Toroidal figures of equilibrium.

A rotating mass in the form of a torus allows an effective balance between the gravitational attraction and the force due to centrifugal acceleration, when the angular momentum is very large. Attention is therefore fixed on a toroidal mass idealized for simplicity to be homogeneous, incompressible, and rotating uniformly about the central symmetry axis. The toroidal figures of equilibrium are determined in a self-consistent manner for various values of the angular momentum. The set of toroidal figures of equilibrium forms a sequence that begins when the angular momentum L is given by (25/12)(4 /3)112L2p112/GM2012 = 0.8437(3). The corresponding Maclaurin spheroid having such a critical angular momentum would have an eccentricity e = 0.9817(1). The toroidal sequence and the Maclaurin sequence do not join on each other directly but are probably connected by an unstable sequence with intermediate shapes. The properties of the toroidal sequence under beaded displacements, in which the toroid becomes thicker in some meridians but thinner in others, are studied for two cases in which the flow pattern is assumed known. Stability against these displacements is found to depend on the flow pattern which is affected by the dissipative mechanism. Whether toroidal figures play any role in the evolution of astrophysical objects is examined from an observational point of view. A hypothesis of toroidal formation and breakup in galaxies and pre-main-sequence stars is postulated and compared with observations. Subject heading: rotation