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

在考林近似的框架下，我们成功地发展了一种新的公式来求解广义相对论和快速自转恒星的r模振荡。按照这个公式，我们开发了一个数值程序，得到了几个快速旋转的广义相对论多项式的平衡序列的r型振荡。为了验证数值程序的正确性，我们计算了牛顿极限下自转恒星和慢转广义相对论恒星的r模振荡，并与用不同方法得到的结果进行了比较。他们同意得很好。我们计算了重力强度为M/R=0.1和0.2的多变指数N=0.5和1.0的均匀旋转多变序列的r模振荡。这里M和R是引力质量和球状恒星的半径。得到了从球形位形到质量脱落极限临界位形的平衡序列的m=2 r模的振荡频率和本征函数。主要研究结果如下：首先，r模的本征频率与未受扰动的自转恒星的角速度之比几乎不依赖于多变指数。这与牛顿的情况是一样的。第二，与此相反，强度T和W分别是转动能和重力能。当M/R=0.1时，a和b分别为a=1.4和b=1.9；当M/R=0.2时，a=1.5和b=1.2~1.3。我们目前对正压星的研究结果表明，对于快速旋转的广义相对论恒星，可能存在r模振荡。

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.