## 2. Instability from second order Sobolev approximationThe 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, is the force due to an optically thin line, with the mass absorption coefficient, and the stellar flux at the line frequency, and the radial optical depth 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 where 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 ## 2.1. Optical depth in second order Sobolev approximationLet and assume 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.), From (6), and again to first order, 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). We separate Here, and Inserting (11) into (10), and keeping only terms linear in and , gives Here, is given modulo the exponential terms from (11), and the dimensionless wavenumber is defined as Since we will concentrate mostly on the case , i.e., the outer wind, we can approximate Finally, we introduced in (12), 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 forceIntroducing the comoving frame frequency, To avoid tedious expressions, we set in the
exponential. Inserting also from (12) we find,
for optically thick lines (subscript ' Note that actually we assumed here; as above, is to be understood modulo the exponential terms. The (small) number is defined as where The integral 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 with from (17). ## 2.3. Dispersion relationBy inserting (17) and (20) into the linearized continuity and Euler
equations for the harmonic perturbations and
We introduce a dimensionless phase speed, , and growth rate, , and set the determinant of (21) to zero, which gives The essential result from this equation system can be found analytically, by assuming that and keeping terms in (see below for a justification). This reduces (23) to The two solution branches correspond to fast growing waves which propagate inward, and slowly decaying waves which propagate outward, By inserting these results into (21), and by using again (24), the ratio of relative perturbation amplitudes is found to be For an interpretation of this result, note that the continuity
equation reads From (26), the The Abbott (1980) termed both these long-scale modes
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 growthYet, 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 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 |