R-modes of neutron stars with the superfluid core

Abstract

We investigate the modal properties of the $r$-modes of rotating neutron stars with the core filled with neutron and proton superfluids, taking account of entrainment effects between the superfluids. The stability of the $r$-modes against gravitational radiation reaction is also examined considering viscous dissipation due to shear and a damping mechanism called mutual friction between the superfluids in the core. We find the $r$-modes in the superfluid core are split into ordinary $r$-modes and superfluid $r$-modes, which we call, respectively, $r^o$- and $r^s$-modes. The two superfluids in the core flow together for the $r^o$-modes, while they counter-move for the $r^s$-modes. For the $r^o$-modes, the coefficient $κ_0\equiv\lim_{Ω\to 0}ω/Ω$ is equal to $2m/[l^\prime(l^\prime+1)]$, almost independent of the parameter $η$ that parameterizes the entrainment effects between the superfluids, where $Ω$ is the angular frequency of rotation, $ω$ the oscillation frequency observed in the corotating frame of the star, and $l^\prime$ and $m$ are the indices of the spherical harmonic function representing the angular dependence of the $r$-modes. For the $r^s$-modes, on the other hand, $κ_0$ is equal to $2m/[l^\prime(l^\prime+1)]$ at $η=0$ (no entrainment), and it almost linearly increases as $η$ is increased from $η=0$. The mutual friction in the superfluid core is found ineffective to stabilize the $r$-mode instability caused by the $r^o$-mode except in a few narrow regions of $η$. The $r$-mode instability caused by the $r^s$-modes, on the other hand, is extremely weak and easily damped by dissipative processes in the star.