z i m u t h a v e r a g e , 7 0 , 4 9 7 c h a r a c t e r i s t i c s , 3 0 3

c h o i c e , 3 0 3 , 3 1 0 e l l ip t i ca l , 3 0 8 e x p a n s i o n

F o u r i e r , 5 2 0 - 5 2 1

in L e g e n d r e f u n c t i o n s , 8 0 - 1 2 3 , 3 0 4 , 3 1 7 - 3 2 4 , 4 8 1 , 5 1 9 - 5 2 0 f inite , 3 0 5 - 3 0 6 , 3 5 6 - 3 5 7

f a m i l y , 3 0 5 , 3 0 8 in tegra l , 7 0

f o r w a r d part , 3 0 5 - 3 0 7 , 3 0 9 i n v e r s e l inear , 3 0 7 , 3 0 8 o f large b o d y , 3 0 8 n o r m a l i z a t i o n , 5 , 7 0 , 3 0 4 s i n g u l a r , 3 0 8 - 3 0 9 o f s p h e r e , 3 0 3 , 3 0 4 s u m

o f e x p o n e n t i a l s , 3 2 0 - 3 2 2 o f G a u s s f u n c t i o n s , 6 7 6

P h a s e in tegra l o f p l a n e t , 6 0 0 - 6 0 1 P h a s e m a t r i x , 4 9 4 - 4 9 8 , 5 0 0 - 5 0 3 , 5 1 3

a z i m u t h - i n d e p e n d e n t part , 3 2 4 , 5 1 3 e f f ec t ive a l b e d o , 5 1 8 e x p a n s i o n , 3 1 7 , 3 2 4 - 3 2 9 , 5 0 9 s y m m e t r y , 5 0 4

XII Index P h a s e shi f t , 3 1 2 , 3 1 7 P h o t o i o n i z a t i o n , 18 P h o t o n

e x i t a n g l e , 6 9 5 p a t h

g e o m e t r i c a l , 5 7 5

m e a n , 3 8 8 , 5 7 7 - 5 7 9 , 5 8 3 - 5 8 7 , 5 8 9 - 5 9 1 n e a r - c o n s e r v a t i v e s ca t t er ing , 5 8 3 - 5 8 4 in s e p a r a t e o r d e r , 5 7 8 in t r a n s m i s s i o n , 5 8 6 - 5 8 7

o p t i c a l , 5 7 5 p r o b a b i l i t y d i s t r i b u t i o n , 5 7 7 - 5 8 2

a s y m p t o t i c , 5 7 9 - 5 8 2 , 5 8 8 - 5 9 0 d i s p e r s i o n , 5 8 5 - 5 8 8 e x a c t , 5 8 5 , 5 8 7 - 5 8 8 m o m e n t , 5 7 7 n o n c o n s e r v a t i v e , 5 8 2 - 5 8 4 s emi - in f in i t e a t m o s p h e r e , 5 7 9 - 5 8 4 , 5 8 9 in s e p a r a t e o r d e r , 5 7 8 - 5 8 0 , 5 8 5 ,

5 8 7 - 5 8 8 s l a b , 5 8 4 - 5 8 5 , 5 8 8 s t a n d a r d c u r v e , 5 8 5 - 5 8 6 , 5 8 8 - 5 8 9

r o o t - m e a n - s q u a r e , 3 8 8 s q u a r e m e a n r o o t , 5 9 4 - 5 9 5 s ta t i s t i c s , 5 7 3 - 5 9 1 , 6 3 4 , 6 3 6 - 6 4 0

for f i x e d p e n e t r a t i o n d e p t h , 6 3 8 - 6 3 9 i n h o m o g e n e o u s s l a b , 5 7 5

P h y s i c a l d e f i n i t i o n , 7 0 - 7 3 P h y s i c a l d e r i v a t i o n , 5 4 , 1 0 1 , 1 7 9 - 1 8 0 , 6 0 3 - 6 0 4 P h y s i c a l m e a n i n g

o f m a t r i x a n d v e c t o r , 1 3 0 o f m a t r i x p r o d u c t , 6 8 - 6 9 , 129 , 1 7 4 , 193 o f N e u m a n n ser ies , 132

P h y s i c a l s y s t e m , l inear , 16 P i l l ar s ca t t er ing , 6 9 7 - 6 9 8 P i o n e e r , 6 2 4 P l a n e a l b e d o , 4

E d d i n g t o n a p p r o x i m a t i o n , 3 7 0 s imi lar i ty , 4 8 2 - 4 8 6 th i ck layer , 4 8 6

P l a n e o f re ference , 2 0 , 4 9 5 - 4 9 7 , 4 9 9 - 5 0 0 c h o i c e , 2 0 , 5 5 5 , 5 6 4 , 5 6 6 r o t a t i o n , 2 2 - 2 3 , 4 9 5 - 4 9 7 , 5 0 0

P l a n e - p a r a l l e l layer , see S l a b P l a n e t , 5 9 9 - 6 4 9 , see also spec i f i c p l a n e t s ;

G e o m e t r i c a l b e d o ; S p h e r i c a l a l b e d o a b s o r p t i o n s p e c t r u m , 3 7 0 , 3 7 2 - 3 7 4 , 5 9 3 ,

