r-modes of neutron stars with superfluid cores

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 that 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 countermove for the r(s)-modes. For the r(o)-modes, the coefficient k(0) = lim(Omega-->0) omega/Omega = is equal to 2m/[1'(l'+1)], almost independent of the parameter eta that parameterizes the entrainment effects between the superfluids, where Omega is the angular frequency of rotation, omega is the oscillation frequency observed in the corotating frame of the star, and l' 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, k(0) is equal to 2m/[l'(l'+1)] at eta = 0 (no entrainment), and it almost linearly increases as eta is increased from eta = 0. The r(o)-modes, for which w' = v'(p) - v'(n) proportional to Omega(3), correspond to the r-modes discussed by L. Lindblom & G. Mendell, where v'(n) and v'(p) are the Eulerian velocity perturbations of the neutron and proton superfluids, respectively. 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 eta. 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.