1. A. Lagg - Abisko Winter School 1  The radiative transfer equation The radiative transfer equation  Solving the RTE Solving the RTE Exercise 1:Exercise 1: forward module for ME-type atmosphere  The HeLIx + inversion code The HeLIx + inversion code  Genetic algorithms Genetic algorithms Exercise II:Exercise II: basic usage of HeLIx + Andreas Lagg Max-Planck-Institut für Sonnensystemforschung Katlenburg-Lindau, Germany  Hinode inversion strategy Hinode inversion strategy Exercise III:Exercise III: Hinode inversions using HeLIx +, identify & discuss inversion problems  SPINOR – RF based inversions SPINOR – RF based inversions Exercise IV:Exercise IV: installation and basic usage  Science with HeLIx + Science with HeLIx + Exercise / discussion time

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3. A. Lagg - Abisko Winter School 3 Set of atmospheric parameters

4. A. Lagg - Abisko Winter School 4 Jefferies et al., 1989, ApJ 343

5. A. Lagg - Abisko Winter School 5 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. 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):

6. A. Lagg - Abisko Winter School 6 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. A. Lagg - Abisko Winter School 7 Choose system of complex unit vectors: clockwise / counter-clockwise around e 0 linear along e 0 Equation of motion: Solution for individual displacement components: Proportionality between el. field and displacement (D=εE):

8. A. Lagg - Abisko Winter School 8 real (absorption, δ) and imaginary (dispersion, κ) part of refractive index n α : absorption coefficient: dispersion coefficient:

9. A. Lagg - Abisko Winter School 9 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. A. Lagg - Abisko Winter School 10 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. A. Lagg - Abisko Winter School 11 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!)

12. A. Lagg - Abisko Winter School 12 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:

13. A. Lagg - Abisko Winter School 13 shift due to LOS-velocity

14. A. Lagg - Abisko Winter School 14 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 2 nd order Taylor expansion

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16. A. Lagg - Abisko Winter School 16 implemented in VFISV (Borrero et al, 2009) VFISV Paper & Download VFISV Paper & Download

17. A. Lagg - Abisko Winter School 17 EM wave in vacuum: conductive media: solution: no absorption without conductivity! absorption & dispersion profiles wave number:

18. A. Lagg - Abisko Winter School 18 Geometry: Observers frame (line-of-sight) ↔ magn. field frame JcdTI, Spectropolarimetry LOS B-field Stokes vector defined in XY plane inclination azimuth

19. A. Lagg - Abisko Winter School 19 Now: define orthonormal complex vectors (frame of abs/disp profiles) ≡ transf. between princ. comp. of vector electric field and Cart. comp.

20. A. Lagg - Abisko Winter School 20 Variation of electric field vector in LOS frame along z contains: absorption / dispersion coefficients geometry (azimuth and inclination) (upper left 2x2 part)

21. A. Lagg - Abisko Winter School 21 Stokes vector:  measurable quantity (real)  energy quantity (time averages) Convenient writing using matrices: Pauli matrices

22. A. Lagg - Abisko Winter School 22 easily transforms to: (RTE = Radiative Transfer Equation)

23. A. Lagg - Abisko Winter School 23 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. A. Lagg - Abisko Winter School 24 R T (1 – Ndz) (T) -1 (R) -1

25. A. Lagg - Abisko Winter School 25 emissive properties of the medium: source function vector

26. A. Lagg - Abisko Winter School 26  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. A. Lagg - Abisko Winter School 27 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. A. Lagg - Abisko Winter School 28 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 (τ c =0) is locatedat observer. Optical depth τ c is the (dimensionless) number of mean free paths of continuum photons between outermost boundary (z 0 ) and point z. RTE is then: with

29. A. Lagg - Abisko Winter School 29 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

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31. A. Lagg - Abisko Winter School 31 absorption and dispersion profiles dashed/solid: weak/strong Zeeman splitting Note: broad wings in ρ V  RT calculations must be performed quite far from line core Q,U only differ in scale

32. A. Lagg - Abisko Winter School 32 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)

33. A. Lagg - Abisko Winter School 33 Position (shift to central wavelength/frequency): B in G, λ in Ǻ Landé factor in LS coupling: strength of Zeeman components:

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37. A. Lagg - Abisko Winter School 37 normalized abs./disp. profiles are now given by:

38. A. Lagg - Abisko Winter School 38 Elements remain formally the same (see slide RTE in Stokes vector)RTE in Stokes vector

39. A. Lagg - Abisko Winter School 39 How useful is this approximation? often used: effective Landé factor g eff Calculation: barycenter of individual Zeeman transitions  2 sigma, 1 pi component (strength unity)  pi component at central wavelength  sigma components:

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42. A. Lagg - Abisko Winter School 42  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. A. Lagg - Abisko Winter School 43  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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45. A. Lagg - Abisko Winter School 45  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. A. Lagg - Abisko Winter School 46 homogeneous equation: define linear operator (=evolution operator) giving transformation of homogeneous solution between two points at optical depths τ’ C and τ C : 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. A. Lagg - Abisko Winter School 47 homogeneous equation: define linear operator (=evolution operator) giving transformation of homogeneous solution between two points at optical depths τ’ C and τ C : multiply RTE by  integration over optical depth  formal solution for τ 1 =0 and τ 0  ∞

48. A. Lagg - Abisko Winter School 48 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. 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. A. Lagg - Abisko Winter School 49 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: 2 nd assumption: Source function vector depends linearly with height: Formal solution then becomes:

50. A. Lagg - Abisko Winter School 50 analytical integration of this equation yields only first element of S 0 and S 1 is non-zero  for Stokes vector we only need to compute first column of K 0 -1 with and the determinant of the propagation matrix

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52. A. Lagg - Abisko Winter School 52 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!

53. A. Lagg - Abisko Winter School 53 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) Gaussiane.g. telescope PSF

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55. A. Lagg - Abisko Winter School 55  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 JU 6302.4936 Fe -1.235 7.50 2.5 2.0 1.0 1.0 2.0 3.0 0.0  2 nd line? ;WL Element LOG_GF ABUND GEFF SL LL JL SU LU JU 6301.5012 Fe -0.718 7.50 1.5 2.0 1.0 2.0 2.0 3.0 2.0