Application of polarized line formation theory to the solar spectrum

Abstract

One of the primary causes for the occurrence of observed features on the Sun is its magnetic field. The traditional way of measuring the solar magnetic fields is through the Zeeman effect. Over the decades, as the precision of the observing instruments has improved, our understanding of the solar magnetic fields has deepened. We now know that there exist fields on the Sun which remain invisible to the Zeeman effect. This apparent invisibility of the fields can be due to their lower strength, orientation, or their tangled nature. However, these fields can still be detected with the help of the Hanle effect. In the absence of magnetic fields, the Sun’s radiation is linearly polarized due to the coherent scattering processes, and the resulting spectrum is known as the “Second Solar Spectrum”. It contains a wealth of richly structured atomic and molecular lines arising from a variety of atomic systems and atmospheric conditions. In the presence of weak magnetic fields, this linearly polarized spectrum gets modified by the Hanle effect. This modification is in the form of a decrease or an increase in the linear polarization (Q/I) depending on the scattering geometry. In addition to this, the Hanle effect also rotates the plane of polarization generating a U/I signal. Understanding the physics of light scattering on atoms and the measurement of weak magnetic fields requires a detailed forward modeling of the spectral line profiles. For this, adequate theoretical tools which can handle the physical processes involving complex atomic systems are needed. Of particular interest are the quantum interferences occurring between the atomic states. There are several lines in the Second Solar Spectrum which are governed by this phenomenon and their analysis demands a proper theoretical treatment of the quantum interferences. This thesis is devoted to exploring the quantum interferences between the fine structure states (J) and hyperfine structure states (F), followed by application of the theoretical derivations to the actual solar observations. The thesis is divided into three parts. In the first part, we derive the redistribution matrix taking account of the J-state interference effects which occurs in atoms with non-zero electron spin S (Chapter 2). For the first time, we derive this matrix semiclassically starting from the Kramers-Heisenberg formula by taking full account of the effects of partial frequency redistribution (PRD) in the presence of magnetic fields,and present the expressions both in the atomic and laboratory frames. The J-state interference acts in the wings between the lines arising from the transitions involving fine structure states. The redistribution matrix relates the Stokes vector of the incident ray and that of the scattered ray. It contains the physics of line scattering. The matrix, derived in Chapter 2, holds good only in the linear Zeeman regime of magnetic fields. In this regime, the magnetic substates (m) belonging to different J-states do not overlap. For the range field strengths found in the solar atmosphere, this theory holds good in most of the cases. The redistribution matrix is then included in the polarized radiative transfer equation and solved for one-dimensional isothermal constant property atmospheric slabs in order to understand the nature of the redistribution phenomenon in the non-magnetic (Chapter 3) and magnetic cases (Chapter 4). We present a heuristic approach of including different types of collisions into the J-state interference theory (Chapter 4). We then apply this theory to the modeling of Stokes (I, Q/I) profiles of the Cr I triplet around 5206 A observed near the solar limb (Chapter ˚ 5). The theoretical (I, Q/I) profiles computed from our theory match closely with the observed profiles, especially at the cross over points in Q/I which are due to the J-state interference effects. We demonstrate the importance of the PRD effects. Clearly, the profiles computed under the assumption of complete frequency redistribution (CRD) fail to reproduce the observed profiles. To obtain a good match at the far wings, we find it necessary to slightly modify the temperature structure of one of the standard one-dimensional (1-D) model atmospheres. With such a small modification, we are able to reproduce the Q/I profile not only at the line core but also in the continuum. In the second part of the thesis, we extend the above formalism to the case of F-state interference which occurs in atoms with non-zero nuclear spin Is. Unlike the J-state interference, the F-state interference acts only in the line core. It causes a decrease in the line core polarization. Like in the previous case, we include the redistribution matrix into the polarized transfer equation and study the nature of the emergent Stokes profiles (Chapter 6), confining our attention to the non-magnetic case. By taking example of the Ba II D2 line profile, we test our theory by reproducing the observed (I, Q/I) profiles (Chapter 7). Barium is a complex atomic system with seven isotopes. Only two out of these seven are odd and exhibit F-state interference. The rest are even isotopes whose atomic levels do not undergo hyperfine structure splitting. The observed (I, Q/I) profiles have contributions from both the odd and the even isotopes in ratio 18:82 (as established from solar abundance studies). The two odd isotopes are treated using the F-state interference theory and the five even isotopes are treated using the simple twolevel atom theory. These are then combined in their respective ratios in the scattering integral while solving the transfer equation. However to obtain a good match with the observed profiles at the line core, we find it necessary to slightly modify the temperature structure of a standard 1-D model atmosphere. We also show that it is not possible to model this line using CRD alone. We consider another example, namely the Sc II 4247 A line which also is governed ˚ by the F-state interference (Chapter 8). The core of this line is composed of thirteen hyperfine transitions which are to be accounted for while modeling it. These thirteen transitions are due to the five upper and five lower F-states resulting from hyperfine structure splitting of the upper and lower J = 2 states. From our efforts so far, we find it difficult to reproduce the triple peak structure seen in the observed Q/I profiles, and also the rest intensity. We get a good match at the wing PRD peaks and at the near wing continuum but the central peak seems to be completely suppressed due to enhanced depolarization by the F-state interference. We suspect that the lower level Hanle effect, which is not accounted for in our treatment, might be playing a role since it can increase the line core polarization when the fields are weak. We are yet to develop the theory of lower level Hanle effect including PRD. This derivation is extremely complex, although in principle it is achievable. Therefore we feel that the Q/I spectra of the Sc II 4247 A˚ line has remained enigmatic to us. It would be very interesting to investigate it in the near future. In the last part of the thesis, we combine the theories of J-state and F-state interference effects and derive a unified redistribution matrix which can handle both these effects simultaneously. In atoms with non-zero S and Is, the atomic states undergo fine structure and hyperfine structure splitting, and exhibit both kinds of interferences. We confine our attention to the collisionless non-magnetic regime, and study the Stokes profiles formed in a 90◦ single scattering event (Chapter 9). Finally, the work presented in this thesis is summarized and the possibilities of extending it to measure the solar magnetic fields is discussed in the last chapter (Chapter 10).