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3/88

LA-3064

UC-34 , PHYSICS

TID-4500

(30th

Ed.)

LOS

ALAMOS SCIENTIFIC

LABORATORY

OF THE U N I V E R S I T Y OF C L I F O R N I O S L M O S

NEW

M E X I C O

REPORTWRITTEN

Februa ry

1964

REPORT DISTRIBUTED:

June

17, 1964

THEORY OF

THE FIREBALL

bY

Hans

A.

Bethe

This report exp ress es he opinions of the author or

authors and does not necessa rily reflect the opinions

or views of the

Los

Alamos ScientificLaboratory.

Contract

W-7405-ENG.

36 with he

U.

S. Atomic

Energy

Commission

1


4/88


5/88

ABSTRACT

The successive stages of the f i re ba l l due t o a nuclear explosion in

ai r a re defined (Sec. 2). Tnis paper i s ch ie fly concerned

with

Stage C,

from

tne,

minimum

in

t h e

ap-parent

f i r e b a l l

temperature t o t h e point where

th e f i r eb a ll becomes transpa rent .

In

the

f i r s t

part of this s tage

( C I ),

the shock

(which

previously was opaque) becomes tra nspa rent due t o

decreasing pressure.

The

ra dia ti on comes from a region

in

which the

temperature distribution i s given es se nt ia ll y by the Taylor solution; the

radiating layer

i s

given by the condition that the

m e a n

free path i s

about

1/50

of theradius (Sec.

3 ) .

The r ad ia ti ng t q e r a t u r e during t h i s

-1/4

stage ncreases about asp , here p is thepressure.

To supply the energy for the rad iation, a coo ling wave proceeds from

j

the outsid e into the hot interio r (Sec. 5 ) . When this wave reaches the

isothermalsphere, he enperature is close t o its second maxmum There-

af ter , th e character

of

the so lu tio n changes; it

i s

naw dominatedby

the

cooling wave (Stage

C

11). m e temperature would decrease s1mi.y (a s

P I6) i f the problem were one-dimensional, but in f a c t it i s probably

nearly constant f o r the hree-dimensional case (Sec.

6 )

The radiating

surface

shrinks

slowly. The

cooling

wave eat s nto he soth erm al sphere

until t h i s

i s

completely used

up.

The i nne r pas t of the sothermalsphere,

3


6/88

i.e., th e p a r t which has not yet been reached by the coo ling wave, con-

ti nu es to expand adiaba tically; it therefore cools veryslowly and

remains opaque

.

Af ter th e ent ir e isothermal sphere i s used up, th e f ir e b al l becomes

transparent and the ra di ati on drops rapidly. The ball

w i l l

therefore be

l e f t

a t

a

rather high empe rature (Sec.

7 ) ,

about 5000 .

The cooling wave reache s the isotherma l sphere at a definit e pres-

sure p, M 5(p1/po) 1/3 bars, where p i s the ambient and po the sea level

1

density. The radiati ng tempe rature at this time

i s

about

10,

OOOo

.

The

sl ig h t dependence of physic al prope rties on yield i s exhibited i n approx-

m t e formulae..

4


7/88

I am

greatly indebted t o Harold Brode of the Rand Corporation for

giving me a patient introduct ion to t h e work on fi r e b a l l dynamics since

World

War

11, fo r sending me t he pertinent Rand publications on the sub-

ject, and

f o r

doing a spe ci al numerical computation f o r

me.

To Burton

Freeman of General Atomic I

am

gratefuloreveralnterest ingiscus-

sions of f i r e b a l l developmentand opacities

of

a i r . I want t o thank

Herman Hoerlin

fo r

much information on the observational material, and

for select in g for me pert inent

Los

A l m s reports. I

am

?specially

gr at ef ul t o Albert Petschek f o r c r i t i c a l reading of th e manuscript

and

for several helpful additions.

My

thanks

are

also due t o

many

other

members

of

Groups

J-10

and

T-12

a t

Los Alamos,

t o J. C, S t u a r t and K.

D.

w a t t a t General

Atomic,

Bennett Kivel

and

J. C.

Keck a t

Avco-Everett,

and t o F. R. Gilmore a t R a n d f o r important in fo ma t on.

.

5


8/88


9/88

1. INTRODUCTION

2. PHASES

IN

DETELOPMENT

3. RADIATIONFRCM BEHIND

THE

SHOCK (Stage

C

I)

a. Role of

H02

b. Temperature Di st rib ut ion behind th e Shock

C. Nean Free Path and Radiating Temperature

d.

Energy

Supply

4.

AESORPTION COiEFFICIENTS

a. Infrared

b.

Visible

c. Ultraviolet

5

.

THE COOLING

WAVE

a.Theory

of

Zel'dovich e t

al.

b. nsideStructure

of

Fireball ,

Blocking

Layer

C. Velocity of

-Coolin@;

Wave

d. Adiabatic Expansion after

Cooling.

Radiating

Temperature

e Beginning of Strong Cooling Wave

f .

Maximum

Ehission

g o Weak Cooling Wave

6.

EFFECT OF

THREE

D ~ E N S I O N S

a. I n i t i a l Conditions and Assumptions

b. Shrinkage

of

Isothemal.

Sphere

c. TheWarm Layer

7.

TRANSPARENT FIREl3ALL (

Stage D)

REFERENCES

7

3

5

9

i2

17

19

24

27

32

32

34

40

44

44

48

51

55

60

64

68

70

72

77

80

84


10/88


11/88

1. INTRODUCTION

Tile radiation- rom t h e

f . i r ~ b ~ . - h a s ~ b ~ ~ s ~ ~ ~ ~ a -

by._ x , .,,

many authors.Already i n t h e Summer Study a t Berkeley

i n 1942,

Bethe

and

Telle r found th a t th e energy transmitt,ed by a nuclear explosion

i n t o a i r

i s

m e d i a t e l y converted into x-rays, and studied the qualita-

1

t ive fea tures of the transmission of the se x-rays.

At

Los Alamos,

Marsnakand others showed that thi s ra di ati on propagates

as

a wave,

175th a sharp front.Hirschfelder

and

Magee gave

the f i r s t

comprehensive

treatment of th is ear ly phase of the fireball development,

and also

studied someof the l a te r phases, esp ec ial ly th e ro le of NO2.

2

3

Ma y optic al observa tio ns have been made

in

the numerous tests of

atomic weapons. Some of tine re su l ts a re contained in Effects of Nuclear

Weapons, ' pp. 70-84 (see a l so pp. 316-368)

A

summary of the spectro -

scopic observations up t o 1956 was compiled by D e W i t t .5 Careful scrutiny

of th e extens ive observ ationa l mater ial would undoubtedly give a wealth

of fu rt he r information.

On

the theore tical side, there has been some analytical

a d

good

deal of numerical work. Ana lyt ical work has concentrated on th e ea rl y

phases. One of the most rec en t analyt ica l papers on the ear ly flow of

9


12/88

6

radiat ion . (Stage

A

of Sec. 2) i s byFreeman.BrodeandGilmore 7 t r e a t

also Stage B, the rad$ation from th e shock fro nt , wi th pa rti cu lar emphasis

on the dependence

on

a l t i tude .

The most complete numerical ca lc ul at io n has been done by Brode 8 on

asea l e v e l megaton explosion.

We

sh al l use h is results extensively,

but f o r convenience we shall tr an sl at e them t o a yield of lmegaton.

Wherever the phenomena are

purely

hydrodynamic,

w e

may simply sc al e th e

linear dimensions and the tim e by the cube root of th e yie ld , and t h i s

is

the

principal.

use

w e

s h a l l make

of

Brode

s

results , e

.g.

Fn

Sec.

3b.

Where radiation i s important, th is sca lin g

w i l l

give only a rough guide.

Brode calculatespressures, emperatures,densities,etc.,asfunctions

of time and radius , f o r scaled (1-megaton) times of about t o 10

seconds. The ca lcula tio ns show clearly he s tages in f i r e b a l l and hock

development, as def ined in Sec. 2, at le as t Stages A

t o

C.

Brode and Meyerott9 have con sidered the physical phenomena involved

in the optical.

"opening"

of the

shock

a f te r t he minimum of radiation,

Stage

C

I

in th e nomenclature of Sec. 2, espe cial lyth e decrease

i n

opacity due t o decreased density and

t o

the dissociat ion of

NO2.

Zel'dovich, KompaneetsandRaizer''have investigated haw the energy

for the radia t ion i s supplied a ft er th e ra d ia ti on minimumand have in tro-

duced the concept of a cooling wave moving in to th e hot f irebdll. The

present report

is

la rg el y concerned with an extension of the ideas of

10


13/88

Much ef fo r t has been devoted t o

t h e

ca lcula tion of f i r eb a l l s a t

various altitudes. Brode'lmade such ca lcu lati on s

in

1958, in connection

with

th e te st se ri e s of t h a t year. Gilmore12made a prediction

of

the

Bluegill explosion i n

1962.

Since then, manymore ref bed cal cul atio ns

of Bluegi l l have been made.

For ny understanding of f i r e b a l l phenomena

it

i s ess ent ia l to have

a

good

equation of s t a te f o r ai r

and

good ta bl es of absorption coeffi-

cients. For the equation of st at e, we have used Gilmore's t ab le s,

although Hilsenrath'sgive more d e ta il

i n

some respects.

