## 3. Radiative transfer in a perturbed medium## 3.1. Basic form of the radiative transfer equationWe consider the is the intensity at the point
in direction
at frequency
,
the mean intensity, the source
function, the true absorption
coefficient and the scattering
coefficient. The negative sign on the l.h.s. of Eq. (1) is
because of the definition of The following basic assumptions are made: where is the grey opacity of the
medium and with the optical depth being defined as The integration in Eq. (3) is carried out from the starting point backward along the ray to the maximum distance where the ray enters the considered volume. This may occur either at an outer boundary, where the irradiation is assumed to be zero, or at an inner boundary where the incident intensity is considered to be known. By integration of Eq. (3) over the solid angle, the following basic form of the radiative transfer equation is obtained: ## 3.2. Consideration of perturbationsWe consider a solution of the radiative transfer problem
Eq. (5) , which has been
calculated by means of simplifying assumptions concerning the
geometry. For example, such an unperturbed problem can be a standard
plane-parallel stellar atmosphere (1D), a spherically symmetric
stellar wind (1D), or an axisymmetric radiative transfer problem (2D).
Next, we introduce small, three-dimensional, where and are the (constant) relative amplitudes of the perturbations which are assumed to be small. The wave vector describes the wavelength and the direction of the perturbation (). The incident intensities are assumed to be unaffected by the perturbations . Strictly speaking, we By inserting Eq. (6) into (4), the optical depth in the perturbed medium is with being the respective optical depth in the unperturbed medium and where and are abbreviations. Since , we find to first order Insertion of Eqs. (6), (7) and (9) into Eq. (5) and truncation to first order terms yields where the arguments of , , and have been omitted. Remembering that satisfies Eq. (5), the order of Eq. (10) cancels out as well as the constant factor . The latter means that the amplitude of the mean intensity variation is in fact phase-independent and that there is no phase-shift between the opacity and the mean intensity variation. After separating the remainder of Eq. (10) into terms containing only and , respectively, the following result for the ratio of perturbation amplitudes can be obtained The resultant ratio of the perturbation amplitudes depends on the overall spatial structure of the unperturbed solution , and on and . It is noteworthy that linear-combinations of opacity perturbations in different directions and with different amplitudes result in the same type of mean intensity perturbations where Eqs. (11) to (14) remain valid for each single mode . In comparison to the results of Spiegel (1957) and Trujillo Bueno & Kneer (1990), the presented formalism (11) to (14) is applicable to arbitrary points and geometries of the unperturbed problem, since it does not rely on the diffusion approximation and accounts for incident radiation fields. © European Southern Observatory (ESO) 2000 Online publication: June 8, 2000 |