Study on the Stability of Rapidly Rotating Stars

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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.