Effect of rotation on quasi-normal modes

3.2 Effect of rotation on quasi-normal modes

In a continuous sequence of rotating stars that includes a nonrotating member, a quasi-normal mode of index l is defined as the mode which, in the nonrotating limit, reduces to the quasi-normal mode of the same index l. Rotation has several effects on the modes of a corresponding nonrotating star:

The degeneracy in the index $m$ is removed and a nonrotating mode of index l is split into $2l + 1$ different $(l,m )$ modes.
Prograde ( $m < 0$ ) modes are now different from retrograde ( $m > 0$ ) modes.
A rotating “polar” l-mode consists of a sum of purely polar and purely axial terms [292 ], e.g., for $l = m$ ,

that is, rotation couples a polar l-term to an axial $l ± 1$ term (the coupling to the $l + 1$ term is, however, strongly favoured over the coupling to the $l − 1$ term [61 ]). Similarly, for a rotating “axial” mode with $l = m$ ,
Frequencies and damping times are shifted. In general, frequencies (in the inertial frame) of prograde modes increase, while those of retrograde modes decrease with increasing rate of rotation.
In rapidly rotating stars, apparent intersections between higher order modes of different l can occur. In such cases, the shape of the eigenfunction is used in the mode classification.

In rotating stars, quasi-normal modes of oscillation have been studied only in the slow rotation limit, in the post-Newtonian, and in the Cowling approximations. The solution of the fully relativistic perturbation equations for a rapidly rotating star is still a very challenging task and only recently have they been solved for zero-frequency (neutral) modes [292 , 295 ]. First frequencies of quasi-radial modes have now been obtained through 3D numerical time evolutions of the nonlinear equations [105 ].

Going further The equations that describe oscillations of the solid crust of a rapidly rotating relativistic star are derived by Priou in [246 ]. The effects of superfluid hydrodynamics on the oscillations of neutron stars have been investigated by several authors, see, e.g., [202 , 67 , 8 , 10 ] and references therein.