Mean radius

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 (${\displaystyle D}$), which is twice the mean radius, is also used. For a non-spherical object, the mean radius (denoted ${\displaystyle R}$ or ${\displaystyle r}$) 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 R is ${\displaystyle {\frac {4}{3}}\pi R^{3}}$. Given the volume of an non-spherical object V, one can calculate its mean radius by setting

${\displaystyle V={\frac {4}{3}}\pi R_{\text{mean}}^{3}}$

or alternatively

${\displaystyle R_{\text{mean}}={\sqrt[{3}]{\frac {3V}{4\pi }}}}$

For example, a cube of side length L has a volume of ${\displaystyle L^{3}}$. Setting that volume to be equal that of a sphere imply that

${\displaystyle R_{\text{mean}}={\sqrt[{3}]{\frac {3}{4\pi }}}L\approx 0.6204L}$

Similarly, a tri-axial ellipsoid with axes ${\displaystyle a}$, ${\displaystyle b}$ and ${\displaystyle c}$ has mean radius ${\displaystyle R_{\text{mean}}={\sqrt[{3}]{a\cdot b\cdot c}}}$.^[1] The formula for a rotational ellipsoid is the special case where ${\displaystyle a=b}$.

Likewise, an oblate spheroid or rotational ellipsoid with axes ${\displaystyle a}$ and ${\displaystyle c}$ has a mean radius of ${\displaystyle R_{\text{mean}}={\sqrt[{3}]{a^{2}\cdot c}}}$.^[4]

For a sphere, where ${\displaystyle a=b=c}$, this simplifies to ${\displaystyle R_{\text{mean}}=a}$.

• For planet Earth, which can be approximated as an oblate spheroid with radii 6378.1 km and 6356.8 km, the mean radius is ${\displaystyle R={\sqrt[{3}]{6378.1^{2}\cdot 6356.8}}=6371.0{\text{ km}}}$. The equatorial and polar radii of a planet are often denoted ${\displaystyle r_{e}}$ and ${\displaystyle r_{p}}$, respectively.^[4]
• 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 ${\displaystyle D={\sqrt[{3}]{360\cdot 294\cdot 254}}=300{\text{ km}}}$.^[5]

• Cubic mean
• Earth ellipsoid
• Geoid
• Geometric mean