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, r⁰- and r^s-modes. The two superfluids in the core flow together for the r⁰-modes, while they countermove for the r^s-modes. For the r⁰-modes, the coefficient κ₀ ≡ lim_Ω→0 ω/Ω is equal to 2m/[l′ (l′ + 1)],almost independent of the parameter η that parameterizes the entrainment effects between the superfluids, where Ω is the angular frequency of rotation, ω 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, κ₀ is equal to 2m/[l′ (l′ + 1)] at η = 0 (noentrainment), and it almost linearly increases as η is increased from η = 0. The r⁰-modes, for which w′ ≡ v′_p - v′_n ∝ Ω³, 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 r⁰-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.