Ellipsoidal figures of equilibrium: Compressible models

We present a new analytic study of ellipsoidal figures of equilibrium for compressible, self-gravitating Newtonian fluids. Using an energy variational method, we construct approximate hydrostatic equilibrium solutions for rotating polytropes, either isolated or in binary systems. Both uniformly and nonuniformly rotating configurations are considered. Compressible generalizations are given for most classical incompressible objects, such as Maclaurin spheroids, Jacobi, Dedekind, and Riemann ellipsoids, and Roche, Darwin, and Roche-Riemann binaries. The validity of our approximations is established by presenting detailed comparisons of our results to those of recent three-dimensional computational studies. Although our treatment is quite different, the presentation of our results follows closely that of Chandrasekhar in his work on the incompressible solutions using the tensor virial method. In the incompressible limit, our equilibrium solutions reduce exactly to those of Chandrasekhar. For binary systems, however, our analysis improves on previous results even in the incompressible limit. Our energy variational method can also be used to study the stability properties of the equilibrium solutions. Both secular and dynamical instability limits can be identified. For an isolated rotating star, we find that, when expressed in terms of the ratio T/\W\ of kinetic energy of rotation to gravitational binding energy, the stability limits for axisymmetric configurations to nonaxisymmetric perturbations are independent of compressibility in our approximation. We also study the effects of rotation and tidal forces on the radial stability of stars against gravitational collapse. Our most significant new results concern the stability properties of binary configurations. Along a Roche sequence parameterized by binary separation, we demonstrate the existence of a point where the total energy and angular momentum of the system simultaneously attain a minimum. A similar minimum exists for Darwin binaries when the polytropic index n of both components is below a critical value n_crit ≈ 2. We show that such a turning point along an equilibrium sequence marks the onset of secular instability. The instability occurs before the Roche limit is reached in Roche binaries, and before the surfaces of the two components come into contact in Darwin binaries. We point out the critical importance of this instability in determining the final evolution of coalescing binary systems.