r-Modes of Neutron Stars with a Superfluid Core LEE, U Astronomical Institute Tohoku University

Abstract R-modes, which are generated by Coriolis force in a rotating star, are driven unstable by emitting gravitational waves. For cold neutron stars, the core is filled with neutron and proton superfluids, which move almost independently to each other. Because of the relative motion between the superfluids, there arises a damping mechanism called mutual friction. In this study, we investigate the modal properties of the r-modes in the core, and the damping effects of the mutual friction on the r-mode instability.

Cold Neutron Star neutron stars cool via neutrino emission etc cold neutron star has Fluid Core + Solid Crust (+ Fluid Ocean) neutron superfluid in the inner crust neutron and proton superfluids in the core

Oscillation Modes of Cold Neutron Star (e.g., McDermott et al 1988) cold neutron stars support a rich variety of oscillation modes p-, f-, g-Modes sound waves propagating in the solid crust interfacial modes If rotation is important inertial modes and r-modes, the restoring force for which is Coriolis force

r-Modes r-modes ( = inertial modes that have dominant toroidal components ) eigenfunction (displacement vector) eigenfrequency

energy loss rates due to gravitational wave emission from oscillation modes are mass multipole moment mass current multipole moment Instability Caused by Gravitational Wave Radiation r-modes are driven unstable by emitting gravitational waves since

Damping Mechanisms for Oscillations viscous damping bulk viscositybulk viscosity shear viscosityshear viscosity where

mutual friction between superfluids (e.g., Tilley & Tilley 1990; Mendell 1991) where

Stability of r-Modes extrapolation formula for r-modes of l=mextrapolation formula for r-modes of l=m

Hydrodynamic Equations for Superfluids --Neutron and Proton Superfluids-- (e.g., Khalatnikov 1965; Mendell 1991) continuity equations (e.g., Andreev & Bashkin 1976) mass densities for neutron and proton superfluidsmass densities for neutron and proton superfluids mass density fluxmass density flux

velocity equations ( Lindblom & Mendell 1994 ) neutron superfluidneutron superfluid proton superfluidproton superfluid electron normal fluidelectron normal fluid

Poisson equation equation of state ( energy density as a function of mass densities ) chemical potentials

Modal Analysis for r-Modes polytropic neutron star model of the index N=1 superfluid core + normal fluid envelope, the boundary of which is at analytic equation of state for the core introducing an entrainment parameter

Numerical Result Two families of r-modes ordinary-fluid-like r-modes --- r^o-modes superfluid-like r-modes --- r^s-modes Two families of inertial modes ordinary-fluid-like inertial modes --- i^o-modes superfluid-like inertial modes --- i^s-modes The two superfluids co-move for r^o- and i^o- modes The two superfluids counter move for r^s- and i^s- modes Frequent mode crossings between r-modes and i -modes Mutual friction is strong for the r^s-, i^o-, and i^s-modes Mutual friction is weak for the r^o-modes and the r-mode instability survives

toroidal components of the displacement vector for the r^o- and r^s-modes with l=m=2 at 

mode crossing between r^o-mode (dashed line) and i^s- modes (solid lines)

mutual friction damping time-scale for the r^o-mode of l=m=2 as a function of the entrainment parameter