non-lte radiative transfer examplesthe lte f -values to be too small due to the neglect of...

Non-LTE Radiative Transfer Examples

Han UitenbroekNational Solar Observatory

Boulder

Solar Spectro-polarimetry and Diagnostic Techniques,Estes Park, Oct 1, 2018

Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples

The Mg i 12 micron emission lines in the solar spectrum

1992A&A...253..567C


Why study these lines?

Pro:

The magnetic sensitivity of line splitting over Doppler width isproportional to gLλ. While gL = 1, the long wavelength makesthese line about five times more sensitive to the magneticfield. In addition, direct measurement of the splitting allowsmagnetic field strength directly, without polarimetry, not onlythe line-of-sight component from Stokes V.

The core of the lines forms at about 400 km above τ500 = 1.

Con:

Line formation is distinctly Non-LTE.

Spatial resolution element if proportional to λ/D, twentytimes worse than in a typical visible spectral line, with thesame telescope.


Temperature stratification of solar models


Mg i Grotrian diagram


Comparison of observed and calculated Mg i 12 µ profiles

1992A&A...253..567C


Formation heights and source function


Divergence of departure coeeficients

Departure Coefficient:

bi =(ni/n

LTEi

)Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples

Departure coefficient ratio and effective temperature

Line Source Function:

S lν =

2hν3

c21

(bl/bu)e(hν/kT ) − 1; Bν =

2hν3

c21

e(hν/kT ) − 1

Source Function Ratio:

S lν

Bν=

1− e−(hν/kT )

(bl/bu)[1− (bu/bl)e−hν/kT

]Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples

Full disk images of He i equivalent width and B‖


Full disk images of He i EQW and Fexii 19.3 nm


Line profiles of the He i 1083 nm triplet

1082.6 1082.8 1083.0 1083.2 1083.4Wavelength [nm]

10−8

10−7

Inte

nsity [J m

−2 s

−1 H

z−

1 s

r−1]

Disk center

1082.6 1082.8 1083.0 1083.2 1083.4Wavelength [nm]

3.94•10−8

3.96•10−8

3.98•10−8

4.00•10−8

4.02•10−8

4.04•10−8

4.06•10−8

Inte

nsity [J m

−2 s

−1 H

z−

1 s

r−1]

Disk center


The He i 1083.0 nm line: chromospheric?

Courtesy Bernhard FleckHan Uitenbroek/NSO Non-LTE Radiative Transfer Examples

The He i 1083.0 nm termdiagram

1SE 1PO 1DE 2SE 2PO 2DE 3SE 3PO 3DE

0

20

40

60

En

erg

y [

eV

]

HE I 1S2

HE I 1S 2SHE I 1S 2S HE I 1S 2PHE I 1S 2PHE I 1S 3SHE I 1S 3S HE I 1S 3PHE I 1S 3DHE I 1S 3DHE I 1S 3P HE I 1S 4SHE I 1S 4S HE I 1S 4PHE I 1S 4DHE I 1S 4DHE II 1S

HE II 2S HE II 2P

HE II 3S HE II 3P HE II 3D

HE III

58.4

353.7

0

388.861083.031083.021082.91318.772058.13501.57706.52471.31

587.56587.56587.56587.56587.56587.60

447.15728.13504.77

667.81492.19 4294.781252.757435.48 2112.00 18617.441700.241954.3195760.172113.20 1908.93 10879.16 43957.16

30.3

825.6

3

164.04

164.05

164.0

4


The He i 1083.0 nm termdiagram, triplet system

3SE 3PO 3DE

20

21

22

23

24

En

erg

y [

eV

]

HE I 1S 2S

HE I 1S 2P

HE I 1S 3S

HE I 1S 3P HE I 1S 3D

HE I 1S 4SHE I 1S 4P HE I 1S 4D

388.

86

1083.031083.02

1082.91

318.

77

706.52

471.31

587.

56

587.56

587.56

587.

56

587.56

587.60

447.

