Rapidly Rotating Polytropes and Concave Hamburger Equilibrium

Abstract

Rotating poly tropes have equilibrium figures of concave hamburger shape, which bifurcates from Maclaurin-spheroid-like figures and continues into toroids. However, two existing numerical computations of the concave hamburgers are quantitatively in contradiction to each other. Reasons for this contradiction are found to lie in the wrong treatments: One of their methods was applied for deformations too strong to be treated within its limit of applicability so that their boundary condition failed in its convergence of the series and in its analytic continuation into the complex plane.A modified method of numerical computation is developed which can not only avoid such problems but is still reasonably efficient. With this method we have recomputed sequences of rotating poly tropes. We have found the following. When the polytropic index N is greater than 0.02, the sequence of the Maclaurin-spheroid-like figures terminates by mass shedding from the equator. When N < 0.02, on the other hand, it continues into a sequence of the concave hamburgers. Contrary to the earlier computation, the Maclaurin spheroids are shown to be the limiting configuration to N = O. Some details are also discussed concerning the bifurcation to the concave hamburgers. § 1. Introduction 191 Ninety years ago, Dyson 1 ),2) obtained a sequence of toroidal figures for a rigidly rotating, self-gravitating, incompressible fluid by means of perturbation method. Recently W ong 3l computed numerically a sequence of toroid-like figures, which is an extension of Dyson's sequence to a region with relatively small angular momentum. Hereafter, we will call them the Dyson-Wong toroids (or the Dyson-Wong sequence).In astrophysical problems containing axisymmetric rotating celestial bodies, many authors quoted Maclaurin spheroids to compare their results with. Therefore, it is important to understand the relation between the Maclaurin sequence and the Dyson-Wong sequence. It is also important to understand the relation between the cases of incompressible and of compressible fluids. For these purposes Fukushima et a1. 4 ) made numerical computations of hydrostatic equilibrium solutions for rapidly and rigidly rotating poly tropes with very small compressibilities and they suggested the following. F1) The Maclaurin spheroid is a special solution and does not represent the limiting case of the rotating polyat

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