Study on the Stability of Rapidly Rotating Stars

我们发展了可用于研究具有可压缩和不可压缩状态方程的快速自转恒星稳定性的数值格式。对于可压缩模型，我们采用旋转多面体，并利用欧拉位移建立了线性化方程，给出了本征值和本征函数。我们分析了多变指数N=1.0、1.5和3.0的旋转多面体。由于稳定性依赖于转动定律，我们研究了三种类型的转动定律。主要结果如下：对于m=2bar模的长期不稳定性的发生，T/Iwi的值不一定是0.14，这一直被广泛认为是“普遍的”，而不考虑旋转律和压缩性。特别是对于常比角动量分布模型，这个值变得很小，这意味着即使在缓慢旋转的情况下，它们也容易在引力辐射下不稳定。对于不可压缩流体，由于密度没有时间导数，我们需要建立不同于可压缩情况的稳定性分析。由基本方程得到的线性化方程不会归结为线性特征值问题，而是非线性特征值问题。由于我们考虑的是轴对称态的变形位形，所以我们可以确定赤道面上表面的欧拉位移值。这样，问题就可以归结为求解一般的非线性微分方程组，可以用牛顿-拉普利森迭代法来处理。将该方法应用于环形构形，发现剪切失稳的主模在自重作用下是稳定的。

We have developed the numerical schemes which can be applied to study the stability of rapidly rotating stars with compressible and incompressible equation of states. Concerning the compressible models we have used rotating polytropes and made use of the Euler displacements to formulate the linearized equations which give the eigen value and eigen functions. We have analyzed rotating polytropes with polytropic indices N = 1.0, 1.5 and 3.0. Since the stability depends on the rotation laws, we have examined three types of rotation laws. The main results are as follows : The value of T/ IW I for the occurrence of the secular instability against the m=2 bar mode is not necessarily 0.14 which has been widely believed 'universal' irrespective of the rotation law and the compressibility. In particular for the constant specific angular momentum distribution models this value becomes rather small, which imp tics that they are apt to unstable against the gravitational radiation even for slowly rotating cases.For the incompressible fluids since there appears no time derivative of the density, we need to formulate the stability analysis different from that for the compressible cases. The linearized equations obtained from the basic equations are not reduced to a linear eigen value problem but become a non-linear eigen value problem. Since we are concerned with the deformed configuration from the axisymmetric states, we can specify the, value of the Euler displacement of the surface on the equatorial plane. By doing so the problem can be reduced to solving ordinary non-linear different lat equations which can be handled by the Newton-Raplison iteration scheme. This developed scheme has been applied to the toroidal con figurations and we found that the principal mode of the shear instability is stabilized by the effect of the self -gravity.