15

4294.781252.75

2112.00

18617.441700.24

1954.3110879.16

43957.16


EUV irradiation from Corona populates triplet levels


Off-limb emission as function of EUV irradiance

Centeno, Trujillo Bueno, Uitenbroek& Collados 2008, ApJ 677, 742Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples

The He i 1083.0 nm contribution function

5.0•10−11

1.0•10−10

1.5•10−10

2.0•10−10

2.5•10−10

3.0•10−10

Contr

ibution function [J m

−2 s

−1 H

z−

1 s

r−1 k

m−

1]

1082.6 1082.8 1083.0 1083.2 1083.4Wavelength [nm]

0

500

1000

1500

2000H

eig

ht [k

m]

Contribution function:

C ≡ S(τ)e−τ dτdh


He i Line contribution function

2•10−13

4•10−13

6•10−13

8•10−13

Con

tribu

tion

func

tion

[J m

−2 s

−1 H

z−1 s

r−1 k

m−1

]

1082.6 1082.8 1083.0 1083.2 1083.4Wavelength [nm]

0

500

1000

1500

2000

Hei

ght [

km]

Line contribution function:

Stot =(ηl + ηc)

(χl + χc)

C =

[ηl

(χl + χc)+

ηc(χl + χc)

]e−τ

dτ

dh


He i Line contribution function with irradiation

1x

2.0•10−13

4.0•10−13

6.0•10−13

8.0•10−13

1.0•10−12

1.2•10−12

1.4•10−12

Con

tribu

tion

func

tion

[J m

−2 s

−1 H

z−1 s

r−1 k

m−1

]

1082.6 1082.8 1083.0 1083.2 1083.4Wavelength [nm]

0

500

1000

1500

2000

Hei

ght [

km]

10x

2•10−12

4•10−12

6•10−12

Con

tribu

tion

func

tion

[J m

−2 s

−1 H

z−1 s

r−1 k

m−1

]

1082.6 1082.8 1083.0 1083.2 1083.4Wavelength [nm]

0

500

1000

1500

2000

Hei

ght [

km]


Source function for the He i 1083 nm triplet

0.0001 0.0010 0.0100 0.1000 1.0000 10.0000 100.0000Column Mass [kg m−2]

10−8

10−7S

, J [

J m

−2 s

−1 H

z−

1 s

r−1]

Stotal

J

BPlanck

Sactive

Sbackgr

1.0820 1.0825 1.0830 1.0835 1.0840λ[micron]


A cylindrical fluxtube with Wilson depression

Stenholm, Stenflo 1977, A&A 58, 273Stenholm, Stenflo 1978, A&A 67, 33

B

∆z

HSRA

HSRA

HSRA

I I

PG PG + PB


Magneto-static Fluxtube model, field structure

0.1

1.0

10.0

100.0

1000.0

log

B [G

auss

]

0.0

0.5

1.0

1.5

2.0

z [M

m]

0

20

40

60

80

incl

inat

ion

[deg

]

0 2 4 6 8 10x [Mm]

0.0

0.5

1.0

1.5

2.0

z [M

m]


Weakening of the Fe i 525.02 nm line

0.4

0.6

0.8

rela

tive

in

ten

sity

0 2000 4000 6000 8000 10000x [km]

525.005

525.010

525.015

525.020

525.025

525.030

525.035

wa

ve

len

gth

[n

m]

2D

0.4

0.6

0.8

rela

tive

in

ten

sity

525.005

525.010

525.015

525.020

525.025

525.030

525.035

wa

ve

len

gth

[n

m]

1.5D FeI 525


UV overionization of Fe i to Fe i284 J. H. M. J. Bruls and O. v. d. Luhe: Photospheric fine structure

the LTE f -values to be too small due to the neglect ofoverionization in above LTE analysis. Nevertheless, thisdoes not automatically imply that the absolute scale ofthe iron oscillator strengths is actually wrong by a factorof 2.5; more likely, the problem lies with the inadequacyof semi-empirical 1D average quiet-Sun models.

