astro.physics.muni.cz · Bibliographic Entry Author: Adam Tichý Masaryk Universit,y acultFy of...

MASARYK UNIVERSITY

FACULTY OF SCIENCE

Polarized Radiative Transfer in Inhomogeneous

Stellar Atmospheres

Ph.D. Dissertation

ADAM TICHÝ

Supervisor: doc. RNDr. Ji°í Kubát, CSc.

Department of Theoretical Physics and Astrophysics

Brno 2019

Bibliographic Entry

Author: Adam TichýMasaryk University,Faculty of Science,Department of Theoretical Physics and Astrophysics

Title of Dissertation: Polarized Radiative Transfer in InhomogeneousStellar Atmospheres

Degree Programme: P°F D-FY4 Physics

Field of Study: P°F TEFY Theoretical Physics and Astrophysics

Supervisor: doc. RNDr. Ji°í Kubát, CSc.Academy of Sciences of the Czech Republic,Astronomical Institute,Ond°ejov, Czech Republic

Academic Year: 2018/2019

Number of Pages: 127

Keywords: radiative transfer; stellar atmospheres;NLTE; polarization; scattering; inhomogeneities;line formation; Hanle e�ect

Bibliogra�cký záznam

Autor: Adam TichýMasarykova univerzita,P°írodov¥decká fakulta,Ústav teoretické fyziky a astrofyziky

Název Práce: Polarizovaný p°enos zá°ení v nehomogenníchhv¥zdných atmosférách

Studijní program: P°F D-FY4 Fyzika

Studijní obor: P°F TEFY Teoretická fyzika a astrofyzika

kolitel: doc. RNDr. Ji°í Kubát, CSc.Akademie v¥d eské republiky,Astronomický ústav,Ond°ejov, eská republika

Akademický rok: 2018/2019

Po£et stran: 127

Klí£ová slova: p°enos zá°ení; hv¥zdné atmosféry; NLTE;polarizace; rozptyl; nehomogenity;formování £ar; Hanleho jev

Abstract

Nowadays, multi-dimensional radiative transfer provides a necessary tool to simulatemany NLTE lines, which are formed at atmospheric regions characterized by geometrical orphysical asymmetries. Physical asymmetries may represent, for instance, an MHD driveninhomogeneities in plasma thermodynamic structure and its magnetism, inhomogeneitiescaused by convective-driven motions, dynamics via various shock-waves or stellar winds,etc.

In the presented thesis we study the problem of the generation and transfer of resonantspectral line intensity and polarization in 3D models of stellar atmospheres, characterizedby horizontal inhomogeneities in plasma kinetic temperature and atomic volume density.For that purpose we solve the NLTE radiative transfer problem using PORTA code byt¥pán and Trujillo Bueno (2013), taking into account resonant scattering polarizationand its modi�cation by magnetic �elds via the Hanle e�ect.

We show that horizontal �uctuations of the thermodynamical structure of stellar at-mospheres can have a signi�cant impact on the intensity and linear polarization of theemergent spectral line radiation, as well as its center-to-limb variation. It applies in bothspatially resolved and spatially averaged cases.

If an interpretation of a stellar spectra is based upon models in plane-parallel geome-tries, it is possible to erroneously deduce the thermodynamic structure of the observedatmosphere; 3D simulations show that horizontal inhomogeneities may in�uence the emer-gent radiation in a similar way as plane-parallel model solutions with speci�c thermalstrati�cation. Ambiguity of the interpretation of observation in terms of stellar atmo-spheric structure thus raises a question on suitability of plane-parallel approximation,which should serve as an average representation of a generally inhomogeneous, realisticstellar atmosphere. Similarly, the polarization signal produced by scattering processes canbe in the presence of horizontal inhomogeneities erroneously interpreted as being due tohorizontal magnetic �elds via the Hanle e�ect.

The knowledge of atmospheric magnetism and thermodynamic structure is crucial forthe proper interpretation of observed stellar spectra; multi-dimensional radiative transferand polarization of radiation thus provides an essential tool for better understanding cer-tain spectral phenomena typical for Solar observations, and for its possible use for stellarobservations.

Abstrakt

V sou£asné dob¥ p°edstavuje multi-dimenzionální p°enos zá°ení nezbytný nástrojk modelování mnoha NLTE £ar formovaných v oblastech atmosfér, charakteristických geo-metrickými £i fyzikálními asymetriemi. Fyzikální asymetrie mohou p°edstavovat nap°íkladMHD efekty v podob¥ nehomogenit v termodynamické struktu°e £i magnetizmu plasmatu,nehomogenity zp·sobené konvektivními pohyby, dynamikou ²okových vln £i hv¥zdnýchv¥tr·, atd.

V p°edloºené práci se zabýváme studiem formování a p°enosu intenzity a polarizacezá°ení v rezonan£ní £á°e ve 3D modelech hv¥zdných atmosfér, charakteristických horizon-tálními nehomogenitami v kinetické teplot¥ a atomové hustot¥ plasmatu. Pro tyto ú£elyhledáme °e²ení problému NLTE p°enosu zá°ení pomocí programu PORTA od t¥pán andTrujillo Bueno (2013), p°i£emº bereme v úvahu rezonan£ní rozptylovou polarizaci a vlivymagnetických polí skrze Hanleho jev.

Ukazujeme, ºe horizontální nehomogenity v termodynamické struktu°e hv¥zdných at-mosfér mohou výrazn¥ ovlivnit intenzitu a polarizaci vystupujícího zá°ení ve spektrální£á°e, a také prom¥nnost okrajového ztemn¥ní. Týká se to jak prostorov¥ rozli²eného, takprostorov¥ pr·m¥rovaného p°ípadu.

Jestliºe se interpretace spektra odvíjí od model· v planparalelní geometrii, je moºnédopustit se chybného ur£ení termodynamické struktury pozorované atmosféry; 3D simulaceukazují, ºe horizontální nehomogenity mohou ovlivnit vystupující zá°ení podobn¥, jako n¥k-terá °e²ení planparalelních model· se speci�ckým teplotním rozvrstvením. Dvojzna£nostinterpretace pozorování tedy klade pochybnosti na vhodnost planparalelních aproximací,které by m¥ly slouºit jakoºto pr·m¥rné p°iblíºení nehomogenních, realistických hv¥zdnýchatmosfér. Podobn¥ polarizace produkovaná rozptylovými procesy za p°ítomnosti horizon-tálních nehomogenit m·ºe být chybn¥ interpretována jako d·sledek p°ítomnosti magnet-ického pole skrze Hanleho jev.

Znalost magnetizmu a termodynamické struktury atmosféry je nezbytná pro správnouinterpretaci pozorovaného hv¥zdného spektra; multidimenzionální p°enos zá°ení a polar-izace zá°ení tak poskytují nástroj nezbytný pro lep²í porozum¥ní speci�ckých jev· typickypozorovaných na Slunci, s moºností vyuºití i pro hv¥zdná pozorování.

c© Adam Tichý, Masaryk University, 2019

Acknowledgements

First of all I wish to thank doc. Ji°í Kubát for all his support, patience and criticismduring my work. He also initialized my cooperation with dr. Ji°í t¥pán, who providedme with PORTA code and helped me to better understand the problematics of polarizedradiation transfer; and consequently with prof. Javier Trujillo-Bueno who initialized thewhole project. I highly appreciate your help.

There are many other people who helped me in various ways during my studies, oneither a scienti�c or a personal level. I highly value the cooperation with Tomá² Henych,Lucie Jílková and other doctors involved in the early days of my studies. Great thanks goesto colleagues and employees at ÚTFA, especially to Mrs. Santarová, prof. Ji°í Krti£ka, PavelKo£í and all the other Ph.D. students. Thanks goes to all cool people I met in Ond°ejov,including it's occasional inhabitants as Mary Oksala (cheers!), as well as to people I met onoccasional meetings such as conferences, summer schools and workshops. Many thanks goesto my personal IT experts Michal Hol£ík and Petr Zikán (also to all guys from PlasmaSolvefor procrastination opportunities). There would be no �ying carpets without Ziky.

On a personal level, �rst of all I wish to thank my family. D¥kuji celé své rodin¥ zaneutuchající d·v¥ru a nezbytnou trp¥livost; d¥kuji svým rodi£·m, sest°e a nete°ím, v²em

z Lázní B¥lohradu a Drahobuz za fand¥ní a podporu. Cheers and thanks goes to guys fromthe Slonokobra �oorball club; always winning without gaining any points. Also to HankaKarberová, who convinced me that crazy ideas are always the best. There are many morepeople I would like to address, but they will know my gratitude as I meet them personally.

Contents

Introduction 16

1 Physics of polarization in spectral lines 19

1.1 Atomic level polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.1.1 Spherical tensor representation . . . . . . . . . . . . . . . . . . . . 22

1.2 Stokes parameters and radiation �eld tensors . . . . . . . . . . . . . . . . . 241.2.1 Radiation �eld tensors . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.3 Statistical equilibrium equations . . . . . . . . . . . . . . . . . . . . . . . . 271.3.1 Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.3.2 Standard ESE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.4 Radiative transfer equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.4.1 Radiative transfer coe�cients . . . . . . . . . . . . . . . . . . . . . 321.4.2 Source function and optical depth . . . . . . . . . . . . . . . . . . . 331.4.3 Radiative transfer without polarization . . . . . . . . . . . . . . . . 34

1.5 Magnetic �eld and the Hanle e�ect . . . . . . . . . . . . . . . . . . . . . . 351.6 NLTE problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.6.1 Local thermodynamic equilibrium . . . . . . . . . . . . . . . . . . . 371.6.2 Transition region and boundary conditions . . . . . . . . . . . . . . 381.6.3 NLTE problem of the 2−nd kind . . . . . . . . . . . . . . . . . . . 38

2 Inhomogeneous atmosphere models: spatially resolved case 41

2.1 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . 412.1.1 Model atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.1.2 Model atmospheres . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.1.3 Horizontal inhomogeneities in temperature and atomic volume density 42

2.2 Optical properties of an inhomogeneous atmosphere . . . . . . . . . . . . . 432.3 Deformation of the τ = 1 surface . . . . . . . . . . . . . . . . . . . . . . . 472.4 Anisotropy and symmetry breaking . . . . . . . . . . . . . . . . . . . . . . 502.5 Fractional linear polarization . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3 Inhomogeneous atmosphere models: spatially integrated case 59

3.1 Stokes vector of the emergent radiation from inhomogeneous models . . . . 593.1.1 Speci�c intensity and radiation �ux . . . . . . . . . . . . . . . . . . 61