6 3 4 - 6 4 0 b r i g h t n e s s d i s t r i b u t i o n , 6 1 5 - 6 1 8 flux, re f lec ted , 6 0 0 full p h a s e , 6 0 6 , 6 0 8 , 6 1 0 i n t e g r a t i o n o v e r d i s k , 3 5 , 5 9 9 - 6 1 4

l i m b , 6 0 2 , 6 1 5 d a r k e n i n g , 132

in frared , 2 2 1 m o d e l c o m p u t a t i o n , 3 4 - 3 6 , 5 9 9 , 6 0 6 - 6 1 0 ,

6 2 1

w i t h g r o u n d , 3 5 , 198

w i t h t h i c k a t m o s p h e r e , 6 0 7 - 6 0 9 near ly b l a c k , 6 0 6 in o p p o s i t i o n , 1 3 2 , 5 2 2 , 5 4 3 - 5 4 4 , 6 1 7 - 6 2 4 a s par t i c l e , 6 0 0 - 6 0 1 , 6 1 0 p h a s e f u n c t i o n , 5 9 9 - 6 1 0

a s y m m e t r y f a c t o r , 6 0 6 - 6 1 1

in tegra l , 6 0 4 - 6 0 5 L e g e n d r e e x p a n s i o n , 6 0 5 , 6 0 7 , 6 0 9 in o p p o s i t i o n , 6 0 7

p o l a r i z a t i o n , 5 4 9 , 5 5 7 - 5 6 3 , 5 9 9 , 6 1 0 - 6 1 4 , 6 2 3 - 6 2 4

n e a r - s y m m e t r y , 5 4 6 - 5 4 7 , 6 1 6 w i t h R a y l e i g h sca t ter ing ,

d i f fuse r e f l e c t i o n , 5 3 7 , 5 5 7 - 5 6 1 finite d e p t h , 6 2 3 - 6 2 4 p o l a r i z a t i o n , 5 4 9 , 5 5 7 - 5 6 3

s u r f a c e r e f l e c t i o n , 196 s y m m e t r y a b o u t e q u a t o r , 6 1 5 - 6 1 6 t h e r m a l e m i s s i o n , 6 4 2 - 6 4 6 w h i t e , 6 0 5 - 6 0 6 , 6 0 8 - 6 0 9

P l a n t c a n o p y , 7 0 5 - 7 0 6 P l a t e s c a t t e r i n g , 6 9 7 - 6 9 8 P o i n c a r é s p h e r e , 4 9 5

P o i n t - d i r e c t i o n g a i n , 4 , 2 9 , 3 7 , 6 9 3 , see also spec i f i c p h a s e f u n c t i o n

b y a d d i n g m e t h o d , 5 6 - 6 0 in A m b a r t s u m i a n ' s m e t h o d , 5 0

i n arb i t rary c o n f i g u r a t i o n , 4 , 2 9 a s y m p t o t i c , 8 4 w i t h g r o u n d s u r f a c e , 6 4 i n h o m o g e n e o u s s l a b , 3 2 , 5 8 - 6 0 r e c i p r o c a l d e f i n i t i o n , 2 9 , 3 2 , 5 6 a t s e p a r a t i o n layer , 6 0 , 4 8 8 at sur face , 3 7 , 4 8 8

P o i s s o n d i s t r i b u t i o n , 5 8 0 P o l a r i z a b i l i t y , 5 3 1

t e n s o r , 3 1 4 , 5 3 2 P o l a r i z a t i o n , 5 , 3 9 , 4 9 3 - 5 1 3 , see also S t o k e s

p a r a m e t e r

c i rcu lar , 4 9 5 - 4 9 6 , 5 0 3 d e g r e e , 4 9 5 e l l ipse , 4 9 5 e l l ip t i ca l , 5 0 1 l inear , 4 9 5 - 4 9 6 , 5 0 1 part ia l , 4 9 6 r e c i p r o c i t y , 1 7 - 2 6 , 3 0 - 3 1

Index XIII

r e p r e s e n t a t i o n , 4 9 5 - 4 9 8 , 5 3 3 C P , 3 2 4 , 5 3 3 t r a n s f o r m a t i o n , 4 9 7 , 5 3 3

s i g n a t u r e , 3 1 5 s ta te , 4 9 4 - 4 9 6

P o l l u t e d w a t e r , 7 1 0 P o l y n o m i a l , 1 5 1 , see also B u s b r i d g e

p o l y n o m i a l ; K u s c e r p o l y n o m i a l ; L e g e n d r e p o l y n o m i a l ; S o b o l e v p o l y n o m i a l

P o l y v i n y l a c e t a t e , 7 1 0

P o w e r l a w s i ze d i s t r i b u t i o n , 6 6 4 - 6 6 5 , 6 7 2 - 6 7 3 P r o p a g a t i o n , see D i f f u s i o n ; R a n d o m m e d i u m

p r o p a g a t i o n Ψ - n o r m , 5 3 9 - 5 4 0 P u l s e d e l a y , 7 1 8 - 7 2 0

s p r e a d in t i m e , 7 1 8 - 7 2 0

Q

Q u a s i - s i n g l e s c a t t e r i n g a p p r o x i m a t i o n , 4 1 , 7 1 3

