R-mode oscillations of general relarivistic and rapidly rotating stars by the Cowling approximation

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We have succeeded in developing anew formulation for solving r-mode oscillations of general relativistic and rapidly rotating stars in the framework of the Cowling approximation. By following this formulation we have developed a numerical code and obtained r-mode oscillations of several equilibrium sequences of rapidly rotating general relativistic polytropes. In order to check the numerical code works properly, we have computed r-mode oscillations of rotating stars in the Newtonian limit as well as those of slowly rotating general relativistic stars and compared our results with those obtained by different methods. they agree very well. We have computed r-mode oscillations of uniformly rotating polytropic sequences with the polytropic indices N=0.5 and 1.0 whose strengths of gravity are M/R=0.1 and 0.2. Here M and R are the gravitational mass and the radius of the spherical stars. Oscillation frequencies and eigenfunctions of m=2 r-modes are obtained for the equilibrium sequences from the spherical configurations to the critical configurations of mass shedding limits. The important results are as follows. First, the rations of the eigenfirequencies of the r-modes to the angular velocityies of the unperturbed rotating stars depend scarcely on the polytropic indices. This is the same as that in the Newtonian case. Second, contrary to this, the strength of where T and W are the rotational energy and the gravitational energy, respectively. The values of a and b are a=1.4 and b=1.9 for M/R=0.1 and a=1.5 and b=1.2~1.3 for M/R=0.2. Our present results for barotropic stars imply that there could exist r-mode oscillations for rapidly rotating general relativistic stars.