# 4. Kern E. Kenyon, Equilibrium and Perturbed Forms of a Rotating Self‐Gravitating Mass

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Volume 15: Pages 290-296, 2002

# Equilibrium and Perturbed Forms of a Rotating Self‐Gravitating Mass

Kern E. Kenyon

4632 North Lane, Del Mar, California 920144134 U.S.A.

A compressible model is adopted for studying theoretically the equilibrium form of a rotating and selfgravitating solid mass as well as certain perturbations of that mass. The mass is assumed to be homogeneous in planes perpendicular to the rotation axis and to stretch radially and uniformly in these planes under the influence of the centrifugal force. Each particle on the surface and within the mass experiences three forces: the inward gravitational attraction and the outward centrifugal and pressure forces. All three forces are shown to depend linearly on the distance from the center of the mass. When the three forces are in balance, a formula for the rotational frequency is produced that is independent of the distance from the center as well as the latitude. It does depend directly on the square root of the mean density times the flattening. If the mean density and rotational periods are supplied by measurements, the model accounts for 35–75% of the observed flattening of the six planets that have nonzero values for their measured flattenings. Increasing density with decreasing radii within the planets qualitatively accounts for the discrepancies. The calculated shape of the equilibrium form is not a spheroid. An infinitesimal perturbation of the equilibrium mass is considered that occurs in planes perpendicular to the rotation axis; it might be caused by a distant gravitating body, for example. Within these planes the perturbation has an elliptical shape such that the axis of rotation passes through the centers of the ellipses, which are concentric, and they all rotate at the same rate as the mass as a whole. Thus there are exactly two high and two low bodily “tides” in one rotation period. This type of perturbation is unable to alter the basic rotation rate of the mass.

Received: March 26, 2001; Published online: December 15, 2008