Study on the Stability of Rapidly Rotating Stars

我们发展了可压缩和不可压缩状态方程的数值格式，可用于研究快速旋转恒星的稳定性。对于可压缩模型，我们采用旋转多面体，并利用欧拉位移来建立线性化方程，给出本征值和本征函数。我们分析了多变指数N = 1.0，1.5和3.0的旋转多面体。由于稳定性取决于旋转定律，我们已经检查了三种类型的旋转定律。主要结果如下：m=2杆模态长期不稳定性发生的T/IWI值不一定是0.14，这一值被广泛认为是“普适的”，与旋转律和压缩性无关。特别是对于常数比角动量分布模型，该值变得很小，这意味着即使在慢旋转情况下，它们也容易在引力辐射作用下不稳定。对于不可压缩流体，由于没有密度的时间导数，我们需要制定不同于可压缩情况的稳定性分析。由基本方程组得到的线性化方程组不是线性特征值问题，而是非线性特征值问题。由于我们关心的是轴对称状态的变形构形，我们可以指定赤道平面上表面的欧拉位移的值。通过这样做，问题可以归结为求解普通的非线性差分方程，可以用牛顿-拉普森迭代格式处理。将该格式应用于环形流场，发现剪切不稳定性的主模态在自重力的作用下是稳定的。

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.