On the Instability of Polytropic Maclaurin and Roche Ellipsoids

Abstract

Abstract The ellipsoidal configurations studied by Lai et al. belong to the special class of adiabatic polytropes, and most results are applicable only in the interval 0 ≤ n < 1 of polytropic indices n. This restricts the applicability of their models mainly to rapidly rotating neutron stars or binaries with a neutron-star component. We show that their results are incorrect for the following three topics: (1) the stability against equatorial mass loss of polytropic Maclaurin and Jacobi ellipsoids, (2) the onset of dynamical instability in polytropic Maclaurin and Roche binary ellipsoids under the influence of the adiabatic exponent, and (3) the onset of secular instability in polytropic Roche binary ellipsoids. For uniformly rotating polytropes of index n = 1.5, 2, and 3, we get precise numerical values of the critical adiabatic exponent at the onset of dynamical instability, in close agreement with the general theoretical formula of Ledoux. In the Appendix, it is shown that for one of the Rosenkilde toroidal modes, secular and dynamical instability in homogeneous Maclaurin ellipsoids are not correlated, occurring in two distinct overlapping eccentricity intervals. For this mode, the Maclaurin ellipsoid can be secularly stable or unstable if it is dynamically stable or unstable. This rectifies previous conclusions of Lyttleton and Chandrasekhar on this subject.