The necessity for large-scale astrostrophic currents also requires that we solve the nonlinear equation (8.69) for the temperature field in the common envelope. Here we have p0 Tu ■ grad S = div(x grad T). (8.87)

By making use of Eqs. (2.11) and (8.83), we can also rewrite Eq. (8.87) in the more convenient form cVp0u ■ grad T = div(x grad T) , (8.88)

where cV is the specific heat at constant volume.

Equations (8.83), (8.86), and (8.88) are the fundamental equations of the problem. They provide a simple but adequate description of the astrostrophic flow and the lateral energy transfer in a common radiative envelope. As usual, appropriate boundary conditions must be prescribed at the inner critical surface and at the outer boundary. Moreover, since we do not expect the astrostrophic flow to penetrate into the Roche lobes, we must ensure that conditions (2.20) and (2.21) are properly satisfied at the inner critical surface. Not unexpectedly, these requirements lead to the formation of a thin viscous boundary layer immediately above the Roche lobes. Since the outer part of the common envelope has to be closely barotropic, we must also require that the baroclinic corrections (i.e., pi, P1, and T1) and the astrostrophic velocity u vanish at a distance from the inner critical surface.

Although a detailed solution of this hydrodynamical problem still lies in the distant future, I hope that the above discussion has made clear the need for a consistent treatment of the astrostrophic balance at the base of the common envelope in a contact-binary star (see Eq. [8.86]). Indeed, there can be little doubt that large-scale currents flowing along (and not across) the equipotentials play an essential role in the problem since, without them, it would be impossible to obtain a solution that satisfies all the basic equations and all the boundary conditions, while being continuous across the inner critical surface.

Admittedly, I have so far considered the astrostrophic balance in static models only, that is, binary systems in which there is no net flux of matter from one stellar component to the other. However, one can show that the necessity of having large-scale astrostrophic currents in the lower part of the common envelope also applies to evolving binary configurations in which one stellar component is losing matter to its companion, so that the shape of their equipotentials is gradually changing in time. In other words, the concept of astrostrophy is equally relevant to both the static and evolving systems. A quantitative study based on Eqs. (8.83)-(8.87) would be most useful, therefore, since these theoretical results could provide considerable insight into the nature of the lateral energy transfer in the common (radiative or convective) envelope of an evolving contact binary.

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