The mean radius (or sometimes the volumetric mean radius) in astronomy is a measure for the size of planets and small Solar System bodies. Alternatively, the closely related mean diameter (), which is twice the mean radius, is also used. For a non-spherical object, the mean radius (denoted
or
) is defined as the radius of the sphere that would enclose the same volume as the object.[1] In the case of a sphere, the mean radius is equal to the radius.
For any irregularly shaped rigid body, there is a unique ellipsoid with the same volume and moments of inertia.[2] The dimensions of the object are the principal axes of that special ellipsoid.[3]
The volume of a sphere of radius Ris. Given the volume of an non-spherical object V, one can calculate its mean radius by setting
or alternatively
For example, a cube of side length L has a volume of . Setting that volume to be equal that of a sphere imply that
Similarly, a tri-axial ellipsoid with axes ,
and
has mean radius
.[1] The formula for a rotational ellipsoid is the special case where
.
Likewise, an oblate spheroidorrotational ellipsoid with axes and
has a mean radius of
.[4]
For a sphere, where , this simplifies to
.