Surface Gravity of Rotating Dumbbell Shapes

TL;DR

This paper models the equilibrium shape of rotating dumbbell-shaped celestial bodies, like asteroids, using cylindrical coordinates and elliptic integrals, providing a new approach to understanding their gravitational and rotational balance.

Contribution

It introduces a novel method for describing the shape of rotating bodies in cylindrical coordinates and solves the equilibrium shape problem using variational techniques.

Findings

Derived a simple elliptic integral formula for gravitational potential

Applied variational method to find equilibrium shapes of dumbbells

Provided numerical solutions for a two-parameter family of shapes

Abstract

We investigate the problem of determining the shape of a rotating celestial object - e.g., a comet or an asteroid - under its own gravitational field. More specifically, we consider an object symmetric with respect to one axis - such as a dumbbell - that rotates around a second axis perpendicular to the symmetry axis. We assume that the object can be modeled as an incompressible fluid of constant mass density, which is regarded as a first approximation of an aggregate of particles. In the literature, the gravitational field of a body is often described as a multipolar expansion involving spherical coordinates (Kaula, 1966). In this work we describe the shape in terms of cylindrical coordinates, which are most naturally adapted to the symmetry of the body, and we express the gravitational potential generated by the rotating body as a simple formula in terms of elliptic integrals. An equilibrium shape occurs when the gravitational potential energy and the rotational kinetic energy at the surface of the body balance each other out. Such an equilibrium shape can be derived as a solution of an optimization problem, which can be found via the variational method. We give an example where we apply this method to a two-parameter family of dumbbell shapes, and find approximate numerical solutions to the corresponding optimization problem.