## 5. Radiative instabilities## 5.1. Radiatively driven sound waves in -inversionsIt is clear from the discussion in Sect. 3and 4that -inversions may exist in stars with high luminosity-to-mass ratios and that such stars are close to being dynamically unstable. Thus, these stars may be vulnerable to radiative instabilities, since the additional destabilization from radiative forces may render them unstable. The existence of radiatively driven sound waves in hot stars due to high has been suggested by Hearn (1972, 1973), and further elaborated by Berthomieu et al. (1976), Spiegel (1976) and Carlberg (1980). The instability is based on the increase in opacity upon compression, since the radiation force is in phase with the velocity perturbation and hence causes amplification. For these high temperatures it requires essentially isothermal perturbations, since Here , while is the
flux weighted mean opacity. Thus only short wavenumbers will be
amplified. The growth rate is especially high as
approaches unity (Berthomieu et al. 1976) but the super-Eddington case
has not been investigated. Carlberg (1980) found that the instability
may provide rapid transfer of momentum from the radiation to the gas
with typical growth rates of s
To study the stability of a -inversion a local, linear, non-adiabatic stability analysis has been carried out. Small, sinusoidal perturbations around the equilibrium values are assumed for the temperature, density and vertical velocity: where and are small. The atmosphere is assumed to be static (), since the outflow velocity must anyway be less than the isothermal sound speed for a -inversion to exist, as demonstrated above. The analysis will be restricted to vertical motion and therefore multi-dimensional instabilities of Rayleigh-Taylor-type will not be found. Carlberg (1980) investigated also such a gradient instability for hot stars but found slow growth rates compared to the radiatively driven sound waves. Such a gradient instability would resemble convective motion which, however, is already present in the layers of interest in the unperturbed model and is thus not specifically searched for here. The equations to be linearized are the material equations for conservation of mass, momentum and energy: (e.g. Mihalas & Weibel Mihalas 1984), where D/Dt, The material equations must be coupled to the equations for the radiation field to account for radiative transfer effects through exchanges of energy and momentum between the photons and the gas. In this exploratory investigation the effects of radiation are included with a time-independent treatment, i.e. the radiation field is assumed to be quasi-static. The radiation momentum and energy equations will then look like: Here For the first relation radiative equilibrium is implicitly imposed,
which is not strictly true in the cases which will be investigated
below. However, the analysis will be restricted only to atmospheric
depths where the convective flux carries of the
total flux (i.e. in the upper part of the convection zone) and
radiative equilibrium can therefore be largely justified. To close the
relations the Eddington approximation () will be
made, which is, like LTE, well justified in the atmospheric layers of
interest. In the following, the radiation energy and momentum
equations will be combined to a single equation, which means that
there is one more variable to perturb, for which Together with the definitions of the isothermal sound speed , the density scale height and an ideal gas for , linearization to first order yields: The notation has been abbreviated according to In order for the above set of equations to have a non-trivial
solution for the perturbations ,
and , the determinant of
the equations must be equal to 0. The three complex eigenvalues
The calculated growth rates are shown in Fig. 5a and b, which
reveals amplification for cm
Thus, radiation-modified sound waves can be amplified in late-type stars with significant radiative forces. However, the perturbations are not very strongly excited with minimum growth times s, contrary to the findings by Carlberg (1980) for hot stars, mainly as a result of the much smaller radiative fluxes (). For the instability to generate large amplification, the sound waves must be reflected at the surface and pass through the region with high radiative forces several times. However, the inwards propagating modes are damped for the same reason as the outgoing are amplified, with the magnitude of the damping rate similar to the growth rate, which suggests that the instability will still not be very efficient. In order to estimate the behaviour of the instability in the global, non-linear regime, a more sophisticated study than presented here must be carried out. ## 5.2. The simplifying case of a isothermal, non-ionizing atmosphereIn order to gain further theoretical insight into which atmospheric properties contribute to the instability presented above, the equations can be simplified by assuming the unperturbed atmosphere to be isothermal and neglecting ionization and scattering. Furthermore, only the limits of optically thin and optically thick disturbances will be studied and hence the combined radiative energy and momentum equation will not be needed. The gas energy equation will thus instead be written as: where denotes the net radiative cooling rate of the gas. Depending on the optical thickness of the disturbance, will be described by either Newtonian cooling or equilibrium diffusion (e.g. Mihalas & Weibel Mihalas 1984). The rate of work done by has been ignored. The linearized continuity equation remains the same but the linearized momentum equation reads as: where and , while the energy equation looks like: Here represents the term arising from the net cooling rate and the ratio of specific heats. For Newtonian cooling and in the diffusion regime is the inverse of the radiative cooling time-scale (e.g. Mihalas & Weibel Mihalas 1984), where the factor takes into account the optical depth effects for optically thick conditions. The simplified dispersion relation then reads: For cm In situations of rapid radiative cooling (), the dispersion relation can be simplified to yield the solutions i.e. isothermally propagating, radiation-modified sound waves. In the limit of slow cooling (), as expected adiabatic acoustic waves are recovered: which travel with the adiabatic sound speed
. For cm For both isothermal and adiabatic perturbations, the sound waves are amplified by the action of the radiative acceleration, as long as the opacity increases with temperature and density (assuming constant radiative flux): Furthermore, in a -inversion is negative and hence tends to excite an initial disturbance. However, the normal momentum conservation counteracts this due to propagation in a density stratified medium; the density inversion will tend to damp the amplitude for outwards propagating waves. In fact, with this simplified atmosphere and the values in Table 1, in both models the last term dominates. Hence, the amplitude of sound waves will decay as they propagate outwards with these assumptions, despite the super-Eddington luminosities and the increase in opacity upon compression. Thus, since the numerical results in Sect. 5.1show amplification for the case without these simplifying assumptions, the effects of e.g. ionization and a temperature gradient tend to work together with the radiative force and its derivatives to overcome the damping contribution from the density gradient. © European Southern Observatory (ESO) 1998 Online publication: January 16, 1998 |