U er Q Qm

Equations (8.91) and (8.92) imply at once that the spin-down time is of the order of (2Q0eT5/R)-', which is the result obtained in Eq. (8.50).

For small Rossby numbers, there is also a great similitude between a geostrophic wind in the Earth's atmosphere and the large-scale circulatory currents in the common envelope of a contact binary having dissimilar components (see Sections 2.5.1,2.5.2, and 8.5). As is well known, the geostrophic balance is a good approximation for the velocity field in the free atmosphere, at some distance above the Earth's surface. It provides for the wind to follow the direction of the surfaces of constant pressure, and for the geostrophic velocity to vary with height according to the thermal wind equation (see Eq. [2.84]). The situation is quite similar in a contact binary with unequal masses in the sense that nonuniform heating at the inner critical surface generates a lateral temperature gradient and, hence, an astrostrophic flow in the bottom layers of the common envelope. In this case, however, because it is impossible to observe the interior of a contact binary, one must solve simultaneously the coupled equations for the large-scale motion and the temperature field in the common envelope (see Eqs. [8.83], [8.86], and [8.88]). This is not expected to be a straightforward task, for the Roche geometry is awkward, to say the least. As far as I can see, the problem can be made more tractable by using the triply orthogonal system of Roche coordinates that is associated with purely tidal distortions (see, e.g., Kopal 1989, pp. 41-44). Obviously, the removal of the centrifugal potential from Eq. (8.62) is a minor approximation because the flattening caused by the centrifugal force can hardly affect the energy transfer between the components in their common envelope. Although the assumed steadiness of the flow is perhaps a more questionable approximation, it should be of no serious concern at this stage, however, since - as was noted in Section 8.5 - static models could prove indispensable to the development of a rational theory of evolving contact-binary stars.

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