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

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3. The disappearance of gas pressure inversions

3.1. Effects of convection

As seen above, a [FORMULA] -inversion must develop at the Eddington limit in hydrostatic equilibrium. One should ask, however, whether strong convection may take place to prevent it, since the ionization zone causing the high opacity and thereby the super-Eddington luminosities also favours convection. In fact, a super-Eddington luminosity in the diffusion regime automatically requires the presence of convection as shown by Langer (1997a). Generalizing Langer's derivation for a non-negligible radiation pressure to also include the effects of ionization, the Schwarzschild criterion [FORMULA] for the onset of convection is equivalent to


(e.g. Mihalas & Weibel Mihalas 1984). Here [FORMULA], while x denotes the ionization fraction of the most abundant element with an ionization potential [FORMULA]. Convection must be initiated as [FORMULA] (r) approaches unity, since the right hand side is always less than 1. Thereby the radiative flux and hence [FORMULA] diminishes when convection is efficient. The Eddington limit can therefore not be exceeded in adiabatic convection zones in the stellar interior and hence [FORMULA] -inversions will not develop. Thus, inversions can only be present in non-adiabatic convection zones close to the stellar surface.

It should be remembered that the location of the Eddington limit is dependent on the convection treatment. A [FORMULA] -inversion will for example disappear if the density scale height is applied instead of the normal pressure scale height for the calculation of the convective flux through the mixing length theory. The mixing length theory is rarely a good description of stellar convection, and in particular for these extreme situations it may be very misleading. Ultimately, hydrodynamical simulations will be necessary to investigate the effects of convection on [FORMULA] -inversions.

3.2. Effects of mass loss

Using the momentum equation one can investigate under which conditions a [FORMULA] -inversion can exist in the presence of a stellar mass loss. Or equivalently: which mass loss rates can one expect from a star if the super-Eddington luminosities would instead drive a stellar wind?

In the presence of a stellar outflow, the equation of hydrostatic equilibrium must be replaced with the momentum equation:


where we have restricted ourselves to the one-dimensional, time-independent case. Also other external forces, such as turbulent forces ([FORMULA]) and centrifugal forces ([FORMULA]), can be contained in [FORMULA] besides radiation. It should therefore be kept in mind that the possibility of an instability may be underestimated when restricting the following discussion to only radiative forces, as other forces also tend to be destabilizing (e.g. Nieuwenhuijzen & de Jager 1995; Langer 1997a,b; Owocki & Gayley 1997).

Together with a constant, spherically symmetric mass loss rate [FORMULA], and the equation of state [FORMULA], it is possible to rewrite the momentum equation into the so called "wind equation":


where the isothermal sound speed defined by [FORMULA] has been introduced. By an inspection of Eq. 6, it is possible to determine under which circumstances [FORMULA] - or [FORMULA] -inversions are allowed. In normal cases, d [FORMULA] /dr [FORMULA] in the surface layers and therefore the parenthesis on the right hand side of Eq. 6is positive as long as a temperature inversion is not present, which is not possible in optically thick layers with a radiative flux. Therefore, if [FORMULA] but [FORMULA], the velocity gradient [FORMULA] must be negative. Since a constant [FORMULA] has been assumed, a density inversion will be present if [FORMULA] and therefore a [FORMULA] -inversion is possible (though not required). If, on the other hand, [FORMULA] while [FORMULA], then [FORMULA] must be positive, and the density gradient cannot be positive: a density inversion cannot exist. Since a [FORMULA] -inversion requires a density inversion, neither inversion can occur in the stellar atmosphere despite the super-Eddington luminosities. Again the possibility of a temperature inversion has been neglected. A sufficient criterion for the disappearance of a [FORMULA] -inversion is therefore that the outflow velocity due to mass loss is high enough ([FORMULA]).

The critical mass loss rate can be estimated by comparing [FORMULA] with [FORMULA]. The values for [FORMULA] and [FORMULA] will be taken from static models, which is justified for these rough estimates, since the atmosphere below the sonic point is normally little affected by mass loss. A model atmosphere with [FORMULA] K, log [FORMULA] [cgs] and solar abundances requires a critical mass loss rate [FORMULA]. Hence [FORMULA] -inversions can exist in stars despite high mass loss rates. Achmad & Lamers (1997) have arrived at similar conclusions and estimates. They also verify the finding numerically with dynamical steady state atmospheres, though not altogether self-consistently.

An even higher critical mass loss rate is found for the R CrB stars. The lack of hydrogen makes the continuous opacity significantly lower than for solar abundances and hence the density at a given optical depth will be correspondingly greater, typically by a factor of 50 (Asplund et al. 1997a). Since the density is higher, the outflow velocity will be lower in order to carry the same [FORMULA]. A similar model as above but with abundances typical for R CrB stars, will therefore require [FORMULA]. The mass loss rates of R CrB stars are poorly known, in particular since it may be episodic, but rough estimates suggest [FORMULA] (e.g. Feast 1986), orders of magnitude smaller than [FORMULA].

It is concluded that [FORMULA] -inversions are not an artifact of hydrostatic equilibrium but may also exist in dynamical steady state atmospheres, even with a high mass loss rate.

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© European Southern Observatory (ESO) 1998

Online publication: January 16, 1998