Appendix A: some comments on the assumptions
We first comment on the assumption of uniform surface pressure (see Sect. 3). This was made principally to preserve uniformity with previous papers (H97, K97 and references therein). Nevertheless it may reasonably be asked whether this simplified boundary condition is a) permissible and b) representative.
Let us see how this simplified boundary condition would fit in with, for example, the geostrophic approximation (see e.g. Tassoul 1992). Here the pressure gradient is regarded as being non-zero on equipotentials ; this is however not in conflict with prescribing a constant pressure on the actual contact binary surface.
Nevertheless we must ask whether a departure from our simple boundary condition would be likely to change the flow pattern so much as to invalidate our general conclusions. To fix matters, let us assume that the specific entropy is uniform over the surface (barotropy); this will give us a situation which is thermally similar to that obtained by setting k = 0 in Sect. 3. If, as previously, the gas is assumed perfect, with constant molecular weight, then the thermal equation of state can be written:
Rather than returning with this equation to Sect. 3, with P now variable over the surface, it is more useful to go directly to the equations of motion. We then find that there is an extra term inside the brackets on the L.H.S. of Eqs. (12) and (13):
where the above triple sum is sometimes referred to as Bernoulli's integral. There is however no change on the R.H.S. of Eqs. (12) and (13).
For small surface pressures and supersonic motions along the surface the pressure-dependent correction term will be quite small so that its influence can be compensated by small changes either in (slight change in surface shape) or in . Hence our conclusions based on the surface flow structure should not be seriously affected by possible variations in surface pressure. The question of the constancy of the Jacobi energy is now replaced by the question of whether Bernoulli's integral (triple sum) is constant over the whole surface, or whether it is just constant along the streamlines. Thus the controversy discussed in this paper still persists, but with slight changes in the details.
Somewhat more serious than the assumption of constant surface pressure is perhaps the neglect of the contribution of the radial velocity component to the velocity divergence; this is our 2-dimensional approximation of setting everywhere on the surface except of course at the singular points where the source and sink are located.
We can improve on this highly idealized situation by allowing both source and sink to have a finite lateral extent aR (a small); we can then `parametrize' this new picture (no longer 2-dimensional) by writing:
where is a small positive constant; the previous idealized case corresponds to , a . The minus sign in Eq. (A3) comes from the consideration that, for example, rising material (source) is decelerated on approaching the surface as far as the radial motion is concerned. Denoting as previously the source strength by A, and going to the limit a we confirm that the results of Sect. 3 are recovered if we identify:
Keeping closely to the methods used in the text, but now assuming a 0, we find that Eqs. (25) and (26) now become replaced by:
It is worth noting that Eqs. (A5) and (A6) predict (for small a ) the presence of stagnation points on the equator as already found in Sect. 5. There are however two further stagnation points very close to the geometrical poles and which serve as a meeting point for those streamlines which do not close up on the surface. The structure of the flow in the region outside the source radius is however the same as for the simple case a = 0 calculated in Sect. 5.
© European Southern Observatory (ESO) 1999
Online publication: December 4, 1998