Mean radius

The 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]

Calculation

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

An oblate spheroid or rotational ellipsoid with axes ${\displaystyle a}$ and ${\displaystyle c}$ has a mean radius of ${\displaystyle R=(a^{2}\cdot c)^{1/3}}$.^[4]

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

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

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 ${\displaystyle R=(6378.1^{2}\cdot 6356.8)^{1/3}=6371.0{\text{ km}}}$. 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 ${\displaystyle D=(360\cdot 294\cdot 254)^{1/3}=300{\text{ km}}}$.^[5]

See also

• Earth ellipsoid
• Geoid
• Geometric mean

References

This page was last edited on 19 June 2024, at 18:58