R-mode oscillations of differentially and rapidly rotating Newtonian polytropic stars - art. no. 024003

For analysis of the r-mode oscillation of hot young neutron stars, it is ne cessary to consider the effect of differential rotation, because viscosity is not strong enough for differentially rotating young neutron stars to be led to uniformly rotating configurations on a very short rime scale after t heir birth. In this paper, we have developed a numerical scheme to solve th e r-mode oscillations of differentially rotating polytropic inviscid stars. This is the extended version of the method which was applied to compute th e r-mode oscillations of uniformly rotating Newtonian polytropic stars. By using this new method, we have succeeded in obtaining eigenvalues and eigen functions of the r-mode oscillations of differentially rotating polytropic stars. Our numerical results show that as the degree of differential rotati on is increased, it becomes more difficult to solve the r-mode oscillations for slightly deformed configurations from a sphere compared to solving the r-mode escalations of considerably deformed stars. One reason for this see ms that for slightly deformed stars a corotation cylinder appears near the stellar surface region if the degree of differential rotation is large enou gh. This is similar to the situation that the perturbational approach of lo w-frequency r-mode oscillations for slowly rotating stars in general relati vity results in a singular eigenvalue problem.