R R a d a r , 6 7 5 R a d i a n c e , 5 , 6 1 5 , 6 5 2 , see also I n t e n s i t y

a t t e n u a t i o n coe f f i c i en t , 7 1 2 u n i t , 6 5 8 - 6 5 9

R a d i a t i o n d e n s i t y , 7 3 , 8 4 , 138 d o s e , 6 5 7 h y d r o d y n a m i c s , 7 0 6 - 7 0 7 pres sure o n s p h e r e , 3 1 3 - 3 1 4 s l ip , 7 0 6

R a d i a t i v e e q u i l i b r i u m , 152 R a d i a t i v e t rans fer , h i s t o r i c a l n o t e , 5 4 R a d i o w a v e s c a t t e r i n g , 7 1 9 - 7 2 0 R a i n , m o d e l , 3 1 6 R a i n b o w , 3 1 2 , 3 1 5 - 3 1 6 , 3 2 2 - 3 2 4 R a n d o m m e d i u m p r o p a g a t i o n , 7 2 0 R a n d o m o r i e n t a t i o n , 2 3 , 5 0 0 , 5 3 2 , 6 0 5 , 661 R a n d o m w a l k , 7 1 8 R a p i d - g u e s s f o r m u l a , 8 3 , 3 6 2 - 3 6 4 R a y l e i g h - G a n s p a t t e r n , 6 7 0 R a y l e i g h p h a s e f u n c t i o n , 3 0 5 - 3 0 6 , 3 1 1 , 3 5 6 ,

4 9 4 , 5 3 1 - 5 3 2

e s c a p e f u n c t i o n , 3 6 2 e x a c t s o l u t i o n , 5 3 7 - 5 3 8 H f u n c t i o n , 5 3 8 in terna l r a d i a t i o n f ie ld , 5 3 8 p l a n e a l b e d o , 4 8 3 - 4 8 6 r e f l e c t i o n f u n c t i o n , 4 8 2 - 4 8 6 , 5 4 3 - 5 4 4 , 6 1 8

s p h e r i c a l a l b e d o , 3 7 0 , 4 8 3 - 4 8 6 t r a n s m i t t e d flux, 5 3 9

R a y l e i g h s c a t t e r i n g , 3 0 5 , 4 9 4 , 5 0 3 , 5 3 1 - 5 6 9 c h a r a c t e r i s t i c f u n c t i o n , 5 3 9 d i f f u s i o n d o m a i n , 5 6 6

e x p o n e n t , 5 3 9 - 5 4 0 , 5 5 4 , 5 6 6 e s c a p e f u n c t i o n , 3 5 8 - 3 5 9 , 3 6 2 , 5 4 0 - 5 4 1 , 5 6 7

n o n c o n s e r v a t i v e , 5 5 4 e x a c t s o l u t i o n , 5 3 6 - 5 3 8 w i t h g r o u n d sur face , 5 5 7 - 5 6 3

r e f l e c t i o n f u n c t i o n , 5 5 7 - 5 6 1 w i t h i s o t r o p i c s c a t t e r i n g , 5 3 2 , 5 3 7 - 5 3 8 , 5 5 4

e s c a p e f u n c t i o n , 3 5 8 - 3 5 9 p h a s e m a t r i x , 5 3 2 , 5 3 4 - 5 3 6

H f u n c t i o n , 5 3 7 , 5 3 9 - 5 4 0 , 5 4 7 M i l n e p r o b l e m , 3 5 6 , 5 3 8 , 5 4 0 - 5 4 1 o f n a t u r a l l ight , 5 3 1 , 5 4 3 , 561 n o n c o n s e r v a t i v e , 5 3 2 , 5 3 7 , 5 5 0 - 5 5 4 p a t t e r n , 5 3 1 - 5 3 2 p h a s e m a t r i x , 5 3 1 - 5 3 6

a z i m u t h - i n d e p e n d e n t par t , 5 3 4

f a c t o r i z a t i o n , 5 3 4 - 5 3 6 p l a n e a l b e d o , 4 8 3 - 4 8 6 , 5 3 8 o n p l a n e t , 5 4 3 - 5 4 4 , 5 5 7 - 5 6 3 , 6 0 7 , 6 1 0 p o l a r i z a t i o n , 3 1 1 , 5 3 1 - 5 3 3

n e a r - s y m m e t r y , 5 4 3 , 5 4 6 - 5 4 7 , 5 6 7 , 6 1 6 in p r i n c i p a l p l a n e , 5 4 1 - 5 4 9

o r d e r s , 5 4 6 , 6 2 2 s i g n , 5 4 2 , 5 4 6 , 561

i n z e n i t h , 5 6 3 - 5 6 7 p u r e , 3 1 1 , 5 3 1 - 5 3 3

r e f l e c t i o n f u n c t i o n , 4 8 2 - 4 8 6 , 5 3 6 - 5 3 8 , 5 4 1 - 5 4 3

