Oscillations and stability of rotating superfluids

Abstract

It is predicted that neutron stars contain a liquid interior of superfluid neutrons and superconducting protons.  The effect of these superfluid components on the various oscillation modes and stability of a rotating neutron star is investigated. We model our superfluid using a simple non-relativistic, two-fluid model, where one fluid consists of the superfluid neutrons and the second fluid contains all the remaining constituents (protons, electrons).  The two fluids are coupled through the equation of state, in particular by entrainment, and are free to rotate at different rotation rates around the same axis.  The initial approach involves Eulerian perturbation theory and subsequently a Lagrangian perturbation framework is developed.  The advantage of the Lagrangian framework is that we can construct a canonical energy for the system allowing us to develop stability criteria fro superfluid stars analogous to the single fluid results by Friedman and Shutz [39].  At present our stability analysis neglects the entrainment effect, and its inclusion is the focus of future work.  However, we do include entrainment in our normal mode investigations.  We consider a self-gravitating, Newtonian, superfluid cylinder.  Numerically, we investigate the normal mode solutions and investigate their dependence on the relative rotation rate and on entrainment.  We observe avoided crossings of modes and the onset of a two-stream instability at a critical relative background rotation rate.

Our investigations are complicated by the presence of various singularities. As a result there exists situations for which we are unable to obtain a numerical solution.  To check our numerics we limit our investigations to situations where these numerical problems are not encountered.  We discover this corresponds to negative values of the entrainment function, α.  Although it is predicted that in the neutrons star core the entrainment will be positive, negative entrainment is not physically unrealistic.  In fact it has been shown [28] that it is what is predicted for neutron star crusts.

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