Strange—Mode Instabilities in Luminous Stars

Scction 9-25 s447

CHERNIGOVSKI, S . , GLATZEL, W., FRICKE, K.J .

Strange - Mode Instabilities in Luminous Stars

If radiation pressure contributes signijicantly to the total pressure in models f o r the envelopes of ho t a n d luminous stars, such as massive objects and Wolf - R a y e t stars, strange - mode instabili t ies wi th growth rates in t h e dynamical range do occur. T h e properties of t hese mechanical instabili t ies are reviewed and a n intui t ive model of t h e underlying in.sta.bility mechan i sm i s discussed. Adop t ing spherical geometry the evolut ion o f the instabili t ies has been followed in to t h e nonl inear regime by numerical s imulat ion. Multiple shocks are f o r m e d and velocity ampli tudes above lo2 km/sec are reached which can imp ly direct m a s s loss of the objects.

1. In t roduct ion

Strange - mode instabilities are a common phenomenon in models for the envelopes of luminous stars with luminosity to mass ratios in excess of x lo3 (in solar units), where the most violent examples have been discovered in models for Wolf - Rayet, stars (see [5] and IS]). The phenomenon can be identified by performing a standard linear, non- adiabatic stability analysis of standard stellar models for the objects mentioned and is then found to be present both within the radial (see [3] and [7]) and nonradial (see [ 6 ] ) spectrum of eigenfrequencies. There is no common precise definition of the term "strange modes". In a loose sense they are additional modes neither fitting in nor following the dependence on stellar parameters of the "ordinary" spectrum. In general they are connected with instabilities having growth rates in the dynamical range.

2 . The mechanism of strange - m o d e instabil i t ies

A useful tool to classify modes and to identify the mechanism of an instability is the NAR approximation, where "NAR" stands for non - adiabatic - reversible (see [I]). It consists of neglecting the time derivative of the entropy in the energy conservation equation for the perturbation. Physically, this can be realized either by adiabatic changes of state or by vanishing heat capacity which is equivalent to very large or very small ratios of thermal and dynamical tiniescales respectively. The second case is considered in the NAR approximation. Consequently, it is justified, if matter cannot store heat efficiently, the thermal timescale in the stellar envelope is short, as a consequence of low densities heat capacities are small, and large deviations from adiabatic behaviour prevail. Such conditions are in fact met in the envelopes of the luminous stars considered here. As a consequence of the first law of thermodynamics the luminosity perturbation vanishes in the NAR approximation. Moreover, zero heat capacity implies disregarding the tlieriiiodyriarriics of the system. in the NAR approximation it is reduced to its mechanical aspects. Thus thermal modes and any instability mechanism relying on a thermal, Carnot - type process, are excluded. Neglecting the time derivative of the entropy implies time reversibility with the consequence that eigenfrequencies come in complex conjugate pairs. Thus the NAR approximation may be used to distinguish between thermal and mechanical origin of both modes and instability mechanisms. As the thermodynamics of the system is essentially disregarded in the NAR approximation one would expect it to be rather unrealistic. However, surprisingly, for Wolf - Rayet stars it provides even quantitatively correct results. In particular, strange - mode instabilities still exist in the NAR approximation. As a consequence, the instability can be described in terms of mechanical quantities only and the luminosity perturbation is not essential for its mechanism. Apart from large deviations from adiabatic behaviour the high luminosity to mass ratios of the envelopes considered also imply the ratio fl of gas pressure and total pressure to be negligible. Therefore the model presented here (see also (21) is based on the NAR approximation and the limit d -+ 0. With these assumptions the mechanical equations (mass and momentum conservation) may be condensed into an acoustic wave equation for pressure (15) and density ( p ) perturbation:

a 2 p a26 at' ar2 I O

It has to be closed by a linear relation between 15 and ,3 which is provided here by the diffusion equation for energy

_.--

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transport. In the N A R approximation and the limit p --t 0 it is given by

Z A M M . Z. Angcw. Math. Mech. 81 (2001) S3

A bar refers to a quantity in the background model, K~ and KT are the logarithmic derivatives of opacity with respect to density and temperature. In an intermediate range of wavenumbers k (k . H p < 1//3; H p : pressure scale height) existing only for sufficiently small /3 the second term on the 1.h.s. (and the second term on the r.h.s.) of the diffusion equation may be neglected. Then the following dispersion relation describing an oscillatory instability is derived:

w4 = - 4 ( g K , k ) 2

( w : frequency, g: gravity.) Physically, a phase lag between pressure and density perturbation is caused in this case by the differential nature of their relation. Thus strange - mode instability may be understood by this phase lag inevitably caused by the differential nature of the diffusion equation in an intermediate range of wavenumbers provided thermal timescales are small and radiation pressure dominates. Apart from low heat capacity and low p non vanishing gravity and K,, # 0 is needed for the instability mechanism to work.

3. The result of strange - mode instabilities

Starting from linearly unstable models in hydrostatic equilibrium numerical simulations of the evolution of strange mode instabilities into the nonlinear regime have been performed in spherical geometry for massive objects and Wolf - Rayet stars (for more details see [4]). In accordance with the validity of the N A R - approximation in the linear analysis luminosity variations are also small in the nonlinear domain, whereas velocities easily reach the order of magnitude of 1021cm/sec. Contrary to Wolf - Rayet stars, where this level is by one order of magnitude smaller than the escape speed, it becomes comparable with it for massive objects. As a consequence, pulsationally driven mass loss was found in massive objects but not in Wolf - Rayet stars. However, as line - driving has not been accounted for, this does not necessarily mean that strange - mode instability is not related to mass loss in Wolf - Rayet stars. Rather the instabilities may provide the initial velocity field necessary for the line - driving mechanism to work. Instead of being strictly periodic the variability of velocity and luminosity in the final nonlinear phase in general exhibits a chaotic or at most quasiperiodic behaviour. The structure of the envelopes is then characterized by the appearance of multiple shock waves lifting off the outer layers of the star. For technical reasons the numerical simulations have so far been restricted to spherical symmetry, whereas strange - mode instabilities do also occur in the spectrum of nonradial modes and even the spherical shock waves are likely to suffer from nonradial instability. Moreover, significant rotation is observed in many of the the objects considered thus requiring an at least two - dimensional treatment of the problem. Work in this direction is in progress.

Acknowledgements

Financial support by the DFG under grants 436 MOL 17/3/98 and 436 MOL 17/4/99 and by the Volkswagen - Stiftang under grant I / Y l 620 is gratefully acknowledged.

4. References

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Address: S. CHERNIGOVSKI, Institute of Mathematics and Informatics, Moldavian Academy of Sciences, Academy Str. 5, 2028 Kishinev, Moldova K . J . FRICKE, W. GLATZEL, Universitats - Sternwarte, Geismarlandstr. 11, D-37083 Gottingen, Germany