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Astron. Astrophys. 358, 665-670 (2000)

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3. Radiative transfer in a perturbed medium

3.1. Basic form of the radiative transfer equation

We consider the time-independent equation of radiative transfer with coherent, isotropic scattering along a ray in direction of the unit vector [FORMULA]

[EQUATION]

[FORMULA] is the intensity at the point [FORMULA] in direction [FORMULA] at frequency [FORMULA], [FORMULA] the mean intensity, [FORMULA] the source function, [FORMULA] the true absorption coefficient and [FORMULA] the scattering coefficient. The negative sign on the l.h.s. of Eq. (1) is because of the definition of s, which denotes the distance backward along the ray [FORMULA] (Mihalas & Weibel Mihalas 1984, p. 345).

The following basic assumptions are made: grey transport coefficients [FORMULA], [FORMULA], and radiative equilibrium [FORMULA]. Frequency-integration of Eq. (1) in this case yields

[EQUATION]

where [FORMULA] is the grey opacity of the medium and I and J are the respective frequency-integrated quantities. The formal solution of Eq. (2) is

[EQUATION]

with the optical depth being defined as

[EQUATION]

The integration in Eq. (3) is carried out from the starting point [FORMULA] backward along the ray to the maximum distance [FORMULA] 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 [FORMULA] 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:

[EQUATION]

3.2. Consideration of perturbations

We consider a solution of the radiative transfer problem Eq. (5) [FORMULA], 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, harmonic perturbations  1 of the spatial opacity and mean intensity structures

[EQUATION]

where [FORMULA] and [FORMULA] are the (constant) relative amplitudes of the perturbations which are assumed to be small. The wave vector [FORMULA] describes the wavelength and the direction of the perturbation ([FORMULA]). The incident intensities are assumed to be unaffected by the perturbations [FORMULA].

Strictly speaking, we assume a perturbation of [FORMULA] as expressed by Eq. (6) and will show that the ansatz for the response of the radiation field (Eq. 7) leads to non-trivial solutions of Eq. (5).

By inserting Eq. (6) into (4), the optical depth in the perturbed medium is

[EQUATION]

with [FORMULA] being the respective optical depth in the unperturbed medium and where [FORMULA] and [FORMULA] are abbreviations. Since [FORMULA], we find to first order

[EQUATION]

Insertion of Eqs. (6), (7) and (9) into Eq. (5) and truncation to first order terms yields

[EQUATION]

where the arguments of [FORMULA], [FORMULA], [FORMULA] and [FORMULA] have been omitted. Remembering that [FORMULA] satisfies Eq. (5), the [FORMULA] order of Eq. (10) cancels out as well as the constant factor [FORMULA]. The latter means that the amplitude of the mean intensity variation [FORMULA] 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 [FORMULA] and [FORMULA], respectively, the following result for the ratio of perturbation amplitudes can be obtained

[EQUATION]

with the abbreviations

[EQUATION]

The resultant ratio of the perturbation amplitudes [FORMULA] depends on the overall spatial structure of the unperturbed solution [FORMULA], and on [FORMULA] and [FORMULA]. It is noteworthy that linear-combinations of opacity perturbations in different directions and with different amplitudes [FORMULA] result in the same type of mean intensity perturbations [FORMULA] where Eqs. (11) to (14) remain valid for each single mode [FORMULA].

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.

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© European Southern Observatory (ESO) 2000

Online publication: June 8, 2000
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