non-lte radiative transfer examplesthe lte f -values to be too small due to the neglect of...
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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
Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples
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.
Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples
Temperature stratification of solar models
Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples
Mg i Grotrian diagram
Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples
Comparison of observed and calculated Mg i 12 µ profiles
1992A&A...253..567C
Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples
Formation heights and source function
Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples
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‖
Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples
Full disk images of He i EQW and Fexii 19.3 nm
Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples
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
Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples
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
Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples
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
Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples
EUV irradiation from Corona populates triplet levels
Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples
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
Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples
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
Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples
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
Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples
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]
Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples
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]
Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples
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]
Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples
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]
Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples
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
Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples
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]
Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples
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
Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples
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
Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples
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
Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples
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
Han Uitenbroek/NSO Non-LTE Radiative Transfer Examples