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 rs-modes. The two superfluids in the core flow together for the r0-modes, while they countermove for the rs-modes. For the r0-modes, the coefficient κ0 ≡ limΩ→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 rs-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 rs-modes, on the other hand, is extremely weak and easily damped by dissipative processes in the star.
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