Scattering theory and radiative transfer in spherically symmetric moving atmospheres

Abstract

Magnetic fields are ubiquitous in astrophysical plasmas. They are responsible for most of the stellar activities and they manifest in numerous ways in the stellar atmospheres. The Sun being our nearest star is the natural laboratory to understand the causes and effects of the magnetic fields. The solar magnetic field couples the solar interior with its atmosphere. It drives the dynamic phenomena such as coronal mass ejections (CMEs) and solar flares. The magnetic field also plays a critical role in heating the solar upper chromosphere and corona as well as in accelerating the solar wind. A variety of techniques are used to infer these magnetic fields and subsequently map them into the layers of the solar atmosphere, from where the concerned observable originates. Studies of polarization properties of spectral lines formed in solar atmosphere serve as one of the best methods to determine the nature of solar magnetic fields (Stenflo 1994). The polarization is the result of breaking of symmetry in the source region. This symmetry breaking in the line forming regions can be attributed to the anisotropic illumination of the atoms and presence of magnetic fields. In the presence of magnetic fields the energy levels of the atoms split into different magnetic m sub-states. The transition from these split magnetic sub-states results in circularly or linearly or elliptically polarized light depending on the strength and orientation of the magnetic field with respect to the chosen line-of-sight (LOS). This effect discovered by Zeeman in the laboratory in 1896, is popularly known as Zeeman effect. It was first seen on Sun by Hale (1908) in sunspots. Using Zeeman effect in spectral lines it became possible to measure both the strength and orientation of the magnetic field vector (especially the strong fields of few kG). Zeeman effect cannot be used effectively in the presence of very weak fields (due to extremely small splitting) and also, it is not suitable to measure the turbulent fields (due to cancellation of the opposite polarities of the magnetic field within the finite spectral resolution). Under such conditions the magnetic field in the solar atmosphere can be determined by the linear polarization measurement of spectral lines with appropriate sensitivity to Hanle effect. Hanle effect is the result of quantum interferences between different magnetic m sub-states of a given atomic level involved in the transition. It is the modification: depolarization or repolarization and rotation of plane of linear polarization, of resonance scattering in the presence of weak magnetic fields. Therefore Zeeman and Hanle effects can be used to diagnose the magnetic field of the Sun in a very different and complementary parameter regimes. The aim of this thesis is to develop pure theoretical tools required for the determination of the solar magnetic fields using polarized spectral line formation theory. The thesis is divided into two parts. In the first part (Chapters 2 and 3) we develop the scattering theory for magnetic dipole (M1) transitions and study the forbidden emission lines formed in solar coronal conditions or any other astrophysical objects with diffuse media. In the second part (Chapters 4–8) we consider the problem of polarized line formulation in spherically symmetric moving atmospheres, for the case of optically thick permitted (electric dipole allowed) lines. Chapter 1 gives a general introduction to the thesis, wherein we describe the basic physical concepts required in both parts of the thesis. Chapter 9 summarizes the work carried out in this thesis and also gives the future outlook on the problems described in both the parts of the thesis. Part-I: Scattering Theory for Magnetic Dipole (M1) Transitions In Chapter 2, we derive the Hanle-Zeeman scattering matrix for M1 transitions. It can be used to study the forbidden emission lines that are formed in any diffuse astrophysical media such as solar corona. Because the Einstein A coefficient is small for the forbidden lines, the magnetic splitting is much larger than the natural line width even for very weak coronal magnetic fields. Thus, we mainly remain in the regime of saturated Hanle effect in forbidden lines, in the solar corona. Thus, Hanle effect in forbidden lines provides an important means of diagnosing the topology of coronal magnetic fields. The earlier formulation of the problem of scattering on forbidden lines in magnetic fields is limited to the regime of saturated Hanle effect. Our aim here is to present a new alternative formulation of the required scattering theory that covers the entire field strength regime. In Chapter 3, we apply the theoretical formalism developed in Chapter 2 to understand the effects of density distributions, magnetic field configurations, and velocity fields on the emergent Stokes profiles of the [Fe xiii] 10747°A coronal forbidden line. We also describe the procedures to perform the integration over the solid angle of incident cone of radiation and the LOS integration, to conduct the above mentioned empirical studies. Part-II: Polarized Radiative Transfer in Spherically Symmetric Moving Atmospheres The plane-parallel approximation of the stellar atmospheres can not be applied to model the formation of several optically thick lines in extended atmospheres. To a good approximation these atmospheres can be represented by a spherically symmetric medium. Furthermore the extended stellar atmospheres are known to be highly dynamic, with low to high speed stellar wind originating in these layers. Such velocity fields present in the line forming regions produce Doppler shift, aberration of photons, and also give rise to advection. All these effects can modify the amplitudes and shapes of the emergent Stokes profiles. Thus our aim here is to develop numerical techniques to solve the polarized line transfer equation in spherically symmetric extended atmosphere with velocity fields and also magnetic fields. In Chapter 4, we develop modern iterative techniques based on operator perturbation to solve the polarized transfer equation in a spherically symmetric static atmosphere. Apart from the Jacobi based polarized accelerated lambda iteration (PALI) method, in this chapter we also develop the fast iterative techniques based on Gauss-Seidel (GS) and successive overrelaxation (SOR). These latter methods are known to be superior over the traditional Jacobi iterative scheme. We describe the numerical steps for the Jacobi, GS, and SOR techniques and study their convergence behavior in the presence of both partial frequency redistribution (PFR) and complete frequency redistribution (CFR) scattering mechanisms. In Chapter 5, we include the effects of radial velocity fields to the problem of polarized line formation in spherically symmetric atmospheres. We describe both observer’s frame and comoving frame (CMF) methods to solve the problem under consideration. We describe the Jacobi based CMF-PALI method in detail as this method is computationally much superior than the observer’s frame method. We also discuss the inclusion of GS and SOR techniques in the CMF-PALI method and study their convergence behavior in the presence of velocity fields for both CFR and PFR. In Chapters 6 and 7, we discuss in detail the numerical results obtained for the problem of polarized line formation in spherically symmetric static and expanding non-magnetic atmospheres. With the help of contribution function and Stokes source vector, we explain the nature of the polarized line profiles formed in both static and moving atmospheres in presence of both CFR and PFR (Chapter 6). We also vary the model parameters both the atmospheric and atomic parameters one at a time (keeping the other parameters as constants) and study the dependence of the linearly polarized line profiles on the model parameters for both static and moving atmospheres (Chapter 7). In Chapter 8, we extend the Jacobi based CMF-PALI method developed in Chapter 5 to include the weak magnetic fields. In the presence of weak fields, Hanle effect comes into play. Unlike the non-magnetic case the physical quantities involved in the problem now become radiation field azimuth dependent. We take into account the approximation-III of Bommier (1997b) to represent angleaveraged PFR in the presence of weak magnetic fields. We also discuss the polarized line profiles formed in the presence of both magnetic and velocity fields for the case of CFR and PFR.