Pl o O R

Because the ratio R2/Dd is of order one in a contact binary, we therefore conclude that, as long as the Rossby number remains much smaller than one, we can let p1 ^ p0 in the common envelope.

Thus, within the framework of our approximations, Eqs. (8.75) and (8.76) become

Similarly, because the flow produces only slight density changes as long as the Rossby number is small, Eq. (8.67) can be approximated by div u = 0, (8.83)

from which it follows that the ratio of the vertical to horizontal speeds is O(d/D) and, hence, much smaller than one.

which describes an approximate balance in the vertical direction between the vertical pressure force and the effective gravity. This is the hydrostatic approximation. It is quite different from the approximation made in Eq. (8.82), in which the horizontal Coriolis force is made to balance the horizontal pressure force that is permanently maintained near the base of the common envelope. This is known as the astrostrophic approximation, and it is the strict analog of the geostrophic approximation discussed in Section 2.5.1.

In order to specify the astrostrophic velocity u in the common radiative envelope of an early-type contact binary, it is necessary to make explicit use of a relation between the horizontal pressure and temperature gradients. In the case of a simple ideal gas, this relation is quite straightforward since Eq. (8.70) then reduces to the linear relation p = R PoT, (8.85)