A brief account of the stability of distorted stars is given. From this appropriate criteria emerge, which are subsequently applied to rotational models of the sun in connection with the solar oblateness problem.

One of the big problems in stellar rotation which has been the object of much debate recently concerns the magnitude of the angular velocity in the central regions of the sun. It is a good example of our general ignorance of the distribution of angular momentum in the interiors of stars. There is good reason, of course, for this ignorance. One can’t make any direct observations and from a theoretical point of view there are many real problems such as the lack of a good theory of convection and meridian circulation, and our ignorance of the structure and magnitude of magnetic fields in the deep stellar interior. These problems among others make it very difficult, for example, to specify a surface condition on the angular velocity. It was pointed out recently (Clement, 1969; this paper is referred to hereinafter as Paper I) that such a condition might enable us to estimate the magnitude of the interior stellar rotation.

Dicke and Goldenberg (1967a) measured the solar oblateness to be σ= (5±0.7) × 10-5 and subsequently interpreted this measurement as evidence that the solar interior rotates with a period of 1.d8. With this interpretation, they then showed that the observed oblateness causes an 8% discrepancy in the Einstein prediction of the perihelion advance of Mercury. The stability analysis of Goldreich and Schubert (1967) seems to preclude such a fast rotation of the solar interior although magnetic field effects could alter their conclusions (Dicke, 1967). More recently Goldreich and Schubert (1968) and Fricke (1969) have calculated upper bounds to the solar oblateness essentially by finding the steepest distribution of angular velocity that is consistent with secular stability at each point in the equatorial plane of the sun. Fricke’s result of σmax=1.4 × 10-5 is based on a stronger stability criterion than that of Goldreich and Schubert who found σmax=1.4 × 10-4; Fricke, however, suggests that this may be in error and should actually be σmax=3.4 × 10-5. In their calculations of σmax the above authors assumed that the outer convective layers of the sun are rotating uniformly and that the angular velocity in the interior is a function of the radial distance from the center of the sun only. We note that while these assumptions are reasonable, neither of them is supported by the observed solar rotation.