The

two cal-

13

14

culations agree.Gilmore's equation was approximated an al yt ic al ly by

Brode

8

For the absor ption coeff ici ent we have

used

the tables by Meyerott

e t al.,

15'16

which extend t o

12,000.

These were supplemented by the

18

workof Kivel and Bai1eyl7

and

more recent work by Taylor

and

Kivel on

the

free-free electron t ransi t ions in the f i e ld of neutral atomsand

molecules. A t higher emperatures, here are calculations byGilmore

and Latter,I9 Karzas

and

LatterY2*and curves by Gilmoregl which are

brought to dateperiodically. The most recentcdcu lation on th e

absorption o f a i r a t about 18,000' and above have been done by Stuart

and Pyatt.*2 This temperaturerange i s not of gre at importance f or th e

problem

of

th i s paper, but is important fo r th e expansion of th e is o-

thermal sphere inside

t h e

shock wave before

it

is

reached by the cooling

wave.

8

Brode has used the average of the absorption coefficient over


14/88

frequency, the opaci ty, which

i s

suff ic ient for t rea t ing t h e in ternal

flaw

of radiation. A re a l i s t i c treatment of the flow t o t h e outside

requires the absorption coefficient as a w c t i o n

of

frequency; Brode

merely wanted t o obtai n reaso nable overa l l resul ts for this flow. He

approximated the opac ity by an analytic expression. Also in most

of

t he

ot'ner work cited above an average opacity has been used. An exception

i s

some of the recent workon

the

radiat ion

flow

i n high al ti t ud e explo-

sio ns (B lueg ill ), where th e frequency dependence

mus

be, and

has

been,

ta ke n nt o account.Gilmore21has ca lculated and made av ai la bl e curves

of

effective opacity,

in

which the radia tion mean fr ee pa th was averaged

(using a Rosseland weighting factor) only over those frequencies f o r

which it

i s

l e s s than 1kilometer.

This

l i s t of references on work on the fireball dynamics

and

opac-

i t y

i s

f a r

from complete.

The energy from

a

nuclear explosion

i s

transmitted through the outer

p a rt s of th e weapon, including i t s case, either by radiation (x-rays) or

by shock o r both. Whichever t h e mode of transmission inside the weapon,

once the energy gets in to th e surrounding

air ,

the energy w i l l be trans-

ported by x-rays. This

i s

because

the

a i r

w i l l

be heated t o s u c h a

high

temperature

(a

millio n degre es or more) th at transp ort by x-rays

i s

much

f a s t e r

than

by hydrodynamics.

This

stage

of

energy transp ort (Stage

A )

has

been extensive ly studi ed by many authors (e.g., Hirschfelder and

12


15/88

Magee

in

Report LA-2000) and w i l l therefore not be further consid,ered . .

here

.

During Stage A, temperatures are very high . The Pianckspectrum

of

t t h e a i r

is

in thex-rayor f a r ult ra v io le t region, and hence is irmnedf-

at el y absorbed by the surrounding a i r . The very hot a i r i s therefore

surrounded by a cooler envelope, and

only t h i s

envelope

i s

v i s i b l e t o

observersat a distance. The obse rvable emperature herefo reh a s l i t t l e

physicalsignificance. It

is

observed that the size

of

the ltrminous

sphere lncreases rapidly, and th e

t o t a l

emission also .Increases,

up

t o

a f i r s t maxmum

When the temperature of the cen tral sphere of a i r has fallen, by

successive emission and re-absorption of x-rays, t o about j0O,00O0,

a

-hydrodynamic shock

forms,

The shock

now moves

f a s t e r than the tempera-

tu re couldpropagate by ra dia tio n tra ns po rt, The shock ther efo re sepa-

rates fromth e very hot, nearly isothenml sphere a t

the

center. This

i s

Stage 13 in th e development. The shock mves by simple

Its front obeys

the

Hugoniot relations, t he density being

Behind the fro nt , the air expands adi aba tica lly, and a t a

of

the

shock

radius, . the density is apt

t o

have fallen.by

o r m r e compared

t o

the shock dens ity, while the temp erature has risen

by a comparable factor, (3.16) Thus th e int er ior

is

a t very

low

density,

and

hence the pressu re m u s t be neasly constant (otherwise there would

be

very large accele rations which soon would equalize the pressure) .

This

greatly simplifies the structure of the shock, and leads t o such simple

. .

13


16/88

re la t ions as

(3.2)

between shock radius and pressure.

Well inside the shock, t h e isothermalspnere pursues i t s separate

his tory .

It

continues t o engulf more material because radiation flow

continues, though

a t

a educed ra te . H. Brode has kindlycalculated

f o r

me th e temperature h is to ri es of se veral ma teri al poi nts, based on hi s

paper RM-2248. These hi sto rie s cle ar ly ex hib it the expansion

of

the i s o -

thermalsphere

in

materialcoordina tes. Tne expansion can

d s o

be treated

by a semi-analytic method which

I

hope t o discuss in a subsequent paper.

The isothermal sphere remains is ol ate d from the out sid e world un t i l

it

i s reached by the cool ing wave, Sec.

5 .

The rad ia ti on to the ou ts id e now comes from th e shock. E a r l y in

Stage B, the shock has

a

precursor of lower temperature, caused by u l t r a -

vi ol et ra di at io n from th e shock, and the observable emperature

i s s t i l l

below t he shock tempera ture (Stage B I ) However, the observableradius

i s verynear he shock radius.Later,

as

the shock frontcools d m ,

t h e shock rad iate s dir ectl y and

i t s

temperature becomes di r ec t ly observ-

able.

(Stage B

11)

.

The f i r s t

m a x i m u m

in vis ible rad iati on probably

OCCUTS

between Stages

B

I a d

B

11. As the shock cools d m , the radi-

at io n from th e shock front decrea ses, and th e observable temperature

decreases t o a m i n i m u m (Ref. 4, par.

2.lI.3,

p. 75) of about 2000.

When the shock i s suff icie ntly cool,

i t s

front becomes transparent,

and onecan look into it to higher emperatures (Stage

C )

.

The central

isothermal sphere, nmev er, remains opaqueand, f o r some time, invisible.

Because higher temperatures a r e now revealed, the to ta l radia t ion increases


17/88

toward a .second maxmum This stage has been very l i t t l e considered

theoretically, except i n numerical calcula tions, and

forms

the subject

of this report.

We shall

show

tha t f o r

some

time In Stage Cy th e ra di at io n comes

fm h e a i r between isothermal sphere and shock fron t (S tag e

C

I ).

During this

time, the rad iati on can be calcula ted essentia lly from the

temperature di str ib ut io n which

i s

se t up by the adi aba tic expansion of

the ma ter ia l behind th e shock (Secs 3 and 5f ,g) The temperature and

t o t a l

intensity

o f

the radiation increase with time toward

the

larger,

second

maxmum

The

energy fort he ra di at i on

i s

largely sqplied

by

a cooling wave

(Sec. 5 ) which gradually ea ts int o th e hot int er ior .

When

this coollng

wave reaches the isothermal sphere, the radiati on temp eratur e reache s

i t s

maximum

(Sec.

5e); it

then declin es again as the coolin g wave e at s

more deeply in to the sothermalsphere(Stage

C

11). This

process ends

by t h e isothermal sphere being comgletely eaten away (Sec. 6 ).

m e r hi s has happened, th e en tir e fi reb all , isothermal sphere and

cooler envelope, i s t ransparent t o

i t s

own thermal. radiati on (St age

D ) .

The

molecular

bands, which previously appeared i n absorption, now appear

i n emission (Stage D I ). Emission

w i l l

lead to further coollng of the

f i r e b a l l ,

though

more

slowly

than before.

Soon,

when th e temperature

f a l l s below about

6000, the

emission becomes very weak,

and

stibsequent

cooling i s almost en tir el y ad ia ba tic (Stage D 11). A t sea level , the

pressure

may

go down t o

1

a r before

the

temperature fa l l s belaw

6000~;


18/88

in th i s case there i s no Stage D 11.. A t higher elevation, there usually

is.

The

f i r e ba l l w i l l

then

remai n

hot,

a t

about

6000~

o r a somewhat

lower temperature due to ad ia ba ti c

expansion

i n

Stage

D

I1

. The

only

process which can now le ad to fu rt h er cooling

i s the rise

of the f i r e -

ball, which leads t o

further

adiabatic

expansion and,

more

important, t o

turbulent

m i x i n g

at the surfa ce with the ambient a i r (Stage

E).

The

t ime required fo r th i s i s ty p ic al ly 10 seconds o r more, being determined

by buoyancy

A t very high alti tud e,

the

shock wave never plays an important

part

but radiation transport continues until the temperature g ets too

low

for

effective emission.

In

other words, Stage A continues t o t h e end.' O f

course, a shock does

form,

but it i s , so to speak, an afterthought, and

it

plays l i t t l e par t in the dis t r ibut ion of energy, A t medium altitude,

l e t

us say,

10

t o

30

kilometers, the st ag es ar e much the same as at se a

le ve l bu t th e shock wavebecomes tra ns pa re nt ea rli er , i.e., a t a higher

temperature,because

t h e

densi ty

i s

l a t e r ;

t h i s means

t ha t

t h e

minimum

emission comes earlier.