15

3.1.2 Photospheric emission . . . . . . . . . . . . . . . . . . . . . . . . . 653.1.3 Fractional linear polarization . . . . . . . . . . . . . . . . . . . . . . 683.1.4 Center-to-limb variation . . . . . . . . . . . . . . . . . . . . . . . . 72

3.2 Weakly magnetized case and the Hanle e�ect . . . . . . . . . . . . . . . . . 733.3 Emergent radiation from a spherically symmetric stellar surface . . . . . . 75

3.3.1 Stellar surface integration . . . . . . . . . . . . . . . . . . . . . . . 763.3.2 Vertical temperature inhomogeneities in plane-parallel models . . . 783.3.3 Variation of the solution with perturbation period . . . . . . . . . . 81

4 MLVRT code 83

4.1 General assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.2 Statistical equilibrium equations . . . . . . . . . . . . . . . . . . . . . . . . 854.3 Structural description of the radiative transfer solver . . . . . . . . . . . . 88

Conclusions 93

The e�ect of horizontal plasma inhomogeneities in 3D NLTE radiation trans-

fer in stellar atmospheres 95

A PORTA 106

A.1 Electronic attachment: inhomogeneous 3D models and data visualizationusing ParaView . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

B Interaction of radiation and matter 110

B.1 Einstein coe�cients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112B.2 Radiative rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

C Vector coupling coe�cients 116

C.1 3−j symbols and coupling of two angular momenta . . . . . . . . . . . . . 116C.2 6−j and 9−j symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

D Spherical tensors and polarimetric quantities 120

E The complex Voigt pro�le 122

16

Introduction

Most of the current stellar spectra diagnosis is based on atmosphere models in plane-parallel or spherical geometries. Thus, the problem is in many cases restricted to 1D or 2Dgeometries (or their hybrids, as 1.5D), which usually serve as an average representationof a real stellar atmosphere. It is su�cient in the case of spatially non-resolved stellarsurface, where the local structures tend to have negligible impact on the emerging radiation(e.g., Auer, 2003a). However, some structures in stellar atmosphere (such as, for instance,inhomogeneities of thermodynamic properties of the plasma driven by convective motions),can modify the emerging radiation, even if it has been spatially averaged over the wholeatmospheric surface. In such a case, we need to investigate the e�ect assuming full 3Dgeometry, i.e., to include the horizontal transfer of radiation throughout the atmosphere,otherwise we are likely to erroneously predict the atmospheric structure (Uitenbroek andCriscuoli, 2011).

The theory of polarization in spectral lines has been described with various details inseveral papers(Landi Degl'Innocenti, 1983a,b, 1984, 1985; Trujillo Bueno, 2003, etc.), andis fully contained in excellent manuscript by Landi Degl'Innocenti and Landol� (2004).The origins of the polarized theory goes back to the �ndings in laser spectroscopy (Cohen-Tannoudji, 1977), while the �rst applications for the radiative transfer and statistical equi-librium in stellar atmospheres were carried out by Bommier and Sahal-Bréchot (1978);Landi Degl'Innocenti and Landi Degl'Innocenti (1972); Sahal-Bréchot (1977). Several otherauthors presented the theory from slightly di�erent points of view (Sten�o, 1976, 1978; Har-rington, 1969, 1970, and references therein), and without accounting polarization by Omontet al. (1972); Dumont et al. (1977); Cooper et al. (1982) for the investigation of partialfrequency redistribution (PRD). In most cases (except the PRD studies), the motivationwas to study the magnetic �elds in Solar atmosphere via the Hanle (or possibly Zeeman)e�ect (e.g., Leroy, 1977), which has, in addition to collisions, depolarizing e�ect in the coreof the line sensible to magnetic �eld.

Chapter 1 is designed as an overview of some basic physical de�nitions and ideas neces-sary to understand the polarization phenomena and its application for equations de�ningthe NLTE problem of the 2−nd kind. Chapters 2 and 3 contain review of the results we haveobtained applying the theory presented in Chapter 1. In Chapter 2 we introduce horizontalplasma inhomogeneities of temperature and atomic volume density on 3D model grids andshow its connection to the polarization of the emerging radiation. Chapter 3 is dedicatedto the sutdy of emergent radiation and polarization in the spatially integrated case.

17

Most of the presented results have been obtained using the multi-level radiative transfercode PORTA by t¥pán and Trujillo Bueno (2013), in order to be able to introduce hori-zontal inhomogeneities of the plasma thermodynamic properties into the 3D atmosphericmodels. PORTA is based on the same physical principles and very similar numerical pro-cedures as described in Chapter 1. All included plots, as well as procedures for data pro-cessing, have been created using the Python libraries Matplotlib and Numpy, except for afew scripts for treating the PORTA output and input �les, which have been written in Cprogramming language. Various data post-processing and 3D visualization is done usingParaView, sketches and diagrams have been created in Inkscape.

We have developed the multi-level radiative transfer Fortran code MLVRT for solving theNLTE problem of the 2−nd kind in one dimension for two-level atoms. It is entirely basedon the theory contained in Chapter 1, and its directive core, the radiative transfer solver, isin some detail described in Chapter 4. It may be used as a programming guide with manypossible applications in stellar astrophysics, whenever the radiative transfer is required. Ithas also been used as a testing tool to check some theoretical predictions and obtain somepreliminary results to the solution of the NLTE problem.

We also include the paper Tichý A., Kubát J.: �The e�ect of horizontal plasma inhomo-geneities in 3D NLTE radiation transfer in stellar atmospheres� in appendix of the currentversion of the thesis.

18

Chapter 1

Physics of polarization in spectral lines

Polarization of light is expected whenever the atom-radiation interaction is characterizedby a process with a broken symmetry, either geometrical or physical. Thus, in the caseof a stellar atmosphere the symmetry-breaking mechanism is related to any anisotropyin the process of line (or continuum) formation, e.g., an anisotropic illumination of theatomic system, collisional interaction with a collimated beam of perturbing particles, orthe presence of an external magnetic �eld.

In this chapter, we aim at the description of interaction of photons and atoms in weaklymagnetized astrophysical plasma, which is usually in the core of calculation of emergingradiation from such environment. First we summarize the basic physical quantities andconcepts of polarized line formation theory, which allow us to de�ne the NLTE problem forthe polarized case. We shall pay attention to the connection with basic laws of quantummechanics and show some examples of consistency with the classical, non-polarized theory.

We do not intend to give the complete description of (polarized) line formation andtransfer theory in stellar atmospheres, as it is a very complex problem involving many�elds of physics, nor do we discuss the coupled problem of (magneto)hydrodynamics andradiation transfer, as it is beyond the scope of the thesis. It is intended to provide someintuitive insight to the whole problematics, and to show advantages in considering thepolarization modeling as a potentially powerful diagnostic tool for the study of stellaratmospheric structure. For this reason, we try to structure this chapter in order to give thenecessary information which is needed to solve the polarized NLTE problem numericallyfor practical applications.

We shall try to stick to the formalism and notation of Landi Degl'Innocenti and Landol�(2004), which has become standard and where all the necessary physics and derivationsare available. For the non-polarized theory we usually follow classical manuscripts Mihalas(1978) or Hubený and Mihalas (2015). First we explain the meaning of atomic polarizationin terms of density matrix elements, followed by the polarization of light in terms of theStokes vector and radiation �eld tensor within the spherical tensor representation in Sec-tions 1.1 and 1.2. In Section 1.3 we describe the time evolution of atomic density matrixas a set of kinetic equilibrium equations in a multilevel approach and transfer equation forthe radiation �eld in terms of the Stokes vector components in Section 1.4. In Section 1.5

19

20 1. Physics of polarization in spectral lines

we show the in�uence of a weak magnetic �elds on the linear polarization of the emergentradiation via the Hanle e�ect. Section 1.6 is dedicated to the description of the NLTEproblem of the 2-nd kind and some basic concepts behind the line formation problem instellar atmospheres, in order to set up the theoretical background for the interpretation ofthe results contained in subsequent chapters.

1.1 Atomic level polarization

For complete description of the atomic system under consideration we use the concept ofthe density operator. Let our atomic system be in statistical ensemble of pure states |ψr〉with probabilities pr, such that

∑r pr = 1, then the density operator is de�ned as the

weighted sum over all possible states (see e.g. Blum, 1981):

ρ =∑r

pr|ψr〉〈ψr|. (1.1)

Density operator is a suitable tool to describe the system which is not in a pure state,since it contains full information on polarization of the atomic system (i.e., populationimbalances and/or quantum interferences between pairs of states).

In a complete, orthonormal basis {|n〉} which satis�es the relations∑

n |n〉〈n| = 1 and〈m|n〉 = δmn, the density matrix elements can be represented as:

ρmn = 〈m|ρ|n〉 =∑r

pr〈m|ψr〉〈ψr|n〉. (1.2)

Interpretation of density matrix elements depends on the particular choice of basis.Mean value of any observable A, that has a representation on the same Hilbert space

as the system is obtained from the density operator as a trace of the product (ρA)1

〈A〉 =∑n

〈n|ρA|n〉 = Tr(ρA). (1.3)

Time evolution of ρ is given by the von Neumann (or Liouville-von Neumann) equation,which is equivalent to the Schrödinger equation for the system in a mixed state:

i~d

dtρ = [H, ρ], (1.4)

1As an example of how to prove similar equalities, we show the proof to Eq. 1.2:

〈A〉 =∑r

pr〈ψr|A|ψr〉 =∑r

pr〈ψr|A

(∑n

|n〉〈n|

)|ψr〉

=∑r

∑n

pr〈ψr|A|un〉〈un|ψr〉 =∑n

〈n|

(∑r

pr|ψr〉〈ψr|

)A|n〉

=∑n

〈n|ρA|n〉 = Tr(ρA).

1.1. Atomic level polarization 21

where the symbol [H, ρ] denotes the commutator of the Hamiltonian of the system and thedensity operator. Equations (1.3) and (1.4), along with some other properties of densityoperator form the basic set of equations, which lead to derivation of statistical equilibriumand radiative transfer equations (the derivation is brie�y summarized in Appendix B).

Here we mention only some of the important properties of the density operator, whichfollow directly from the de�nition (1.1). From Hermicity of the density operator, i.e., ρ = ρ†

the symmetry property of the density matrix ρmn = ρ∗nm follows. Since the density operatoris positive de�nite, the diagonal terms of the density matrix are real and non-negative.

