r‐Modes of Neutron Stars with Superfluid Cores

We investigate the modal properties of the $r$-modes of rotating neutronstars with the core filled with neutron and proton superfluids, taking accountof entrainment effects between the superfluids. The stability of the $r$-modesagainst gravitational radiation reaction is also examined considering viscousdissipation due to shear and a damping mechanism called mutual friction betweenthe superfluids in the core. We find the $r$-modes in the superfluid core aresplit into ordinary $r$-modes and superfluid $r$-modes, which we call,respectively, $r^o$- and $r^s$-modes. The two superfluids in the core flowtogether for the $r^o$-modes, while they counter-move for the $r^s$-modes. Forthe $r^o$-modes, the coefficient $\kappa_0\equiv\lim_{\Omega\to0}\omega/\Omega$ is equal to $2m/[l^\prime(l^\prime+1)]$, almost independent ofthe parameter $\eta$ that parameterizes the entrainment effects between thesuperfluids, where $\Omega$ is the angular frequency of rotation, $\omega$ theoscillation frequency observed in the corotating frame of the star, and$l^\prime$ and $m$ are the indices of the spherical harmonic functionrepresenting the angular dependence of the $r$-modes. For the $r^s$-modes, onthe other hand, $\kappa_0$ is equal to $2m/[l^\prime(l^\prime+1)]$ at $\eta=0$(no entrainment), and it almost linearly increases as $\eta$ is increased from$\eta=0$. The mutual friction in the superfluid core is found ineffective tostabilize the $r$-mode instability caused by the $r^o$-mode except in a fewnarrow regions of $\eta$. The $r$-mode instability caused by the $r^s$-modes,on the other hand, is extremely weak and easily damped by dissipative processesin the star.Comment: 22 pages, 22 figures, accepted for publication in the Astrophysical Journa