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Abisko Winter School:
Inversion of the Radiative Transfer Equation
Andreas Lagg
Max-Planck-Institut für Sonnensystemforschung
Katlenburg-Lindau, Germany
 The radiative transfer
equation
 Solving the RTE
Exercise 1: forward module for
ME-type atmosphere
 The HeLIx+ inversion code
 Genetic algorithms
Exercise II: basic usage of
HeLIx+
 Hinode inversion strategy
Exercise III: Hinode inversions
using HeLIx+, identify &
discuss inversion problems
 SPINOR – RF based
inversions
Exercise IV: installation and
basic usage
 Science with HeLIx+
Exercise / discussion time
A. Lagg - Abisko Winter School 1
The radiative transfer
equation
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Goal of this lecture
Set of atmospheric
parameters
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Physical basis of the problem
Jefferies et al., 1989, ApJ 343
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Absorption and Dispersion profiles
JCdTI, Spectropolarimetry
Medium: made of atoms (electrons surrounding pos. Nucleus)
 individual displacements can be thought of as electric dipoles:
vector position of e- motion induced by ext. field
e- charge
polarization of single dipole
N = number density
 electric polarization vector P
overall electric displacement (4π
accounts for all possible directions
of impinging radiation):
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Classical oscillator model
Classical computation using Lorentz electron theory.
Electron can be seen as superposition of classical oscillators:
time dependent, complex
amplitude of motion
Oscillators are excited by force associated with external field:
quasi-chromatic, plane wave
restoring force (quasi-elastic):
force constant:
damped by resisting force:
damping tensor (diagonal)
e- mass
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Equation of motion
Choose system of complex unit vectors:
linear along e0
clockwise / counter-clockwise
around e0
QM-picture: corresponds to 3 pure
quantum states mj=+1,0,-1
 linked to left circular, linear and
right circular
square of complex
refractive index nα2
Equation of motion:
Solution for individual displacement components:
Proportionality between el. field and displacement (D=εE):
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Absorption / dispersion coefficients
real (absorption, δ) and imaginary (dispersion, κ)
part of refractive index nα:
absorption coefficient:
dispersion coefficient:
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Absorption / dispersion profiles
JCdTI, Spectropolarimetry
Absorption profiles:
account for the drawing of electromagnetic energy by the medium
Dispersion profiles:
explain the change in phase undergone by light streaming through the medium
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Quantum-mechanical correction – continuum
Medium has many resonances (atoms, molecules).
 bound-bound transistions (spectral lines)
 bound-free transisiton (ionization&recombination)
 free-free „transitions“ (zero resonant frequency)
 continuous absorption takes place
Assumption: negligible anisotropies for continuum radiation:
 all for Stokes parameters are multiplied by same factor:
if continuum radiation is unpolarized on input it remains
unpolarized on output.
Note: within limited range of spectral line the continuous abs/disp profiles remain
esentially constant  dropped frequency dependence
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Quantum-mechanical correction – line formation
Lorentz results are exact for electric dipole transitions when
compared with rigorous quantum-mechanical calculation.
Exception:
(1) frequency-integrated strength of the profiles is
modified:
oscillator strength (proportional to
square modulus of the dipole matrix
element between lower and upper
level involved in the transition)
(2) more complex splitting than normal Zeeman triplet is
necessary
(3) a re-interpretation of the damping factor (not well
understood quantitatively in either classical or QM
case!)
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Thermal motions in the medium
Every atom in the medium has a non-zero velocity component.
Assumption: Maxwellian velocity distribution:
Doppler width
micro-turbulence velocity
(ad-hoc parameter), takes
into account motions on
smaller scales than mean
free path of photons
 absorption / dispersion profiles must be convolved with a Gaussian
use reduced variables:
or in wavelength:
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Abs./disp. & thermal motions
JCdTI, Spectropolarimetry
shift due to LOS-velocity
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Faraday & Voigt functions
important: fast algorithm for efficient computation
Hui et al. (1977):
H & F are the real and imaginary parts of the quotient of a
complex 6th order polynomial. Slow but accurate.
Borrero et al:
Fast computation using 2nd order Taylor expansion
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Fast computation of Faraday&Voigt
Borrero et al. (2008)
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Fast computation of Faraday&Voigt
implemented in VFISV (Borrero et al, 2009)
VFISV Paper & Download
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Light propagation through low-density
weakly conducting media
EM wave in vacuum:
no absorption
without conductivity!
conductive media:
wave number:
solution:
absorption & dispersion profiles
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The radiative transfer equation
JcdTI, Spectropolarimetry
Geometry: Observers frame (line-of-sight) ↔ magn. field frame
LOS
B-field
inclination
azimuth
Stokes vector
defined in XY
plane
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Coordinate transformations (1)
Now: define orthonormal complex vectors (frame of abs/disp profiles)
≡ transf. between princ. comp. of vector electric field and Cart. comp.
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Coordinate transformations (2)
Variation of electric field vector in LOS frame along z
(upper left
2x2 part)
contains:
• absorption / dispersion coefficients
• geometry (azimuth and inclination)
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Transformation to Stokes vector
Stokes vector:
 measurable quantity (real)
 energy quantity (time
averages)
Convenient writing using
matrices:
Pauli matrices
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RTE in Stokes vector
easily transforms to: (RTE = Radiative Transfer Equation)
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The propagation matrix
absorption: energy from all
polarization states is withdrawn
by the medium (all 4 Stokes
parameters the same!)
dichroism: some polarized
components of the beam are
extinguished more than others
because matrix elements are
generally different
dispersion: phase shifts that take place
during the propagation change different
states of lin. pol. among themselves
(Faraday rotation) and states of lin. pol.
with states of circ. pol. (Faraday pulsation)
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Similar approach: see also Jefferies et al., 1989, ApJ 343
R
T
(1 – Ndz)
(T)-1
(R)-1
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Emission Processes
emissive properties of the medium: source function vector
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Local Thermodynamic Equilibrium





