Ellipsoid - Dynamical Properties

Dynamical Properties

The mass of an ellipsoid of uniform density ρ is:

The moments of inertia of an ellipsoid of uniform density are:

For a=b=c these moments of inertia reduce to those for a sphere of uniform density.

Ellipsoids and cuboids rotate stably along their major or minor axes, but not along their median axis. This can be seen experimentally by throwing an eraser with some spin. In addition, moment of inertia considerations mean that rotation along the major axis is more easily perturbed than rotation along the minor axis.

One practical effect of this is that scalene astronomical bodies such as Haumea generally rotate along their minor axes (as does the Earth, which is merely oblate); in addition, because of tidal locking, moons in synchronous orbit such as Mimas orbit with their major axis aligned radially to their planet.

A relaxed ellipsoid, that is, one in hydrostatic equilibrium, has an oblateness a − c directly proportional to its mean density and mean radius. Ellipsoids with a differentiated interior—that is, a denser core than mantle—have a lower oblateness than a homogeneous body. Over all, the ratio (b–c)/(a−c) is approximately 0.25, though this drops for rapidly rotating bodies.

The terminology typically used for bodies rotating on their minor axis and whose shape is determined by their gravitational field is Maclaurin spheroid (oblate speroid) and Jacobi ellipsoid (scalene ellipsoid). At faster rotations, piriform or oviform shapes can be expected, but these are not stable.