For reasons of limiting the computational effort we usehydrogenic photoionization cross sections, which do notdiffer markedly from the experimental cross sections listedby Lites (1972). However, they do differ considerably fromthe most reliable computed cross sections (Bautista 1997),which include numerous resonances and even show an in-crease towards shorter wavelengths. Fortunately, due tothe strong wavelength dependence of the intensities in thesolar photosphere, only the part immediately shortwardof the bound-free edge is relevant and there the differ-ences between the computed cross sections, with the res-onances smoothed, and hydrogenic cross sections are onlymoderate.

The large uncertainties in the electron collision rates,for which we employ the impact approximation (Seaton1962), don’t play a crucial role in most of the photosphere,where the the population departures of all levels are al-ready kept close together by the radiative coupling. Weneglect neutral hydrogen collisions, but note that theywould only constrain the population departures yet closertogether but not produce essential changes.

The validity of the model atom has been tested by com-paring its results for 1D and 2D model atmospheres withthose obtained for a significantly more complete modelatom. We finally note that it is not our intention to ex-actly model the Fe i 5247 and 5250 A lines for compari-son with observations–which would indeed require the useof the most accurate atomic data available–but to showthe behavior of typical photospheric lines in a flux sheetenvironment.

For the purpose of comparison, we also compute thestrongly scattering Ca iiH&K lines for some of the fluxsheet models. We employ the standard 6-level modelfor calcium (see, e.g., Uitenbroek 1989, and referencestherein), and also include (angle-averaged) partial redis-tribution of the line photons. Polarization not being animportant diagnostic for these lines, we restrict ourselvesto Stokes I profiles for a few inclination angles.

2.3. The radiative transfer computations

All non-LTE radiative transfer computations are carriedout by means of the new radiative transfer code RH de-veloped by Uitenbroek (1999) under the assumption thatthe field-free approximation is valid, i.e. that the magneticfield does not have a noticeable influence on the popu-lations. As shown by Bruls & Trujillo Bueno (1996) thefield-free approximation is a valid assumption for all situ-ations where the equilibrium is not dominated by strongmagnetically-sensitive long-wavelength lines.

3

1

2

LTE POPULATIONS

1

3

2

NLTE POPULATIONS

neutral iron

population depletion

Fig. 3. Properties of the iron ionization balance in the solarphotosphere; the 3 levels in the iron schematically representthe low- and mid-excitation neutral levels and the once ion-ized stage. The ionization equilibrium is set by the ultravio-let radiation field. At wavelenghts shorter than about 2000 A(upper left), responsible for the photoionization out of thelow-excitation levels, Jν lies close to Bν , whereas at longerwavelengths, contributing to the photoionization out of themid-excitation levels, Jν exceeds Bν (top panels). For LTEpopulations (bottom left) this means that for level 1 ioniza-tion and recombination are in balance (bottom left), whereasfor level 2 the photoionization exceeds the recombination. Thatthen leads to depletion of the population of level 2, that isshared with level 1 through strong lines and collisions (middlepanel), until there is a balance between the net ionization fromlevel 2 and the net recombination into level 1 (bottom right)

Given the unpolarized non-LTE solution, for the ironlines we also perform a formal solution by means of theSPSR code (Rees et al. 1989; Murphy & Rees 1990;Murphy 1990) to obtain the Stokes I and V profiles for8000 equidistant lines of sight in the x − z-plane, withinclination α w.r.t. the vertical.

3. Results: Iron

3.1. Statistical equilibrium

The main deviation from LTE of the iron populationscomes from the ionization balance. We therefore discussthat in some detail before continuing with the line profiles.

Bruls & vd Luhe 2001, A&A 366, 281


Source function of the Fe i 525.02 nm line

10−8

10−7

so

urc

e f

un

ctio

n

4000 4500 5000 5500 6000x [km]

−100

0

100

200

300

he

igh

t [k

m]

2D FeI 525

core

wing


Ionization at the ground-level edge

−12

−11

−10

−9

−8

log

Sou

rce

func

tion

[J m

−2

s−1

Hz−

1sr−

1 ]

4500 5000 5500x [Mm]

0

500

1000

1500

z [M

m]

core

edge

lambda = 156.8


non-lte radiative transfer examplesthe lte f -values to be too small due to the neglect of...

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