F o u r i e r t e r m s , 5 1 9 , 5 4 3 - 5 4 4

o r d e r s , 5 5 2 - 5 5 3 , 5 6 1 - 5 6 3 in pr inc ipa l p l a n e , 5 4 1 - 5 4 4

r e p r e s e n t a t i o n , 5 3 3 s e m i - i n f i n i t e a t m o s p h e r e , 5 4 0 - 5 5 0 s l a b , 5 5 5 - 5 6 7

a s y m p t o t i c , 5 5 9 - 5 6 1 e i g e n v a l u e s , 5 6 2 - 5 6 3

s p h e r i c a l a l b e d o , 4 8 3 - 4 8 6 t a b l e s , 5 3 7 - 5 3 8 t r a n s m i s s i o n f u n c t i o n , 5 3 6 - 5 3 8 , 5 6 4 - 5 6 7

R e a c t o r , 2 8 , 6 9 2 - 6 9 3 R e c i p r o c i t y , 1 6 - 3 3 , 5 9 , 3 9 3

arb i trary c o n f i g u r a t i o n , 2 4 - 2 7 o f d e t e c t o r a n d s o u r c e , 135 , 6 1 5 o f e s c a p e a n d i n j e c t i o n , 7 2 i n F m a t r i x , 2 1 - 2 2 in p h y s i c a l m e a n i n g , 17 , 100 , 1 3 5 , 2 1 3 w i t h p o l a r i z a t i o n , 2 7 , 5 4 6 , 5 6 7 , 6 1 6

XIV Index

R e c i p r o c i t y (com.) p r i n c i p l e , 1 6 - 1 7 , 8 5 , 129 re f l ec t i on f u n c t i o n , 1 9 3 , 6 1 4 - 6 1 6 s l ab , 3 0 - 3 1 sur face , 3 0

t r a n s m i s s i o n f u n c t i o n , 193 u s e i n p h o t o m e t r y , 6 1 4 - 6 1 6

R e d i s t r i b u t i o n e q u a t i o n , 5 0 3 o v e r f r e q u e n c i e s , 9 4 f u n c t i o n , 4 9 , 6 9 , 4 9 7

H e n y e y - G r e e n s t e i n p h a s e f u n c t i o n ,

3 3 2 - 3 3 3 , 3 9 7 , 6 5 2 in tegra l , 7 0 i s o t r o p i c s ca t t er ing , 150

m a t r i x , 130 , 1 3 1 - 1 3 4 R e d u c t i o n t o H f u n c t i o n s , 3 6 2 R e d u n d a n t p a r a m e t e r , 8 6 R e f e r e n c e , see P l a n e o f re ference R e f e r e n c e b a n d , 641 R e f l e c t a n c e in h y d r o l o g i e o p t i c s , 7 1 2 R e f l e c t i o n , see also spec i f i c p h a s e f u n c t i o n s ;

P l a n e a l b e d o ; S p h e r i c a l a l b e d o coe f f i c i en t , 2 0 2 o f d i f f u s i o n s t r e a m , 7 2 f u n c t i o n , 5 - 6 , 3 6 - 3 8 , 5 1 , 7 2 , 1 0 2 - 1 0 3 , 4 9 8

b y a t m o s p h e r e o v e r c l o u d , 6 2 7 - 6 2 9 o v e r g r o u n d , 6 3 , 6 4 , 6 2 5 - 6 2 6 , 6 2 8 - 6 2 9 o v e r sea , 6 3 , 6 2 6 - 6 2 7

b i m o m e n t , 8 0 - 8 1 , 701

first o r d e r , 5 1 9 c o m p u t a t i o n , 9 1 , 1 1 8 - 1 2 0 first o r d e r , 3 6 7 - 3 6 8 , 3 9 7 , 4 8 4 , 4 8 7 ,

5 1 8 - 5 1 9

F o u r i e r e x p a n s i o n , 5 0 4 - 5 0 5 , 5 2 1 - 5 2 3 h i g h o r d e r t e r m s , 3 8 8 - 3 9 3 i n h o m o g e n e o u s s l a b , 3 1 , 5 8 - 5 9 , 61 as m a t r i x , 6 9 m o m e n t , 1 1 6 - 1 1 7 , 5 1 9 near ly c o n s e r v a t i v e , 7 9 - 8 0 , 3 8 9 - 3 9 2 ra t io o f t o t a l t o first o r d e r , 4 5 4 - 4 5 5 , 521 r e d u c e d t o H f u n c t i o n , 9 1 - 9 2 , 109 ,

1 1 5 - 1 1 7 s imi lar i ty , 3 7 2 - 3 7 4 , 4 8 0 th ick layer , 8 2

m a t r i x , 4 9 8 , 5 0 3 - 5 0 6 , 5 1 3 t h i n s l a b , 5 0 3 - 5 0 4

n e b u l a , 6 9 3 , 7 1 4 - 7 1 5 m u l t i p l i c i t y , 7 1 5 ra t io o f t o t a l t o first order , 7 1 4 - 7 1 5

R e m o t e s e n s i n g a c t i v e , 6 7 5 - 6 7 6 p a s s i v e , 6 7 5

R e s o l v e n t , 120

R e s t r a i n t , see C o n s t r a i n t

R e v e r s e e x p e r i m e n t , 2 8 - 2 9 , 3 2 , 2 0 7 R o a d m a p , 1 1 8 - 1 1 9 R o t a t i o n m a t r i x , 4 9 6 , 5 0 0 , 5 3 3

