Rapidly Rotating Polytropes and Concave Hamburger Equilibrium

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.