Natural basis for the density operator of an atomic system is the basis formed by theeigenvectors of angular momentum. For the density matrix elements we have:

ρ(αJM,α′J ′M ′) = 〈αJM |ρ|α′J ′M ′〉, (1.5)

where we have used the substitution |n〉 → |αJM〉, J being angular momentum quantumnumber, M the quantum number corresponding to the projection of angular momentumonto the z− axis and α is the set of quantum numbers corresponding to Hamiltonian H0(so called unperturbed Hamiltonian). Thus:

H0|αJM〉 = EαJ |αJM〉J2|αJM〉 = J(J + 1)|αJM〉Jz|αJM〉 = M |αJM〉, (1.6)

where Jz is the projection of angular momentum J onto the z−axis. Diagonal termsρ(αJM) ≡ ρ(αJM,αJM) represent populations of sub-levels M and M ′. The total popu-lation of the level αJ with energy EαJ and angular momentum J is given by:

nαJ =∑M

ρ(αJM,αJM) (1.7)

whereas non-diagonal terms ρ(αJM,α′J ′M ′) correspond to coherences (or phase inter-ferences) between di�erent sub-levels M and M ′. We assume a multi-level approach,which means that we neglect any coherences between sub-levels not belonging to the sameαJ−level, so that

〈αJM |ρ|α′J ′M ′〉 = δα′αδJ ′J〈αJM |ρ|αJM ′〉. (1.8)Level is characterized by the state vector in the form |αJ〉 =

∑M |αJM〉 with energy

EαJ and angular momentum quantum number J with all possible quantum numbers M ={−J,−J + 1, · · · , J − 1, J} (i.e., if J > 0, the level is degenerated with respect to energy(2J+1)−times), while sub-level is characterized by the state |αJM〉 with the unique valueM . If an atom is polarized, the degeneracy of with respect to energy is removed and itgives rise to nonzero coherences among sub-level with di�erent M within the |αJ〉 state.It follows that only the levels with J > 0 are allowed to be polarized.

Degeneracy of sub-levels pertaining to the level with angular momentum quantumnumber J can be also removed by the action of an external magnetic �eld. Eigenvalues ofthe Hamiltonian

H = H0 +HB, (1.9)


whereHB =

e0~2mc

( ~J + ~S) +BD (1.10)

is the magnetic �eld Hamiltonian obtained from the non-relativistic Dirac equation (e.g.,Messiah, 1961), where e0 is the electron charge, m the mass of the electron, c the speedof light, ~J and ~S are the total angular momentum and electron spin, respectively and BDis the so-called diamagnetic term (which is about 10−10B [G] lower that the �rst term ofHB, and can be thus neglected if the magnetic �elds are not extremely strong), can beexpressed via the perturbation theory as

(H0 +HB)|αJM〉 = (EαJ + µ0gαJBM)|αJM〉, (1.11)

where µ0 = e0~/(2mec) = 9.27 × 10−21 erg G−1 corresponds to the intrinsic magnetic mo-ment of the electron having spin S = 1/2, also called the Bohr magneton (see, e.g, Landauand Lifshitz, 1977, 111) and gαJ is the Landé factor of given transition, de�ned as

gαJ = 1 +〈αJ ||~S||αJ〉√J(J + 1)

. (1.12)

Reduced matrix elements 〈αJ ||~S||αJ〉 can be evaluated in the so-called L − S couplingscheme to obtain

gLS = 1 +1

2

J(J + 1) + S(S + 1)− L(L+ 1)J(J + 1)

, (1.13)

which is due to the selection rules of transitions caused by a magnetic �eld (M−M ′) = ±1or 0 identically zero if J = 0. Equation (1.11) holds if the energy separation of transitionsdue to the action of the magnetic �eld is much lower than transitions due to the unperturbedHamiltonian H0, and if the magnetic �eld vector points in the same direction as the atomicz−axis (otherwise |αJM〉 would be the eigenstate of H0, but not HB). The second termof Eq. (1.11) causes energy separation of sub-levels characterized by quantum numbers M(often referred to as magnetic sub-levels) according to the value of B and the Landé factor(1.13).

1.1.1 Spherical tensor representation

For convenience and better physical interpretation of density matrix elements, the densityoperator is usually expressed in the basis of spherical tensors, also referred to as the mul-tipole moments of the density matrix (Sahal-Bréchot, 1977; Landi Degl'Innocenti, 1984):

ρKQ (αJ, α′J ′) =

∑MM ′

(−1)J−M√

2K + 1

(J J KM −M ′ −Q

)ρ(αJM,α′J ′M ′), (1.14)

where 0 ≤ K ≤ 2J de�nes the rank of the tensor with components −K ≤ Q ≤ K.The sum goes over all possible values of magnetic quantum number −J < M < J and

1.1. Atomic level polarization 23

−J ′ < M ′ < J ′. The symbol in brackets is the so called 3−j symbol, which is connectedto vector coupling coe�cients when coupling two angular momenta J and J ′ and can becalculated via easily performed numerical procedures (see Appendix C, or any textbookon quantum mechanics or Racah algebra, e. g. Messiah (1961)).

Elements with K = 0 correspond to populations of the |αJ〉 level by relation2

nαJ =√

2J + 1ρ00(αJ), (1.15)

while the other elements are directly related to the polarization of a given atom. Forinstance, elements with K = 1 refer to orientation of an atom (or average values of itsangular momenta in x, y and z axes) and are responsible for the circular polarization ofthe emitted radiation by the atomic system. Density matrix elements with K = 2 refer toalignment of an atom (or average values of bilinear combinations of its angular momentacomponents in all axes), and are similarly responsible for the linear polarization of theemitted radiation. As an example, for a level α with angular momentum quantum numberJ = 1, we can �nd exact expressions for the density matrix elements in terms of standardrepresentation elements (1.5), i.e. ρ(M,M ′) using Eq. (1.14) and rules for Wigner couplingcoe�cients summarized in Appendix C:

ρ10(J = 1) =1√2ρ(1, 1)− 1√

2ρ(−1,−1) (1.16a)

ρ11(J = 1) =−1√2ρ(0,−1)− 1√

2ρ(1, 0). (1.16b)

ρ20(J = 1) =1√6ρ(−1,−1) + 1√

6ρ(1, 1)−

√2

3ρ(0, 0) (1.16c)

ρ21(J = 1) =1√2ρ(0,−1)− 1√

2ρ(1, 0). (1.16d)

ρ22(J = 1) = ρ(1,−1). (1.16e)

The remaining elements with Q < 0 can be calculated from the conjugation relation

ρK−Q = (−1)Q[ρKQ ]∗. (1.17)

Elements with Q = 0 are real and are linear combinations of the populations of magneticsub-levels pertaining to level with angular momentum J . The elements of ρKQ with Q 6= 0are complex numbers and represent the linear combinations of coherences between themagnetic sub-levels whose magnetic quantum numbers M di�er by Q, i.e., M −M ′ = Q3.Totally we have (2J + 1)2 density matrix elements for the level with angular momentumJ (e.g. Trujillo Bueno, 2003).

2Notice that populations are de�ned relatively, while many authors work with number densities perunit volume.

3This relation can also be used as a clue on which 3−j coe�cients in Eq. (1.14) are zero withoutdirectly calculating them for all possible combinations of (M,M ′), which can be quite tedious and/orcomputationally costly. There is a number of selective rules for Wigner 3−j, 6−j and 9 − j coe�cients,that can be easily implemented to numerical procedures.


1.2 Stokes parameters and radiation �eld tensors

The Stokes vector can be de�ned in several ways. Maxwell equations describe the lightas oscillating electric and magnetic �elds, whose vectors oscillate perpendicular to eachother and to the direction of propagation ~Ω. Electric �eld vector of the monochromaticwave propagating along the z−axis can be decomposed into x− and y− components as(neglecting the z−component):

Ex(~r, t) = <[ε1e

i(kz−ωt)]Ey(~r, t) = <

[ε2e

i(kz−ωt)] , (1.18)where ε1,2 = E1,2eiφ1,2 . The choice whether we use the electric or magnetic �eld vectoris arbitrary. The Stokes parameters in units of electric �eld intensity (PI , PQ, PU , PV ) arede�ned as bilinear combinations of ε1,2:

PI = ε∗1ε1 + ε

∗2ε2

PQ = ε∗1ε1 − ε∗2ε2

PU = ε∗1ε2 + ε

∗2ε1

PV = i(ε∗1ε2 − ε∗2ε1). (1.19)

Parameters PQ, PU and PV de�ned by Eqs. (1.19) are nonzero if amplitudes E1,2 orphases φ1,2 of x− and y− components of the electric �eld (1.18) di�er. In such generalcase, the tip of the electric �eld vector rotates in the x− y plane and creates the so-calledpolarization ellipse, and the Stokes parameters (1.19) describe its geometrical properties.

The de�nition of Stokes parameters (1.19) is, however, not very useful for practicalpurposes, as the measured quantity is usually the �ux, or energy carried by the beam.Moreover, the monochromatic wave is replaced by a concept of quasi-monochromatic wave,having spread in frequency dν and solid angle dω.

Let us de�ne unit vectors ~ex(~Ω) and ~ey(~Ω), such that they form an orthogonal basis inthe plane perpendicular to the direction of propagation ~Ω. From a macroscopic point ofview, the speci�c intensity I(~r, ~Ω, ν, t) at position ~r, frequency ν at time t carries energyδA0 across an element of area dS into a solid angle dω in time interval dt and frequencyrange (ν, ν + dν), such that (Mihalas, 1978):

δA0 = I(~r, ~Ω, ν, t) dS µ dω dν dt, (1.20)

where µ = cos θ, θ being the angle between ~Ω and the normal to the surface dS.Imagine an idealized linear polarization �lter, which is totally transparent along its

transmission axis and non-transparent (totally opaque) along its perpendicular axis. Ifwe orient the transmission axis successively along the directions ~ex(~Ω), ~ey(~Ω), (~ex(~Ω) +~ey(~Ω))/

√2 and (−~ex(~Ω) + ~ey(~Ω))/

√2, and denote the measured energies as δA1, δA2, δA3

and δA4, respectively, we can de�ne Stokes parameters Q and U in units of energy as follows

1.2. Stokes parameters and radiation �eld tensors 25

(the position, direction, frequency and time dependence is dropped, as it is assumed to bethe same as in Eq. (1.20)):

δA1 − δA2 =Q dS µ dω dν dt (1.21)δA3 − δA4 =U dS µ dω dν dt. (1.22)

Similarly we de�ne the Stokes parameter V , which is proportional to the energy dif-ference between the signals transmitted by the left-handed (δA5) and right-handed (δA6)circular polarization �lters:

δA5 − δA6 = V dS µ dω dν dt. (1.23)

Thus we have the complete Stokes vector as:

~I =

I0I1I2I3

=

IQUV

= k〈PI〉〈PQ〉〈PU〉〈PV 〉

[ erg cm−2 s−1 srad−1 Hz−1 ], (1.24)where k is a dimensional constant, originating from the fact that Stokes vector (1.19) is inunits of a squared electric �eld, while Eqs. (1.20) � (1.23) are in units of intensity. We donot need its actual value, as we are working with ratios Q/I, U/I and V/I. The values of~P in Eq. (1.24) has to be averaged over the time interval ∆t and an area of the detector∆S, in order to describe the quasi-monochromatic wave.