Stage C

proceeds si mi la rly to sea level,but

a t th e second maximum of radiation the pressure

i s

s t i l l much above

ambient, there fore Stage C I1 involves a gr ea te r ra di al expansion of the

isothermal sphere than a t

sea

l e v e l which proceeds simultaneously

with

th e inward motion of he coo ling wave.Moreover, there

i s

much adia-

16


19/88

W believ e that the theory developed i n t h i s paper w i l l be useful

jn

studying the dependence of phenomenaon al ti tu d e (ambient pre ssu re),

but we have not yet explo ited it

f o r

t h i s purpose.

3. RADIATION FRaM BEHDTD THE

SHOCK

(Stage C I)

a., Role of NO2

The diatomic species in equilib rium air, both neutra ls and ions,

show very l i t t l e absorption

i n

th e vi si bl e at temperatures

Up

t o about

16

Iioooo~. This i s shown clearly

in

theab le s by Meyerott e t We

define the vis ib le fo r th e purposes of t h i s paper, ar bi tr ar il y

and

incorrect ly, as the frequency range

hv

= 1/2 o 2-3/4 ev

i

= 4050 t o 22,300 an

Then,even a t a density as high as lopo p, = density of a i r a t HIT

= 1.29

x

w/cm ), the mean freepath i s never less han 100 meters

at k0OO0, 1000 meters a t

3000,

and s t i l l longer at luwer temperature.

3

Tnese values refer

to

hv

= 2-5/8

ev;

f o r

lawer frequencies, the mean

f r e e

path

is

even longer.

I n

-the sea -level shock wave from a megaton explosion, t h e temper-

a t w e range.

from 3000

t o

4000'

occupies

a

distance of about 10

meters

(

see Sec. 3b).

Thus

this region i s definit ely ranspa rent, even

at

the


20/88

highestpossiblede nsi ty of about l o p For explosions at nigher a l t i -

tude, t h i s conclusion i s even more true.

0.

Tne ta bl es by Meyerott e t

al..

o not include absorption by tria tom ic

(and more complicated)molecules, O f these, NO2

i s

known t o have strong

absorption bands i n th e vi si bl e. Af te r th is paper was completed, I

received new ca lcu lati ons by Gilmoreg3 which include t he ef fe c t of U02.

The effect

i s

very s t r ik ing as

i s

shown

by

Table

I,

which g ive s the

abso rption coef ficient for the typic al frequency hv

=

2-1/8 ev, and

f o r

several

temperatures and dens ities (the abso rption

i s

strong from

PIP.

T

Without

With

Table I. Absorption Coefficients a t hv

=

2-l/8 ev

with and without

NO2

about 1-314 o 2-3/4 ev).

Note: For

each value, th e power of 10 i s indicated by a strperscript.

10 10

10 10 1

Because NO

i s

triatomic, i t s absorption depends strongly on density.

2

As

the shocked gas expands, the NO2 dissociates and the gas becomes trans-


21/88

I

b.

Temperature Distribution behind the Shock

We wish to ca lcula te the temperature distribution behind the shock.

We can do

t h i s

because th e ma te ria l which has gone through

the

shock

expands

very

nearly adiabatically, as

long

as it i s not engulfed by the

internal ,

hot

isothemal sphere. We

are

interested in the pe riod when

the shock temperature goes from about

4000

t o a

few hundred

degrees, i.e.,

un t i l th e strongcooling wave (Sec. 5 ) s tar t s . Fora 1-megaton explosion,

t h i s corresponds roughly t o

t

= 0.05 t o

0.25

second.

A t

a given

time,

the pressure

i s

near ly constant over most of

the

volume ins ide the shock, except f o r t h e Me d i a t e neighborhood of the

shock;

the

shock pressure

i s

roughly twice

th is

constant, inside pressure.

C o n r p a r i n g

two material elements in the inside regio n,

we

may calculate

their relative temperatures i f we

knm.

t.heir . t q e r a t u r e s when t he

she-ck

.

I1

traversed them,andassume ad iaba tic. expansion from there

on.

A

material element which

i s

i n i t i a l l y a t po i n t

r

w i l l

be

shocked

when the shock radius i s r. The shock pressure a t t h i s t ime

is*

Ps( )

= 20(

7Av -

lw 3

where

Y =

yi el d i n megatons

r

=

radius

in

kilometers

P pressure

pE

energy perunit volume

- 1 z - z 3 . 3 )

~

?Notations

f o r

themodp amic quantities

similar

t o those ofGilmore

(Report IN-1543)

.,

19


22/88

and the average of y i s taken over the volume insi de th e shock wave.

Tne

bas is

of

(3.2) i s th a t th e to ta l energy in the shockedvolume, Y,

i s the volume times th e average energy pe r unit volume, t h e la tt e r i s

t h e average pressu re divid ed by y r -

1,

and the average pressure i s

cl os e t o one-half of the shock pressure.

Tne d en si ty at th e shock i s

.. -

\

wnere the sub scri pt

s refers to

shock conditions. Now

an

examination of

Gilmore's tables13 shows t h a t 7 does h o t change very much

along

an adi-

abat.

As

an example, 'we l i s t i n Table I1 certain quanti t ies referring

t o some adiabats which w i l l be pa rti cu lar ly important fo r our theory.

Tnese ar e the adi aba ts for which t h e temperature T i s between

4000

and

12,000 at a density of O . l p o . In th e second l i n e we

list

the temper-

ature Ts fo r th e same entropy

S

at a density ps =

lopo.

This i s close

enough to the shock density ( 3 . 4 ) so w e may consider Ts as th e temper-

ature of the

same

material when the shock wave went through

it.

(Adia-

\

batic expansion,

i.e.,

no radiatio n ransp ort, i s assumed.)The third

line gives y r

- 1

o r the bresent conditions,

p

= 0 . 1 ~ ~nd T, the

fourth l ine

i s the same quantity for the shock conditions. It

i s

evident

t'nat

7

-

1

i s

nearly constant for T

=

4000

and

6000,

not

so

constant

f o r 8000 and

12,000.

On th e average y '

-

1

0.18.

The last

two

l i n e s

i n Table

I1

give the number

of

par t i c le s

(

atoms, ions, electrons) per

o r ig ina l a i r molecule under present If and shock conditions.

20


23/88

Table

11.

Adiabats

( 'Present density, 0 .1-p shock density,

l opo )

0;

T 4,000 6,000 8, ooo 12,000

T

9,000 10,500

18,000

28,000

y - 1

0

213

0.194

0.144

0 1-53

S

Y; - 1 0.208

0

190 0.190 0.20

z 1-13 1.27 -

1.68 2.06

=S

1-03

.

1.6 2.06

3

Assming

an

adiabat

of

constant

7

=

y ,

the den sity of

a

mass

element i s

P =

PS ( k ) 1 / Y

3.5)

Now

ps i s

a constant, and a t

any

given time,

p i s

t he s&e

for a l l

mass

elements except those very close

t o

the

shock,

hence

4 7

P - Ps

w means proportional- t o )

If

we

now

introduce the abbreviation

3

m = r

(3 .7)

which i s proportional to the mass ins ide the

mass

element considered,

and i f we use

(3.2),

we

f i n d

t h a t a t

any

given time

21


24/88

The radius R

i s

given

R3

=

I -

P

J ;+ =

m

- 1 const.

7

3 . 9 )

We sha l l set t h e constant equal t o zero whichamounts

to

the ( incorrect)

assmption that (

3.8)

holds d a m t o m = 0. Actually, the isot'nermal

sphere gives the constant a fi ni te , po si ti ve value.

To fin d the tem peratu re dist rib uti on, we note t h a t the enthalpy

H

7 E

Y 1 - l P

We are

us i ng

th e en thalpy rather than the intern al energy because the

in te ri or of th e shock

i s

a t constant pressure, not constant density.

The hermodynamic eqmti on fo r H is

T d S = d H - ~ d p

At givenpressure,

i.e.,

given ime, 3.8) t o ( 3 . 1 0 ) give

22


25/88

Now the approximate relation between internal energy and temperature i s

given by Gilmore, Fig. 5 ,

viz.,*

4

where T i s the temperature in u ni ts of 10 degrees.

using p =

from

3 . 3 ) and setting y =

1.18,

which i s a reasonable

*

average (see Table 11)w e may

rewrite

(3.13) :

where p i s the pressure in

bars.

Since

H

i s proportional

t o E,

3 . 1 2 )

and (3.14) give

%e thermodynamic relation

@) =

T

g) P

T

V

leads t o a r ela tion between y -

1

and the exponents in the re la t ion

E = &

T

x y

Since we have chosen y = 1.18

and

y = 1.5 th i s re la t ion g ives x = 0.09.

This

i s

i n

sufficient agreement with x

=

0.1 as used in (3.13)

.


26/88

T - R

-10

(3 .16)

8

The numerical calculations

o f

Brode a re i n good agreement

w i t h

t h i s a t

the relevant times,

from

about 0.05 t o 0.5 second.

It

w i l l be noted that (3.16) was obtained without integration of

the hydrodynamic equations; it

follows

simply from the equati on of st at e.

The weakest assumptions are 1) he relation between E and

T, (3.13), and

( 2 ) the neglec t of th e constant

i n ( 3 .9 ) . B u t in

any case, T

w i l l

be a

very hign power of

R.

C. Mean FreePath and Radiating Temperature

The emission of radiation from

a

sphere of va ria ble temperature i s

governed by the absorp tion coeffi cient. For vi si bl e li gh t, th e absorption

coefficient ncreases apidlywith temperature. For any given wave

6

length, tine emission w i l l then come from a layer which i s one optical

mean free path inside the hot mate ria l.*

*Actually the

m a x i m u m

emission comes from deeper insi de the fir eba ll.

