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 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 counter-move for the r ^ { s } -modes .
For the r ^ { o } -modes , the coefficient \kappa _ { 0 } \equiv \lim _ { \Omega \rightarrow 0 } \omega / \Omega is equal to 2 m / [ l ^ { \prime } ( l ^ { \prime } +1 ) ] , almost independent of the parameter \eta that parameterizes the entrainment effects between the superfluids , where \Omega is the angular frequency of rotation , \omega the oscillation frequency observed in the corotating frame of the star , and l ^ { \prime } 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 , \kappa _ { 0 } is equal to 2 m / [ l ^ { \prime } ( l ^ { \prime } +1 ) ] at \eta = 0 ( no entrainment ) , and it almost linearly increases as \eta is increased from \eta = 0 .
The r ^ { o } -modes , for which \mbox { \boldmath$w$ } ^ { \prime } \equiv \mbox { \boldmath$v$ } ^ { \prime } _ { p } - \mbox { % \boldmath$v$ } ^ { \prime } _ { n } \propto \Omega ^ { 3 } , correspond to the r -modes discussed by Lindblom & Mendell ( 2000 ) , where \mbox { \boldmath$v$ } ^ { \prime } _ { n } and \mbox { \boldmath$v$ } ^ { \prime } _ { 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 .