Mean radius (astronomy)

For any irregularly shaped rigid body, there is a unique ellipsoid with the same volume and moments of inertia. In astronomy, the dimensions of an object are defined as the principal axes of that special ellipsoid.

Calculation

The dimensions of a minor planet can be uni-, bi- or tri-axial, depending on what kind of ellipsoid is used to model it. Given the dimensions of an irregularly shaped object, one can calculate its mean radius:

An oblate spheroid, bi-axial, or rotational ellipsoid with axes ${\displaystyle a}$ and ${\displaystyle c}$ has a mean radius of ${\displaystyle R=(a^2\cdot c{)}^{1/3}}$.

A tri-axial ellipsoid with axes ${\displaystyle a}$, ${\displaystyle b}$ and ${\displaystyle c}$ has mean radius ${\displaystyle R=(a\cdot b\cdot c{)}^{1/3}}$. The formula for a rotational ellipsoid is the special case where ${\displaystyle a=b}$.

For a sphere, which is uni-axial (${\displaystyle a=b=c}$), this simplifies to ${\displaystyle R=a}$.

Planets and dwarf planets are nearly spherical if they are not rotating. A rotating object that is massive enough to be in hydrostatic equilibrium will be close in shape to an ellipsoid, with the details depending on the rate of the rotation. At moderate rates, it will assume the form of either a bi-axial (Maclaurin) or tri-axial (Jacobi) ellipsoid. At faster rotations, non-ellipsoidal shapes can be expected, but these are not stable.