In a previous paper (Casini & Landi Degl'Innocenti 1993, hereafter referred to as CL93), we attacked the problem of the formation of hydrogen lines in the presence of simultaneous electric and magnetic fields. There we gave the relevant formulae needed for the construction of the propagation matrix, which enters the vector radiative-transfer equation (VRTE) for polarized radiation.
In principle, the simultaneous solution of the VRTE and of the statistical equilibrium equations for the density-matrix elements of the atomic system would determine the Stokes profiles of any given spectral line at any optical (or geometrical) depth in the plasma structure under investigation (e.g., Landi Degl'Innocenti 1983). In practice, however, this goal is very seldom achieved because of the enormous complexity of the problem, especially when non-LTE effects resulting in atomic polarization (e.g., anisotropy of the radiation field) are accounted for.
In general, the presence of external fields further complicates the problem, if it is expected that the fields may vary within the plasma structure under investigation. In fact, the calculation of the propagation matrix for one configuration of the external fields may be very time consuming, and in general this calculation must be repeated at each of the intervals which subdivide the integration path of the VRTE.
In our first work on this subject (CL93), we considered a somewhat trivial application of the results there obtained, since we simply neglected atomic polarization and assumed uniform electric and magnetic fields, and a constant source function (i.e., uniform temperature), throughout the plasma structure. In addition, we assumed that the plasma were optically thin--so the integration of the VRTE boiled down to the mere calculation of the emission vector--and that the external fields were the only line-broadening mechanism present. The introduction of thermal-Doppler broadening in subsequent work (see Casini & Landi Degl'Innocenti 1995; Casini & Foukal 1996, hereafter referred to as CF96) enabled us to consider less academic, yet still limited, applications of the original results.
In the present paper, we intend to further relax the constraints imposed in our previous applications. So we will consider the formation of polarized hydrogen lines in a bounded, optically-thick slab, assuming different analytical dependences of the source function on the optical depth. The other constraints will be maintained instead, in particular the severe limitation of neglecting non-LTE effects on the populations of the atomic levels. This is, in fact, a major reason for discrepancy between observed and calculated polarization profiles, whenever our calculations are applied to cases in which, for instance, anisotropy of the radiation field cannot be neglected, such as in prominences. However, having considered different analytical forms of the source function should enable one to parameterize departures from the LTE approximation, limitedly to the problem of the validity of Boltzmann's and Saha's equations.
We also assume that the temperature of the slab and the external fields
are slowly-varying functions of the optical depth, so we can safely
approximate the propagation matrix with a constant matrix
to be evaluated
once, on a set of sample frequencies within the spectral range of the
line. This approximation then enable us to explicitly integrate
the VRTE, thus obtaining analytical expressions for the emerging Stokes
vector.
Actually, the constraint of a constant propagation matrix could be relaxed without introducing any further conceptual involvement in the calculation of the polarization profiles, as long as non-LTE effects are still neglected. However, the integration of the VRTE could no longer be performed analytically, and one should rather approach the problem through the evolution-operator formalism (Landi Degl'Innocenti & Landi Degl'Innocenti 1985; Landi Degl'Innocenti 1987).
Our results are then applied to the calculation of the Stokes profiles
of hydrogen lines emerging from a slab of different total optical depths.
This is done in the case of H
, and of the 7-6 and 12-8
transitions, which are of relevant interest in the investigation of solar
plasmas. In particular, the 7-6 and 12-8 transitions have been proved
particularly promising as diagnostic lines for electric-field measurement
in solar coronal structures (Casini 1996; Foukal & Behr
1995, CF96).