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, easily-modified nuclear equation of state which contains all of the thermodynamic information required to compute the coupled fluid oscillations, with parameters that are constrained by nuclear physics and the requirement that the maximum mass of a neutron star is $\geq 2M_{\odot}$. Using four parametrizations of this equation of state with nuclear compressibilities $K=230$-$280$ MeV, 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, nuclear compressibility and strength of neutron-proton entrainment, and compare to previous calculations of $g$-mode frequencies due to leptonic buoyancy. We also compute the $p$-mode spectra, confirming previous findings that the two fluids behave as if uncoupled except the case of large entrainment, and show the existence of nearly resonant mode pairs which could lead to nonlinear $p$-$g$ instabilities even at zero temperature.