To see t h i s we compute the ernigsion normal t o t h e surface, J =

jdr@(R)eD(R)dE? where e =-

i s the emissivity, and the remain-

hv3

e -hv@

C2.e

ing notation

i s

a s i n t h e

text

(below). The integrand has a maximum a t

R"/$* =

+

cuhv/fl*. The op ti ca l depth a t th e maximum is

D* = D(R*) =

n u

@nv'kir*

which i s about 2, rather than 1

in

t h e blue.

B y

steepest

a - 1

descents and with occasio nal use of

1/m

= 0

it turns out t h a t J =

4

B( v,T*),

grea ter by about

a

fac tor 2 than the J used in

e

the t ex t .

24


27/88

Assume that th e mean f ree path

is

& = G I T

8

-n

where the exponent n

and

the coeff icient 81 depend on the wave length.

Then the optical. depth for a given R

i s

=

I

.e(R')

R

R

R

R

D(R)

= 1

we need

25


28/88

Since the interesting values of R are of the order of a few hundred

meters, the emitt ing layer w i l l be def ined by t he fa c t th at t he mean

free path

i s

about 5 t o 10 meters.

Equations

(3.18) t o (3.21)

hold f o r emission in the exact ly radial

direct ion.

A t

an angle 8 with he radiu s we get nstead

The apparent emperature of the

emitting

layer i s then, according

t o

where

T ( R , O ) i s

t h e emission emperature f o r forwardemission. The

in te ns ity of emission i s a

known

(Planck) function

of

the wave length

and the emperature .Since th e apparent emperaturedecreases though

slowly)with 0 , according t o (3.23) th er e w i l l be limb darkening.

On

t h e many photographs of atomic explosions, it shouldbe possible t o

observe t h i s Limb darkening and thus check the va lue of

n.

The relation (3.17) needs

t o

hold only in the neighborhood of t he

value

(

3

-21)

of

4

and

i s

therefore quite general as

long

as

4,

decreases

with ncreasing emperature. The exponent nshouldbedetermined at

constantpressure.Forcertain wave leng ths, espe cially in t he u l t r a -

26


29/88

vio le t , (3.17) i s not valid; these wave len gths are strongly absorbed by

cold or cool a i r (Sec

.

kc)

.

For example, a t p = po and T = 2000, the

mean free path i s less than 1meter f o r all l ight16 of hv > 4.7 ev

(A < 0.26 p ) . Since the emission of l i gh t

of

such short wave length from

such cool a i r i s negl ig ib le , the f i reba l l w i l l not emit such rad iati on at

al.

A detailed discussion of the absorption coefficient i n the vis ib le

w i l l

be given in Sec. 4b. As can

be

seen from th e ta bl es ofMeyerott

e t

al.

and from

our

Table

VI,

f o r a density p = po the mean fr ee pat h

i s of the order of 5 meters a t about 6000~ . This corresponds t o

a

pres-

sure13 of about

25

bars.

For p = O.lpo t he re qu is it e mean f re e pa th of

a few meters i s obtained f o r about 10,OOOo, with p M 7 bars. Thus fo r a

re la ti ve ly modest decrease in pressure, the effe ctiv e temperature of

radiation increases from 6000 t o 10,OOOo, corresponding t o a very sub-

16

J

st an ti al increase

in

radiatio n ntensity . This

i s

the mechanism

of

the

increase in radiation tarard the second

maxmum

A more detailed

d i s -

cussion

w i l l

be given in Sec. 5f.

d. Energy Supply

As l ong

as the radiating temperature i s low, not much energy w i l l

be emitted as radiation, and th i s emission

w i l l

o n l y sl ig ht ly modify the

cooling of the ma teria l due to ad iaba t ic expansion. However,when the

radiating temperature increases, t he radiation cooling w i l l exceed the

adiaba tic cooling t o an increasing extent. It then becomes necessary t o

supply energy from the int er io r to th e radi at ing su rf ac e.


30/88

Most of th e ra di at io n comes from a tnickness ofone o p t ic a l mean

free pa tn near the radiu s a t which ( 3.21) i s sa t i s f i ed .

Let

J be the

radi ation emi tted per unit area per second (which

w i l l

be

of

the order

of the bla ck body radia tion; see belowand Sec. 4 c )

;

nen the

loss of

enthalpy due t o rad iat i on, per gram per second, w i l l be

-

= .ep

J

Adizbaticexpansion,

(%)adi -

A s

long as tne shock

tne

ambient pressure

according t o (3.11),

w i l l

give an entnalpy change

i s

strong, .e. ,

pl,

the pressure

as long as

p

is la rg e compared t o

behaves as

vhere t i s t he time from t h e explosion; herbfore,

,.-

@)adi

- t

1.2 p

Tine rzdia-tion

will be

a r e l a t ive ly smail perturbat ion as long a s

( j. 2h.) -is smaller Clan (3.27) . This

will

st op be% the cas e when

J

P

= 1.2 ,c

28


31/88

- p - = J

.2 R

50

t ,

(3.29)

Because

of

the steep dependence

of

ternperatwe on

R, (3.16)~

the radi-

a t i ng

surface

R

w i l l

be

close

t o

the shock front

Rs. NOW

the Hugoniot

re la t ionss ta te hat orc lose

to

1

i s

(?)I/*

(3.30)

where p i s the ambient density and

p = 2p, tne

shock' pres sure .Further-

more, f o r the

strong

shock case,

1 S

Rs -

t

0.4

4

The black body ra d ia ti on a t temperature LO

T ' i s

=

5.7

X

~ ~

erg/cm sec

JO

2

3 . 3 3 )

Actually, only tine ra diat ion

up t o

about hvo

=

2.75

ev

can be

emitted t o

large dista nce s because tne absorption

i s

too great for radiat ion

of


32/88

higher requency (above, and Sec.

L C ) .

The

f r a c t i o n of

theblack body

spectrum which can be emitted i s given, in su ff ic ie n t approximation, by

*.

where

U,

0

h V

o w

0

lJ = -

(3.34)

1

1

3 . 3 5 )

'

I n ( 3 . 3 4 ) we have ne gle cte d the fac t tha t the

i nf r ar ed, bel ow

about

1/2 ev, a l s o cannot be emi tted (Sec . 4a),

and

nave approximated

(

eU - 1)

i n

tjne Planck spectrum bye-U; bothcorrectionsaresmall.

To

=

8OWo

has

been chosen as a reasonable average emperature

(

see Sec.

5 ) .

Near

t h i s temperature, va rie s about as T

,

so thatne ctua l adiat ion

to large dis tances i s about

-1.5

Solving

(3.32) f o r p,

with

po =

1-29

(normal ai r

density)gives


33/88

For

p1 =

po,

T = 0.8,

t h i s

i s

14

bars.

A t higher temperature ( T '

> 1)

he ul tra vi ol et can be transported

as

easily as the vis ible a l though i t

w i l l

not escape t o large dis tances

(Sec. kc ). Then

it

i s reasonable t o use the

f u l l

black body radiation

( 3 .33 ) for he emission. Ins ert in g hi s nt o ( 3 .32 ) gives

(,),,

8/3

1

O,

t h i s

i s 40 bars.

( 3 .39 )

Thus for

p

greater than 1 4 t o

40

bars , the radiat ion

i s only

a

f r ac t ion of the adiabat ic coolin g, for lower pressure radiatio n cooling

i s

more important.

A t

the

lower pres sure s hen , energy

m u s t

be supplied

from

the ins i de to maintain the radi at io n. Thi s give s r ise to a cooling

wave moving

inwards

as

w i l l

be discussed in Sec.

5 .

It i s interes t ing that the condit ion

( 3 . 38 ) refers

to the p r e s su re

alone.Neither the oca lden sity nor theequation of sta te en ter s. The

opacity of air ente rs only insofar as

i t

determines th e rad iat in g temper-

a tu r e

T

'

through the condition

(

3

*21)

A more accurate expre ssion for the limit ing pressure w i l l be derived

in

Sec. 5e.

It w i l l

tu rn

out

t o be considerably lower, about

5 bars.


34/88

4.

ABSORPTI ON coEFF1c1ms

a. Infrared

/ .

Themain abso rption in the infra red i s due to free-f ree elect ron

t ran s i t i on s . These are t rea ted incor rec t ly i n th e paper by Meyerott e t

al . , i n whicn it i s assumed that such transitions occuronly i n the

f i e ld of ions .

A t .

tk e important emperatures

of

8000 or les s , the degree

of ionizat ion i s

o r

le ss . Therefore, th eree - f r eerans i t i ons

i n

th e f i e ld of n eut ral atomsand molecules ar e muchmore important than

th os e in th e fi el d of ions, even thougheach indiv idua l atom cont ribute s

fa r le ss th an each ion.

16

The effectiveness

of

neut ra ls in inducing fr ee- fre e t ra nsi t io ns has

18

been measured and int erp ret ed by Taylor and Kivel a t

laboratory. A s compared t o one' ion, theeffectiveness

tant neut ra l spec ies i s

N2

:

2.2

f

0.3

X

N:

0.9

f

0.h x

lo-=

0: O e 2 f

0.3

X lo-=

t h e Avco-Everett

of the most mpor-

Thus nit rog en giv es about t h e same contrib ution wnether molecular o r

atomic, and th econtr ibut ion of oxygen i s verysmall. A s a result, one

atom

of

a i r i s equivalent

t o

about

a

=

0.8

X ion

(of

unit charge).

