R-mode oscillations of differentially and rapidly rotating Newtonian polytropic stars

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Abstract
For 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 led to uniformly rotating configurations on a very short time scale after their birth. In this paper, we have developed a numerical scheme to solve the r-mode oscillations of differentially rotating polytropic inviscid stars. This is the extended version of the method which was applied to compute the r-mode oscillations of uniformly rotating Newtonian polytropic stars. By using this new method, we have succeeded in obtaining eigenvalues and eigenfunctions of the r-mode oscillations of differentially rotating polytropic stars. Our numerical results show that as the degree of differential rotation 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 oscillations of considerably deformed stars. One reason for this seems that for slightly deformed stars a corotation cylinder appears near the stellar surface region if the degree of differential rotation is large enough. This is similar to the situation that the perturbational approach of low-frequency r-mode oscillations for slowly rotating stars in general relativity results in a singular eigenvalue problem.
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