Abstract:
The discovery of Zeeman effect opened a new window to observe the Sun. It enabled
to understand the magnetic field which is one of the basic features that govern
different physical processes on the Sun. These magnetic fields leave their signatures
in the splitting and polarization of spectral lines which can be extracted using
spectro-polarimetry. With the advent of increasingly powerful telescopes, like with
the SOT/SP instrument on-board the Japanese HINODE satellite, which achieved
a spatial resolution of 0.3 arcsec (or 200 km on the Sun), the small-scale structure
of solar magnetic fields have been explored via the Zeeman effect. It is however
now understood that the fundamental building blocks of solar magnetism are present
on scales much smaller than those resolved by HINODE. The polarization of line
radiation which is caused by resonance scattering on bound atomic levels acts as
a tool to measure these small scale magnetic fields. A modification of this process
by external magnetic fields is called the Hanle effect. Comparison between
the available constraints from the Hanle effect with the magnetic fluxes resolved by
the HINODE spacecraft reveals that about two thirds of the total magnetic flux remains
invisible at the HINODE 200 km resolution, since this flux is tangled on very
small scales, possibly 10-100 m, which is nearly 4 orders of magnitude smaller than
the HINODE resolution limit. While Hanle effect constraints on the properties of
this “hidden” magnetic flux exist, very little is known about its depth dependence.
One of the main objectives of the thesis is to explore the center-to-limb variation
(CLV) of the Stokes profiles which is in turn governed by the height variation of the
temperature-density structure in the solar atmosphere. Apart from this we study one
of the primary physical quantity needed to carry out the analysis of the scattering
process i.e., the redistribution matrix. These studies are motivated by the existence
of many unexplained signatures in the Second Solar Spectrum (SSS), which is the
linearly polarized spectrum of the Sun caused by coherent scattering process.
The thesis is divided into three parts. The first part is dedicated to the observations
and modeling of the CLV of the well known Ca I 4227 Å line. Not much
progress is done in the literature in the area of modeling the CLV of different lines in
the SSS because of the complexity of the problem. The main challenge is to obtain a
single model atmosphere which can provide a simultaneous fit to the CLV of the (I,Q=I) spectra. From the theoretical perspective, the line radiative transfer modeling
of the observed data in the magnetically quiet regions using the scattering theory is
essential. To this end, we have to solve the polarized line transfer equation that governs
the absorption, emission and scattering of radiation in the stellar atmosphere.
We use the one-dimensional (1D) modeling approach to study the CLV of the Ca
I 4227 Å line, which exhibits largest scattering polarization of all the lines in the
visible spectrum of the Sun. For the purpose of our studies we have observed the
Ca I 4227 Å line at 14 positions from the center-to-limb on the Sun using Zurich
Imaging Polarimeter-3 at Istituto Ricerche Solari Locarno in Switzerland. We have
modeled the CLV of this line using different realistic solar atmospheres (Chapter 2).
From the studies in Chapter 2 we concluded that no single 1D model attempted by us
helps us in providing a simultaneous fit to the CLV of the Stokes profile. The solar
atmosphere is too complex to be represented by a 1D model atmosphere. We have
to go beyond 1D modeling like multi-dimensional modeling to represent the actual
solar atmosphere. However this does not represent an impediment to the use of the
Ca I 4227 A line for solar magnetic field diagnostics. To demonstrate this we have
carried out a model independent analysis to determine the turbulent weak magnetic
fields in the solar atmosphere.
The second and third part of the thesis is dedicated to the theoretical studies of
the fundamental physical quantity called the redistribution matrix which is required
to study the physics of scattering processes. The redistribution matrix contains all
the information of the scattering process, and hence is an important parameter to be
studied in detail. In this regard, there are certain approximations made in different
theoretical approaches developed in the literature to enable reduce the complexity of
the problem. However, such approximations point towards the existence of spectral
features in the SSS which cannot be explained using the available standard theories.
Our efforts in the second and third part of the thesis is to relax some of the approximations
and study its effects on the emergent Stokes profiles by considering different
examples.
In second part of the thesis we study a series of problems concerning the polarized
line formation with the angle-dependent partial frequency redistribution (PRD).
The liner polarization of the strong resonance lines are sensitive to the type of frequency
redistribution used. The PRD matrix is dependent on the incoming and outgoing
frequencies and angles. In order to reduce the computational efforts angleaveraged
PRD functions are used in most of the studies in the literature. In Chapters
3, 4 and 5 we consider different problems and relax this approximation and study
the effects of using angle-dependent PRD on emergent Stokes profiles. In Chapter 3
we study the effect of electron scattering redistribution on atomic line polarization in
non-magnetic regime. We use angle-dependent electron scattering and atomic redistribution functions and present efficient numerical technique to solve this problem.
In Chapter 4 we study the combined effects of angle-dependent PRD and quantum
interference phenomena arising between the fine structure (J) states of a two-term
atom or between the hyperfine structure (F) states of a two-level atom by restricting
our attention to the case of non-magnetic and collisionless line scattering on atoms.
From the studies in Chapters 3 and 4 in the non-magnetic regime we conclude that
the effect of angle-dependent PRD are sensitive to the optical thickness of the slab
used for the radiative transfer studies. In Chapter 5 we study the effect of using
angle-dependent PRD in the presence of weak magnetic field, i.e. the Hanle effect.
We present efficient decomposition techniques to solve the problem at hand. We
point towards the necessity to use angle-dependent PRD to solve the Hanle transfer
problem accurately, especially for the Stokes U parameter by taking the example of
vertical magnetic fields and turbulent magnetic fields.
In the third part we have attempted to relax another common assumption made
in our previous calculations, i.e. assuming the polarization of the lower-level of the
atom involved in the scattering process is zero. In Chapter 6 we formulate a general
theory for magnetized media to handle the problem of radiative transfer including
the effects of PRD and polarization of the lower-level of the atom involved in the
transition, starting from Kramers-Heisenberg scattering formulation. We then obtain
the radiative transfer equation starting from the well established quantumfield theory
approach for the problem at hand. Further we apply this theory to two case studies
in the non-magnetic regime which leads us to the conclusion that the effects of lower
level polarization are significant only in the line core. Based on our conclusion
we also propose a simplified numerical approach to solve the problem of polarized
radiative transfer with PRD and lower-level polarization.