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

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5. Radiative quantities in the diffusion limit

We now turn to the description of the expressions for the flux and the radiative acceleration in a differentially moving medium with velocities [FORMULA] and gradients w.

The monochromatic flux in the diffusion limit for a linearized source function (Eq. 39) now reads

[EQUATION]

In order to emphasize the effects of the motions we write the flux in the form

[EQUATION]

and hence the monochromatic acceleration as

[EQUATION]

where [FORMULA] and [FORMULA] are the static quantities (18) and (22), respectively. According to Eq. (41), the "w correction factor" is given by

[EQUATION]

Analogously to the monochromatic expressions, we give the corresponding wavelength-integrated quantities in the form

[EQUATION]

[EQUATION]

with [FORMULA] and [FORMULA] being the corresponding integrated static quantities. Then

[EQUATION]

[EQUATION]

with G as defined by Eq. (8). We point out that in our formalism - due to the additional factor [FORMULA] - the w-correction for the total radiative acceleration differs from that of the total flux although the corresponding monochromatic correction factors are identical. Note that - although G is independent of n - [FORMULA], [FORMULA], and [FORMULA] do depend on the direction via the n -dependence of w.

We may now introduce a generalized Rosseland opacity [FORMULA] for a differentially moving medium so that the total flux can be described analogously to the static case. We define [FORMULA] by the relation

[EQUATION]

Note that [FORMULA] comprises, for [FORMULA], the conventional Rosseland mean [FORMULA]. According to Eq. (45) the generalized Rosseland mean is then given by

[EQUATION]

When applying [FORMULA] one should keep in mind that it has been defined specifically for expressing total fluxes. However, it is not the appropriate generalization for the static mean opacities, which e.g. enter the local radiative energy balance, and are frequently also replaced by a Rosseland mean in the literature.

We note that our formulae which express the radiative quantities in terms of their static values can - in a straightforward manner - be applied only to deterministic extinction coefficients. The evaluation for stochastic line distributions is somewhat more involved since then the expectation values of all radiative quantities (including the static ones) are to be considered. This case will be discussed in a subsequent paper of this series.

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© European Southern Observatory (ESO) 2000

Online publication: July 7, 2000
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