Inertial modes of neutron stars with a superfluid core

We investigate the modal properties of inertial modes of rotating neutron stars with a core filled with neutron and proton superfluids, taking account of entrainment effects between the superfluids. In this paper, the entrainment effects are modelled by introducing a parameter so that there is no entrainment state at eta = 0. We find that inertial modes of rotating neutron stars with a superfluid core are split into two families, which we call ordinary fluid inertial modes (i(o)-modes) and superfluid inertial modes (i(s)-modes). The two superfluids in the core counter-move for the i(s)-modes. For the i(o)-modes, kappa(0) = lim(Omega-->0) omega/Omega is only weakly dependent on the entrainment parameter, where and are the angular frequency of rotation and the oscillation frequency observed in the corotating frame of the star, respectively. For the i(s) modes, on the other hand, |kappa(0)| increases almost linearly as eta increases. Avoided crossings as functions of eta are therefore quite common between i(o) - and i(s)-modes. We find that some of the i(s)-modes that are unstable against the gravitational radiation reaction at eta = 0 become stable when eta is larger than eta(crit), the value of which depends on the mode. Since the radiation-driven instability associated with the current multipole radiation is quite weak for the inertial modes and the mutual friction damping in the superfluid core is strong, the instability caused by the inertial modes will be easily suppressed unless the entrainment parameter eta is extremely small and the mutual friction damping is sufficiently weak.