S

S a t u r n g e o m e t r i c a l b e d o , 6 2 0 - 6 2 1 M i n n a e r t p l o t , 6 1 6 , 6 1 9

S c a l a r , 6 8 S c a l a r d e n s i t y , 4 8 8 , 6 9 3 S c a t t e r i n g

a n g l e , 4 9 9 , 601

b a c k w a r d , 3 0 5 , 3 0 7 , 3 8 4 , 4 8 9 - 4 9 1 c o h e r e n t , 1 8 - 2 0

c o n s e r v a t i v e , 7 8 , 8 1 - 8 3 d e p t h , 4 8 1

far-f ie ld , 1 8 - 2 0 , 2 2 i n c o h e r e n t , 2 0 , 2 7

l o c a l , 4 9 4 , 4 9 9 - 5 0 3 m a t r i x , 4 9 6 , 5 0 0 , 5 1 9

S c h w a r z s c h i l d - M i l n e in tegra l e q u a t i o n , 127 S e a r c h l i g h t , 6 9 3 - 6 9 4 S e a s o n a l c h a n g e , 6 1 6 S e a w a t e r , 7 0 9 - 7 1 0 S e e i n g , 7 2 0

S e m i - i n f i n i t e a t m o s p h e r e , 9 0 , 3 5 5 - 3 7 6 , 3 8 8 - 3 9 4

S h e l l , s c a t t e r i n g , 6 9 2 S h o r t h a n d n o t a t i o n , 6 9 - 7 0 , 1 2 8 - 1 3 0 , 5 1 2 S i m i l a r i t y , 3 0 3 , 3 1 7

in d i f f u s i o n p a t t e r n , 3 7 5 in e s c a p e f u n c t i o n , 3 5 7 - 3 5 9 in e x t r a p o l a t i o n l e n g t h , 3 7 4 - 3 7 6 n o t in a z i m u t h - d e p e n d e n t t e r m s , 6 1 8 p a r a m e t e r , 7 5 , 3 6 9 , 3 7 2 , 4 7 9 i n re f l ec t ion f u n c t i o n , 3 9 7 - 3 9 9 r e l a t i o n , 2 2 3 - 2 2 4 , 4 7 7 - 4 8 1 , 6 4 6 , 6 8 3

a c c u r a c y , 4 8 2 a l t e r n a t i v e s , 4 7 8 - 4 7 9 c o n s e r v a t i v e s ca t t er ing , 3 5 5 - 3 6 2 ,

4 0 3 ^ 0 5 , 4 0 8 , 4 7 9 d e g e n e r a t e d , 3 5 5 n o n c o n s e r v a t i v e , 3 6 4 , 3 7 2 - 3 7 4 , 4 7 7 - 4 7 9 for s l a b , 4 0 3 ^ 0 5 , 4 7 7 - 4 8 1

in spher i ca l a l b e d o , 3 7 0 - 3 7 1 test , 3 7 0 , 3 7 2 - 3 7 6

c l o u d s , 4 8 7 ^ 8 9 b y t r u n c a t i o n , 3 6 6 - 3 6 9

S i n g l e s c a t t e r i n g a l b e d o , see A l b e d o S i n g u l a r e i g e n f u n c t i o n e x p a n s i o n , 4 3 , 6 7 , 7 4 ,

9 0 , 2 2 2

Index XV

w i t h p o l a r i z a t i o n , 5 0 8 R a y l e i g h sca t t er ing , 5 3 6

S i z e d i s t r i b u t i o n , 6 6 4 - 6 6 5 o f a e r o s o l ,

o f d r o p s , 3 1 5 - 3 1 6 , 3 1 8 , 3 2 0 , 3 2 3 S i z e p a r a m e t e r , 3 1 8 - 3 2 2 S k y , see also B l u e s k y

c l e a r f r a c t i o n , 6 8 3 d a y l i g h t , 3 1 0 - 3 1 1 , 6 5 1 - 6 6 1

m u l t i p l e s ca t t er ing , 6 5 5 - 6 5 7 p o l a r i z a t i o n , 5 5 7 , 6 6 4 , 6 6 6 - 6 6 8

e l l ip t ica l , 6 6 7

w i t h g r o u n d re f l ec t ion , 5 5 5 n e u t r a l p o i n t , 5 5 5 - 5 5 7 , 6 6 6 at z e n i t h , 5 6 3 - 5 6 7

r a d i a n c e , 6 6 4 , 6 6 6 - 6 6 8 n e a r h o r i z o n , 6 6 7 R a y l e i g h sca t ter ing , 5 6 4 - 5 6 5 s i n g l e s c a t t e r i n g f r a c t i o n , 6 5 5 - 6 5 7 z e n i t h - h o r i z o n r a t i o , 6 5 1 - 6 5 4

b e l o w c l o u d s , 5 0 7 , 6 5 1 - 6 5 2 w i t h s n o w c o v e r , 6 5 3 - 6 5 4

red, 6 5 1 w h i t e , 6 5 1

S l a b , see also s p e c i f i c p h a s e f u n c t i o n i n h o m o g e n e o u s , r e c i p r o c i t y , 3 0 - 3 1 th i ck , a s y m p t o t i c e x p r e s s i o n , 7 6 - 7 8

