Astron. Astrophys. 359, 788-798 (2000)

4. Limit of small w

With the assumption that is sufficiently smooth we may - keeping the line width fixed - expand the exponential in Eq. (5) in terms of w to second order,

and obtain for the "w correction factor" to second order in w

Replacing by etc. again, the flux in the diffusion limit becomes

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

With Eq. (17) we obtain for the wavelength integrated (total) flux (I:45)

and correspondingly for the generalized Rosseland mean opacity (I:50)

with the coefficients

Here and are the corresponding static quantities, and is the weighting function defined by Eq. (7).

The total radiative acceleration (I:46) becomes

with being the static value and

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

Let us start with a continuum obeying a power law in wavelength, i.e.

In this case the static Rosseland opacity is . With these expressions we find for the coefficients of the flux (18) and the radiative acceleration (22)

and

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.

Fig. 1. First-order coefficient of the total flux for the power-law continuum as a function of the index n. Curves for temperatures of and K.

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.

Fig. 2a and b. Examples for the behavior of the extinction coefficients (top row ) and the derivatives of the corresponding free mean paths , , (2nd row to bottom ) contributing to the velocity-dependent terms of Eqs. 20, 24, 21 as functions of : a  Spectral edge of shape, see Eq. (29), with , (full curve), , (crossed curve), , (diamonds) on a continuum ; b  single Lorentzian line with , (full curve), , (crossed curve), , (diamonds) also on a continuum , see Eq. (32).

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

In order to evaluate the effect of the motions (in the limit of small w) on the wavelength-integrated radiative flux and acceleration, the derivatives

and

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 subsequent discussion of several examples it is convenient to introduce the abbreviation

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 - by using Mathematica - we find for the second-order coefficient (21)

with being defined by (37),

and

and refer to the line center ().

In the limit we get for a fixed finite value of the line strength A

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.

Fig. 3a-d. Relative accuracy (cf. Eqs. (37), (43)) as a function of the line strength A and the "integration range" m for a  , b  , c  , and d  . The lower curves indicate an accuracy of 10%, the central ones of 1%, and the upper ones of 0.1%.

The integrals contributing to the coefficients of the total radiative acceleration (23, 24) are

and furthermore

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

and hence

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).

Fig. 4. Absolute value of the contribution for a single Lorentzian line on a continuum as a function of A and . It is seen that strongly increases with decreasing line width .

Fig. 5. Contribution of two overlapping Lorentz lines to as a function of their relative strength and of their separation (in units of ) for , , and - for both lines - .

Fig. 6. Contribution of two overlapping Gaussian lines (cf. Eq. (11)) to as a function of their relative strength and of ther separation (in units of W) for , , and - for both lines - a thermal width of .

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.

Fig. 7. Same as Fig. 2 but for two ensembles of 100 lines whose line centers are randomly distributed in the interval and whose strengths follow a logarithmic series distribution which is very similar to a power law with index . All lines have the same damping width .

Fig. 8. Same as Fig. 7 but for Note the strongly changed vertical scales.

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).

Fig. 9. Distribution of for 100 Lorentzian lines on a continuum with randomly chosen central wavelengths (100 realisations) and a logarithmic series distribution of strengths. The bins of which each has a width of cover the range . The mean value is with a standard deviation of 748.

Fig. 10. Distribution of with for the lines described in Fig. 9, but with the integrations extended over subintervals of . The bins with a width cover the range . The mean value is with a standard deviation of 416.

© European Southern Observatory (ESO) 2000

Online publication: July 7, 2000
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