only radiation (and not matter) is allowed to deviate from
thermodynamic equilibrium
all thermodynamic properties of matter are governed by the
thermodynamic equlibrium equations but at the local values
for temperature and density
 local distribution of velocities is Maxwellian
local number of absorbers and emitters in various quantum
states is given by Boltzmann and Saha equations
Kirchhoff‘s law is verified (emission = absorption)
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RTE for spectral line formation
Propagation matrix K must contain contributions from
• continuum froming and
• line forming
processes:
frequency-independent absorption coefficient for continuum:
frequency-dependent propagation matrix for spectral line:
contains normalized absorption
and dispersion profiles
line-to-continuum
absorption coefficient ratio
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Optical depth
Convenient: replace height dependence (z) by optical depth (τ)
Note: optical depth definied in the opposite direction of the ray path
(i.e. –z), origin (τc=0) is locatedat observer.
Optical depth τc is the (dimensionless) number of mean free paths of
continuum photons between outermost boundary (z0) and point z.
RTE is then:
with
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Switch on magnetic field
Lorentz model of the atom (classical approach):
assume:
medium is
isotropic
Now: apply a magnetic field:
 interpretation of angles as
azimuth and inclination
Lorentz force acts on the
atom:
take component α:
results in shift of
abs/disp profiles:
 red, central and blue component
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Absorption of Zeeman components
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normal Zeeman triplet
absorption and
dispersion profiles
dashed/solid:
weak/strong Zeeman
splitting
Note: broad wings in ρV
 RT calculations must
be performed quite far
from line core
Q,U only differ in scale
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Quantum mechanical modifications
Simple Lorentz model explains
only shape of normal Zeeman
triplet profiles
 quantum mechanical
treatment mandatory
Assumption:
LS-coupling (Russel Saunders)
Changes compared to Lorentz:
• number of Zeeman sublevels
• strength of Zeeman
components
• WL-shift for splitting
Unchanged:
• computation of abs/disp
coefficients
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Computation of Zeeman pattern
Position (shift to central wavelength/frequency):
Landé factor in LS coupling:
B in G, λ in Ǻ
strength of Zeeman
components:
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Examples of Zeeman patterns
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Examples of Zeeman patterns
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Examples of Zeeman patterns
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The elements of the propagation matrix (1)
normalized abs./disp. profiles are now given by:
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The elements of the propagation matrix (2)
Elements remain formally the same (see slide RTE in Stokes vector)
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Effective Zeeman triplet
often used: effective Landé factor geff
Calculation: barycenter of individual Zeeman
transitions
 2 sigma, 1 pi component (strength unity)
 pi component at central wavelength
 sigma components:
How useful is this approximation?
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Effective Landé factor – example 1
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Effective Landé factor – example 2
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Summary: RTE in presence of a magnetic field
 Continuum radiation is unpolarized
 medium is assumed to be isotropic as far as continuum
formation processes are concerned
 thermal velocity distribution is Maxwellian (Doppler width can
include microturbulence)
 Absorption processes are assumed to be
 linear
 invariant against translations of variable
 continous
(=basis for dealing with line broadening and Doppler shifting
through convolutions)
 material properties are constant in planes perpendicular to a
given direction (plane parallel model, stratified atmosphere)
 absorptive, dispersive and emissive properties of the medium
are independent of the light beam Stokes vector
 radiation field is independent of time
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Summary: RTE in presence of a magnetic field (cont‘d)
 effects of refractive index gradient on EM wave equation are
ignored
 all thermodynamic properties of matter are assumed to be
governed by thermodynamic equilibrium equations at the local
temperatures and desnities (LTE hypothesis)
 scattering takes place in conditions of complete redistribution
 no correlation exists between the frequencies of the incoming
and scattered photons
 all Zeeman sublevels are equally populated and no coherences
exist among them
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Solving the RTE
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Model atmospheres



