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Astron. Astrophys. 359, 780-787 (2000)

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2. Static case and nomenclature

The aim of our diffusion calculations is to derive simple expressions for the radiative flux and radiative acceleration at positions far away from the surface in a medium

  • (i) that is optically very thick,

  • (ii) in which the extinction coefficient hardly varies over a photon mean free path, and

  • (iii) in which the variation of the source function can be approximated by a linear function in the neighborhood of the point [FORMULA] where the radiative quantities are to be calculated.

For this purpose we start with the general form of the transfer equation for a static 3D medium for the monochromatic intensity [FORMULA] at a wavelength [FORMULA] along a ray described by [FORMULA] or the unit vector [FORMULA], respectively,

[EQUATION]

Here S is the source function and [FORMULA] the monochromatic extinction coefficient comprising absorption as well as scattering. In cartesian coordinates Eq. (1) reads

[EQUATION]

with [FORMULA] (cf. Oxenius 1986).

We introduce logarithmic wavelengths

[EQUATION]

which are mostly used throughout this paper in place of the usual wavelengths [FORMULA] in order to obtain Doppler shifts that depend only on spatial coordinates and angles. Then [FORMULA], [FORMULA].

In the diffusion limit, the mean intensities are very close to the Planck function [FORMULA] of the local temperature [FORMULA]. This implies that LTE conditions prevail and the source function is identical to the Planck function,

[EQUATION]

In the following we need also the wavelength-integrated Planck function denoted by

[EQUATION]

with [FORMULA] being the Stefan-Boltzmann constant, the spatial derivatives of [FORMULA] and [FORMULA], respectively,

[EQUATION]

with [FORMULA], and the weighting function

[EQUATION]

This weighting function, which enters the Rosseland mean opacity (Eq. (19)), is normalized according to [FORMULA] and is independent of n . It decreases exponentially with [FORMULA] for very large as well as for very small [FORMULA] (Fig. 1).

[FIGURE] Fig. 1. Weighting function [FORMULA] as function of [FORMULA] for [FORMULA].

In terms of these quantities the source function in the neighborhood of [FORMULA] can be written - see assumption (iii) above - as

[EQUATION]

Here [FORMULA] and

[EQUATION]

are sufficiently slowly varying functions of [FORMULA] over a few mean free photon paths [FORMULA] (in the continuum).

In order to clarify the nomenclature, we present the relevant static radiative quantities first (cf. Cox & Giuli 1968, Mihalas 1978, Mihalas & Weibel-Mihalas 1984).

2.1. Radiative flux

The total flux

[EQUATION]

is obtained by integration over all wavelengths from the vector of the monochromatic flux,

[EQUATION]

It is evident from Eq. (12) that the essential physics is contained in the antisymmetric average of the specific intensities [FORMULA] which has a flux-like character (cf. Mihalas 1978). In fact, it is the (monochromatic) flux in the two-stream approximation. In the following we therefore will concentrate on this average and - in order to simplify the notation - subsequently write

[EQUATION]

and

[EQUATION]

2.2. Radiative acceleration

The vector of the net radiative acceleration (force per unit volume) has the same direction as that of the flux so that the total radiative acceleration is

[EQUATION]

with

[EQUATION]

being the monochromatic acceleration. Here the role of the antisymmetric average of the intensities is the same as that occurring in the flux. Analogously we write

[EQUATION]

2.3. Static radiative quantities in the diffusion limit

For a static medium we find from the well-known solution of Eq. (1) along a ray at a position [FORMULA] well inside the medium

[EQUATION]

The wavelength-integrated flux can be expressed in terms of the (static) Rosseland mean opacity [FORMULA], which is defined by

[EQUATION]

with [FORMULA] being the weighting function given in Eq. (8). Then we obtain from Eqs. (14) and (18)

[EQUATION]

and from Eq. (12)

[EQUATION]

the classical result of Rosseland (1924).

According to Eq. (17) the expressions for the radiative acceleration become

[EQUATION]

and according to Eqs. (15) and (16)

[EQUATION]

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© European Southern Observatory (ESO) 2000

Online publication: July 7, 2000
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