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We investigate the compressional modes of cold neutron stars with coresconsisting of superfluid neutrons, superconducting protons and normal fluidelectrons and muons, and crusts that contain superfluid neutrons plus a normalfluid of (spherical) nuclei and electrons. We develop a two-fluid formalism forthe core that accounts for leptonic buoyancy, and an analogous treatment forthe crust. We adopt the Cowling approximation, neglecting gravitationalperturbations, but include all effects of the background space-time. Weintroduce a phenomenological, easily-modified nuclear equation of state whichcontains all of the thermodynamic information required to compute the coupledfluid oscillations, with parameters that are constrained by nuclear physics andthe requirement that the maximum mass of a neutron star is $\geq 2M_{\odot}$.Using four parametrizations of this equation of state with nuclearcompressibilities $K=230$-$280$ MeV, we calculate the Brunt-V\"{a}is\"{a}l\"{a}frequency due to leptonic buoyancy, and find the corresponding $g$-modefrequencies and eigenfunctions. We find that the WKB approximation reproduces$g$-mode frequencies closely. We examine the dependence of $g$-mode frequencieson stellar mass, nuclear compressibility and strength of neutron-protonentrainment, and compare to previous calculations of $g$-mode frequencies dueto leptonic buoyancy. We also compute the $p$-mode spectra, confirming previousfindings that the two fluids behave as if uncoupled except the case of largeentrainment, and show the existence of nearly resonant mode pairs which couldlead to nonlinear $p$-$g$ instabilities even at zero temperature.

Compressional modes in two-superfluid neutron stars with leptonic buoyancy