Mean radius

A sphere (top), rotational ellipsoid (left) and triaxial ellipsoid (right)

The mean radius in astronomy is a measure for the size of planets and small Solar System bodies. Alternatively, the closely related mean diameter ($D$), which is twice the mean radius, is also used. For a non-spherical object, the mean radius (denoted $R$ or $r$) is defined as the radius of the sphere that would enclose the same volume as the object. 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. The dimensions of the object are the principal axes of that special ellipsoid.

Calculation

Given the dimensions of an irregularly shaped object, one can calculate its mean radius.

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

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

For a sphere, where $a=b=c$, this simplifies to $R=a$.

Examples

• For planet Earth, which can be approximated as an oblate spheroid with radii 6378.1 km and 6356.8 km, the mean radius is $R=(6378.1^{2}\cdot 6356.8)^{1/3}=6371.0{\text{ km}}$. The equatorial and polar radii of a planet are often denoted $r_{e}$ and $r_{p}$, respectively.
• The asteroid 511 Davida, which is close in shape to a triaxial ellipsoid with dimensions 360 km × 294 km × 254 km, has a mean diameter of $D=(360\cdot 294\cdot 254)^{1/3}=300{\text{ km}}$.

See also

• Earth ellipsoid
• Geoid
• Geometric mean