ar X iv : a st ro - p h / 02 07 41 5 v 1 1 9 Ju l 2 00 2 The r - mode instability of neutron stars with the superfluid core

We investigate the modal properties of the r-modes with l = m in rotating neutron stars with the superfluid core, where l and m are the indices of the spherical harmonic function representing the angular dependence of the r-modes. For the modal analysis, we employ simple neutron star models, which are composed of the normal fluid envelope and the core filled with neutron and proton superfluids and a normal fluid of electron. The stability of the r-modes against gravitational radiation reaction is examined by taking account of viscous dissipations due to shear and a damping mechanism called mutual friction in the superfluid core. At a given rotation frequency Ω, we find three r-modes, for which the values of the coefficient κ0 ≡ ω/Ω in the limit of Ω → 0 are numerically indistinguishable, corresponding to 2m/[l(l + 1)], where ω is the oscillation frequency observed in the corotating frame. We find the damping due to the mutual friction in the superfluid core is effective to stabilize the r-mode instability in cold neutron stars, in the sense that the r-modes become unstable only for neutron stars rotating at angular frequencies Ω̃ ≡ Ω/√πGρ̄ > ∼Ω̃L where ρ̄ is the mean density of the star. The lower limit Ω̃L is dependent on the entrainment effects between the neutron and proton superfluids in the core, and Ω̃L > ∼0.4 for the models considered in this paper. The r-modes we found in this paper have the scaling given by w ≡ vp − vn ∝ Ω, where vp and vn are the Eulerian perturbations of the velocities of the proton and neutron superfluids. On the other hand, the r-modes that Lindblom & Mendell (2000) considered in their perturbative treatment with Ω being regarded as the infinitesimal expanding parameter have the scaling of w ∝ Ω, and hence are not necessarily the same as the r-modes discussed in this paper. We suggest that the existence of the r-mode solutions with the scaling of w ∝ Ω depends on the approximations we employ to formulate the dynamics of the superfluids in the core. Subject headings: instabilities — stars: neutron — stars: oscillations — stars : rotation