1.2.1 Radiation �eld tensors

For the purpose of calculating radiative rates and transfer coe�cients, the radiation �eldis described by a set of tensors JKQ (νul) of rank K = 0, 1, 2 with components denoted byQ = −K,−K + 1, · · · , K, where subscripts u and l correspond to upper and lower level,respectively, i.e., |u〉 = |αuJu〉 and |l〉 = |αlJl〉. The Stokes vector is transformed into theJKQ as

JKQ (νul) =

∫ ∞0

dνΦ(ν − ν0)∮

dΩ

4π

3∑i=0

TKQ (i, ~Ω)Ii(ν, ~Ω), (1.25)

where Φ(ν− ν0) is the complex pro�le function (Voigt, Lorentz or Gauss), Ii (i = 0, 1, 2, 3)are components of the Stokes vector (1.24), ν is the frequency and ~Ω is the unit vector inthe direction of propagation of the radiation beam (spherical coordinates). Tensor TKQ (i, ~Ω)simpli�es rotation of coordinate system, and is therefore used to de�ne spherical tensor rep-resentation for polarimetric quantities. For exact values see Appendix D, for all necessaryderivations see Landi Degl'Innocenti and Landol� (2004, ch. 5.11).

Radiation �eld tensors JKQ (νul) given by Eq. (1.25) are used directly to calculate radia-tive rates for statistical equilibrium equations. For convenience, we de�ne the operator ofthe integration over solid angle and frequency:

Ξ̂ ≡∫ ∞

0

dνΦ(ν − νul)∮

dΩ

4π, (1.26)


where dΩ = dµ dφ and µ = cos θ, θ ∈ [0, π/2] being the angle between line of sight andz−axis and φ ∈ [0, 2π] azimuth. If the radiation �eld is purely isotropic, the only nonzeroJKQ is of the rank K = 0, J

00 , which is equal to the frequency and solid angle averaged

mean intensityJ00 (νul) = Ξ̂ I(ν, ~Ω). (1.27)

It is the only relevant radiation component taking place in a non-polarized rate equations.In a case of cylindrical symmetry of the radiation �eld about the z−axis, the additionalnonzero radiation term is J20 (here Q denotes the Stokes vector component I1):

J20 (νul) =1

2√

2Ξ̂[(3µ2 − 1)I − 3(1− µ2)Q

]. (1.28)

Radiation �eld component (1.28) is often used to de�ne the so-called anisotropy factor

A =J20 (νul)

J00 (νul), (1.29)

which quanti�es the anisotropy of the radiation �eld with respect to inclination µ; giventhat the intensity I is in Eq. (1.29) dominant (or, if the radiation is assumed to be non-polarized) it has negative values for predominantly horizontal radiation, and positive ifvertical. Components of J2Q with Q 6= 0 quantify breaking of the cylindrical symmetrythrough the azimuth exponentials appearing in the spherical tensor TKQ (see Table D.1 inAppendix D). Radiation tensor components with Q 6= 0 are complex quantities, and thuscan be decomposed into its real and imaginary parts. Components of rank K = 1 tensorare related to the circular polarization, which is irrelevant in cases studied further; theirexact expressions can be straightforwardly obtained from Eq. (1.25).

Radiation components of J2Q (alignment components, in analogy with the sphericalcomponents of density matrix with K = 2) are tied to the linear polarization, their exactexpressions being

J21 (νul) =−√

3

2Ξ̂{√

1− µ2 exp iϕ (µ I + µQ− iU)}

(1.30a)

J22 (νul) =−√

3

4Ξ̂{

exp 2iϕ[(1− µ2) I − (1 + µ2)Q+ 2µiU

]}(1.30b)

Radiation �eld tensors JKQ thus provide a complex information about any anisotropy thatcan arise when dealing with the radiation transfer through an inhomogeneous stellar at-mosphere.

1.3. Statistical equilibrium equations 27

1.3 Statistical equilibrium equations

Atom is considered devoid of hyper�ne structure (energy shifts due to the interaction ofnucleus with the magnetic �eld generated by moving bound electrons), its energy levelsdescribed by quantum numbers α and J , where J = L+S (LS−coupling scheme, where L isorbital momentum and S stands for spin of the electron) and α is a set of all other quantumnumbers (for a plasma composed of hydrogen atoms, α would stay for a main quantumnumber n, denoting the energy level with respect to the ground state). The magnetic �eldvector is assumed to point in the direction of the atomic z−axis.

Consider an atom undergoing a single interaction with electromagnetic �eld. Thenthe time evolution of the atomic system is described by statistical equilibrium equations(ESE)4. In the general form valid for a multilevel atom (Landi Degl'Innocenti, 1983a):

d

dtρKQ (αJ) =− 2πiνLgαJQρKQ (αJ)

+∑αlJl

∑KlQl

ρKQ (αlJl)TA(αJKQ,αlJlKlQl)

+∑αuJu

∑KuQu

ρKuQu (αuJu)[TE(αJKQ,αuJuKuQu)

+ TS(αJKQ,αuJuKuQu)]

−∑K′Q′

ρK′

Q′ (αJ)[RA(αJKQK′Q′) +RE(αJKQK

′Q′)

+RS(αJKQK′Q′)], (1.31)

whereνL =

µ0B

h= 1.3996× 106B [cgs] (1.32)

is the Larmor frequency, quantifying the characteristic action time of a magnetic �eld astB = 1/νL.

Physical meaning of Eq. (1.31) is that it describes the time evolution of the densitymatrix elements in terms of the so-called radiative transfer rates. Given the radiation �eld,solution of Eq. (1.31) gives the ρKQ elements of all levels involved, i.e., populations andcoherences between pairs of magnetic sub-levels, while the coherences are the smaller, thelarger is the energy di�erence between the considered states |αJM〉 and |αJM ′〉. The lefthand side is usually set to zero (stationary case), thus achieving the statistical equilibrium.

Coe�cients TA, TS, TE and RA, RS, RE are, respectively, coherence-transfer andcoherence-relaxation rates. Former are responsible for increase of a given ρKQ (αJ) element,later for its decrease. Subscripts A, S,E denote "Absorption", "Stimulated emission" and"spontaneous Emission". They represent probability of the 'energy transition' within theatomic system from quantum state |αJKQ〉 to the state |α′J ′K ′Q′〉, which refers to 'upper'or 'lower' state in the terms of the energy di�erence between the two considered states.

4Or kinetic equilibrium equations, as the statistical equilibrium usually refers to the stationary case,i.e. d/dt = 0


As an example, coherence-transfer (`incoming` transition) rate for absorption reads asfollows

TA(αJKQ,αlJlKlQl) = (2Jl + 1)B(αlJl → αJ)

×∑KrQr

√3(2K + 1)(2Kl + 1)(2Kr + 1)

× (−1)Kl+Ql

J Jl 1J Jl 1K Kl Kr

(

K Kl Kr−Q Ql −Qr

)JKrQr (ν̄), (1.33)

where JKrQr (ν̄) is calculated using eq. (1.25), B(αlJl → αJ) is Einstein coe�cient (see Sect.B.1) and terms in brackets are Clebsch-Gordan or vector-coupling 9-j and 3-j coe�cients(brie�y summarized in Appendix C)5. The simplest rate is relaxation rate due to sponta-neous emission:

RE(αJKQK‘Q‘) = δKK′δQQ′∑αlJl

A(αJ → αlJl).

Therefore this relaxation radiative rate is equal to sum of Einstein coe�cients for spon-taneous emission to all other lower levels considered, as one can guess intuitively. Ratesresponsible for spontaneous emission don't depend on the radiation �eld. Complete list ofradiative rates is given in Appendix B.2.

However, since the absorption and stimulated emission rates do depend on the radiation�eld, we have actually additional set of unknowns, as the radiation �eld tensor JKQ (ν̄) itselfis a�ected by the populations of the levels and sub-levels of the emitting atom. Thuswe need another set of coupled equations to estimate the radiation �eld arising from thestatistical ensemble of atomic states given by the density matrix ρKQ (αJ). This additionalset of equations is often referred to as radiative transfer equations or evolution equationsfor the radiation �eld.

Given the known values of JKQ (ν̄) at given point of the atmosphere, Eq. (1.31) becomesthe set of linear equations for the unknown ρKQ (αJ), which can be expressed in the matrixnotation:

R~x = ~0, (1.34)

whereR represents the matrix composed of the radiative rates and ~x is the vector composedof the density matrix elements. Such equations are linearly dependent, so in order to getphysically relevant solution, we use the normalization condition∑

J

nαJ =∑J

√2J + 1ρ00(αJ) = 1, (1.35)

which characterizes the equivalent of the conservation of particles in classical case (Mihalas,1978), to close the system of equations (1.34).

5There is a nice calculator available at http://www-stone.ch.cam.ac.uk/wigner.html

1.3. Statistical equilibrium equations 29

Given the most important mechanisms that in�uence the atomic system are the radia-tion (subscript R), collisions (C) and magnetic �eld (M), the resulting kinetic equilibriumequations can be symbolically expressed as

d

d tρKQ (αJ) =

d

d tρKQ (αJ)|R +

d

d tρKQ (αJ)|C +

d

d tρKQ (αJ)|M , (1.36)

where the collisional and magnetic �eld terms are described in Sections 1.3.1 and 1.5,respectively.

Equations (1.31) contain only radiative rates for bound-bound transitions, therefore it isnecessary to include bound-free, free-bound transitions. This is done ad-hoc, where bound-free and free-bound rates are taken from classical (non-polarized) theory (Mihalas, 1978).Continuum transitions are not studied here, as we usually give the continuum opacity aconstant value in models under investigation.

