MacGregor et al. (1979) and Carlberg (1980) derived the linear growth rates of this so-called line-driven instability by assuming flow perturbations are optically thin. Contrary to their results, Abbott (1980) found that growth rates of long-scale perturbations, for which the Sobolev approximation could be applied, are zero. Owocki & Rybicki (1984; OR in the following) 'bridged' these opposing results by showing that harmonic perturbations of wavelength shorter than one third to one half the Sobolev length, (primes indicate spatial differentiation), are highly unstable at the constant rate given by MacGregor et al. (1979) and Carlberg (1980); for , on the other hand, the growth rate drops as , implying marginal stability for .
Subsequent analytic work on the instability was concerned with: (1) the influence of scattering, e.g., flow stabilization due to line drag (Lucy 1984; Owocki & Rybicki 1985); (2) growth rates for non-radial velocity perturbations (Rybicki et al. 1990); and (3) growth rates for flows with optically thick continuum (Owocki & Rybicki 1991; Gayley & Owocki 1995).
Numerical wind simulations by Owocki et al. (1988) and Owocki (1991) showed the occurence of broad rarefaction regions over which the gas is highly accelerated, and is eventually decelerated again in a strong reverse shock, which heats the gas to X-ray temperatures. Subsequent radiative cooling compresses this gas into dense and narrow clouds. The idea that such clouds should correspond to the observed discrete absorption features in P Cygni profiles is meanwhile mostly ruled out (cf. Puls et al. 1993; Cranmer & Owocki 1996; and the volume edited by Moffat et al. 1994). On the other hand, after some initial difficulties (Cooper & Owocki 1992, 1994), it seems now plausible (Feldmeier et al. 1997) that the observed X-ray emission from O stars (cf. Hillier et al. 1993, and references therein) stems from instability-generated shocks.
The present paper returns to the issue of linear stability analysis, with a twofold aim: (1) to present an easy and straightforward derivation of growth rates, which, for the physically most interesting regime of moderate long-scale perturbations, complements the more elaborate analyses of OR and Lucy (1984); and (2) to show how the second order Sobolev approximation implies an unstable growth of inward propagating radiative-acoustic waves, where the latter were firstly described by Abbott (1980) and result already from a first order Sobolev treatment.
A related second order expansion was already performed earlier by Owocki (1991, priv. comm.) to show that the line force in (first order) Sobolev approximation is to one order in L more accurate for pure scattering lines than for pure absorption lines. The aim of this investigation was to explain certain difficulties which are encountered when one tries to reproduce the stationary wind solutions of Castor et al. (1975; CAK in the following) and Pauldrach et al. (1986) - which both apply first order Sobolev approximation - by using instead the exact line force for the case of pure line absorption (Owocki et al. 1988; Poe et al. 1990). In another context, the second order Sobolev approximation was also considered by Sellmaier et al. (1993).
Notice that the derivation given below is for pure absorption lines in a purely radial flow from a point source of radiation. According to Lucy (1984) and Rybicki et al. (1990), the growth rates are then maximum ones.
© European Southern Observatory (ESO) 1998
Online publication: March 10, 1998