Abisko Winter School: Inversion of the Radiative Transfer Equation

1. Abisko Winter School:Inversion of the Radiative Transfer Equation Andreas LaggMax-Planck-InstitutfürSonnensystemforschungKatlenburg-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

2. The radiative transfer equation

3. Goal of this lecture Set of atmospheric parameters

4. Physical basis of the problem Jefferies et al., 1989, ApJ 343

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):

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

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 nα2

8. Absorption / dispersion coefficients real (absorption, δ) and imaginary (dispersion, κ) part of refractive index nα: absorption coefficient: dispersion coefficient:

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

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

11. Quantum-mechanical correction – line formation Lorentz results are exact for electric dipole transitions when compared with rigorous quantum-mechanical calculation. • Exception: • 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) • more complex splitting than normal Zeeman triplet is necessary • a re-interpretation of the damping factor (not well understood quantitatively in either classical or QM case!)

12. Thermal motions in the medium Every atom in the medium has a non-zero velocity component.Assumption: Maxwellian velocity distribution: micro-turbulence velocity(ad-hoc parameter), takes into account motions on smaller scales than mean free path of photons Doppler width  absorption / dispersion profiles must be convolved with a Gaussian or in wavelength: use reduced variables:

13. Abs./disp. & thermal motions JCdTI, Spectropolarimetry shift due to LOS-velocity

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

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

16. Fast computation of Faraday&Voigt implemented in VFISV (Borrero et al, 2009)VFISV Paper & Download

17. Light propagation through low-densityweakly conducting media EM wave in vacuum: no absorption without conductivity! conductive media: wave number: solution: absorption & dispersion profiles

18. The radiative transfer equation Geometry: Observers frame (line-of-sight) ↔ magn. field frame JcdTI, Spectropolarimetry LOS B-field inclination azimuth Stokes vector defined in XY plane

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.

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)

21. Transformation to Stokes vector Stokes vector: • measurable quantity (real) • energy quantity (time averages) Pauli matrices Convenient writing usingmatrices:

22. RTE in Stokes vector easily transforms to: (RTE = Radiative Transfer Equation)

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)

24. Similar approach: see also Jefferies et al., 1989, ApJ 343 R T (1 – Ndz) (T)-1 (R)-1

25. Emission Processes emissive properties of the medium: source function vector

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)

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

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

29. Switch on magnetic field Lorentz model of the atom (classical approach): assume:medium is isotropic  interpretation of angles as azimuth and inclination Now: apply a magnetic field: Lorentz force acts on the atom: take component α: results in shift of abs/disp profiles:  red, central and blue component

30. Absorption of Zeeman components

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

32. 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

33. Computation of Zeeman pattern Position (shift to central wavelength/frequency): • B in G, λ in Ǻ Landé factor in LS coupling: strength of Zeeman components:

34. Examples of Zeeman patterns

35. Examples of Zeeman patterns

36. Examples of Zeeman patterns

37. The elements of the propagation matrix (1) normalized abs./disp. profiles are now given by:

38. The elements of the propagation matrix (2) Elements remain formally the same (see slide RTE in Stokes vector)

39. 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?

40. Effective Landé factor – example 1

41. Effective Landé factor – example 2

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

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

44. Solving the RTE

45. 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:

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

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∞

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).

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:

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