1.3.1 Collisions

Atoms in a realistic astrophysical plasmas are subject to various collisions with surroundingparticles (for instance other atoms, ions, free electrons, etc., hereafter perturbers) whichcan change the atomic state, including polarization. Collisional rates can be divided intoelastic and inelastic, according to the e�ect they have on an atomic system having energyeigenvalues in the form |αJM〉. Inelastic collisions are, in analogy with radiative rates,such that they induce transitions |αJM〉 → |αuJuMu〉 (transfer rates) with a correspondingenergy loss of the perturbing particles; and as a superelastic counterpart there are the ratesinducing transitions of the form |αJM〉 → |αlJlMl〉, increasing the kinetic energy of theperturber. Elastic collisions conserve the kinetic energy of both interacting particles, andinduce transitions in the form |αJM〉 → |αJM ′〉, shifting populations of magnetic sub-levels within the αJ−level (angular momentum changing its projection onto the z−axis).

It is assumed that impact approximation is valid, i.e. the characteristic collision time tcis much lower than the natural relaxation time tc � 1/Aul, and much lower than the timebetween successive collisions tc � 1/f , where f = σv̄rNP , σ being the cross section, NPthe density of perturbers and v̄r the mean velocity of the perturber relative to the atom.Perturbing particles are also assumed to have Maxwellian distribution on velocities, as thecase of interaction of atomic system with a collimated beam of perturbers (or collisionswith particles having any other non-isotropic velocity distribution) is not considered here.

Contribution of various rates to the time evolution of the density matrix can be ex-pressed in a similar form as the contribution of radiative rates. It is possible to deriveanother set of (collisional) ESE and simply add it to radiative ones in eq. (1.31), as indi-cated by (1.36). Collisional rate equations can be written as follows (Landi Degl'Innocenti


et al., 1990):

d

dtρKQ (αJ) =

∑αlJl

√2Jl + 1

2J + 1C

(K)I (αJ, αlJl)ρ

KQ (αlJl)

+∑αuJu

√2Ju + 1

2J + 1C

(K)S (αJ, αuJu)ρ

KQ (αuJu)

−

[∑αuJu

C(0)S (αuJu, αJ) +

∑αlJl

C(0)I (αlJl, αJ)

]ρKQ (αJ)

+D(K)(αJ)ρKQ (αJ), (1.37)

where D(K)(αJ) is the depolarization rate due to elastic collisions

D(K)(αJ) = C(0)E (αJ)− C

(K)E (αJ), (1.38)

giving D(0) = 0, which shows that elastic collisions do not a�ect population of the(αJ)−level. In analogy with classical treatment of collisions (e.g., Mihalas, 1978), wehave (denoting for simplicity (αuJu, αlJl) = (u, l))

C(0)I (u, l) = Ne

∫ ∞v0

σul(v)f(v)dv

and

C(0)S (l, u) =

2Jl + 1

2Ju + 1exp

(Eu − ElkBT

)C

(0)I (u, l).

For the time being we consider only K = 0 ranks of collisional rates. In above expressions,f(v) is (Maxwellian) velocity distribution, σul(v) is the cross section, Ne free electrondensity and v0 threshold velocity (nonrelativistic)

v0 =

√2(Eu − El)

m.

Cross sections σul(v) can be found throughout the literature mostly in the form of approx-imate expressions, suitable for various applications, or in the form of tabular values forparticular transitions.

Equations (1.37) are valid for an atomic system in a multi-level approach and if thereis no additional physical agent, such as magnetic �elds. However, the typical collision timetc = 1/f in solar-like atmosphere is usually much smaller than characteristic time of anaction of magnetic �eld 1/(2πνL), as the tc varies approximately in range 10−13 − 10−14 s(assuming the most relevant collision atom�free electron), giving the condition for the mag-netic �eld B � 106 G, satis�ed in solar-like and most magnetic stars (Landi Degl'Innocentiand Landol�, 2004).

1.4. Radiative transfer equation 31

1.3.2 Standard ESE

An important note should be made on the limiting case of Eq. (1.31), if we neglect polariza-tion. Density matrix elements with K > 0 vanish and the atomic system is described onlyby its populations ρ00(αJ) ≡ ρm. The �rst term on the rhs. of Eq. (1.31) disappears as wellas the summation over K and Q. Moreover, as we neglect polarization of the interactingradiation, the radiative rates are also greatly simpli�ed.

After some algebra, one can rewrite Eq. (1.31) into the form (Landi Degl'Innocenti andLandol�, 2004):

d

dtρm =

∑l

ρlB(l→ m)I(νml)

+∑u

ρu [A(u→ m) +B(u→ m)I(νum)]

− ρm∑u

B(m→ u)I(νum)

− ρm∑l

[A(m→ l) +B(m→ l)I(νml)] , (1.39)

where I(νul) is equal to J00 (νul) from Eq. (1.25) with the Stokes vector ~S = (S0, 0, 0, 0).The �rst term on the rhs. of Eq. (1.39) is responsible for increase of the population of levelm via absorption of radiation at frequency νml from all lower levels l. The second termis similarly proportional to the number of energetic transitions from all upper levels u tolevel m via spontaneous and stimulated emissions. The last two terms are proportional tothe number of the same transitions from the level m, hence the minus sign.

Except the slightly di�erent notation, equations (1.39) are analogous to the classicalstatistical equilibrium equations (cf. Mihalas, 1978; Hubený and Mihalas, 2015).

1.4 Radiative transfer equation

Similarly to ESE, one can de�ne the density operator of the radiation �eld and using theMaster equation approach, as suggested by Eq. (B.2), and deduce the evolution equationsfor the radiation �eld.

Following Landi Degl'Innocenti and Landi Degl'Innocenti (1972) andLandi Degl'Innocenti (1983a), for the radiation beam speci�ed by the Stokes vector~S = (I,Q, U, V )T propagating through medium we have the radiative transfer equation(RTE):

(1

c

∂

∂t+

dds

)IQUV

=

�I�Q�U�V

−

ηI ηQ ηU ηVηQ ηI ρV −ρUηU −ρV ηI ρQηV ρU −ρQ ηI

IQUV

. (1.40)


Here ~� represents creation of photons via emission for all Stokes parameters. The 4× 4propagation matrix consists of the absorption part KA and stimulated emission part KS,such that K = KA −KS and can be decomposed into three parts:

K =

ηI 0 0 00 ηI 0 00 0 ηI 00 0 0 ηI

+

0 ηQ ηU ηVηQ 0 0 0ηU 0 0 0ηV 0 0 0

+

0 0 0 00 0 ρV −ρU0 −ρV 0 ρQ0 ρU −ρQ 0

,where the �rst term, which consists only of absorption coe�cients ηI (lowering the mag-nitude of I by absorbing part of the energy from the beam) on the diagonal of K, isproportional to classical absorption and is denoted to as absorption matrix. The secondterm composed of ηQ, ηU and ηV is the dichroism matrix, responsible for selective ab-sorption of radiation in the particular polarization state. The third antisymmetric termcomposed of ρQ, ρU and ρV is called the dispersion matrix and represents the dephasingbetween the di�erent polarization states.

1.4.1 Radiative transfer coe�cients

Once we have the density matrix elements, we can estimate the radiative transfer coe�-cients in the following form (Landi Degl'Innocenti and Landol�, 2004):

ηAi (ν, ~Ω) =Nhν

4π

∑l

∑u

(2Jl + 1)B(l→ u)∑KQ

√3(−1)1+Jl+Ju+K{

1 1 KJl Jl Ju

}TKQ (i, ~Ω)ρ

KQ (l)

1.4. Radiative transfer equation 33

An important note should be made on the complex pro�le function Φ(νul−ν) = φ(νul−ν) + iψ(νul − ν) (for more details see Appendix E), which enters the expressions (1.41),(1.42) and (1.43). These expressions have been derived assuming that Bohr frequencies ofthe involved transitions do not depend on magnetic quantum numbersM andM ′. However,in the presence of an inclined external magnetic �eld, such that we are operating in the socalled Zeeman e�ect regime (∆νD < νL � νul, Landi Degl'Innocenti (1985)), where ∆νD isthe Doppler width, the eigenvalues of the total Hamiltonian H take the form (see Section1.1 for more details):

H|αJM〉 = (EαJ + µ0gαJBM)|αJM〉. (1.45)

Thus, the resulting Bohr frequencies are:

ναJ,α′J ′ =EαJ − Eα′J ′

h→ ναJM,α′J ′M ′ = ναJ,α′J ′ + νL (gαJM − gα′J ′M ′). (1.46)

In such a case, the line pro�le would have been split into the Zeeman components accord-ing to the magnetic �eld strength via the Larmor frequency νL as in Eq. (1.32) and thedi�erence gαJM − gα′J ′M ′ for the considered transition.

The fact that we are ignoring the M , M ′ dependence of Φ(νul − ν) means that weare operating in much weaker magnetic �eld regimes starting from the zero �eld (νL �∆νD) up to the so called Hanle e�ect regime (Aul ≈ νL � ∆νD) (Landi Degl'Innocenti,1983b; Bommier et al., 1991). We are speaking about the weakly magnetized plasma, whichsimpli�es the radiative transfer coe�cients, but we are giving up the Zeeman splitting ofthe line.

1.4.2 Source function and optical depth

For convenience, Eq. (1.40) is rewritten into the form (denoting for simplicity ~I =~I(~r, ~Ω, ν, t)): (

1

c

∂

∂t+

dds

)~I = ~�−K~I.

Assuming the stationary case (∂/∂t = 0) and dividing by ηI , we obtain RTE in slightlydi�erent, but for most applications more convenient notation (Rees et al., 1989; Tru-jillo Bueno, 2003):

ddτ~I = K̃~I − ~S, (1.47)

where ~S = ~�/ηI is the source function, reduced propagation matrix K̃ = K/ηI and thespatial coordinate ds is replaced by elementary optical length, or optical length along theray dτ = −ηI ds.

The source function ~S measures the number of photons created in the unit optical depthinterval. Using Eq. (1.44), one can write the source function for the Stokes I in the form:

SI =2hν3

c2ηSI

ηAI − ηSI=

2hν3

c21

ηAI /ηSI − 1

. (1.48)


The dimensionless optical distance ∆τ = −∫ s2s1ηI ds corresponds to the number of mean

free paths of the photon at frequency ν in the direction of propagation ~Ω between the pointss1 and s2. It also quanti�es the escape probability p(τ) of the photon traveling throughthe optical distance ∆τ as p(∆τ) = exp (−∆τ); thus, for large optical distances ∆τ � 1,the material is non-transparent (escape probability is small) and we are talking aboutthe optically thick medium, while for the the values ∆τ � 1, the material is completelytransparent (lim∆τ→0 p(∆τ)→ 1), and we are talking about the optically thin medium.

