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Astron. Astrophys. 330, 641-650 (1998)

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5. Radiative instabilities

5.1. Radiatively driven sound waves in [FORMULA] -inversions

It is clear from the discussion in Sect. 3and 4that [FORMULA] -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 [FORMULA] 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

[EQUATION]

Here [FORMULA], while [FORMULA] is the flux weighted mean opacity. Thus only short wavenumbers will be amplified. The growth rate is especially high as [FORMULA] 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 [FORMULA] s-1 for [FORMULA] K. It is possible that a similar instability is exhibited also in late-type supergiants, with the important differences being the super-Eddington luminosities in the [FORMULA] -inversions and that isothermal perturbations are not necessary, since both ([FORMULA]) [FORMULA] and ([FORMULA]) [FORMULA] in the upper layers of the [FORMULA] -inversions (Fig. 4).

[FIGURE] Fig. 4. The dependence of the Rosseland mean opacity on temperature and density for a solar abundances and b R CrB abundances. The atomic data for the opacities are taken from the Opacity Project (Seaton et al. 1994 and references therein). The top of the [FORMULA] -inversion for a model atmosphere with [FORMULA] K and log [FORMULA] occurs at log [FORMULA] for solar abundances and at log [FORMULA] for an R CrB composition

To study the stability of a [FORMULA] -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:

[EQUATION]

where [FORMULA] and [FORMULA] are small. The atmosphere is assumed to be static ([FORMULA]), since the outflow velocity must anyway be less than the isothermal sound speed for a [FORMULA] -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:

[EQUATION]

(e.g. Mihalas & Weibel Mihalas 1984), where D/Dt, e, B and J denote the Lagrangean co-moving derivative, the specific internal energy of the gas, the wavelength integrated Planck function ([FORMULA]) and mean intensity ([FORMULA]), respectively. The absorption and Planck mean opacities have been replaced by the Rosseland mean opacity for absorption [FORMULA], which is a good approximation for the relevant optical depths. The scattering contribution to the total extinction is here denoted by [FORMULA]. Furthermore LTE ([FORMULA]) has been assumed.

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:

[EQUATION]

Here H and K denote the wavelength integrated Eddington flux ([FORMULA]) and radiation pressure ([FORMULA]) respectively. In the diffusion regime for the unperturbed atmosphere, [FORMULA] is valid. The equilibrium radiation quantities are related by:

[EQUATION]

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 [FORMULA] 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 ([FORMULA]) 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 J is chosen:

[EQUATION]

Together with the definitions of the isothermal sound speed [FORMULA], the density scale height [FORMULA] and an ideal gas for [FORMULA], linearization to first order yields:

[EQUATION]

The notation has been abbreviated according to
[FORMULA] and [FORMULA]. Derivatives with respect to v and J have been ignored, which is justified in optically thick layers where line opacity is unimportant. True enough, photoionization edges introduce some velocity dependence on [FORMULA] but this minor effect is neglected.

In order for the above set of equations to have a non-trivial solution for the perturbations [FORMULA], [FORMULA] and [FORMULA], the determinant of the equations must be equal to 0. The three complex eigenvalues w for a given real wavenumber k then correspond to the allowed perturbations. Two of the modes will represent radiation-modified gravity-acoustic waves while the third is a non-propagating, heavily damped, thermal wave. Here two different cases will be investigated numerically, one corresponding to a [FORMULA] -inversion in a supergiant with solar abundances and the second the same but for an R CrB star. As before, both model atmospheres have [FORMULA] K and log [FORMULA]. Values for some of the physical parameters at some atmospheric layer where the instability may occur are found in Table 1.


[TABLE]

Table 1. Local values of the atmospheric structure. Both the model with a solar and an R CrB composition have [FORMULA] and log [FORMULA]


The calculated growth rates are shown in Fig. 5a and b, which reveals amplification for [FORMULA] cm-1 (solar composition) and [FORMULA] cm-1 (R CrB composition). For large k, the perturbations are as expected damped due to the negligible optical depths of the disturbances ([FORMULA]), which make the sound waves little affected by the radiative forces. Instead, the smoothing effects from radiation energy exchange due to large gradients in the perturbed variables dominate. The maximum growth rates are slightly smaller for a solar composition than for a H-deficient composition: [FORMULA] respectively [FORMULA] s-1, which corresponds to an e-folding time on the order of the sound crossing time of the atmosphere. For the largest wavenumbers with such growth rates, the real parts are significantly higher, which implies that the sound waves are only slightly amplified per wavelength. For smaller k, however, the growth rate is comparable with the oscillation rate. In fact, for the model with solar composition, the wave is essentially non-propagating for the smallest k. However, the use of a local analysis is not justified when the wavelengths of the perturbations are comparable with the scale heights of the variations for the variables. For [FORMULA] cm-1, the propagation speed is slightly less than the isothermal sound speed in both cases.

[FIGURE] Fig. 5. The numerical solutions to the full set of linearized equations for the gravity-acoustic wave modes for a solar and b R CrB compositions. The real parts are the dotted curves. The solid lines correspond to amplification of the imaginary parts, while the dashed curves represent damping. For wavenumbers with damping, the absolute magnitude of the imaginary part is shown

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 [FORMULA] s, contrary to the findings by Carlberg (1980) for hot stars, mainly as a result of the much smaller radiative fluxes ([FORMULA]). 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 atmosphere

In 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:

[EQUATION]

where [FORMULA] denotes the net radiative cooling rate of the gas. Depending on the optical thickness of the disturbance, [FORMULA] will be described by either Newtonian cooling or equilibrium diffusion (e.g. Mihalas & Weibel Mihalas 1984). The rate of work done by [FORMULA] has been ignored.

The linearized continuity equation remains the same but the linearized momentum equation reads as:

[EQUATION]

where [FORMULA] and [FORMULA], while the energy equation looks like:

[EQUATION]

Here [FORMULA] represents the term arising from the net cooling rate [FORMULA] and [FORMULA] the ratio of specific heats. For Newtonian cooling

[EQUATION]

and in the diffusion regime

[EQUATION]

[FORMULA] is the inverse of the radiative cooling time-scale (e.g. Mihalas & Weibel Mihalas 1984), where the factor [FORMULA] takes into account the optical depth effects for optically thick conditions. The simplified dispersion relation then reads:

[EQUATION]

For [FORMULA] cm-1, [FORMULA] is much smaller than [FORMULA] for the two investigated models and can be ignored.

In situations of rapid radiative cooling ([FORMULA]), the dispersion relation can be simplified to yield the solutions

[EQUATION]

i.e. isothermally propagating, radiation-modified sound waves. In the limit of slow cooling ([FORMULA]), as expected adiabatic acoustic waves are recovered:

[EQUATION]

which travel with the adiabatic sound speed [FORMULA]. For [FORMULA] cm-1, the propagation speed will be modified by the neglected terms [FORMULA], though the growth/damping rates will remain the same.

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):

[EQUATION]

Furthermore, in a [FORMULA] -inversion [FORMULA] 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.

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© European Southern Observatory (ESO) 1998

Online publication: January 16, 1998
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