a. lagg - abisko winter school 1 the radiative transfer equation the radiative transfer equation ...
TRANSCRIPT
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
A. Lagg - Abisko Winter School 2
The radiative transfer equation
A. Lagg - Abisko Winter School 3
Goal of this lecture
Set of atmospheric parameters
A. Lagg - Abisko Winter School 4
Physical basis of the problem
Jefferies et al., 1989, ApJ 343
A. Lagg - Abisko Winter School 5
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):
A. Lagg - Abisko Winter School 6
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
A. Lagg - Abisko Winter School 7
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
A. Lagg - Abisko Winter School 8
Absorption / dispersion coefficients
real (absorption, δ) and imaginary (dispersion, κ) part of refractive index nα:
absorption coefficient:
dispersion coefficient:
A. Lagg - Abisko Winter School 9
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
A. Lagg - Abisko Winter School 10
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
A. Lagg - Abisko Winter School 11
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!)
A. Lagg - Abisko Winter School 12
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:
A. Lagg - Abisko Winter School 13
Abs./disp. & thermal motions JCdTI, Spectropolarimetry
shift due to LOS-velocity
A. Lagg - Abisko Winter School 14
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
A. Lagg - Abisko Winter School 15
Fast computation of Faraday&Voigt Borrero et al. (2008)
A. Lagg - Abisko Winter School 16
Fast computation of Faraday&Voigt
implemented in VFISV (Borrero et al, 2009)VFISV Paper & Download
A. Lagg - Abisko Winter School 17
Light propagation through low-densityweakly conducting media
EM wave in vacuum:
conductive media:
solution:
no absorption without conductivity!
absorption & dispersion profiles
wave number:
A. Lagg - Abisko Winter School 18
The radiative transfer equation
Geometry: Observers frame (line-of-sight) ↔ magn. field frame
JcdTI, Spectropolarimetry
LOS
B-field
Stokes vector defined in XY plane
inclination
azimuth
A. Lagg - Abisko Winter School 19
Coordinate transformations (1)
Now: define orthonormal complex vectors (frame of abs/disp profiles)
≡ transf. between princ. comp. of vector electric field and Cart. comp.
A. Lagg - Abisko Winter School 20
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)
A. Lagg - Abisko Winter School 21
Transformation to Stokes vector
Stokes vector: measurable quantity (real) energy quantity (time
averages)
Convenient writing usingmatrices:
Pauli matrices
A. Lagg - Abisko Winter School 22
RTE in Stokes vector
easily transforms to: (RTE = Radiative Transfer Equation)
A. Lagg - Abisko Winter School 23
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)
A. Lagg - Abisko Winter School 24
Similar approach: see also Jefferies et al., 1989, ApJ 343
R
T
(1 – Ndz)
(T)-1
(R)-1
A. Lagg - Abisko Winter School 25
Emission Processes
emissive properties of the medium: source function vector
A. Lagg - Abisko Winter School 26
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)
A. Lagg - Abisko Winter School 27
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
A. Lagg - Abisko Winter School 28
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
A. Lagg - Abisko Winter School 29
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
A. Lagg - Abisko Winter School 30
Absorption of Zeeman components
A. Lagg - Abisko Winter School 31
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
A. Lagg - Abisko Winter School 32
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 splitting
Unchanged:• computation of abs/disp
coefficients
Assumption:LS-coupling (Russel Saunders)
A. Lagg - Abisko Winter School 33
Computation of Zeeman pattern
Position (shift to central wavelength/frequency):
B in G, λ in ǺLandé factor in LS coupling:
strength of Zeeman components:
A. Lagg - Abisko Winter School 34
Examples of Zeeman patterns
A. Lagg - Abisko Winter School 35
Examples of Zeeman patterns
A. Lagg - Abisko Winter School 36
Examples of Zeeman patterns
A. Lagg - Abisko Winter School 37
The elements of the propagation matrix (1)
normalized abs./disp. profiles are now given by:
A. Lagg - Abisko Winter School 38
The elements of the propagation matrix (2)
Elements remain formally the same (see slide RTE in Stokes vector)
A. Lagg - Abisko Winter School 39
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:
A. Lagg - Abisko Winter School 40
Effective Landé factor – example 1
A. Lagg - Abisko Winter School 41
Effective Landé factor – example 2
A. Lagg - Abisko Winter School 42
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
A. Lagg - Abisko Winter School 43
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
A. Lagg - Abisko Winter School 44
Solving the RTE
A. Lagg - Abisko Winter School 45
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:
A. Lagg - Abisko Winter School 46
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
A. Lagg - Abisko Winter School 47
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∞
A. Lagg - Abisko Winter School 48
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).
A. Lagg - Abisko Winter School 49
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:
A. Lagg - Abisko Winter School 50
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
A. Lagg - Abisko Winter School 51
Milne-Eddington - Demo
A. Lagg - Abisko Winter School 52
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!
A. Lagg - Abisko Winter School 53
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
A. Lagg - Abisko Winter School 54
Macroturbulence
A. Lagg - Abisko Winter School 55
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