1.4.3 Radiative transfer without polarization

The non-polarized equation of radiation transfer can be obtained from Eq. (1.40) by as-suming that only the diagonal terms of the propagation matrix K are nonzero, emissionis set to ~� = (�I , 0, 0, 0)T and the propagating radiation beam is considered non-polarized,i.e., ~I = (I, 0, 0, 0)T: (

1

c

∂

∂t+

dds

)I = �I − ηI I. (1.49)

Similarly, for the radiative transfer coe�cients (1.41), (1.42) and (1.43), we can easilyobtain the non-polarized limit. For instance, for the absorption coe�cient ηA0 (ν, ~Ω), wehave by considering K = Q = 0 and T 00 (0, ~Ω) = 1 (see Tab. D.1):

ηA0 (ν, ~Ω) = Nhν

4π

∑l

∑u

(2Jl + 1)B(l→ u)√

3(−1)1+Jl+Ju{

1 1 0Jl Jl Ju

}ρ00(l)φ(νul − ν).

(1.50)The 6-j symbol can be evaluated as (see Appendix C):{

1 1 0Jl Jl Ju

}= (−1)1+Ju+Jl 1√

3 (2Jl + 1).

Assuming two level atom and taking into account that N√

2Jl + 1 ρ00(l) = nl is the popula-

tion (number density per unit volume) of the lower level, we can further simplify Eq. (1.50):

ηA0 (ν, ~Ω) = nlhν

4πB(l→ u)φ(νul − ν), (1.51)

which is the classical extinction coe�cient without correction for stimulated emission.The stimulated emission coe�cient can be evaluated in a similar way, and the emissioncoe�cient is straightforwardly obtained from Eq. (1.44).

The special case of source function de�ned by Eq. (1.48) without accounting polarizationcan be obtained. The ratio ηAI /η

SI in the denominator is evaluated by stating that the

absorption pro�le of (1.41) is exactly the same as the emission pro�le of (1.44) � which hasalready been assumed (complete frequency redistribution). After some easy algebra, onecan get for the source function

SI =2hν3

c21

(2Ju+1)ρ00(l)

(2Jl+1)ρ00(u)− 1

=2hν3

c21

gu nlgl nu− 1

, (1.52)

1.5. Magnetic �eld and the Hanle e�ect 35

where gl and gu are statistical weights of the lower and upper level, respectively. In thespecial case of LTE, where the Boltzmann distribution of bound levels holds (see Ch. 1.6.1),we see that the source function is equal to Planck distribution:

SI =2hν3

c21

exp hνkBT− 1

= BP (T ). (1.53)

This is an important feature of the source function and will be repeatedly reminded inChapters 2 and 3 (cf., e.g., Mihalas, 1978; Hubený and Mihalas, 2015; Rutten, 2003).

1.5 Magnetic �eld and the Hanle e�ect

Hanle e�ect is the mechanism in�uencing the scattering polarization of radiation emittedby an atom embedded in an external magnetic �eld.

The easiest way to illustrate the mechanism is to assume that the scattering two-levelatom having Jl = 0 and Ju = 1 is treated as a damped oscillator. The damping constantfor such atom is γ = Aul = 1/tl, where tl is the lifetime of the excited level and Aul isEinstein coe�cient for spontaneous emission. The external magnetic �eld acts on movingelectron via the Lorentz force ~FL = e~v × ~B, modifying the linear polarization signals Qand U with respect to the non-magnetic case by rotating the polarization plane. The linearpolarization plane is rotated according to the ratio νL/Aul.

When the lifetime of the excited level is short (i.e. high value of Aul) with respectto 1/νL, the atom is deexcited without much in�uence of the magnetic �eld, thereforethe density matrix elements ρKQ (u) as well as the Stokes vector ~S = (I,Q, U, V ) of thescattered radiation remain unchanged, corresponding to the zero �eld regime (νL � Aul).In the opposite limit νL/Aul � 1, the polarization plane can rotate multiple times aroundthe propagation axis before the atom is deexcited, causing complete depolarization for thesaturated magnetic �eld intensity. Thus, we assume that the Larmor frequency νL satis�esthe following inequalities for the Hanle e�ect regime:

Aul ≈ 2π νL � ∆νD.

From the quantum mechanical point of view, the Hanle e�ect arises from the magneticterm of the perturbed Hamiltonian (1.45) and corresponding Bohr frequencies (1.46), whichis in the statistical equilibrium equations represented by the �rst term on the rhs. of Eq.(1.31), namely (cf. Eq. (1.36))

d

d tρKQ (αJ)|M = −2πiνLgαJQρKQ (αJ). (1.54)

In the magnetic �eld reference system (i.e., with the magnetic �eld vector pointing inthe same direction as the atomic z− axis), the density matrix elements with Q = 0 areuna�ected by the magnetic �eld, while the coherences (ρKQ elements with Q 6= 0) arere-phased and reduced according to the value of B.


In the case of a two level atom, neglecting collisions and stimulated emission, thestatistical equilibrium equations can be solved analytically, separating the Hanle e�ectsuch that:

d

d tρKQ (αJ) =

1

1 + iQHu

[d

d tρKQ (αJ)

]B=0

, (1.55)

whereHu =

2πνLgαJAul

(1.56)

is the coe�cient quantifying the sensitivity of the atomic system on a magnetic �eld via theHanle e�ect, and it can be used to estimate relevant values of B for which the consideredtransition is sensitive.

2πνLgαJ = 8.79× 106B gαJ ≈ Aul = 1/tl. (1.57)

The sensitivity of the line transition to the Hanle e�ect varies with the lifetime of theexcited level tl and the Landé factor, restricting the magnetic �eld intensities to approxi-mately from 10−3 up to 102 gauss.6

The Hanle e�ect is signi�cant in the so-called weak �eld regime, according to the clas-si�cation proposed by Landi Degl'Innocenti (1983b). It means that the magnetic �eld issupposed to be su�ciently weak for the Zeeman splitting to be negligible with respect tothe natural broadening of the spectral line, i.e., νL � ∆νD, where νL is given by Eq. (1.32)and ∆νD is the Doppler line width due to thermal motions.

Thus, for a properly chosen spectral line, the Hanle e�ect can be used as a diagnostictool when interpreting linear polarization observations, allowing the detection of magnetic�elds which are too weak to produce a measurable Zeeman e�ect either in intensity orcircular polarization.

1.6 NLTE problem

Even if the atomic and atmospheric models are both relatively simple (for instance astationary, homogeneous, non-magnetized atmosphere composed of two-level atoms), theNLTE problem itself is complex enough to be a considerable challenge to solve by modernnumerical methods. Allowing the atomic and radiation polarization, we are forced to usegeneralized versions of statistical equilibrium and radiative transfer equations, as was de-scribed in Sections 1.3 and 1.4. However, by solving the NLTE problem of the 2−nd kind,we obtain information on the full Stokes vector of the emergent radiation.

In order to describe the matter in astrophysical plasmas in terms of statistical thermo-dynamics (i.e., using the set of well de�ned quantities such as pressure, temperature or anythermodynamic potential, etc.), we have to assume some sort of equilibrium state for this

6The sensitivity on sub-gauss magnetic �eld magnitude is usually corresponding to the so-called lowerlevel Hanle e�ect, taking place when Jl 6= 0, since the lifetime of the ground level in solar-like atmospheresis usually several orders higher than that of the upper level (Trujillo Bueno, 2001).

1.6. NLTE problem 37

matter. The microscopic requirement for the matter to be in thermodynamic equilibrium(TE) is the detailed balance argument, which states that the net rate of every energetictransition within the system is strictly balanced by the net rate of its counterpart. It isan analogy to the energy conservation in the closed system. In the case we describe theplasma by its microstates rather than macrostates, we are interested in the distributionfunctions of the radiation �eld and atoms (molecules, ions, etc.).

1.6.1 Local thermodynamic equilibrium

In local thermodynamic equilibrium approach, we describe the state of the matter at eachpoint of the medium ~r by equilibrium values of chosen physical quantities � usually thetemperature T (~r) and populations of all possible atomic states Ni(~r). In other words, thereservoir of heat and particles for the considered ensemble of atoms represented by atomicstates Ni(~r) is composed by the surrounding particles and the system is with this reservoirin thermodynamic equilibrium; however, in LTE we assume that this reservoir is position-dependent and as a consequence, the values of T (~r) and Ni(~r) can change with position ~rthroughout the atmosphere as well.

The state of the matter in stellar atmospheres is fully described by distributions ofenergetic states of the matter and radiation. Kinetic energy (or speed) distribution isusually assumed to be Maxwellian. Standard form in terms of speeds is

f(v)dv =m1/2

(2πkBT )1/2exp

(− mv

2

2kBT

)4πv2dv, (1.58)

and gives the probability, that a particle of mass m at temperature T has the speed inrange (v, v + dv).

The distribution material degrees of freedom (populations of atomic states, ioniza-tion degrees, etc.) is in LTE given by Saha-Boltzmann statistics (see, e.g., Mihalas, 1978;Hubený and Mihalas, 2015). For instance the population of the bound level i of atomicspecies k in ionization state j, relative to the total number of atoms and ions Njk =

∑i nijk

is given by: (nijkNjk

)LTE

=gijk

Zjk(T )exp

(− χijkkBT

), (1.59)

where gijk is the degeneracy factor (how many states with (nearly) the same energy), χijkis the energy di�erence between the state |ijk〉 and the ground state |0jk〉. Zjk(T ) ≡∑

i gijk exp (−χijk/kBT ) is the partition function, which comes from normalization condi-tions of the distribution function (see, e.g., Hummer and Mihalas, 1988). The equilibriumdistribution of radiation is given by Planck function:

Bν(T ) =2hν3

c2

[exp

(hν

kBT

)− 1]−1

. (1.60)

Equations (1.58) and (1.59) are functions of (local) temperature T (~r) and can be un-derstood as conditions for the LTE to hold; the LTE assumption cannot be applied, if any


of these conditions is not met. However, the radiation �eld is allowed to depart from thePlanck distribution (1.60), in which case we do not solve the statistical equilibrium, andonly calculate the emitting radiation from the �xed populations (1.59).

