The equilibrium and the stability of the Darwin ellipsoids

Abstract

Darwin's problem is concerned with the equilibrium and the stability of synchronously rotating homogeneous masses under their mutual gravitational and tidal interactions. The problem is solved consistently, in a method of approximation due to Jeans, in two special cases: the case when one of the two components is of infinitesimal mass compared to the other and the case when the two components are of equal mass and congruent. In the former case, the problem insofar as the equilibrium and the stability of the infinitesimal mass is concerned, is hardly distinguishable from Roche's simpler problem in which the distorting mass is treated as a rigid sphere. However, in Darwin's formulation, the distorting mass (in the case considered) is a Maclaurin spheroid; and a principal result is that Darwin's problem has no solution when the eccentricity of the spheroid exceeds a certain maximum value (=0.40504). In the case of the congruent components, the maximum angular velocity of orbital rotation, the distance of closest approach, and the Roche limit (where the equilibrium ellipsoid can be deformed into a neighboring equilibrium ellipsoid by a quasi-static, infinitesimal, solenoidal, ellipsoidal displacement), all occur at different points along the sequence; and instability, by a mode of natural oscillation of either component by itself, sets in at a still different point. It appears, moreover, that of the two figures of equilibrium one obtains (at each separation) those with the greater elongations overlap; all the physically realizable equilibrium ellipsoids are therefore stable with respect to their individual natural oscillations. The bearing of these results on the concepts of "limiting stability" and "partial stability" due to Darwin and Jeans is briefly examined.