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

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4. Solution in the limit of high optical depth

According to Baschek et al. (1997b) the solution of the transfer equation (Eq. 32) for constant w, a depth-in dependent extinction coefficient [FORMULA], and no incident radiation at the two boundaries [FORMULA] and [FORMULA] of the layer considered, yields in the positive direction of [FORMULA] the intensity

[EQUATION]

and - due to symmetry - in the opposite direction

[EQUATION]

Here and in the following we introduce w as additional variable in the argument list of all quantities referring to moving media, while static quantities are denoted without w.

In order to apply the solution of Baschek et al. (1997b) to the diffusion limit we here need to demand only that [FORMULA] - similarly as [FORMULA] - does not change significantly over a mean free photon path in the continuum, i.e. we may abandon their strict assumption of [FORMULA] being independent on s.

The solutions for [FORMULA], [FORMULA], and hence for [FORMULA] can conveniently be written in terms of the spectral thickness (Baschek et al. 1997b)

[EQUATION]

with [FORMULA] being an arbitrary logarithmic reference wavelength, a formalism which we will also use in this paper for the derivation of radiative quantities in moving media.

In terms of [FORMULA] the flux for the linearized source function reads (cf. Sect. 2)

[EQUATION]

For a fixed value of [FORMULA] we then obtain in the limit [FORMULA], [FORMULA] @

[EQUATION]

(see also Fig. 2).

[FIGURE] Fig. 2. The dependence of the last integral in Eq. (38) on [FORMULA], calculated for a single Lorentzian lines on a continuum for various line strengths A, demonstrates the approach towards the diffusion limit; upper curve: very weak line, lower curve: strong line.

Eq. (39) is the key equation we derive in this paper. It contains the effects of the differential motions within the medium and allows - as will be shown subsequently - to calculate the flux, the radiative acceleration, and a generalized expression for the Rosseland mean opacity.

From the definition (37) of the spectral thickness follows [FORMULA] since the extinction coefficient [FORMULA] is always positive. Hence [FORMULA] increases monotonically with [FORMULA] and [FORMULA] for either sign of w. If in addition [FORMULA] increases sufficiently fast with increasing [FORMULA] and hence with [FORMULA], the integrals (39) exist. For this to occur, already a wavelength-independent continuum, resulting in a linear dependence of [FORMULA] on [FORMULA], suffices. In this case there is a sufficiently large optical depth at all wavelengths - as is required by the diffusion approximation - and hence the validity of Eq. (39) is guaranteed. Furthermore we point out that in the limit of large optical depths the integrals over [FORMULA] in (35) and (36) formally extend over an infinite interval; due to the effective cutoff properties of the exponential functions, however, they extend in practice - due to the assumption of a linearized depth-dependence of the source function - only over a distance of the order of [FORMULA].

At this place we convince ourselves that the argument of the exponential in Eq. (39) reduces to Eq. (18) in the limit [FORMULA] since

[EQUATION]

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© European Southern Observatory (ESO) 2000

Online publication: July 7, 2000
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