4. Limit of small w
and obtain for the "w correction factor" to second order in w
From this expression we realize that, while the static part of the monochromatic flux is determined only by the (monochromatic) extinction coefficient or, equivalently, by the mean free path , the effect of the differential motions is - to second order in w - determined by the first two wavelength derivatives of the mean free path of the photons and, of course, by the gradient w itself.
4.1. Wavelength-integrated radiative quantities
and correspondingly for the generalized Rosseland mean opacity (I:50)
Here and are the corresponding static quantities, and is the weighting function defined by Eq. (7).
Since the and are functions of temperature, density, and chemical composition only, they may be precalculated once for all so that we have - for small velocity gradients w - derived a convenient form of the radiative quantities for actual radiation-hydrodynamical calculations.
4.2. Effects of various extinction distributions
In order to understand the various terms given above in some detail we now discuss several special cases of the extinction coefficient which may be regarded as basic building blocks for more general extinction distributions.
4.2.1. Power-law continuum
Surprisingly, the coefficient of the linear term () for the radiative acceleration reduces to
Thus in a differentially moving configuration with a power law extinction the flux increases super-exponentially with n (Fig. 1) and the acceleration only linearly.
Note that a direct evaluation of from (4) and (I:39) leads to divergences for the power-law wavelength distribution of the extinction coefficient (25) since at some wavelength the configuration starts to become optically thin and our basic assumptions are no longer valid. However, already the addition of an arbitrarily small wavelength independent extinction removes this problem.
4.2.2. Spectral edge
Here we consider an edge superimposed on a wavelength independent continuum and restrict ourselves here to the first order terms since they already give non-negligible contributions. We approximate the edge by the tangent hyperbolic function
i.e. we consider a "jump" of finite width and total height (Fig. 2a). We note that although the edge is symmetric with respect to , its derivatives are not.
In order to determine the contribution of the edge to the wavelength integrated quantities we integrate from to (). Note that the weighting function G varies only very little over the edge and therefore can be approximated by a constant. The relevant integrals are found to be
for the monochromatic flux and
for the monochromatic radiative acceleration; the corresponding coefficients and are negative.
These expressions show that both the flux as well as the radiative acceleration may be reduced or increased depending on the sign of w and that the amount of the change is independent of the sharpness of the edge.
4.2.3. Single narrow line
Next we treat the effects of a narrow single line of Lorentzian shape, a case which we have discussed already in a different context in Paper I, writing the extinction coefficient (I:51) in the form
have to be determined. Examples are shown in Fig. 2b. The dependence of these derivatives on the line strength A is for faint lines (), whereas very strong lines () exhibit a proportionality for (33), for (34), (35), and for (36). As the same also holds - for any finite m - for the corresponding integrated contributions, this behavior indicates that the effect of lines of moderate strength will be most important one.
for the (second order) contribution to the flux integrated over the interval . Note that - in contrast to the coefficient (Eq. (21)) - does not contain the weighting function G.
The contribution of a single Lorentzian line (on a continuum) to the wavelength-integrated quantities is obtained by integrating the derivatives over the line, i.e. over from to where on the one hand has to be sufficiently large, and on the other hand the weighting function G should not vary significantly with . For the evaluation we utilize that the Lorentz profile is symmetric around , i.e. , and hence also . Then for the integral in the first-order coefficient (20) for the total flux obviously
and refer to the line center ().
Note that in detail the dependence of X on A and m is rather involved. Its evaluation yields the following limiting cases
In Fig. 3 we show the dependence of on A and m for four values of the damping constant in order to give a guideline to which distance from the line center, i.e. to which m, the integration over the line has to be carried out so that a sufficient accuracy of the second-order coefficient of the flux is achieved. It is seen that in every case one has to integrate far into the lines wings, i.e. to values where the line contribution to the extinction coefficient has decreased to less than . A minimal value of m can be used for lines with , for stronger and for weaker lines the relative contribution of the wings becomes more and more important and the integrations have to be extended further out.
The integrals contributing to the coefficients of the total radiative acceleration (23, 24) are
since , and decreases for .
The results derived in this subsection show that spectral lines - as long as they are isolated and have a symmetric shape - only contribute to the total flux in second order and not at all to the total acceleration (the latter being in strong contrast to the situation in optically thin or moderately thick situations). This implies that the differential motion in an optically very thick medium reduces via the effect of the (isolated, symmetric) lines or, equivalently, in creases the effective opacity . This effect is independent of the direction of the flow (cf. Eq. (18)). Although effective only in second order, the influence of the lines on the flux may still be quite strong since usually there are very many lines. Note also that narrow lines of medium strength should be the most influential and that the importance decreases with increasing continuum opacity.
4.2.4. Many spectral lines
We first consider the case that the spectrum is dominated by isolated symmetric lines on a wavelength-independent continuum, i.e. that the line spacing is sufficiently large so that the lines do not essentially overlap and that the integration over each line in the limit , as described in Sect. 4.2.3, can effectively be carried out without interference with the neighboring lines. We furthermore assume that the weighting function G does not vary significantly over the integration range for each line. In this case the result of Sect. 4.2.3 holds that the only effect in a moving medium is to decrease the total flux in second order of w.
If on a flat continuum there are L (Lorentzian) lines, denoted by the index l, of strength and width at the position , only those spectral regions that are influenced by the lines contribute to the relevant integral in the coefficient (Eq. (21)). Furthermore, for isolated lines the line-dominated portion of the spectrum is negligible compared to the line spacings so that the static Rosseland mean can be replaced by and hence approximately
with the abbreviation
as given by Eq. (43), and and defined by (41) and (40), respectively.
In the special case of L spectral lines with identical and (and hence , , and ) Eq. (47) reads
This expression is valid for any number L of lines as long as they do not overlap. If, however, in addition we assume that there are many lines, i.e. , distributed over the spectrum in such a way that the trapezoidal integration rule gives a sufficiently accurate result for the integral , we can further evaluate the sum. For this it is required that the line density , i.e. the number of lines per unit -interval, or, equivalently, the line separation is roughly constant over the relevant part of the spectrum. Then
since the weighting function is normalized according to .
An extinction coefficient which is more realistic than that for isolated lines comprises overlapping spectral lines and hence asymmetric features. The complexity of the effects already shows up already in the very simple case of two overlapping lines if their relative strength and separation are varied (cf. Figs. 5, 6).
As can be seen from from Figs. 7 and 8, the situation gets even more intricate whenever several lines overlap. However, it is obvious that the strong dependence on the damping - seen already for a single line - is maintained and that the contribution to the -integrals are largest from regions of weak absorption with large gradients.
In order to get some impression of the variations of resulting from line shifts we generated 100 random realisations of central wavelengths for 100 lines in the interval and determined numerically. As is seen from Fig. 9 the resulting values of may vary strongly with an asymmetric distribution. However, all values stay negative as is expected from the discussion of single lines. This need not be correct if the integration is restricted to smaller spectral regions (cf. Fig. 10) covering a smaller number of lines (on the average 10 in our case): then the fluctuations are much larger and even positive values of may occur (though with low probability).
© European Southern Observatory (ESO) 2000
Online publication: July 7, 2000