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Astron. Astrophys. 324, 344-356 (1997)

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4. Concluding remarks

We have seen how the physical processes that shape the second solar spectrum ([FORMULA]) are quite different in nature from those of the ordinary intensity (Stokes I) spectrum. Examples have been given of prominent spectral signatures of quantum interference, hyperfine structure, and fluorescent scattering in the observed polarized spectrum, effects that are invisible in the normal solar intensity spectrum. The observed degree of polarization is determined by the atomic polarizability factor [FORMULA], the ratio between the line and continuum opacities, the anisotropy of the incident radiation field, depolarizing collisions, magnetic fields (Hanle effect), etc. In general one would need to account for all these effects in a full treatment of radiative transfer with polarized scattering and partial redistribution for a numerically given model atmosphere. For initial, exploratory purposes we have introduced an idealized, parametrized model that bypasses the radiative transfer problem, ignores collisional and Hanle depolarization, and treats the effect of the anisotropy of the radiation field (geometric depolarization) as a frequency-independent scale factor.

In spite of these quite drastic idealizations it has been possible to obtain surprisingly good fits to some of the more prominent observed polarization features and to identify the underlying physics. Thus the sign reversals of the polarization curve around the Na I D1 and D2 lines are shown to be due to quantum interference between the [FORMULA] and [FORMULA] excited states. The apparently anomalous relative polarization amplitudes of the 318 Fe I 4957 Å doublet lines are found to be due to fluorescent contributions within the multiplet. The triplet structure of the observed polarization profile of the Ba II 4554 Å line is explained in terms of hyperfine structure splitting and relative isotope abundances.

At the same time we have identified observed polarization features that we are not yet able to explain within the framework of the present theory. While we believe that the triplet polarization peak around the core of the Na I D2 line is primarily the result of partial redistribution effects coupled to radiative transfer, which could be modeled if a full radiative-transfer treatment with a realistic model atmosphere would be carried out, the narrow polarization peak at the center of the Na I D1 line is an enigma. As a [FORMULA] transition it should be intrinsically unpolarizable at the resonant frequency (line center), and hyperfine structure in the D1 line does not seem to help much.

The reality of the polarization peak in the Na I D1 line is supported by observations of an even more pronounced, narrow polarization peak in the core of the Ba II 4934 Å line (Stenflo & Keller 1996b ), which is the "D1 line" of multiplet No. 1 of Ba II, for which the 4554 Å line is the "D2 line". Since for this multiplet the fine structure splitting is as large as 380 Å , quantum interferences between the D2 and D1 lines are insignificant. Furthermore, the Ba II D1 line is less strong and deep in Stokes I as compared with the Na I D1 line. The D1 core polarization is thus an enigma not only for Na I, but even more so for Ba II.

One aspect that we have left out of the present treatment is the frequency redistribution problem, which needs to be dealt with before one can incorporate the present theory (which here only has been given in frequency-coherent form) in a radiative-transfer formalism with full physical realism for quantitative modelling of solar structures. Well-defined formulations of partial frequency redistribution of polarized radiation exist for single-transition Rayleigh scattering, but the generalization to multiple excited levels with quantum interferences is far from straightforward. It is for instance not clear how collisional redistribution works in the case of a mixed quantum state, like the [FORMULA] and [FORMULA] coherent superposition of the excited states of the Na I D1 - D2 scattering transition. Such mixed quantum states are very common.

A major simplification of the theory has been achieved by ignoring any atomic polarization of the initial state. This allows us to discuss the polarization properties in terms of Mueller scattering matrices or phase matrices that are decoupled from the statistical equilibrium problem. The neglect of initial-state polarization should be a very good approximation for almost all cases that will be encountered in the solar spectrum, since for spectral lines with scattering as a significant contributor to the line emission the initial state generally has such a long life that it has plenty of time to be depolarized by collisions and weak magnetic fields. The life time of the initial state with respect to radiative absorption is longer than the life time of the excited state (which is determined by the spontaneous emission rate) by approximately the Boltzmann factor [FORMULA], where T is the radiation temperature in the solar atmosphere (about 6000 K). In the visible part of the spectrum this factor is on the order of 100. Our theory could in principle readily be extended to include initial-state polarization, but the advantage that this would bring would be minor in comparison with the great technical complications it would entail. The generally far larger effects of magnetic fields, collisions, partial redistribution, and radiative transfer need to be dealt with first.

Although our theory for polarized Raman scattering with contributions from entire multiplets in principle allows for the presence of magnetic fields of arbitrary strength and direction, we have only expressed it in explicit form for the case of zero magnetic field. A major future task will be to extend the theory to provide an explicit framework that is suited for calculations that include the Hanle and Zeeman effects as well as the mixed regime of intermediately strong fields. We need to be able to handle arbitrary fields for the interpretation of the next generation of vector polarimetric observations, which will be produced by ZIMPOL II, the second generation of our imaging Stokes polarimeter. Then it will be possible to explore the local spatial fluctuations of the scattering polarization due to magnetic fields and the spectral signatures of the mixed Zeeman-Hanle regime in active regions.

The inclusion of magnetic fields in the theory leads to great technical complications because of the complex geometries (which involve the four spatial directions of the incident and scattered radiation, the magnetic field vector, and the local vertical), so certain idealized regimes will first be dealt with, like microturbulent magnetic fields (Stenflo 1982 ; Faurobert-Scholl 1993 ; Faurobert-Scholl et al. 1995 ). Another theoretical challenge will be to develop a sufficiently fast computer code for general multi-level polarized radiative transfer, which is flexible enough to incorporate our Raman scattering theory with magnetic fields. Since we with good reason may disregard the initial-state atomic polarization it is sufficient to treat the statistical equilibrium part of the multi-level problem with standard techniques that ignore the polarization, and then use the resulting level populations to solve the vector radiative transfer equation with the polarized scattering matrix and the absorption Mueller matrix, both of which contain the Zeeman effect. When such a tool for the solution of general polarized radiative transfer problems will become available, we will be in a position to begin to more systematically exploit the rich diagnostic potential of the second solar spectrum.

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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998