Compressional modes in two-superfluid neutron stars with leptonic buoyancy

We investigate the compressional modes of cold neutron stars with cores consisting of superfluid neutrons, superconducting protons and normal fluid electrons and muons, and crusts that contain superfluid neutrons plus a normal fluid of (spherical) nuclei and electrons. We develop a two-fluid formalism for the core that accounts for leptonic buoyancy, and an analogous treatment for the crust. We adopt the Cowling approximation, neglecting gravitational perturbations, but include all effects of the background space-time. We introduce a phenomenological nuclear equation of state which contains all of the thermodynamic information required to compute the coupled fluid oscillations, with parameters that are constrained by the nuclear physics and the requirement that the maximum mass of a neutron star is $\geq 2M_{\odot}$. Using this equation of state, we calculate the Brunt-V\"{a}is\"{a}l\"{a} frequency due to leptonic buoyancy, and find the corresponding $g$-mode frequencies and eigenfunctions. We find that the WKB approximation reproduces $g$-mode frequencies closely. We examine the dependence of $g$-mode frequencies on stellar mass and strength of neutron-proton entrainment, and compare to previous calculations of $g$-mode frequencies due to leptonic buoyancy. For $p$-modes, we find that the usual WKB treatment often breaks down because the displacements of the two superfluids may have large phase differences. The $p$-mode spectra may permit the existence of nearly resonant mode pairs which could lead to nonlinear $p$-$g$ instabilities.