2. Instability from second order Sobolev approximation
The reason that Abbott (1980) found no indication of wind instability is that he used the Sobolev approximation in lowest order, even for the flow perturbations. E.g., consider an optically thick line, with line force per unit mass (Sobolev 1960; Castor 1974). By assuming harmonic perturbations, this implies a phase shift of between velocity perturbations, , and the response of the line force, . Hence, the line force does no net work on the velocity perturbation over a full cycle, and the perturbation does not grow (OR).
To see how the instability arises, we consider at first the expression for the exact line force, before the Sobolev approximation is applied. Let be the frequency displacement from line center, , in Doppler units, ( the ionic thermal speed, c the speed of light), as measured in the observers frame. For radially directed photons, the force per unit mass due to photon absorption in a single line is,
with the stellar radius. In first order Sobolev approximation, , , and are assumed to be constant over the narrow region, i.e., the Sobolev zone, over which photons of given frequency can be absorbed in the line transition. This implies
Introducing then a velocity perturbation, , into (1) and (3) leads to a rather complex expression. To allow a further progress, MacGregor et al. (1979) and Carlberg (1980) assumed that the exponential term is not affected by the perturbation, i.e., the perturbation is optically thin.
As will be discussed in Sect. 4, the velocity jumps which are caused by the corresponding short-scale perturbations are rather small, namely of order the thermal speed, and cannot explain, e.g., the observed X-ray emission from OB stars.
We are therefore primarily interested in long-scale perturbations, which, despite of their reduced growth rates (cf. below), can still grow into saturation and give rise to large velocity, temperature, and also density jumps. Notice that can then no longer be assumed. However, as will be shown in the following, the growth rates are then easily derived from applying a second order Sobolev approximation.
The bridging length where this long-scale limit breaks down, and the increase of the growth rate bends over to the constant, maximum rate given by Carlberg (1980), is set by the Sobolev length. This can be seen from the fact that the first order Sobolev approximation does not lead to an instability (Abbott 1980), while the second order approach leads to the correct growth rates for long-scale perturbations. The second order approximation differs from the first order one by terms in L.
2.1. Optical depth in second order Sobolev approximation
and assume small perturbations, so that the wind velocity field remains monotonic. Then (3) can be transformed to a frequency integral,
where the integration variable is defined as . As before, is assumed to be constant over the Sobolev zone, which is a reasonable assumption for resonance lines and transitions from metastable levels, both dominating the line force. Performing a Taylor series expansion of to first order (and abbreviating , etc.),
We consider now only the Doppler core of the line, where de-shadowing effects are most pronounced, and therefore the growth rates are largest (OR). With , and since for a Doppler profile, the optical depth in second order Sobolev approximation is,
Remember that is the optical depth from a first order treatment.
The divergence of the expression in (10) for is an extrapolation artefact of the linear expansions performed in (8) and (9), and is compensated for by higher order terms in the Taylor series. In any case, this term enters the force response only via the combination , which vanishes for (see below).
Here, and k are assumed to be real, while and are complex to allow for arbitrary phase shifts between velocity and density perturbations, and for unstable growth, respectively. The expression , where asterisks refer to an arbitrary, however fixed location in the stellar rest frame, accounts for the stretching of perturbations in the accelerating velocity field (i.e., is independent of radius). For perturbations which originate in the photosphere, e.g., one could choose . Note that if the sound speed is small compared to any other speed (flow speed and wave speeds), pressure forces can be neglected, and the above expression for wave stretching is exact.
where , and the second equalities hold for the CAK velocity law for a wind from a point source, , with terminal wind speed .
2.2. The perturbed line force
The integral E, which is easily evaluated numerically, is of order unity for , and depends only weakly on , namely (Castor 1974).
The decisive fact in (17) is the occurence of a positive feedback between velocity and force perturbations, . First, note the dependence of growth rates , in accordance with the results by OR. Moreover, also the quantitative values agree well with those from the long-wavelength limit of the bridging law by OR, cf. their Eq. (28). E.g., for a line of optical depth we find a growth rate which is 20% smaller than that given by OR; in view of the different approximations performed by them and in our derivation, this is completely admissible.
