Inertial modes of neutron stars with the superfluid core

We investigate the modal properties of inertial modes of rotating neutron stars with the core filled with neutron and proton superfluids, taking account of entrainment effects between the superfluids. In this paper, the entrainment effects are modeled by introducing a parameter $\eta$ so that no entrainment state is realized at $\eta=0$. We find that inertial modes of rotating neutron stars with the superfluid core are split into two families, which we call ordinary fluid inertial modes ($i^o$-mode) and superfluid inertial modes ($i^s$-mode). The two superfluids in the core counter-move for the $i^s$-modes. For the $i^o$-modes, $\kappa_0=\lim_{\Omega\to 0}\omega/\Omega$ is only weakly dependent on the entrainment parameter $\eta$, where $\Omega$ and $\omega$ 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, $|\kappa_0|$ almost linearly increases 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.