1.6.2 Transition region and boundary conditions

The matter in the deepest layers of an atmosphere is dense enough for the collisional ratesto dominate over the radiative rates (collisional rates depend on density, see Eq. (1.37),contrary to the radiative ones). Since collisional rates satisfy the detailed balance require-ment in most regions of interest, we can assume that the plasma is approximately in (local)thermodynamic equilibrium in these regions. Most of the energy of locally created photonsis transformed into the kinetic energy of perturbing particles, and the radiation is said tobe thermalized. Source function is equal to the Planck function and depends only on thelocal value of temperature T (~r):

Bν(T (~r)) = Sν , (1.61)

as in Eq. (1.53). The populations are distributed according to Eq. (1.59) and we can safelyassume LTE for the material in such regions, which serves us as a lower boundary conditionfor radiative transfer problem. Also, in a collision dominated regime, the depolarization rateD(K)(αJ) de�ned by Eq. (1.38) is much larger than any illumination anisotropy, so that theatomic polarization is completely removed and phase relations between magnetic sub-levelsare not present.

In the opposite limit of sparse outer atmosphere layers, the collision rates are dueto low densities ine�ective in modifying the atomic state (including radiation inducedpolarization), and the radiation itself is strongly anisotropic (illumination from the loweratmosphere layers is stronger than the radiation coming from higher layers). These partsof an atmosphere are thus sometimes referred to as the anisotropy-dominated regime.

The radiation �eld in the intermediate layers between the collision-dominated and theanisotropy-dominated regions, usually referred to as the transition region, is strong enoughfor the radiative processes to become more dominant compared to collisions, so that wecan no longer assume LTE and must deal with non-equilibrium physics, including thedetailed knowledge of collisional rates. We must solve the kinetic equation of radiation toestimate its spatial variation when propagating throughout the transition region � and weexpect deviations from distributions de�ning the LTE (except the Maxwellian distributionof velocities).

The three regions (collision-dominated, transition and anisotropy-dominated) corre-spond roughly to the photosphere, chromosphere and corona for a solar-like atmosphere,respectively (Landi Degl'Innocenti, 1983b).

1.6.3 NLTE problem of the 2−nd kindThe NLTE problem is stated by assuming that distributions (1.59) and (1.60) are not valid,and that the actual state of the atomic system and radiation �eld is given by self-consistent

1.6. NLTE problem 39

solution of the radiative transfer and statistical equilibrium equations. The case where theradiation and atomic system polarization is included is usually referred to as the NLTEproblem of the 2−nd kind, and is a natural generalization of the classical NLTE problem(e.g., Mihalas, 1978).

Radiative transfer equation and formal solution

The formal solution to Eq. (1.47) is (Trujillo Bueno, 2003):

~IO = ~IM exp (−τMO) +∫ τMO

0

dt [~S −K′(t)~I(t)] exp (−t), (1.62)

where the reduced propagation matrix K′ = K/ηI − 1 (1 being the 4 × 4 unit matrix) isused for convenience. Subscripts M and O corresponds to the two successive points alongthe path of the radiation beam (the radiation at point M is assumed to be known). Bysolving Eq. (1.62), we can obtain full Stokes vector of radiation at each point and discretizedfrequency ν and direction ~Ω.

To solve the Eq. (1.62) we apply the short characteristic method proposed by Kunaszand Auer (1988) and for the polarized case generalized by Rees et al. (1989) and Tru-jillo Bueno (2003), where the three successive points M, O, P are used to specify the localvalues of the radiative transfer coe�cients and radiation �eld. The propagation matrixK′ is interpolated linearly, while for the source function ~S we use the monotonic Bézierinterpolation method (Auer, 2003a; t¥pán and Trujillo Bueno, 2013).

Iteration scheme

As was mentioned in Section 1.3, the two sets ESE and RTE are coupled, and can be solvediteratively, at least in principle, by the so-called Λ−iteration. In this approach we set someinitial values of populations, for instance, to the LTE values according to (1.59), calculateall necessary transfer coe�cients (Sect. 1.4.1) and solve the radiative transfer (1.62) toobtain intensity Iν . Using non-polarized analogy to the Eq. (1.25), we calculate the meanintensity, insert into the radiative rates (such as the absorption rate (1.33)) and update thedensity matrix elements ρKQ at each point of the atmosphere (see schematic Figure 1.1).The new density matrix elements are used to update the radiative transfer coe�cients, andwe continue iteratively as long as the self-consistent solution is reached.

One Λ iteration can be interpreted as the transfer of energy that is carried by a photonby one mean free path. In the NLTE approach, we thus assume that di�erent parts ofthe considered region are simultaneously a�ecting each other. It corresponds to the energytransport by radiation throughout the plasma, in contrast to the LTE assumption, whichstates that the matter is at each point ~r isolated (i.e., in equilibrium). This can lead toenormous number of iterations before the convergence is reached, depending on the totaloptical thickness of the line. To overcome the di�culties inherently tied to the Λ−iterationand lower the number of iterations, it is possible to apply the accelerating method ofoperator splitting and preconditioning ESE by Rybicki and Hummer (1991), which has


Figure 1.1: NLTE loop.

been further modi�ed for the polarized case by Trujillo Bueno and Manso Sainz (1999)and t¥pán (2008).

Chapter 2

Inhomogeneous atmosphere models:

spatially resolved case

In this chapter we self-consistently solve the NLTE problem of the 2−nd kind in full 3Dgeometry in order to model the formation and transfer of spectral line radiation emergingfrom an optically thick atmosphere. By solving the NLTE problem in three dimensions, weare enabled to introduce horizontal inhomogeneities of thermodynamic properties of theatmospheric plasma.

In order to solve the NLTE problem of the 2−nd we use the radiative transfer codePORTA of t¥pán and Trujillo Bueno (2013), taking into account resonant scatteringpolarization and its possible modi�cation by magnetic �elds via the Hanle e�ect.

This chapter is structured as follows. In Section 2.1 we formulate the problem by choos-ing a model atom and a reference model atmosphere. Further we introduce horizontalplasma inhomogeneities in temperature and atomic volume density of the atmosphericmodel and choose the set of free parameters to the problem. In Sections 2.2 and 2.3 weexamine consequent changes in optical properties of an inhomogeneous atmospheric modelsand show its main implications on the formation and transfer of spectral line radiation.Sections 2.4 and 2.5 are devoted to the simulated observations of linear polarization pat-terns and its connection to inhomogeneity-related asymmetries along the spatially resolvedmodel surfaces.

2.1 Formulation of the problem

2.1.1 Model atoms

For the purpose of this study, we consider an academic spectral line at λ0 = 5000 of a two-level atom. The Einstein rate of spontaneous deexcitation is Aul = 108 s−1. A collisionaldestruction probability � = 10−4 is used as a constant throughout the atmosphere.

We allow the upper level polarization (transition Jl = 0→ Ju = 1, where Jl and Ju theangular momentum quantum numbers of the lower and upper level, respectively), which

41

42 2. Inhomogeneous atmosphere models: spatially resolved case

is a typical example of resonance scattering. In terms of density matrix, the lower levelis de�ned only by its population-related element ρ00(l). The upper level is characterizedby ρKQ (u) elements with K = 0 and 2, i.e. ρ

00(u) and ρ

2Q(u) with Q ∈ {−2,−1, 0, 1, 2},

related to population (K = 0 rank) and alignment (K = 2 rank). Density matrix elementsof the upper level ρ2Q(u) are in the absence of another physical mechanisms in�uencingthe atomic polarization (for instance a magnetic �eld) proportional to the correspondingradiation �eld multipole moments J2Q.

Other principal choices include the lower level polarization, i.e. Jl = 1 → Ju = 0 andJl = 1→ Ju = 1. In the presented thesis we do not discuss e�ects of lower level polarization(i.e., Jl > 0), as it is left for future studies. Preliminary calculations suggest that, for theconsidered academic line, the intensity and polarization of the radiation emerging fromatmospheric models introduced further is only slightly a�ected by the choice whether weassume the lower level polarization or not. However, the lower-level polarization plays acrucial role in some speci�c cases of interpretation of observations, as it is connected tovarious mechanisms (selective absorption, lower-level Hanle e�ect, etc.) in�uencing theformation and transfer of radiation (e.g., Trujillo Bueno and Landi Degl'Innocenti, 1997;Trujillo Bueno et al., 2002; Manso Sainz and Trujillo Bueno, 2003).

2.1.2 Model atmospheres

As a reference model we use a semi-in�nite isothermal (T0 = 6000K) exponentially strati-�ed atmosphere:

N(z) = N0 exp

(−z − z0

β

). (2.1)

The 3D model grid with 100 × 100 × 120 points along the x, y, and z axes, respectively,is a box with the dimensions of Dx × Dy × Dz = 1000 × 1000 × 2000 km3 with periodicboundary conditions in the horizontal directions. The plasma at the very bottom of a modelat z = z0 ≡ 0 is assumed to be thermalized, i.e. the source function is determined only bythe thermal emissions and is equal to the Planck function at local temperature S(x, y, z0) =B(T (x, y, z0)). We assume that the atmosphere is not illuminated by any external sources,thus the upper boundary condition (z = zmax ≡ 120) is given by S(x, y, zmax) = 0.

Equation (2.1) re�ects the vertical strati�cation of an atmosphere due to equilibriumbetween gravity and buoyancy forces acting on a material at each geometrical height. Forthe strati�cation parameter we choose β = 75 km and N0 = 1012 cm−3.

2.1.3 Horizontal inhomogeneities in temperature and atomic vol-

ume density

We introduce inhomogeneities in the form of perturbations of temperature and atomicvolume density. Perturbations sinusoidally �uctuate in the horizontal plane along x and y

2.2. Optical properties of an inhomogeneous atmosphere 43

axes. For the atomic density N(x, y, z) we use:

N(x, y, z) = N(z)

[1 + αN sin

(2π

x

Dx

)sin

(2π

y

Dy

)], (2.2)

where Dx and Dy are the sizes of the domains in the x− and y−directions, respectively,and N(z) is the mean atomic density at the height z given by Eq. (2.1). The dimensionlessparameter αN ∈ [0, 1] quanti�es the amplitude of the perturbation: the case αN = 0corresponds to the 1D iso-density atmosphere while in the case αN = 1, the perturbationis maximum. The quantity N(z) of Eq. (2.1) also contains information about the averagedensity at each height of an atmosphere, independently of the perturbation amplitude αN .

