Abisko Winter School: Inversion of the Radiative Transfer Equation

A. Lagg - Abisko Winter School 1

Abisko Winter School:Inversion of the Radiative Transfer Equation

The radiative transfer equation

Solving the RTEExercise 1: forward module for

ME-type atmosphere

The HeLIx+ inversion code Genetic algorithmsExercise II: basic usage of

HeLIx+

Andreas LaggMax-Planck-Institut für SonnensystemforschungKatlenburg-Lindau, Germany

Hinode inversion strategyExercise 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




Goal of this lecture

Set of atmospheric parameters


Physical basis of the problem

Jefferies et al., 1989, ApJ 343


Absorption and Dispersion profiles

Medium: made of atoms (electrons surrounding pos. Nucleus) individual displacements can be thought of as electric dipoles:

JCdTI, Spectropolarimetry

vector position of e- motion induced by ext. fielde- chargepolarization of single dipole

N = number density electric polarization vector P

overall electric displacement (4π accounts for all possible directions of impinging radiation):


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


Equation of motion

Choose system of complex unit vectors:

clockwise / counter-clockwisearound e0

linear along e0 QM-picture: corresponds to 3 pure quantum states mj=+1,0,-1 linked to left circular, linear and right circular

Equation of motion:

Solution for individual displacement components:

Proportionality between el. field and displacement (D=εE):

square of complex refractive index nα

2


Absorption / dispersion coefficients

real (absorption, δ) and imaginary (dispersion, κ) part of refractive index nα:

absorption coefficient:

dispersion coefficient:


Absorption / dispersion profiles JCdTI, Spectropolarimetry

Absorption profiles:account for the drawing of electromagnetic energy by the mediumDispersion profiles:explain the change in phase undergone by light streaming through the medium


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


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!)


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:


Abs./disp. & thermal motions JCdTI, Spectropolarimetry

shift due to LOS-velocity


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


Fast computation of Faraday&Voigt Borrero et al. (2008)


Fast computation of Faraday&Voigtimplemented in VFISV (Borrero et al, 2009)VFISV Paper & Download

http://arxiv.org/abs/0901.2702


Light propagation through low-densityweakly conducting media

EM wave in vacuum:

conductive media:

solution:

no absorption without conductivity!

absorption & dispersion profiles

wave number:



Geometry: Observers frame (line-of-sight) ↔ magn. field frame

JcdTI, Spectropolarimetry

LOSB-field

Stokes vector defined in XY plane

inclination

azimuth


Coordinate transformations (1)

Now: define orthonormal complex vectors (frame of abs/disp profiles)

≡ transf. between princ. comp. of vector electric field and Cart. comp.


Coordinate transformations (2)

Variation of electric field vector in LOS frame along z

contains:• absorption / dispersion coefficients• geometry (azimuth and inclination)

(upper left 2x2 part)


Transformation to Stokes vector

Stokes vector: measurable quantity (real) energy quantity (time

averages)

Convenient writing usingmatrices:

Pauli matrices


RTE in Stokes vector

easily transforms to: (RTE = Radiative Transfer Equation)


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)


Similar approach: see also Jefferies et al., 1989, ApJ 343

R

T

(1 – Ndz)

(T)-1

(R)-1


Emission Processes

emissive properties of the medium: source function vector


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)


RTE for spectral line formation

Propagation matrix K must contain contributions from• continuum froming and• line formingprocesses:

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


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


Switch on magnetic field

Lorentz model of the atom (classical approach):

assume:medium is isotropic

Now: apply a magnetic field:

Lorentz force acts on the atom:

take component α:

results in shift of abs/disp profiles:

interpretation of angles as azimuth and inclination

red, central and blue component


Absorption of Zeeman components


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


Quantum mechanical modifications

Simple Lorentz model explains only shape of normal Zeeman triplet profiles quantum mechanical treatment mandatory

Changes compared to Lorentz:• number of Zeeman sublevels• strength of Zeeman

components• WL-shift for splittingUnchanged:• computation of abs/disp

coefficients

Assumption:LS-coupling (Russel Saunders)


Computation of Zeeman pattern

Position (shift to central wavelength/frequency):

B in G, λ in ǺLandé factor in LS coupling:

strength of Zeeman components:


Examples of Zeeman patterns






The elements of the propagation matrix (1)

normalized abs./disp. profiles are now given by:


The elements of the propagation matrix (2)

Elements remain formally the same (see slide RTE in Stokes vector)


Effective Zeeman triplet

How useful is this approximation?

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:


Effective Landé factor – example 1


Effective Landé factor – example 2


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


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


Solving the RTE


Model atmospheres

Medium specified by physical parameters as a function of distance

this determines the local values foroptical depthpropagation matrixsource function vector

set of such parameters:


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


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∞


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 MatricesNumerical 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. 805-817.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).


The Milne-Eddington solution

In special cases an analytic solution of the RTE is possible.Most prominent example: Milne-Eddington atmosphere (Unno Rachkowsky solution)

Unno (1956), Rachkowsky (1962, 1967)

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:


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


Milne-Eddington - Demo


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!


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


Macroturbulence


Exercise Iforward (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 GEFF SL LL JL SU LU JU6302.4936 Fe -1.235 7.50 2.5 2.0 1.0 1.0 2.0 3.0 0.0

2nd line?;WL Element LOG_GF ABUND GEFF SL LL JL SU LU JU6301.5012 Fe -0.718 7.50 1.5 2.0 1.0 2.0 2.0 3.0 2.0

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Abisko Winter School: Inversion of the Radiative Transfer Equation