The fre e-f ree abs orp tio n co eff icie nt in cm'l

i s

then

cLf

=

0.87

x 10

(%)

(e - ) (hv)-3

=

7.0

T

'-I/*(b)

(

e- ) (hv)

3


35/88

where e - i s th e number of el ec tr on s pe r ai r atom, the qu ant ity tab ula ted

i n Gilmore,l3 and hv

i s

the quantum energy i n ev. Table I11 gives some

numbers f o r hv = 1ev, four emperatu res

and

three dens i t ies .

A t

80007

the free-free absorption i s subs t an t i a l , a t

lawer

temperature s negligible .

A t 12,000 the transitions aremostly in th e f i e ld f -7ons i.e.,

Meyerott's numbers need

only

s l ight cor rec t ion , and the absorption i s

l a rge

Table

111.

T

47 OOo

6,000

87 000

12,000

Free-FreeAbsorption Coefficients

c m

)

fo r hv

=

1ev

-1

5.5-&

1.05'5

1.9 7

Note: f o r each value, th e power

of 10

i s

indicated by a superscript.

Another cause of absorption in the infrared i s the vib rat ion al bands

of NO, which have

an

osc i l l a to rt r eng th of

and

hv = l/k ev.

The resu lting abso rption coef ficien t i s about

where (N O ) i s the

percent at p/po

=

cm e While this

1

number of NO molecules

per

a i r atom. Tnis i s

a

few

1

and

T = 4000 t o

8000,

giving p

= 2 X l om3to

i s of the order of magnitude relevant

f o r

emission,

33


36/88

(3.21),

it

i s

small compared to th e fre e-f ree ab sor pt ion at' th i s

low

frequency,except at T

lom3

(mean freepath

less

than 10 meters) o r I L > lom2(4

1 meter).

The

tab le shows that strong absorption covers a partic ula rly

wide

spect ra l

region a t 4000,

shrinking

substant ial ly

a t

6000~. Very strongabsorption,

p > LOm2

m-', occurs i n quite a l w g e spectral . region fo r T

=

4000

but

shrinks to pra ct ic al ly nothing

a t 6000~

nd

t o nothing a t

8000.

A t

12,000, very strong absorption occurs again

but

i s now in the vis ible .

The

strong absorption

i n

t he u l t r av io l e t (hv

> 3.5

w) eans f i rs t

of

al l that

the W i s not emitted t o l a rge distances and can therefore

not be observed; e.g., a t T

=

4000 and

hv

= 3.5 ev, we have

%e should

also

add the photoelectric absorption from exc ited states of

NO,

which should be especially noticeable at

8000.

41


44/88

The fra ct ion of th e Planck s p e c t m beyond u = 10 i s only about Y$, so

t h a t d s s i o n of these frequencies i s negligible.

Table

VII.

Ul tra vi ol et Absorption: Sp ectr al Regions {

hv

in ev)with

Large Absorption as a Function of Temperature for p = O.lpo

T

I L > 10-3 LL > ml

2,000 4.7 - 7.2

5.5

- 7.2

3,000 3.9 .. 7.2 4.7 - 7.2

4,000

3.5

-

7.2

4.6

-

7.2

6,000

4.0

-

7.1

5.8 - 6.0

8,000

2.7

-

6.3

None

12,000

All

2.7 -

3.5

The u lt ra vi o le t can, hawever, be tra nsp orte d qui te easi ly at 8 0 0 0 ~

and even more e a s i l y a t

12,000

i f there i s a temperature gradient

.

Such

a gradient i s always av ai labl e, whether we have adiab atic condit ions

(Secs. 3 5f) or a strong coo ling wave (Sec.5d) Therefore, here w i l l

be a

flow

of ultraviolet radiat ion at the radiat ing temperature, defined

in Sec.

5 ,

which

w i l l

be shown ( S e c s . 5d,

5f ) t o

be about 10,OOOo o r

s l i gh t l y

less . To ca lcu la te th is fl ow we shoulddetermine th e tempera-

tu re gr ad ie nt from consideration s such as Sec. 5d o r

5f,

and then insert

t h i s

into

theradiat ion

fluw

equation.

This

is

similar

t o

(5.3)

except

that

only

the ul traviolet contribution to the flow should be taken into

account.

42


45/88

When t h i s

i s

done i n a case of constan t

(

frequency-independent

)

absorption coefficient, the ul t ravio le t t ranspor t w i l l be r e l a t e d t o t h e

vis ib le radia t ion t ranspor t as the respec t ive in tens i t ies

i n

the Planck

spectrum. This condit ion seems t o be ne ar ly fu lf i l le d at 12,000. A t

8000, the absorption in the near u l t ravio le t (2.75 t o 4.2 ev) is about

t h r e e t i n e s

that

in the v is ib le ; then the W t ransport

w i l l

be one-third

of t h a t corresponding t o

the

Planck inten sity. Sinc e

the

radiat ing

temperature near the second rad iat ion

maximum is

between

8000 t o 10,OOOo,

t he ac tua l

W

transport w i l l be between one-third and the full Planck

value, relative

t o

the vis ib le radia t ion. According t o

(3 .36) ,

the

W

contains about 43

,of

the Planck i n t e n s i t y a t

8000 ;

hence th e to ta l

radiat ion t ransport a t

t h i s

temperature i s about

of

the

black

body radiation.

A t 12,000

we get

t h e

f u l l

black body value.

For simplicity w e have assumed

the

full

black body

radiat ion in Sec.

5 ,

even though the W i s not emit ted to large dis ta nces . But t h i s problem

could,. and should, be tre at ed more accurately.

Having

discussed the influence of the W on the t o t a l r a d i a t i o n

flow,

we nuw examine w h a t happens t o the W radia t ion after it has gone through

t h e

radiating

layer, i.e., th e la ye r which

emits

t h e v i s i b l e l i g h t t o

large distances. The very near ultrav iolet, 2.75 t o

3.5

ev,

w i U .

be

partially absorbed a t 4000 t o 6000, especial ly i f t he l aye r of matter

a t t h e s e intermediate emperatures becomes th ic k, 0.3 t o 0.5

gm cm

o r SO.

I f

2

43

/


46/88

The W beyond 3.5

ev

w i l l be strong ly absorbed a t 4000'. Thus the layers

of a i r a t intermediate temperatures get additional heat

which

counteracts

and

may even exceed th e ad ia ba ti c cooling.This

w i l l

tend t o increase

the thickn ess of the medium-temperature laye r. Th is in

turn w i l l (mod-

e ra te ly ) lower th e ra di ati ng temperature, but three-dimensional effects

act the op po sit e way (Sec. 6 ).

a

.

Theory of Z e l dovich

e t

al .

Zelsdovich, Kompaneets,and Raizer

lo

quoted as Z ) have considered

t h e

loss

of ra dia tio n by hot mater ial when th e ab sorptio n coeffic ient

fo r th e ra di ati on in cr ea ses monotonically with temperature. Theyhave

shown t h a t

i n

t h i s case a cooling wave proceeds into the hot material

from the outside . Tnis i s

t o

say, t h e cool emperatureoutsidegradually

ea t s

i t s

way into

the

hot material, while the material

in

the center

remains

unaffected and merely expands adiabatically.

For

simplicity, ZelPdovich

e t al.

consider

a

one-dimensionalcase.

They fu rt he r as,sume th at the sp ec ifi c heat i s constant and express their

theory

i n

terms of the temperature.This

i s

notnecessary; we sh a l l

merely assume that both the enthalpy

H

and the absorp tion coeffic ient

for radia t ion are a rbi t ra ry but monotonically increasing functions

of

the

temperature. Like

2,

we sh a l l assume,

in

t h i s subsection

only,

tha t he

ra di at io n tr an sp or t can be described by an opacity (Rosseland mean) rat he r

than considering each wave len gth separat ely.

44


47/88

The

fundamental statement

of 2 i s

t ha t the cooling wave keeps

i t s

shape, i.e., that the enthalpy (and other functions of the temperatme)

the cooling wave proceeds tmards smaller x,

i.

.

, nwards.

H

is, of

i s

given by

H = H(x +

Here t i s the time,

Lagrangian velocity

x i s

the Lagrangian coordinate,

and

u

i s

t he

of

th e cooling wave. We have w rit te n

x +

ut

so

t ha t

course,

a

decreasiw function

of x +

ut .

The

Lagrangian coordinate i s

best

measured

in gm/cm

and i s defined by

where X i s the geometrical(Eulerian)coordinate.For given pressure p,

the

density

p i s

a function

of

H

so

that

X x )

can

be

calculated

from

( 5 0 2 )

The Lagrangian vel oci ty

u,

measured i n

gm/cm

sec,

i s

a'constant.

For any given

If,

we

know

the temperature T, hence the opac ity

K

and

the radiat ion flow

where a

i s

the Stefan-Boltzmann constant,

2

4

a

= 5.7 X lom5

erg/cm se c deg

45


48/88

The equation

of

continuity i s

a H

a J

== a x

5*4)

The

energy in radiati on has been neglected, which

is

jus t i f i ed i n

all practical cases . Using (5.1), (5.4) can be ntegra ted over x t o

give

u H + J = C

5

* 5 )

where

C

is a constant. This i s the fundamental result of 2,

If

the opacity increases monotonically w i t h

H,

then in t h e i n t e r i o r

J w i l l be very nearly zero, and therefore

c =

uH*

5.6)

where

Ho

i s the enthalpy. in th e undisturbed interior, hot region.

