r-MODE Oscillations of Differentially and Rapidly Rotating Newtonian Polytropic Stars

For the analysis of the r-mode oscillation of hot young neutron stars, it is necessary to consider the effect of differential rotation, because viscosity is not strong enough for differentially rotating young neutron stars to be lead to uniformly rotating configurations on a very short time scale after their birth. In this paper, we have developed a numerical scheme to solve r-mode oscillations of differentially rotating barotropic stars. This is the extended version of the method which was applied to compute r-mode oscillations of uniformly rotating stars. By using this new method, we have succeeded in obtaining eigenvalues and eigenfunctions of r-mode oscillations of differentially rotating polytropes with the polytropic index N = 1 for the following rotation law. Ω = {Ω_c A² }/(R/R_{eq})² + A² }, where Ω is the angular velocity, R is the distance from the rotation axis, R_eq is the equatorial radius of the star and Ω_c is the central angular velocity. The quantity A is a parameter which represents the degree of differential rotation.

The characteristic features of r-mode oscillations for differentially rotating polytropes can be summarized as follows: 1) The eigenvalues decrease as the central angular velocity increases. 2) If the degree of differential rotation is large, only the fluid near the surface region can oscillate appreciably but the bulk of the star almost remains at its original position. 3) The eigenfrequencies of r-modes locate well below the critical curve when the degree of differential rotation is small. Here the critical curve consists of states where corotation points of the mode appear inside the stars. When we increase the degree of differential rotation, the eigenfrequencies approach to the critical curve. It implies that corotation points are likely to appear for configurations if the degree of differential rotation is large enough.

In this paper, we have succeeded in developing a new scheme which can be used to solve r-mode oscillations even for extremely young stars which rotate differentially. This is a necessary development in order to study the scenario that neutron stars have lost their angular momentum via gravitational waves just after their birth by the r-mode instability.

By analyzing the r-mode oscillations of differentially rotating polytropes, we have shown that the eigenvalue problem becomes singular for configurations with large degree of differential rotation. This explains that the perturbational approach of r-mode oscillations for slowly rotating stars in general relativity results in a singular eigenvalue problem.