From (18), the growth rate is almost constant for all moderately optically thick lines, except for a dependence on (atomic species), and the above weak dependence on . This allows us to estimate the response of the total line force on perturbations, which is needed below for the derivation of dispersion relations. Following CAK, we assume that the total line force is the simple sum of individual contributions from all lines, i.e., line overlap is neglected; and furthermore that the ratio of the force due to all thin lines to the force from all thick lines is , where , and typically for O supergiants. Since, by Eq. (2), the force per unit mass due to an optically thin line is not affected by perturbations, one obtains . For the CAK wind in the limit of vanishing sound speed, the total line force can be written , with g the gravitational acceleration. Hence, the Euler equation reads , and the response of the total line force to a perturbation is
with from (17).
2.3. Dispersion relation
We introduce a dimensionless phase speed, , and growth rate, ,
and keeping terms in (see below for a justification). This reduces (23) to
For an interpretation of this result, note that the continuity equation reads in the comoving frame and after applying the WKB approximation (i.e., mean flow gradients neglected), , or, for harmonic perturbations, , with phase speed .
From (26), the unstable waves propagate inward at a phase speed in the observer's frame, or in the comoving frame, i.e., they stand with respect to the star. Eventually, these waves should steepen into reverse shocks, as is indeed found from numerical simulations. The phase shift between density and velocity fluctuations is , similar to the case of ordinary, inward propagating sound waves.
The damped waves, on the other hand, propagate outward at a phase speed in the stellar rest frame, or in the comoving frame, i.e., they stand with respect to the wind. (In order to derive this from the amplitude relations (27), one has to set .) By including pressure terms, Abbott (1980) found more exactly in the comoving frame. Also, the phase shift of these waves is , instead of being for ordinary, outward propagating sound waves.
Abbott (1980) termed both these long-scale modes radiative-acoustic waves.
Finally, (24) remains to be justified. By assuming , we are restricted to the highly supersonic, outer wind. Then, , since can be neglected in , so that , and . The latter quantity is small compared to unity in the considered long wavelength limit. Finally, with we are restricted to perturbations for which the WKB approximation is valid, i.e., to wavelengths shorter than the wind scale height, .
2.4. Limiting wavelength for unstable growth
Yet, since we did not apply the WKB approximation to derive (21), this system contains more information regarding the long wavelength regime than does the analysis of OR. Especially, one can derive an upper limiting wavelength, , above which even inward propagating waves are no longer unstable, but decay instead.
Assume for the moment that curvature terms of the velocity field can be neglected, i.e., . For not too small , the numerical solution of (23) shows that for the inward mode, and can be neglected in (23). With , and setting the growth rate , we find from (23),
Assuming again that , one finds that (as was to be expected) the critical wavelength is set by the scaleheight of the wind, . E.g., for the CAK velocity law,
Except close to the star, is rather large. An easy calculation shows that, for , even the growth rates from (17) are so small that such perturbances would not grow significantly over a wind flow time. Hence, from (29) is of not much practical interest. (Note also that for such large values of , one would have to include sphericity terms in the above derivation.)
Finally, the numerical solution of (23) shows that curvature terms, , are rather unimportant in the outer wind, and lead to a small downward revision of only. Interestingly, however, curvature terms imply that also outward propagating waves can become unstable within a certain wavelength regime. Yet, since the corresponding growth rates are very small, this issue is of academic interest only.
We close this section with a perspective. Present numerical simulations of winds subject to the line-driven instability are cpu-time expensive because the line force is integrated directly, without applying any Sobolev approximation. Our above results imply, however, that the quantitatively correct, linear growth rates can be alternatively obtained from a second order Sobolev treatment. It is furthermore known (Owocki et al. 1988) that these linear rates hold over almost the full growth regime, until they quickly drop to zero when saturation is reached, i.e., when the thermal band becomes optically thin. It seems therefore plausible to perform instability simulations using the cheap Sobolev line force in second order instead of a more elaborate line transfer.
One has to keep in mind, however, that such a method, based on higher-order extrapolation of local conditions, does not incorporate the inherently nonlocal physics that occurs within the nonlinear growth of the instability. Furthermore, in order that a meaningful comparison with non-local integral methods, especially the Smooth Source Function method of Owocki (1991), can be done, it may be necessary to first develop the second order Sobolev forms for the diffuse force terms.
© European Southern Observatory (ESO) 1998
Online publication: March 10, 1998