Equation (5.5) becomes

J = u(H0 -

H)

(5.7)

which

can

be integrated t o give x(T), since H(T)

md

K( T , p ) are

known

fun ctio ns. We have

put i n

evidence the f a c t

that K

depends on pressure

in addition

t o T.

Overmost of the range of

T,

K(T)

i s

the most

r ap id ly


49/88

varying (increasing) function

o f T

and the variat ion of H

-

H is

less

important; therefore,

T x)

becomes s te ep er as

T

increases;

but

when

H

g e t s

veryclose t o

II

th e most rap idlyvaryingfunction-

in (5.8)

i s

Ho

- H, and

H

approaches

Ho

exponentially as ea f o r

s m a l l X.

The

qual-

i ta ti v e behavior of T(x)

i s sham

in

Fig.

1, in accord with

Z.

To obtain

t h i s shape

it

i s essent ia l

that

K( T ) increase much fa st er th an

T .

0

0

3

*O

Fig.

1.

Temperature di s tr ib u ti on in ' Cooling wave.

On the outs ide, we fi n a l l y come t o

a

point xlwhere

on l y

one

optical

mean f r e e path

i s

outside xloFrom th i s po in t we get black body emission,

4

J(xl)

J1 =

aTl

5 . 9 )

Using

t h i s

in 5.7)

we

f ind

47


50/88

J,

U

A

Ho

- H(T1)

To

determine

u

w e therefore

1. Find the temperature T

1

have t o proceed as follows:

a t which the opacit y

K(T1)

is such th at

there

i s one optical mean freepathoutside x i.e.,

1'

For t h i s purpose we

m u s t haw

of course,

the

temperature distribution

T(x)

f o r T

0. An in teres t ing inter -

mediate state

i s

when the cooling wave has just reached the blocking

la ye r. Tnen, using Tb

J1 -

Jb,

"b = 1% -

H1

= 1.8 and 5.14),

Clear ly th i s m a k e s sense only i f the subtracted term in the numerator i s


55/88

smaller than th e f i r s t term, i .e , when the pressure i s suff ic ient ly

high.

A s

the pressure decreases

below

about

10

bars, he blocking ayer

opens

up

and ceases t o block t'ne

flow

of radiation.

In th e very beginning, when 'ne cooling wave ju s t s ta r t s , t'ne top

temperature

of

the cooling wave,

i s

close to the rad iat in g tempera-

ture

T hen

(5.18)

becomes

TO'

1'

1 =

-

(EX

1

If

we use

(5.12)

fo r J, assume

d

log T/d

log R and R t o

be constant,

wd

use

(3.14),

tinen

For further

discussion,see Sec. 5f.

A s

T increases,

u

decreases from (5.23) via

(5.21) to

(5.19).

Af'ter

th e cooling wave has penetrated t o t h e isothermal sphere, u

i s

apt t o

increase aga in because T i n t h e denominator of

(5.19)

w i l l decrease due

t o ad iabati c expansion

of

the sothermalsphere. Thus th e ve lo cit y u i s

apt t o be a m i n i m w n vhen the cooling wave has just reached the isotherma l

sphere

C

The variation of u with time

i s

not very grea t. Likewise, t he shape

of the cool ing wave changes onlyslowlywith ime. Tne shape i s obtained

53


56/88

by integrating

5 . 8 ) ; it

depends on timebecause K

i s

a (rather slowly

variable) function

of

thepressure, and

H

i s a (slow) function of time.

This

jus t i f ies

approximately th e bas ic assumption

( 3 .l) of 2:

While the

0

cooling wave does not preserve i t s shape exactly, it does so approximately.

The picture i s th en th at at any given

time

t there a re up t o f ive

regions behind th e shock. I n Fig. 2 th e temperature

i s

plotted schemat-

icallyagainst heLerangecoordinate r. Starting from thecenter,

there i s

f i r s t t h e isothermal sphere I

n

Fig.

2 ).

Fig.

2.

Schematic temperaturedistribution. I, isothermal

sphere;

111,

cooling wave; VI, undisturbed air;

E,

shock front

11,

IY,

and

V

are expanding adiabat

-

ical ly.

This may be followed by a region I1 i n which the temperature di str ib u-

t i o n

i s

ess en tia lly th at est ab lis he d by adiab atic expansion behind th e


57/88

shock, Eq. (3.15)

.

Nextcomes 'ne cooling wave I11 in whicn t e emper-

ature f a l l s more steeply,according t o (3.8)

.

( A t

la t e t imes,region I1

i s wiped out and I11 follows immediately upon

I

.

Region IV includes

th e mat er ia l which has gone through tn e cooling wave,and now cools adi-

abatically; nence the temperature falls

slowly

with r (Sec

.

jd)

D

i s

the ma teri al poi nt from whicn t h e cooling wave sta rt ed or ig in all y; region

V, outside hat point, i s also expanding adi aba tica lly, but

f r o m

snock

conditions; hus it is the continuation of region 11. Finally region VI

i s t h e a i r not yet shocked. A s time goes on, th e cooling wave I11 moves

inward, wiping

out

region

I1 and

then eating nto region I. Region IV

accordingly g r o w s toward the inside, but i t s outer end D stays fixed.

Region

V

expands into V I by shock.

We

note once more that u i s the ve loc i ty in Lagrange coordinates,

and in

gm/cm

sec. The problem i s xnade omewhatmore complicated by the

th re e dimensions and th e ad ia ba ti c expansion, cf . Sec.

6,

but tne

pr in -

cipal features remain the same

d. Adiabatic Expansion a f t e r

Cooling.

Radia ting Temperature

When a given mater ia l element has gone through t he cooling wave, it

i s l e f t at he rad iati ng temperature

T1.

Thereaf'ter, it

w i l l

expand

adiabatically. Equation (3.25) shows that

f o r

adiabatic expansion

or,

using (3.10)

55


58/88

a l o g I1

- y -

1

l og p

t y ' a t

A s long as the

snoek

is strong, (3.26)

holds

md therefore

GiI-more'sl3

Tables 1.1 and

13

shoy that

f o r T = 5000 to

8000 and

p =

1 o 1 0 bars, y var ies f romabout,

1.13

t o 1.20, so

a log-1

=

- 0.lh t o -

0.20

A t l a t e tjm-es, p no longerdecreases as

fast

a s t , o

H

nlso decreases

more s l o ~ , ~ l - yut (

5.25)

remains valid.

-1.2

We use now t h e

approximate

equation

of

s t a t e 3 o l k ) (together with

B =

y

'E) and find

= P

w L t h

= 0.136

t o 0.167


59/88

using 5 .25 )

m d y

= 1.13 t o 1.20. A given material element vllich went

through the cooling wave

a t

temperature

T '

and pressure

p T r i l l

now ( a t

pressure

p)

have the temperature

I

m m

T t = oe

m

If

w e

assume Tmt o be independent

of time,

th en at a given tfme in Cne

adiabatic region

(T v ,

of

Fig. 2

as

defined in Sec. l;b. Since weow use materialcoordinates

i n gm/cm

,

we should use mean free paths n he same unit. Table V I and (4. 10) m d

(lk.12)

give tine required nformation. We m i t e (4.12)

in

t he form

57


60/88

2

vitn A =

0.1G

q/ cm and n = 6.3.

The optical depth

is,

according t o (5 .32) and (7 .33)

x1

4

n

1 u t Y

* P -

1

0

(5. 34)

Set t ing

D

= 1,

y

= 1. 15, A = 0.10, and

n

= 6.5 gives

Tnis

i s

m

expl ici t express ion f o r

tne

radicting temperature

i n terms

of

tne ve loc i ty

of the

cooling wave.Tae l a t t e r depends i n

turn on

the

radiating temperature,

i ncreasi ng

with T p 4 so t n a t T

'

occurs altogether

in t n e

10.2

power

and t'nus

can be determined very accu rate ly.

1'

1

Equation 3.s)

an

be fu rt he r reduced by usi ng tne rel at ion between

pressure

and time

wIxLch

i s , f or

a st rong shock, approximately


61/88

where

t i s i n

seconds, p in bars , a d Y i n megatons, a d

pl i s

the den-

s i t y of tine mbient,undisturbed aLr. Then ( > 3 6 ) becomes

Tne

right hand side of

5 . 3 8 )

gi ve s th e co&plete dependenceon

Y

and

p1

since,deriving

(5.36),

we have only used

tne

opacity

'law

3 . 3 3 ) and

th e ad isb ati c cooling of a i r,

5 . 3 2 ) ,

both of which zre independent

of

the explosive yield

Y

and of

t ne

ambient density of t h e a i r .

iY0r.r insek-t u from

5 .

X,9); hen w e obtain

(5 .39)

Tnis

equation gives he radiating emperature i n

t e a s

of

t ne cen t r a l

temperature

T

and

of

thequant i t ies

on

ther igh t

hand

side. The ra di -

a t

n2

temperature

i s

proportional t o a low power

of

th e cent ral temper-

e t u r e

(

about

tne

1/6 '

pover); tnus as th e ins ide coo1s, tne radi atio n

dec reases see Sec. 5e

f o r

d e t a i l s ) .

