Astron. Astrophys. 332, 245-250 (1998)

3. Why are radiative-acoustic waves unstable?

We give here some simple, heuristic arguments for the occurence of unstable, radiative-acoustic waves in line-driven winds.

Assume that at some location, r, in the wind, the velocity field experiences an accidental perturbation, and becomes slightly steeper, as is the case, e.g., at the zero crossing of a harmonic perturbation. Since the first order Sobolev force per unit mass due to a thick line is , the perturbed gas experiences a larger force, and is accelerated to higher velocities. From the left panel of Fig. 1, one sees that then the node around which the band is tilted is shifted inward. Correspondingly, if the velocity field becomes shallower at some node, the line force drops, the gas is decelerated, and the node is also shifted inward. Thus, long-scale harmonic perturbations of the velocity field, for which the Sobolev approximation can be applied, induce inward propagating waves.

Fig. 1. Left: the occurence of long-scale, inward propagating waves in first order Sobolev approximation. Right: the line-driven instability in second order Sobolev approximation. Upward (downward) pointing arrows indicate that the corresponding quantity rises (drops) as consequence of a flow perturbation.

The instability arises then from second order (curvature) terms. The optical depth from (10) drops for a concave deformation () of the thermal band, i.e., an elevation. (We have assumed here that the 'mean' over the perturbation remains constant). In consequence, the line force rises, and the perturbed gas is accelerated to higher speeds. This enhances the elevation, and the curvature increases further. The right panel of Fig. 1 displays this feedback between perturbations in v and . Similarly, for a convex deformation, i.e., a trough, the line force drops, and the trough becomes deeper.

In later stages, when fast gas starts to overtake slow gas, the original sine-wave perturbation is transformed into a sawtooth. Here, the gas is highly accelerated over a broad velocity elevation, and subsequently decelerated in a narrow, reverse shock front. The corresponding kind of behavior is found for the solutions of the inviscid Burgers equation, .

© European Southern Observatory (ESO) 1998

Online publication: March 10, 1998
helpdesk.link@springer.de