S m a l l - a n g l e s c a t t e r i n g , m u l t i p l e , 6 6 0 - 6 6 1 ,

7 1 7 - 7 2 0 S m o g , 6 6 6 S n o w , 179

a l b e d o , 6 5 3 c o v e r , 6 5 3 - 6 5 4 flake, 3 1 4 p a c k , 7 0 3 - 7 0 5

re f l ec tance , 7 0 3 - 7 0 4 S o b o l e v m e t h o d , 1 1 8 - 1 2 0 S o b o l e v Φ f u n c t i o n , 1 3 8 - 1 3 9 , 2 0 9 , 2 2 0 S o b o l e v p o l y n o m i a l , 4 3 S o b o u t i f u n c t i o n , 2 2 6 - 2 2 7 , 2 3 0 - 2 3 1 S o f t part ic le , 6 7 0 - 6 7 3 S o l a r r a d i a t i o n t o g r o u n d , 6 5 4 - 6 5 5 S o u r c e

h o m o g e n e o u s d i s t r i b u t i o n , 2 1 8 - 2 2 1 layer

in terna l , 6 8 , 8 3 - 8 4 , 1 3 7 - 1 3 8 , 2 1 3 i s o t r o p i c s c a t t e r i n g , 1 5 3 - 1 6 1 , 2 0 6 u n i d i r e c t i o n a l , 8 5

at sur face , 3 2 , 5 0 , 129 in u n b o u n d e d m e d i u m , 1 5 3 - 1 6 1 z e r o order , 4 8 , 193

S o u r c e f u n c t i o n , 4 9 4 , 4 9 9 a s y m p t o t i c , 1 6 0 - 1 6 1 in d i f f u s i o n d o m a i n , 9 7

e x p a n s i o n , 1 3 0

first o r d e r , 138

as f o u r - v e c t o r , 4 9 8 - 4 9 9 , 501 i s o t r o p i c s ca t t er ing , 1 3 3 - 1 3 7 , 1 5 0 , 179 , 188 ,

190 , 2 0 6 , 2 0 9 , 2 1 1 - 2 1 3 , 2 2 2 o r d e r s , 137 , 1 6 0 - 1 6 1 at sur face , 1 9 3 , 2 1 3

s t a r t i n g t e r m , 4 8 , 130 , 134

t o t a l , 130 , 138 S o u r c e m a t r i x , 5 0 2 S p e c t r a l l ine , see A b s o r p t i o n l ine S p e c u l a r re f l ec t ion , 6 2 6 - 6 2 7 S p h e r e , as scat terer , 3 1 1 - 3 1 4 , 511 S p h e r i c a l a l b e d o , 4 , 2 6 , see also spec i f i c p h a s e

f u n c t i o n ; R e f l e c t i o n , f u n c t i o n , b i m o m e n t

a p p r o x i m a t i o n , 3 6 9 - 3 7 1

e x p a n s i o n , 3 6 9 - 3 7 0 first o r d e r t e r m , 3 7 0

o f p l a n e t , 6 0 0 - 6 0 1 , 6 0 3 - 6 1 1

ra t io t o t a l t o first o r d e r , 3 7 0 , 4 5 4 - 4 5 5 s imi lar i ty , 3 6 9 - 3 7 1 , 4 8 2 ^ 8 6 o f s n o w , 7 0 4

S p h e r i c a l g e o m e t r y , 6 0 2 , 6 9 2 a p p r o x i m a t i o n , 6 6 7

S q u a r e m e a n r o o t p a t h l e n g t h , 5 9 4 - 5 9 5 S t a n d a r d h a z e , s i ze d i s t r i b u t i o n , 6 6 5 - 6 6 6 ,

6 6 9 S t a n d a r d p r o b l e m , 3 8 - 3 9

w i t h p o l a r i z a t i o n , 3 9 S t e l l a r a t m o s p h e r e , 6 9 2

e m i s s i o n , 6 4 3 S t o k e s p a r a m e t e r , 2 0 - 2 2 , 3 1 6 , 3 2 4 , 4 9 3 - 4 9 8 ,

5 0 0 , 5 0 3 , 5 0 5 , 5 3 3 - 5 3 5 S t r a t o s c o p e , 6 2 2 S t r a t o s p h e r e , h e a t i n g , 6 8 2 S t r u v e f u n c t i o n , 6 7 2 S u b c r i t i c a l d o m a i n , 198 , 2 3 4 S u c c e s s i v e o r d e r

a z i m u t h - d e p e n d e n t t e r m s , 5 1 4 - 5 1 8 , 5 2 2 - 5 3 0 F o u r i e r c o m p o n e n t s , 4 9 f r o m i n v a r i a n c e , 5 0 m e t h o d , 4 2 , 4 5 - 4 6 , 6 7

c o n v e r g e n c e , 1 4 1 , 3 9 3 , 5 1 4 - 5 1 5 H e n y e y - G r e e n s t e i n p h a s e f u n c t i o n ,

4 0 6 - 4 0 8

i s o t r o p i c s ca t t er ing , 1 3 5 - 1 3 7 , 1 4 1 - 1 4 2 s e m i - i n f i n i t e a t m o s p h e r e , 3 8 8 - 3 9 3

h a l f - s t e p , 4 7 - 4 9 , 4 9 4 s t a r t i n g f u n c t i o n , 4 8

S u l f u r i c a c i d , 6 1 2 S u n

c o r o n a , 6 5 8 - 6 5 9 d i sk , 129 , 6 5 8 v i s ib i l i ty t h r o u g h c l o u d , 6 5 4 - 6 5 5 , 6 6 0 - 6 6 1

