3. Continuity of motions in the surface layer
In hydrodynamical considerations concerning the outer layers of a star it is practicable to take a mechanical rather than an optical definition of the surface. Consistent with both H 97 and K 97 we shall assume the surface to be an isobar:
where the `constant' in Eq. (1) has some small (normally very small) value.
Assuming the material to be an ideal gas we have:
so that the continuity equation:
where we have assumed constancy of µ.
Now the concept of `surface streamlines' used by both authors referred to above is clearly only meaningful if these streamlines actually lie in the surface; in view of Eq. (1) this means that the vectors and grad P must be taken to be mutually perpendicular. Hence Eq. (4) simplifies to:
The assumption underlying Eq. (5) should perhaps be stated in more detail. It is that the `surface' of the system can be taken to be one on which the pressure is constant and that, at points on that surface, the velocity of the flow is parallel to the surface and, hence, perpendicular to the pressure gradient. We comment on this assumption in the Appendix A.
In the approximation used in this paper we regard the motions as becoming essentially 2-dimensional "motions on a sphere" as the surface is approached. Here a new assumption is involved, namely that the radial component of the velocity does not contribute significantly to the divergence of the velocity on the surface. This approximation could be more serious than the one involving the constancy of surface pressure; it is also discussed in the Appendix A.
Introducing a polar coordinate system, we then have on the surface :
We next introduce the temperature distribution over the surface; we take for this the simple form:
(k = constant) where the unit vector is always assumed to point away from the source and towards the sink in the simple model of Sect. 2.
Substituting now from Eqs. (6) and (7) into Eq. (5) we find:
Now the thermal capacity of the surface layers is extremely low so that k will be (mainly) determined by energy transport in the deeper layers. For good contact (as assumed in Sect. 2) we can expect k to be quite small. The question we therefore have to answer is whether we can simply drop the k-term in Eq. (8) or whether we must solve the full equation first and then go to the limit in the resulting full solutions.
Although we can not answer this question in general, we can answer it in particular cases. A particular solution of Eq. (8) of special interest here is:
which corresponds to a source-sink pair with the source and sink located at and respectively. Thus we see not only that the source and sink are diametrically opposite but that we can now define the polar axis of coordinates much more precisely as being the axis which joins source and sink. We also note that the k-term modulates the solutions but does not change their essential character.
Going now to the limit we find
so that, at least in this particular case, the `short cut' of simply dropping the k-term in Eq. (8) seems to work. We shall assume without proof that this procedure is also viable for , or at least for those cases which we shall treat later in this paper.
Finally, we note that the use of the strictly two-dimensional approach of this section introduces the disadvantage that the velocity becomes arbitrarily high (see Eqs. (9) and (10)) in the immediate neighbourhood of the source or sink centre. This behaviour can however be avoided by using a more general approach. We shall discuss this aspect in Sect. 7.
© European Southern Observatory (ESO) 1999
Online publication: December 4, 1998