It

alsodecreases slowly with time

C

due t o t h e pressure factor on tile righ t nand side ,

T

1 - P

For given

p and T the radiation emperature

i s

higher

f o r

lower yield,

T

- Y

and for n igher a l t i tude ,

Ti

-

p,

-0.032

C'

1

9

-1./21

.

For sea level , for

Y = 1,

and for

p

=

5

bar s (cf . Sec. >e

f o r

t h i s

choice)Brode'scalculationsgive T' M 3.6; then

5 . 3 9 )

yields Ti

=

1.08,

C

59


62/88

o r

a radiating emperature

of

10,800.

Tnis

i s a

reasonableresult,

tho wh appre cia bly hig he r ha n he eqera tur es us ua lly observed. How-

ever,

as we

shall show in Secs.5e

md f ,

our calculation gives he

max-

imum

temperature reached,

md i s

l i k e l y to be somewhat too hig h due t o

OUT

approxinmtions

.

e. Bewinning

of

S t r o w Coolim Wave

2

Equation

5-19)

ives the inward speed

of

t he cooling wave in

@/a

sec. Precisely,

t h i s

i s

th e speed a t which the poin t

of

t e q e r z h r e

T

(radiating temperature ) moves rel ativ e to the ma ter ial , once fne cooling

1

wave

i s

ful ly est ab lish ed . But even

i f

there

i s

no cooling wave, i.e.,

i f we have simply ad iaba tic expansion behind th e shock,

a

point of given

temperature

T1

w i l l move intrzrd. Th is ad iaba tic motion''

i s

t he minimum

velocity vlnich thepoint T can nave. Therefore , i f tneadiabatic speed

i s

greater than (5.19)

, it

w i l l be tne correct velocity. O f course, there

1

w i l l

s t i l l

be

a.

cooling wave because t h i s i s needed

t o

supply the energy

for

the radiat ion;

tills

'be&< cooling wave

will

be described i n Sec 5 g .

But

i t s

inward motion, more acc ura tel y the ve loc ity of i t s foot (point

C

i n Fig. 2 ) , w i l l not be determined by the requirement of sufficient energy

flow,

(3.19), bu t

by the ad iab atic speed which we sha ll de riv e from

( 3 . 1 7 ) Region

I V of

Fig.

2

w i l l nar be absent, Tnus, outside he

cooling wave, at po in t

C y

region

V

will

begin Med iately , with the

tem pera ture distri butio n given by (3.15).

It i s the ref ore important t o determine th e time ta (and pressme p

)

a

60


63/88

a t which

u

of ( 5

.IS)

becomes larger than t'le ad iab at ic speed of temper-

ature T Before th i s time

ta,

we 'have

a

weak cooling wave

.

We c a l l

this Stage C

I;

afterwards, the

cooling

wave

i s

strong, and

the

radiation

i s ess ent ial ly described by the theory of Sec. 5d (Stage C 11). To

determine tine point of separation p between these two stages i s

a

refine-

ment of the considerat ions of Sec.

3d.

1'

a

Tne temperature in th e ad ii ib at ic al ly expanding mate rial behind t he

shock

i s

a

function of time and posi tion , given by nydrodynamics

and

equation of s ta te . The imrard motion of a point of given emperature

rel-ative t o the material i s given by

tfnere the subscript

r

means that

the

pa r t i a l de r iva t ive

must

be talcen at

given material

point

r,

not a t given geometrical radius

R .

W

have

from

(3.15)

61


64/88

assuming

t he s t rong

shock

r e l a t ion p

-

-1 2

. The ratio (5.40) i s tnen

I-Iere w e

use the rela tion (3.10)

P -

yIt - 1 eE

and f ind

(5 945

where

H

has been labeled

3

ecause

it

ref ers to th e ra dia t in g tempera-

tu re . % re we

may

i n s e r t (3.30)

and (3.31-),

viz.,

We may then compare t he r e su l t w i th 5 IS),

the

veloc i ty of t h e cooling

vave (set t ing J = 0 )

0

u =

Ho

-

5.

The comparison gives

62


65/88

Insert ing

(3.33) f o r JIJ and

changing the un it

of p

fromdynes/cm 2 t o

bars = LO dpes/cm

we

get

2

Setting nar

p1

= po and C ~ O O S ~ ~ ,s

in

Sec. 5d, Tc =

3.6

and T1 =

1.08

yields

p1 = 5.0 bars

Thus the cr i t ica l pressure i s 5 bars, which was tne reason f o r the choice

of t h i s , number a t

t'ne

end of Sec. 5d. Tnere i t vas

shown

t h a t 5 bars and

T '

=

3.6 l eads t o Ti = 1.08 so t h a t our numbers are cons ist en t.

C

We

had to rely on Brode ' s so lu t ion to f i n d Tc for a given

p;

t h i s

could

only

be avoided by obtaining an ana lytic solutio n

fo r the

i s o t n e m l

sphere which w i l l be discussed i n a subsequent paper. Apart from this

our treatment i s analytical, making us of the equation

of

st at e and the

absorption characteristics of a i r .


66/88

pressure i s larger than pa, 5 .50)

.

Then the radiatin g surfa ce i s i n

the adiab atic regio n desc ribed n Sec. 3c.

The

condition for i t s posi-

tion is, (3.21), t =

R/5O. From Brode's curves, th e pos it io n of

a

point

of

temperature T

'

near

1 s

given approximately by

R = 0.78 Y T

P

1/3 r-0.10 -1/4

This expression comes pu rely

from

the numerical calcu lation , except th c t

th e co rr ec t dependence on yi el d

i s

inserted. Y

i s

i n megatons, p in ba rs ,

R in kilometers. Equation (5.51) holds

from p =

5 t o 100 bars within

about

5%.

The absorptioncoefficient

i s

given in (4.10), which yields

using t'ne equation

of

s t a t e (4.11). Equating th i s t o 1/50 of

(5.51)

gives

Tr408 = 4.5 y

-1/3

P 1.20

Thus

the radiating temperature increases

as

the pre ssu re decreases.

b

64


67/88

Since p

- ,

- 12

9

en t ire ly due t o

tihe

opening up

of

the shock,

i.e.,

the decrease in

absorption with decreasing d ensity (pressure) .

For

given pressu re,

the

radiating temperature

(5.54)

is s l i gh t l y

higher

f o r

smaller yield, an ef fe ct which

has

been observed. A f ac tor

of

1000 i n

yield corresponds t o

a

f ac tor

1.6

i n

temperature, hence

a

f ac tor

7 i n

radiation per

unit

area.

The

tot al rad iat ed ' power(assuming

blac k body) a t given p

i s

proport ional t o

The

re la t ively law power

of

t he y i e ld i s remarkable i n t h i s formula,

vnichdescribes Stage

C I.

Since th e to ta l energy

radiated

shouldbe

approximately proportional t o

Y,

the dura t ion

of

Stage C

I

i s then pro-

por t iona l t o Yo*6o . The observed time t o th e second radiat ion maximum

i s

abotrt proportional t o

Y . O u r

theoretical dependence

of

T

on yield

.7 1

i s there fore sornewnat to o strone;. The time dependence of

(3.56)

i s quite

strong, about

as t

2

A s ve nave snam in Sec.

5e,

Stage

C

I

ends when

the

pressure reaches

5 bars.

For

th i s value

of p,

and

f o r Y

= 1, ( 7 S ) gives

T

=

0.92.

Tnis

i s

s l i zh t ly l e s s

t han

t he

T ' = 1.08

deduced

i n

Sec. 5d f o r

t he

same

1

p

m d

Y f r o m

the cooling

??me.

The discrepancy must

be

due

t o

a small


68/88

inconsistency in our apyroximations .

In our one-diniensional neoqy, tne end

of

Stage C

I

narks -the

m a x -

Snrrm.

of

rad iati on, bot h n temperature and t o t a l emission. I n Stage C I

because th e mat er ia l becomesmore transparent. I n Stage C I1 the reverse

i s

the case, (5.39), because t he ma te ri al which has gone through the

cooling wavebecomes th ic ke r wi th time,(5,.32).Tnis ma ter ial provides

opaci ty for the v is ib le l ight from the f i reba l l ; s ince it becomesmore

opaque, the rad ia t io n must nowcome from a layer

of

smaller absorption

coefficient , (5.35), and therefore of lower temperature.

A s (5.36)

s h a ~ s ,

the increase of thickness t

2n

the denominator) i s more important than

th e continued decrease of de ns ity (f ac to r p1/3

).

Tnese results w i l l be

modified in t h e three-dimensional heory, Sec. 6.

k

The maxi mum temperature has been calc ula ted as T = 0.92 or 1.08,

from

o m

two

calculations;

it

i s

clearly close

t o

1,

i.e.,

~ O , O O O ~ .

This number i s not to o mucn out of l i n e with observation considering

t n a t we have calc ula ted a

maximum.

In fac t , tne t rans i t ion

from

Stage

C I

t o

C

I1 cannot be sudden as

we

have assumed; the cooling wavemust

beg in gradua lly, and tn er ef or e th e temperature peak which we have calcu-

la ted w l l actua l ly

be

cut off (Fig.

3 ) .

The observable

maximum

may

easi ly be 1000 lower than our calcula tion.

66


69/88

f

CALCULATED

LOG

t

Fig.

3.

The calculat ed ncrease n empe rature i n Stage C

I,

and decrease in Stage

C I1

( s t r a igh t l ine s ) ,

and

the

estimated actual behavio r.