XVI Index

S u n (cont.) h a z y , 6 6 0 - 6 6 1 l i m b , 6 5 8

S u p e r i o r c o n j u n c t i o n , 6 0 8 S u r f a c e

b r i g h t n e s s , 6 1 5

c o r r e c t i o n f u n c t i o n , 1 8 3 - 1 8 7 deta i l , v i s ib i l i ty , 6 2 5 , 6 2 9 - 6 3 2 re f l ec t ion , 6 , 4 9 1 , 6 0 6

S y m b o l i c e q u a t i o n , 5 0 6 S y m m e t r y , see also R e c i p r o c i t y

a s s u m p t i o n , 4 9 8 , 501 in a z i m u t h , 4 9 9 c h e c k , 17 m e r i d a n , 6 1 5 r e l a t i o n , 17 r o t a t i o n a l , 4 9 3

S y n t h e t i c s p e c t r u m , 6 3 4

Τ

r v e c t o r , 128 , 130 T a y l o r e x p a n s i o n , 5 7 7 T e r m i n a t o r , 6 0 2 , 6 1 5 T h e r m o d y n a m i c s

e q u i l i b r i u m , 17

n o n e q u i l i b r i u m , 17 T h i c k n e s s , r e d u c e d , 4 0 5 T h o m s o n sca t t er ing , 531 T h o u g h t e x p e r i m e n t , 17 , 9 9 , 6 9 3 T h r e e b y three m a t r i x , 5 0 3 T i m e d e l a y

a l o n g p a t h , 5 7 3 , 5 7 4 u p o n sca t ter ing , 5 7 4

T i m e d e p e n d e n t p r o b l e m , 153 T i m e - r e v e r s a l , i n v a r i a n c e , 17 T i t a n , g e o m e t r i c a l b e d o , 6 2 0 - 6 2 1 T r a n s f e r e q u a t i o n , 4 1 , 4 7 - 4 8 , 7 5 , 9 2 - 9 3 ,

1 2 7 - 1 2 8 , 7 1 2

a n i s o t r o p i c , 3 7 6 - 3 8 8 b o u n d a r y c o n d i t i o n , 111 f o r m a l s o l u t i o n , 4 8 h o m o g e n e o u s , 5 0 8

T r a n s f e r f u n c t i o n , 2 - p o i n t , 2 7 - 2 8 , 3 3 T r a n s i t i o n r e g i o n , 1 7 8 - 1 7 9 , 2 0 9 T r a n s l a t i o n

o f K u b e l k a - M u n k f o r m u l a s , 7 0 0 o f n o r m a l i z a t i o n , 7 8 o f s y m b o l i c f o r m u l a s , 5 0 6 , 5 1 2 o f t e r m i n o l o g y , 6

n u c l e a r sca t ter ing , 6 9 3 T r a n s l u c e n t a t m o s p h e r e , 6 2 5 - 6 2 6 , 6 2 8 - 6 2 9

o v e r i d e n t i c a l a t m o s p h e r e , 6 2 9 o v e r L a m b e r t sur face , 6 2 8 - 6 2 9

T r a n s m i s s i o n , 3 6 - 3 8 , see also spec i f i c p h a s e f u n c t i o n

by a d d i n g m e t h o d , 61 d i f fuse , 6 1 , 5 0 3

first o r d e r , 3 9 7 , 4 8 7 , 5 1 8 - 5 1 9 m o m e n t , 5 1 9

T r a n s m i s s i o n f u n c t i o n , 6 , 3 7 , 1 0 2 - 1 0 3

F o u r i e r e x p a n s i o n , 5 0 4 , 5 2 1 - 5 2 2 i n h o m o g e n e o u s s l a b , 3 1 , 5 8 - 5 9 , 61 a s m a t r i x , 6 9 th ick layer , 8 2 , 6 5 4 z e r o o r d e r , 3 7 , 5 2 , 5 5 , 61

T r a n s m i s s i o n m a t r i x , 5 1 3 z e r o order , 5 0 4

T r a n s p a r e n t a t m o s p h e r e , 6 2 5 - 6 2 6 Tr ia l f u n c t i o n , 4 1 0 T r u n c a t i o n , 3 6 6 - 3 6 9 , 4 8 1 ^ 8 7 T u r b i d i t y , 3 1 5 , 6 6 2

a f f e c t i n g s k y b r i g h t n e s s , 6 5 5 - 6 5 8 s p e c t r u m , 3 1 6

T u r b u l e n t s p e c t r u m , 7 0 7 T w i l i g h t , 6 9 2

T w o b y t w o m a t r i x , 5 0 8 - 5 0 9 , 5 1 3 , 5 3 3 - 5 3 6 T w o - s t r e a m a p p r o x i m a t i o n , 4 1 , 3 0 9 , 4 9 1 , 6 9 9 ,

