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Astron. Astrophys. 341, 902-911 (1999)

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2. Theoretical approach

2.1. Physical processes involved

In 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 [FORMULA] 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 [FORMULA] 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 radiation

We 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 [FORMULA] and V, as defined for example in Stenflo (1994). If we choose the coordinate system such that Stokes Q represents linear polarization with respect to the direction parallel to the nearest solar limb the above assumptions imply that Stokes U and V are intrinsically zero:

[EQUATION]

Hence, we can exclude Stokes U and V from our considerations and define the Stokes vector as

[EQUATION]

We introduce [FORMULA], where [FORMULA] is the angle between the direction normal to the surface and the line of sight. The optical depth is defined as

[EQUATION]

z being the geometric height, [FORMULA] the continuum absorption coefficient, and [FORMULA] the scattering coefficient. Polarized radiative transfer in the absence of magnetic fields then is described by the equation

[EQUATION]

with the total source function

[EQUATION]

The first term in Eq. (5), associated with pure absorption, is determined by the Planck function [FORMULA] and is given by

[EQUATION]

The second term in Eq. (5) contains all radiative sources associated with scattering and can be written as

[EQUATION]

where [FORMULA] represents the direction of the incident radiation within the differential solid angle [FORMULA], and [FORMULA] is the Rayleigh phase matrix, which accounts for the angular dependence of Rayleigh and Thomson scattering. This is given by (Stenflo 1994)

[EQUATION]

where [FORMULA] denotes a matrix with the only non-vanishing element [FORMULA], representing unpolarized, isotropic scattering, whereas the matrix [FORMULA] accounts for linear-polarization scattering and is given by

[EQUATION]

2.3. Numerical technique

The 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, [FORMULA] and [FORMULA], 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 [FORMULA] and [FORMULA] 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 [FORMULA] and [FORMULA].

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 considered

The 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.

  • AYFLUXT1, AYP2, AYCOOL8: These model atmospheres are based on models introduced by Ayres et al. (1986) and Solanki et al. (1994). AYFLUXT1 is the magnetic component of a plage model, AYP2 a plage model, and AYCOOL8 the non-magnetic component of a plage region.

  • MACKKL6: 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.

  • FALA7, FALC5, FALF4, FALP3: These are models of Fontenla et al. (1993). FALA7 corresponds to their model A, FALC5 to model C, FALF4 to model F, and FALP3 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: FALA7 the supergranular cell center, FALC5 the average quiet Sun, and FALF4 the bright network. FALP3 is a plage model. FALC5 is based on the MACKKL6 atmosphere, but in FALC5 the temperature in the chromosphere has been raised to account for the UV emission lines (see Fig. 1).

  • AND9: 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.

[FIGURE] Fig. 1. Temperature as a function of geometric height for the nine solar model atmospheres considered.

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© European Southern Observatory (ESO) 1999

Online publication: December 16, 1998
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