## 2. Theoretical approach## 2.1. Physical processes involvedIn order to quantitatively describe radiative transport (see Sect. 2.2) the physical processes have to be understood. Traditionally a distinction is made between pure absorption and scattering (Mihalas 1978). Here we focus on the processes causing the continuous spectrum. In pure absorption part of the energy of the radiation field is converted into kinetic energy of the gas and thus thermalized. As was first proposed by Wildt (1939), the negative ion of hydrogen dominates the continuous absorption in the solar photosphere, where the visible continuum is formed. The scattering coefficient in our computer code contains the combined effects of both Thomson scattering at free electrons, which is wavelength independent, and Rayleigh scattering at neutral hydrogen, which obeys the well-known law. Both processes are coherent. In the scattering process an incident, anisotropic radiation field gets polarized. The anisotropy, a necessary prerequisite for the scattering polarization, is mainly a consequence of the CLV of the intensity, i.e., the limb darkening. ## 2.2. Formulation of the transfer problem for polarized radiationWe consider a plane-parallel, static atmosphere with homogeneous
layers. No magnetic field is included in the calculations. The
polarized radiation field is described by the four Stokes parameters
and Hence, we can exclude Stokes We introduce , where is the angle between the direction normal to the surface and the line of sight. The optical depth is defined as
with the total source function The first term in Eq. (5), associated with pure absorption, is determined by the Planck function and is given by The second term in Eq. (5) contains all radiative sources associated with scattering and can be written as where represents the direction of the incident radiation within the differential solid angle , and is the Rayleigh phase matrix, which accounts for the angular dependence of Rayleigh and Thomson scattering. This is given by (Stenflo 1994) where denotes a matrix with the only non-vanishing element , representing unpolarized, isotropic scattering, whereas the matrix accounts for linear-polarization scattering and is given by ## 2.3. Numerical techniqueThe structure of the computer code is as follows: The most important input is the solar model atmosphere. In the first step the absorption and scattering coefficients, and , are calculated while neglecting polarization. This is accomplished by solving the hydrogen radiative transfer problem in non-LTE using the code MULTI by Carlsson (1986). In MULTI and are obtained with the opacity package of Gustafsson (1973), taking non-LTE aspects into account. The second step then consists of solving the polarized transfer equation (4) using the previously computed and . For the solution of Eq. (4) a standard technique is used, the so-called Feautrier method. The idea of the Feautrier method is to write the transfer equation for each depth point in the form of a system of second-order differential equations subject to two boundary conditions, one at the top, the other at the bottom of the atmosphere. Integrals are approximated by Gaussian quadrature and differentials by difference formulae, which leads to a set of linear equations with block tridiagonal structure that remains to be solved. A detailed description of the Feautrier method is given by Mihalas (1978). The following boundary conditions are used: No radiation is entering the atmosphere from above, whereas at the bottom of the atmosphere the diffusion approximation (Mihalas 1978) is applied. ## 2.4. Solar model atmospheres consideredThe nine solar model atmospheres that we have considered are labeled by abbreviations. The number in the subscript indicates the atmospheric model when ordered from the hottest to the coolest atmosphere, as may be seen from Fig. 1, which shows the temperature as a function of geometric height. -
**AYFLUXT**These model atmospheres are based on models introduced by Ayres et al. (1986) and Solanki et al. (1994). AYFLUXT_{1}, AYP_{2}, AYCOOL_{8}:_{1}is the magnetic component of a plage model, AYP_{2}a plage model, and AYCOOL_{8}the non-magnetic component of a plage region. -
**MACKKL**This semi-empirical solar model atmosphere, constructed by Maltby et al. (1986), is representative of the average quiet Sun. The temperature as a function of height has been derived from observed CLV curves of the continuous intensity spectrum over a wide wavelength range from X-rays to radio waves._{6}: -
**FALA**These are models of Fontenla et al. (1993). FALA_{7}, FALC_{5}, FALF_{4}, FALP_{3}:_{7}corresponds to their model A, FALC_{5}to model C, FALF_{4}to model F, and FALP_{3}to model P. All models are semi-empirical and include effects of particle diffusion in the transition region to explain UV emission lines of hydrogen and helium correctly. The first three models describe the quiet Sun: FALA_{7}the supergranular cell center, FALC_{5}the average quiet Sun, and FALF_{4}the bright network. FALP_{3}is a plage model. FALC_{5}is based on the MACKKL_{6}atmosphere, but in FALC_{5}the temperature in the chromosphere has been raised to account for the UV emission lines (see Fig. 1). -
**AND**This is model 2 of Anderson (1989). It is a theoretical, non-LTE solar model atmosphere in hydrostatic and radiative equilibrium with plane-parallel geometry. Line blanketing has been included. There is no temperature rise in the chromosphere because non-thermal heating mechanisms are not included in the model. Although on the real Sun the temperature does increase in the chromosphere, this model atmosphere is useful for reference purposes to study the chromospheric influence on polarization._{9}:
© European Southern Observatory (ESO) 1999 Online publication: December 16, 1998 |