7 0 6

T w o - v e c t o r , 5 0 8 - 5 1 3 , 5 4 0

U

U m a t r i x , 4 9 7 U n d e r w a t e r a t m o s p h e r e , 6 2 7 U n i f o r m i n c i d e n c e , 3 9 9 , 4 0 1 , 4 6 8 , 4 7 2 , 4 7 6 U n i q u e n e s s , 7 4 , 1 0 3 , 105 , 162 , 2 2 4 , 6 1 0 U n p o l a r i z e d l ight , see N a t u r a l l ight U o p e r a t o r , 4 9 7 U r a n u s

g e o m e t r i c a l b e d o , 6 2 0 - 6 2 1 l i m b d a r k e n i n g , 6 2 2 m o d e l a t m o s p h e r e , 6 2 2 p o l a r i z a t i o n , 5 5 9

U v e c t o r , 6 9

V

V a r i a t i o n pr inc ip l e , 132

V e c t o r , see μ v e c t o r ; τ v e c t o r ; S h o r t h a n d n o t a t i o n

V e c t o r p r o d u c t , 6 8 , 128 V e n e r a e i g h t , 6 4 5 - 6 4 6 V e n u s

a b s o r p t i o n l ine , 5 9 8 , 6 3 4 - 6 3 6 p h a s e ef fect , 6 3 5 - 6 3 6

a t m o s p h e r e e n e r g y b a l a n c e , 6 4 4 - 6 4 6 c a r b o n d i o x i d e b a n d s , 6 3 5

Index

c l o u d

m u l t i p l e layer , 6 4 5 - 6 4 6 part i c l e , 6 1 1 - 6 1 3

re fract ive i n d e x , 6 1 2 - 6 1 3 s i ze d i s t r i b u t i o n , 6 1 1 - 6 1 3

p r e s s u r e a t t o p , 6 1 3 g r o u n d a l b e d o , 6 4 5 in frared l i m b d a r k e n i n g , 6 4 2 - 6 4 3 p h a s e f u n c t i o n , 6 1 0 - 6 1 2 p o l a r i z a t i o n , 196 , 3 1 5 , 5 0 7 , 6 1 0 - 6 1 4

m e a s u r e m e n t , 6 1 1 R a y l e i g h s c a t t e r i n g c o n t r i b u t i o n , 6 1 3 in s e c o n d o r d e r , 6 1 4 t h e o r y , 6 1 1 - 6 1 4

s p h e r i c a l a l b e d o , 6 1 0 - 6 1 1 t e m p e r a t u r e , 6 4 2 - 6 4 6 t h e r m a l e m i s s i o n , 6 4 2 - 6 4 6 u l t r a v i o l e t m a r k i n g , 6 1 3

V i r t u a l a n g l e , 1 6 6 , 1 8 1 , 2 3 0 - 2 3 1 V i s i b i l i t y o f o b j e c t i n w a t e r , 7 0 7 V o l c a n i c e r u p t i o n , 6 6 2 , 6 6 6 , 6 8 2

W

W a l l i s f o r m u l a , 3 9 0 , 5 2 8 W a t e r

c l o u d , 4 8 7 ^ 8 8 , 5 0 0 d r o p ,

a s y m m e t r y f a c t o r , 3 1 3 p h a s e f u n c t i o n , 3 1 5 - 3 2 1 , 4 9 6

W a v e p r o p a g a t i o n i n t u r b u l e n t m e d i u m , 7 0 7

W v e c t o r , 6 9

XVII

X

X f u n c t i o n , 4 3 - 4 4 , 91 i s o t r o p i c s ca t t er ing , 136 , 139 , 1 9 3 , 2 0 8 - 2 1 0 ,

2 2 4 - 2 2 7 m o m e n t , 2 1 9 , 2 2 5 - 2 3 0

e x p a n s i o n , 2 2 8 n o n l i n e a r r e l a t i o n , 2 3 2

n u m e r i c a l v a l u e , 2 2 5 - 2 2 7 , 2 9 2 - 2 9 9 R a y l e i g h p h a s e f u n c t i o n , 5 3 8 R a y l e i g h sca t ter ing , 5 5 5 , 5 6 6

X ray

e x t i n c t i o n , 7 1 8 h a l o , 7 1 8 - 7 1 9 pu l sar , 7 1 9 s c a t t e r i n g , 7 1 7

Y

y f u n c t i o n , 4 3 - 4 4 , 91 i s o t r o p i c s c a t t e r i n g , 136 , 139 , 1 9 3 , 2 0 8 - 2 1 0 ,

2 2 4 - 2 2 7 c o n t o u r d i a g r a m , 2 2 5 - 2 2 6 m o m e n t , 2 1 9 , 2 2 5 - 2 3 0

e x p a n s i o n , 2 2 8 n o n l i n e a r r e l a t i o n , 2 3 2

n u m e r i c a l v a l u e , 2 2 5 - 2 2 7 , 2 9 2 - 2 9 9 v i r t u a l a n g l e s , 2 2 5 - 2 2 6

R a y l e i g h p h a s e f u n c t i o n , 5 3 8 R a y l e i g h sca t t er ing , 5 5 5 , 5 6 6

Ζ

Z e n i t h , see S k y Z o d i a c a l l ight , 6 6 8 , 7 1 3

multiple light scattering. tables, formulas, and applications. volume 1

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