The

radius of

the radiat ing surface increases w i t h time i n Stage

C

I.

Inser t ing

( 5

.?4)

i n t o

(5.51)

gives

R - Y

P

0.34

-0.225 t0.27

5057)

In Stage

C 11,

Yne surface moves rath er rap idl y inward re l a t i v e t o .t h e

material, due

t o

thecooling wave. Inaddition,

f o r

sea evelexplosions

at leas t , the pressure

i s

no longer much above ambient,

so

t h a t t h e out-

ward

motion

o f

the material

slows

d m . Thus the geometr icradius

of

the

rad iat ing su rface no longe r inc rease s much, and

soon

begins

t o

decrease.

Tnerefore, the tot a l rad iat io n

w i l l

reach

i t s

maximum

a t t h e

same

time

as, o r very soon af ter , the maxi mum of the temperature.


70/88

G . We&

Cool-ing Wave

In

Stage

C I1

tne coo lin g wave dominates th e ra d ia ti o n (Sec. 5d)

.

I n

Stage C

I t h e

cooling

mme als o ex is ts , because t he energy

of

t h e

radiztj-on

m u s t

be provided. Hovever, t h e speed a t wnich th e trave proceeds

inward i s nov governed by tn e ad ia bat ic expansion, (3 .LO

a

I n order t o

obtain he correct f lux of radiat ion

J a t

the radiat ing surface, we

must therefore use (5.18) i n reve rse: Tine tem perature

T

a t he nne r

edge

of

Cne

cool i ng

77ave (point

B

in Fig. 2) will reg ula te i t se l f in such

a

vay

t h a t

(5*18)

i s sa t i s f ied , wi th ugiven by 5.lJr5). Using 5.461,

w e

thus

get the condit ion

1

0

or solving for

To and

in se rt in g numbers:

13

y - 0.9

P

3/2

T i 4

5

059)

PlTeglecting

Jo ,

we f in d th at Ho increases rapidly as p decreases.

Wemay in s e r t (5.31 -); then the r ight nand sid e varie s as p

T h u s

the cooli ng wave s t a r t s very weak and the n rapid ly incre ases i n

strengtn. Its head (point B inFig. ' 2 )

i s

a t

f i r s t

c lose to

i t s foot

(point C).

A s

time goes on, it movesmore deeply in to t'ne ho tmaterial,

ea t ing up region

I1

o f Fig. 2. The energy which i s made available

f o r

5/2 - t3*

68


71/88

radiat ion i s essent ial ly the difference H

: Tne

material drops

suddenly

i n

temperature from

T

t o

T

as hecool ing wave weeps over

0 -

0

1

i t ,

and the energy difference

i s

s e t f r e e

f o r

radiation.

The term J depends

on H

on p, and on th e temperatu regradient

i n th e region nside th e

cooling

wave (region 11). If th i s reg ion

is

adia bat ic, he temperaturegradient can be ca lcu la ted from (3.15). Since

0 0

J 1 i s also related t o the temperature gradient in the adiabatic region

I V, t he r a t io J

/J

tends to uni t y as Ho -.

H1.

Thus the

l e f t

hand side

of

(7 .39)

will have a certain

m i n i m w n

value.Tnis seems to in di ca te

tha t t he re i s no cooling wave a t all until the pres sure has f al le n below

0 1

a ce rt ai n cr it ic al value. We have not in vestigate d thi 's

point

in deta i l .

It i s poss ible that it

i s

simply

r e l a t e d t o tne break-away of the lumi-

nous front from the shock wave,

i.e.,

the beginning

of

Stage

C

.

Equation (5 .99) de sc ribe s the weak cooling wave

in

Stage

C I

no

matter what t he di str ibu tio n of tempe,rature i n region

11.

m e s t a g e

comes t o an endwhen H reaches the maximum possible value,

Hc .

There-

af te r, th e wave described

by 5 .39 )

becomes inadequate

t o

supply the

0

ra di at io n energy: Since th eentha lpycm no longer ncreas e, he speed

of t he cdoling wave has to increase. Tni s speed i s

tnen

given

by ( 5.l9),

and Stage

C

I1 has begun.


72/88

6.

EFFECT OF THREE DDENSIOUS

a. I n i t i a l Conditions and Assumptions

The f ac t t na t t he f i r eb a l l i s spherical causes some deviations f r om

the one-dimensional tiieory of Sec. 5 . In th is se ct io n we shal l

di scuss

Stage C 11, tine penetration of the coolin g wave in to th e isothermal

sphere, i n t h r e e dimensions. I n it i a l l y , we assume tn at he cooling wme

has ju st reached the sotnermal sphere; we c a ll t h e corresponding ti n e

t

=

ta,

and the pressure

i s

t he c r i t i c d p r e s s u r e

p

a

=

5

bars derived

i n

Sec. 5do Subsequently, t h e mass of tne sothermal sphere decreases due

t o tine coo ling wave.

We assume that the ma ter ial , tfnich i s a t temperature T > Tm = +OCOo

a t t he i n i t i a l time t

w i l l

s tay in t h i s temperature ange t'hroughout

Stage C 11. Tnis i s reasonable because th is ma ter ial w i l l be heated by

t h e

ul tra vi ole t ra dia tio n coming

from

the insi de, because

of

tne strong

absorption of a i r

of

medium temperature

(3000

t o

6 0 0 0 ~ )

o r

W

(Sec. 4c)

The W heating i s expected t o compensate approximately th e cooling due

t o a diaba tic expansionof th i s mater ia l ; tn i s i s confirmed by

rough

estimates

of

the he ati ng and cooling.

We

do not knowhow tine temperature

i s di st ri bu te d in th e '%arm layer between the rad iati ng temperature T1

and Yne temperature Tm

=

4000'; tnis could

o n l y

be determined by a detailed

calcu lation of the

UV

radiat ion

flow

i n

t h i s region. We assume th at th e

d i s t r ibu t ion i s smooth, as

it

i s i n Stage C 1, and tha t the refo re the

thickness of the 'harm layer in gm/cm i s d i r e c t l y proportional t o the

a'

2


73/88

required mean free path of v isi bl e lig nt at th e ra di at in g temperature

(a ls o in gm/cm

)

which in tu rn determines

T~

i t se l f .

2

8

We take he i n i t i a l conditions fromBrode, using h is e r n e s

a t

a

time when the ns ide pre ssu re i s

4 .1

bars , th i s being c los es t to p = 5

bars

of all

the curves he naspublished.Thiscorresponds

t o

a scaled

(1-megaton) time

tc =

Oo32> sec. A t t h i s time, mportantp'hysical

guan-

tit es are as given i n Table

V I 1 1

(dimensions scaled t o 1megaton) o The

l a s t column

o f

the table

i s

not given by Brode but

w i l l

be explained i n

Sec.

6c. In

the table , we have defined a quantity proportion al t o th e

a

mass,

0

J

where

R is

measured

in

hundreds

of

meters. It

i s

in teres t ing that the

mass

of the w a r m layer i s much (4.6 times) larger than that

of

the isothermal

sphere. I t s

volwne

i s about

2.4

tfmes la rg er in Brode

's

calculations.

Table V I I I . Conditions When Cooling Wave Reaches Iso thermal

Sphere, According t o Brode

1

Isothermal Warm Layer

Sphere Brode Sec. 6c

Outer radius, meters

38.5 517 438

Mean temperature

T '

3.0

0.70


74/88

6 . Sinrinkage of I s o t n e m a l Sphere

We denote t h e 'tmass''

of

the isothermal sphere,

as

defined by

(6. 1)~

by 5 Then t h i s

mass

w i l l decrease, due t o t h e progress

of tne

cod ing

wave inward, according t o

d l

d t =

- PO Rf

Note the po n the denominator and the absence of the fac tor

& f l y

both

due t o the de f in i t io n

(6.1)

The

density

of

t h e isothermal sphere

i s

nearly uniform

and vi11

bedenoted by

p

I n i t i a l va lues a t time tar

w i l l bedenotedby

a

subscript a. The isothermal sphere expands adia-

bat ically, hence

i s

and th er ef or e at any time t

The speed of t h e

cooling

wave i s given by 5.LO), thus

5

u =

-

Hl

(6.4)

where J has been

set

equal t o zero because there

i s

no appreciable flow

0


75/88

of radiati on nside he sphere.

H i s

th e enthalpy inside ne sphere,

0

which decreases adiabatically

H

i s th e enthalpy at he radiat ing surface .

It w i l l

be shown i n See. 6c

th at th e temperature T

i s

nearly constant with

tlme,

so t n a t in good

approximation and J in (6.5) ar e constant.

We

shall a lso

make

the

poorer approximation that

3

Ho. Then

(6.5)

and

(6.6)

give

1

1

1

We define

x = E -

t

a

73

J


76/88

Pa

f ( x )

= -

P

and obta in the

simple

di f fe re nt ia l equation

ay

- - Af(x)

- l / 3 Y

,

dx-

vni ch yie lds , tog eth er wit n the boundary conditions,

v

(6.10)

(6.11)

1

The constant A can bedetermined from Sec. ?e. Tne condition here

i s

dr

u = - p E

a

(6.12)

where p is th e de ns it y ou ts id e t'ne cooling wzve. According t o ( 3

&)

Tnen

u s i n g

(5.43)

R

a

E

a

(6.14)

74


77/88

which i s a pure number, and sma

fireball 1

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