The Tensor Virial Method and Its Applications to Self-Gravitating Superfluids

Abstract

This review starts with a discussion of the hierarchy ofscales, relevant to the description of superfluids inneutron stars, which motivates a subsequent elementaryexposition of the Newtonian superfluid hydrodynamics.Starting from the Euler equations for a superfluid and anormal fluid we apply the tensor virial method to obtainthe virial equations of the first, second, and third orderand to compute their Eulerian perturbations. Specialemphasis is put on the computation of perturbations of thenew terms due to mutual gravitational attraction and mutualfriction between the two fluids. The oscillation modes ofsuperfluid Maclaurin spheroids are derived from the firstand second order perturbed virial equations. We discuss twogeneric classes of oscillation modes which correspond tothe co-moving and relative oscillations of two fluids.These modes decouple if the normal fluid is inviscid. Wealso discuss the mixing of these modes (when the normalfluid is viscous) and its effect on the dynamical andsecular instabilities of the co-moving modes and theirdamping.