Transfer of polarized radiation in astrophysical context a thesis submitted for the degree of Doctor of Philosophy in the faculty of science, Indian Institute of Science, Bangalore S. K. Sengupta [Ph.D Thesis]

Thesis Supervisor A. Peraiah

Doctor of Philosophy Indian Institute of Science, Bangalore 1996

Resonance line polarization is sensitive to the magnetic field (Hanle effect), the collisional rates, interference between atomic sublevels, the frequency redistribution, the degree of angular anisotropy of the incident radiation and the geometry of the line forming region. The collisional rates are determined by various physical causes. The above mentioned processes are greatly affected by the radial expansion of the atmosphere. On the other hand it is well-known from observations that early type stars, giant and supergiant stars, symbiotic stars) luminous late type stars possess differential velocities along the radial direction in their outer layers with high velocities of tens of mean thermal units and sometimes even hundreds of mean thermal units. These objects show degree of polarization ranging from 0.1 % to 5 % in the optical band. Therefore in modelling the atmospheres of these objects it is essential to include velocity fields which may play crucial role in the formation of polarization profile.
The present study was undertaken with the goal of understanding the behavior of line polarization in an extended and radially expanding stellar atmosphere. In this thesis detail study of the effect of differential radial velocity on the distribution of line intensities and line polarization of an extended stellar atmosphere has been made with the two level atom approximation. They are described in the following chapters of this thesis. The contents of these chapters are briefly presented here.
The first chapter contains a brief survey of the investigations on the problem of transfer of polarized radiation in stellar atmospheres that has already been done by various authors. In the second chapter the relevant equation for the transfer of polarized radiation in an extended and expanding stellar atmosphere has been formulated. In general a linearly polarized beam is represented by the three Stoke's parameters I, Q and U. In a plane parallel or spherically symmetric medium, because of the axial symmetry of the radiation field, only two parameters are required to define the polarization state of the radiation field: the specific intensity of the radiation field I (= 1/ + Ir) and the polarized intensity Q (= I, - Ir) where 1/ and Ir are the components which are perpendicular and parallel to the surface respectively. The degree of linear polarization p = Q/I gives the measure of the angular anisotropy of the diffuse radiation field.
In the rest frame calculations one can use velocities up to one to two mean thermal units only. Beyond two mean thermal units it is very difficult to get correct solution. The difficulties arise because of the fact that absorption co-efficient changes continuously due to Doppler shifts and therefore the angle frequency mesh required to solve the equation becomes enormously large and one has to take a mesh of infinite size. The calculation becomes quite involved simply because photons can be redistributed from any given point to any other point in the interval vo(l- Vma~/c) to vo(l + Vma:r:/c) where Vo is the central frequency of line and Vma~ is gas velocity. Hence the transfer of radiation has been considered in the comoving frame. In comoving frame it is easy to handle such large velocities. The absorption co-efficient in the comoving frame is constant as the Doppler shifts do not create problems of frequency changes in the line absorption. Therefore one can reduce the size of the angle frequency mesh considerably and the number of angles and the number of frequency points are considerably reduced in the case of comoving frame and this can easily give exact solution for the atmosphere of stars in which the velocities are as large as 100 mean thermal units and even more. Further the phase function for the static case which gives the angular distribution of the polarized radiation can well be applied in the comoving frame calculations. The transfer equation for polarized radiation in comoving frame is presented in this chapter.
In the third chapter the methodology for the numerical solutions of the transfer equation for polarized radiation is described in detail. For solving the plane parallel and the spherical radiative transfer problems in cormoving frame the method due to Peraiah has been employed. In general the following steps are taken in obtaining the solution. (i) The medium is devided into a number of 'cells' whose thickness are less than or equal to the critical thickness that is determined on the basis of the physical characteristics of the medium. The critical thickness ensures stability and uniqueness of the solution. (ii) The integration of the transfer equation is performed on the 'cell' which is a two dimensional grid bounded by radial and angular points. (iii) The discrete equations are compared with the canonical equation of the interaction principle and the transmission and the reflection operators of the 'cell' which contains all the physical informations in the problem under consideration are obtained. (iv) Lastly, all the cells are combined by internal field algorithm and the diffuse radiation field is obtained. For a thick layer, the so called star product is used. The generalization of all the steps of this finite difference method to include the polarization state of radiation is presented in this chapter. Since a two dimensional vector {III,.)T has been employed to represent the specific intensity vector, the matrices appearing in the computational algorithm get dimensions twice as much for the corresponding scalar line transfer problem.
The method can handle the problem arising out of the coupling of the comoving points across the line profile and the local velocity gradients. The stability of the solution is achieved by controlling the step size which arises in the discretization in radial, angle and frequency integrations. Choosing a proper step size, stable solution can be found. Thus the stability and the uniqueness are maintained.
Finally, the frequency-independent 1- and r- components of the source vectors that have been computed in the comoving frame are used to obtain the corresponding line profiles and the polarization line profiles along the line of sight of an observer at infinity.
In the fourth chapter the effect of differential radial velocity on the distribution of line intensities and line polarization in a stellar atmosphere stratified in parallel planes has been presented in detail. The medium is assumed to be homogeneous and isothermal. Two different types of velocity rules have been adopted in this case. The velocity at the outermost layer is taken to be 5, 10 and 20 mean thermal units with zero velocity at the innermost layer. The results have been compared with that of the static case. The line intensity profile and the polarization profile in the comoving frame as well as in the observer's frame are discussed in detail. Two types of media have been considered: (i) purely scattering medium and (ii) partially scattering medium through which the role of the thermalization parameters is investigated.
As the stellar radius increases the curvature effect plays a dominant role and a spher- ically symmetric geometry becomes more relevant in that case. In the fifth chapter the effect of differential radial velocity in the distribution of line intensities and line polarization for a spherically symmetric, inhomogeneous and isothermal medium is presented in detail. The atmospheric models could represent the outer layers of early type stars, giant and supergiant stars as well as luminous late type stars. In the fourth chapter the role of non-zero thermalization parameter has been discussed. In the case of a spherically symmetric stellar atmosphere, a fixed value of the thermalization parameter is taken in all models and the effect of differential radial expansion to line polarization under different optical depth as well as sphericity of the medium has been discussed in detail.
Although the models employed are highly idealized, they can provide a reasonably good physical insight into the polarized line formation problem in a radially expanding and extended spherical stellar atmosphere.
Finally, in the sixth chapter the results of the investigation described in the thesis are summarized with specific conclusions.