Medium specified by physical parameters as a
function of distance
this determines the local values for
 optical depth
 propagation matrix
 source function vector
set of such parameters:
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Formal solution of the RTE
homogeneous equation:
define linear operator (=evolution operator) giving transformation of
homogeneous solution between two points at optical depths τ’C and τC:
multiply RTE by
 integration over optical depth
I of light streaming through the
medium (no emission within medium)
contribution from emission,
accounted for by KS
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Formal solution of the RTE
homogeneous equation:
define linear operator (=evolution operator) giving transformation of
homogeneous solution between two points at optical depths τ’C and τC:
multiply RTE by
 integration over optical depth
 formal solution for τ1=0 and τ0∞
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Actual solutions of the RTE
RTE has no simple analytical solution (in general). In most
instances, only numerical approaches to the evolution operator can
be found.
Details of this numerical solution:
Egidio Landi Degl'Innocenti:
Transfer of Polarized Radiation, using 4 x 4 Matrices
Numerical Radiative Transfer, edited by Wolfgang Kalkofen.
Cambridge: University Press, 1987.
Bellot Rubio et al:
An Hermitian Method for the Solution of Polarized Radiative Transfer
Problems, The Astrophysical Journal, Volume 506, Issue 2, pp. 805817.
Semel and López-Ariste:
Integration of the radiative transfer equation for polarized light: the
exponential solution, Astronomy and Astrophysics, v.342, p.201-211
(1999).
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The Milne-Eddington solution
Unno (1956), Rachkowsky (1962, 1967)
In special cases an analytic solution of the RTE is possible.
Most prominent example: Milne-Eddington atmosphere (Unno
Rachkowsky solution)
 all atmospheric parameters are
independent of height and direction
In this case, the evolution operator is:
2nd assumption: Source function vector depends linearly with height:
Formal solution then becomes:
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ME-solution: Stokes vector
analytical integration of this equation yields
only first element of S0 and S1 is non-zero
 for Stokes vector we only need to compute first column of K0-1
with
and the determinant of the propagation matrix
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Milne-Eddington - Demo
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Symmetry properties of RTE solution
transform propagation matrix:
assume: no changes in LOS velocity throughout atmosphere
consequence: net circular polarization of a line is always zero in the
absence of velocity gradients:
in other words: if the NCP≠0  velocity gradients must be present!
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Line broadening
Observed profiles are often wider than synthetic profiles of same
equivalent width (i.e. profiels absorbing the same amount of energy
from the continuum radiation).
Effect can be caused by:
 macroturbulence: unresolved motions within spatial resolution
element (turbulence larger than the mean free path of the
photons). Ad-hoc parameter (no actual physical reasoning)
 assumed to be height independent
 instrumental broadening of the line profiles (limited resolution of
telescope and limited resolution of spectrograph, filter profiles)
Gaussian
e.g. telescope PSF
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Macroturbulence
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Exercise I
forward (synthesis) module

write computer code to compute
elements of propagation matrix
 write forward module for Stokes profile
calculation in ME type atmosphere
 display results for various atmospheric
parameters

suggested spectral line:
;WL
Element LOG_GF ABUND
6302.4936 Fe
-1.235 7.50

GEFF
2.5
SL
2.0
LL
1.0
JL
1.0
SU
2.0
LU
3.0
JU
0.0
GEFF
1.5
SL
2.0
LL
1.0
JL
2.0
SU
2.0
LU
3.0
JU
2.0
2nd line?
;WL
Element LOG_GF ABUND
6301.5012 Fe
-0.718 7.50
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