For the temperature perturbations we use an analogous formula:

T (x, y, z) = T0

[1 + αT sin

(2π

x

Dx

)sin

(2π

y

Dy

)], (2.3)

where the dimensionless parameter αT has similar meaning as the parameter αN inEq. (2.2). The reference temperature T0 also refers to the mean horizontal temperature av-eraged over the x− y plane of the same geometric height z, i.e. T (z) = 〈T (x, y, z)〉x,y = T0.It is independent of αT .

If αN and αT have the same sign, the �uctuations are said to be in phase, if otherwiseαN and αT have di�erent sign, they are said to be in anti-phase. It is also important to notethat period of perturbations introduced by Eqs. (2.2) and (2.3) is equal to unity, thus theinhomogeneity oscillates just once per both horizontal domains along the x− and y− axeshaving dimensions Dx = Dy = 1000 km. We shall treat the case of di�erent perturbationperiods in Chapter 3.

2.2 Optical properties of an inhomogeneous atmosphere

Radiative transfer equation in the form (1.47) quanti�es changes of Stokes parameters ~I(ν)of the radiation beam propagating through an optically active medium in terms of gener-alized, position-dependent source and propagation coe�cients, i.e. ~S(ν) = ~�(ν)/ηI(ν) andK̃ = K(ν)/ηI(ν), respectively (de�nitions of all quantities are summarized in Section 1.4).After we have self-consistently solved the NLTE problem of the 2−nd kind for the con-sidered atmospheric model, we possess the detailed knowledge of their values at each gridpoint. It allows us to calculate the emergent radiation for an arbitrary line-of-sight speci�edby ~Ω = (µ, ϕ), obtained from the �nal formal solution of the radiative transfer equation inthe form (1.62).

Horizontal inhomogeneities of thermodynamic properties indicated by Eqs. (2.2) and(2.3) cause corresponding changes in optical properties of the model with respect to homo-geneous, plane-parallel solution, which can have a signi�cant impact on spectral featuresof the intensity and polarization of the emergent radiation.


Source function is at each point of the medium given by contribution of thermal emissionand locally scattered radiation, which is in the case of a plasma composed of two-level atomsgiven by a simple formula:

S = (1− �)J00 + �B(T ), (2.4)whereB(T ) is the Planck function, J00 is the mean intensity of the locally scattered radiationand � = 10−4 is the collisional destruction probability.

Both αN and αT can be expressed in terms of the Planck function and opacity �uctu-ations, which directly enter the radiative transfer equations. Relative changes we obtainfrom expressions of ηI(N, T ) by Eq. (1.51) and B(T ) by Eq. (1.60) in terms of N and T .Opacity is proportional to T and N as ηI ∼ N/

√T through the lower-level population and

line half-width due to thermal motions

∆νD =ν0c

√2kBT

m.

Let us de�ne changes in B(T ) and ηI(N, T ) at each grid node (x, y, z) where the per-turbation reach its extremal values by ratios

δB

B=B(T0 + δT )

B(T0)(2.5a)

δηIηI

=ηI(T0 + δT,N + δN)

ηI(T0, N), (2.5b)

where δT = T0αT and δN = NαN , giving δB/B = δηI/ηI = 1 if αT = αN = 0 or if itis calculated at points with T (x, y, z) = T0 and N(x, y, z) = N(z). Resulting formulas forextremal changes in opacity and Planck function due to perturbation amplitudes αT andαN are (frequency dependence dropped, assuming the line-center frequency):

δB

B=

exp[hνulkT

]− 1

exp[

hνulkT (1+αT )

]− 1

(2.6a)

δηIηI

=(1 + αN)√

1 + αT. (2.6b)

If we observe a point on the top of the model where the density perturbation reachesits minimum value (Fig. 2.1, top panels), we see deeper into the atmosphere, becausethe matter becomes more transparent, and therefore these point are the brightest due tolarger local values of the source function at the height of formation (see Fig. (2.1), bottompanels). However, temperature perturbations have exactly opposite e�ect: at the minimumof perturbation we see the coolest parts of the atmosphere (top panels in Fig. 2.2), andtherefore the faintest as well, as shown in Fig. (2.2, bottom panels).

Table 2.1 contains approximate values of ratios δηI/ηI and δB/B at points where theatomic density and temperature perturbations reach its maximal (positive and negative)values. Maximal change in the Planck function due to temperature perturbations with am-plitude αT = 0.6 is about δB/B ≈ 6.3 when the perturbation is at the positive maximum,

2.2. Optical properties of an inhomogeneous atmosphere 45

0.0

0.2

0.4

0.6

0.8

1.0

y [Mm]

Density ratio N(x,y,z)/N(z) Density ratio N(x,y,z)/N(z)

0.0 0.2 0.4 0.6 0.8 1.0x [Mm]

0.0

0.2

0.4

0.6

0.8

1.0

y [Mm]

log10I [cgs]

0.0 0.2 0.4 0.6 0.8 1.0x [Mm]

log10I [cgs]

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

−6.30

−6.25

−6.20

−6.15

−6.10

−6.05

−6.00

−5.95

−6.30

−6.25

−6.20

−6.15

−6.10

−6.05

−6.00

−5.95

Figure 2.1: Maps of perturbations in atomic volume density (αN = −0.6 to the left andαN = −0.99 to the right) along the horizontal x − y plane. Bottom panels show corre-sponding line-center intensities. Inclination µ = 1.0 (disc center).

and about δB/B ≈ 7×10−4 when the perturbation is at its minimum. Since the variation ofB(T ) with temperature is strong and nonlinear, we expect that horizontal inhomogeneitiesgive rise to asymmetries in irradiation of certain points across the line-forming region. Itleads to modi�cation of intensity of the emergent radiation in both spatially resolved andspatially integrated cases.

At surface points with low temperatures with respect to average T0 = 6000 K theopacity grows up, while the Planck function drops, and we obtain highly absorptive areasin the atmospheric model. In the opposite limit of high temperatures with respect toaverage T0, the opacity drops, while the Planck function grows up and we recover bright,more transparent areas with high local values of the source function.


Table 2.1: Changes in opacity and Planck function calculated from perturbations of atomicvolume density and temperature at its extremal values. For the given coe�cients αN andαT , we show approximate values of ratios δηI/ηI , δB/B and temperature at the pointswhere the sine functions of both equations (2.2) and (2.3) have the values −1 (min, x =3/4Dx) and 1 (max, x = Dx/4), at a �xed coordinate y = Dy/4.

Perturbation coe�cients δηI/ηI δB/B Temperature [K]αN αT min max min max min max0.0 0.4 1.3 0.9 4×10−2 4.0 3600 84000.0 0.6 1.6 0.8 7×10−4 6.3 2400 96000.6 0.0 0.4 1.6 0 0 6000 60000.99 0.0 1×10−2 2.0 0 0 6000 60000.6 0.4 0.5 1.4 4×10−2 4.0 3600 8400-0.6 0.4 2.1 0.3 4×10−2 4.0 3600 84000.99 0.6 2×10−2 1.6 7×10−4 6.3 2400 9600-0.99 0.6 3.1 8×10−3 7×10−4 6.3 2400 9600

0.0

0.2

0.4

0.6

0.8

1.0

y [Mm]

Temperature [K] Temperature [K]

0.0 0.2 0.4 0.6 0.8 1.0x [Mm]

0.0

0.2

0.4

0.6

0.8

1.0

y [Mm]

log10I [cgs]

0.0 0.2 0.4 0.6 0.8 1.0x [Mm]

log10I [cgs]

2400

3200

4000

4800

5600

6400

7200

8000

8800

9600

2400

3200

4000

4800

5600

6400

7200

8000

8800

9600

−6.30

−6.25

−6.20

−6.15

−6.10

−6.05

−6.00

−5.95

−6.30

−6.25

−6.20

−6.15

−6.10

−6.05

−6.00

−5.95

Figure 2.2: Maps of perturbations in temperature (αT = 0.4 to the left and αT = 0.6 tothe right) along the horizontal x− y plane. Bottom panels show corresponding line-centerintensities. Inclination µ = 1.0 (disc center).

2.3. Deformation of the τ = 1 surface 47

2.3 Deformation of the τ = 1 surface

Eddington-Barbier approximation estimates the radiation I(ν, µ) at frequency ν, emergingfrom an optically thick atmosphere in the direction µ = cos θ, θ being the inclinationangle of the line-of-sight, measured from the normal to the atmospheric surface. The maincontribution to the emergent intensity is from atmospheric layers where the optical depthfor given frequency and line of sight is around unity, and its magnitude is approximatelygiven by the local value of source function (e.g., Rutten, 2003):

I(ν, µ) ≈ S(ν, τν = µ). (2.7)

Thus, if we identify the τν = µ surface with the formation region of the radiation atfrequency ν for the given line-of-sight, we can use the relation (2.7) as a connection betweenspectral features of an emergent line radiation with the locally calculated source function(usually normalized to B(T ), cf. Eqs. (1.61) and (1.53)).

As frequency departs from the line-center ν = ν0, the area of formation of ~I(ν, ~Ω)moves to lower atmospheric layers due to the frequency dependence of the absorptionpro�le. Moreover, it is additionally deformed due to the dependence of opacity on varyingtemperature and atomic volume density. Thus, the radiation is at each frequency formedat di�erent depths of an atmospheric model, experiencing di�erent physical conditions(sources and opacities, irradiation anisotropies, geometrical asymmetries, etc.); or inversely,the intensity and polarization at each frequency of the emergent spectral line radiationcarries information about physical conditions at region of its formation. If an atmosphericmodel is also horizontally inhomogeneous (as is the case of a real stellar atmosphere), thespectral features of the emergent radiation also non-trivially depend on the inclination andazimuth angles of the direction of propagation, making practically impossible to reliablyapproximate stellar atmosphere by a plane-parallel model (Uitenbroek and Criscuoli, 2011).

Figures 2.3 and 2.4 show corrugation of the τ = 1 surface due to inhomogeneities in themodel with temperature and atomic volume density perturbation amplitudes αT = 0.6 andαN = −0.99, respectively. The particular model has been chosen to illustrate separatelythe e�ect of both temperature and density inhomogeneities. The colormap correspondsto source function normalized to the mean value of the Planck function S/〈B(T )〉, where〈B(T )〉 is the value of the source function at the deepest layers of the model, averaged overx− y plane (corresponding to the value of intensity at continuum).

Models are horizontally cut by the τν = 1 surface for frequencies such that the corre-sponding wavelength shifts (λ0 − λ) [] cover the emerging line intensity pro�le from line

astro.physics.muni.cz · Bibliographic Entry Author: Adam Tichý Masaryk Universit,y acultFy of...

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