Mean radius explained

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 (

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.^[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

2D

The area of a circle of radius R is

\pi R ^{2} . Given the area of an non-circular object A, one can calculate its mean radius by settingA = \pi R ^{2} _{mean} or alternativelyR _{mean} = \sqrt{ (A)/(\pi) }

For example, a square of side length L has an area of

L ^{2} . Setting that area to be equal that of a circle imply thatR _{mean} = \sqrt{ (1)/(\pi) } L \approx 0.3183 La and semi-minor axis b has mean radius R _{mean=\sqrt{a} ⋅ b} .

For a circle, where

a = b , this simplifies to R _{mean=a} .

3D

The volume of a sphere of radius R is

(4)/(3) \pi R ^{3} . Given the volume of an non-spherical object V, one can calculate its mean radius by settingV = (4)/(3) \pi R ^{3} _{mean} or alternativelyR _{mean} = \sqrt[ 3 ]{ (3 V)/(4 \pi) }

For example, a cube of side length L has a volume of

L ^{3} . Setting that volume to be equal that of a sphere imply thatR _{mean} = \sqrt[ 3 ]{ (3)/(4 \pi) } L \approx 0.6204 L

Similarly, a tri-axial ellipsoid with axes

a , b and c has mean radius R _{mean=\sqrt[3]{a} ⋅ b ⋅ c} .^[1] The formula for a rotational ellipsoid is the special case where a = b .

Likewise, an oblate spheroid or rotational ellipsoid with axes

a and c has a mean radius of R ^{2} _{mean = \sqrt[ 3 ]{a} ⋅ c } .

For a sphere, where

a = b = c , this simplifies to R _{mean=a} .

Examples

• For planet Earth, which can be approximated as an oblate spheroid with radii and, the mean radius is

R = \sqrt[ 3 ]{ 6378 . 1 ^{2} ⋅ 6356 . 8 } =6371 . 0 km . The equatorial and polar radii of a planet are often denoted r _{e} and r _{p} , respectively.^[4]

• The asteroid 511 Davida, which is close in shape to a triaxial ellipsoid with dimensions, has a mean diameter of

D = \sqrt[ 3 ]{ 360 ⋅ 294 ⋅ 254 } =300 km .^[5],

• US hat size is the circumference of the head, measured in inches, divided by pi, rounded to the nearest 1/8 inch. This corresponds to the mean diameter.^[6]
• Diameter at breast height is the circumference of tree trunk, measured at height of 4.5 feet, divided by pi. This corresponds to the mean diameter. It can be measured directly by a girthing tape.^[7]

See also

• Cubic mean
• Earth ellipsoid
• Geoid
• Geometric mean

Notes and References

1. Distorted, nonspherical transiting planets: impact on the transit depth and on the radius determination. J.. Leconte. D.. Lai. G.. Chabrier. Astronomy & Astrophysics. 528. A41. 2011. 9. 10.1051/0004-6361/201015811. 1101.2813 . 2011A&A...528A..41L .
2. Book: https://perso.math.u-pem.fr/pajor.alain/recherche/docs/Mil-Paj-isot.pdf . V. D.. Milman. A.. Pajor. Geometric Aspects of Functional Analysis . Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space . . 65–66 . 1987–88. 1376 . Springer. Berlin, Heidelberg. 10.1007/BFb0090049. 978-3-540-51303-2 .
3. High precision model of precession and nutation of the asteroids (1) Ceres, (4) Vesta, (433) Eros, (2867) Steins, and (25143) Itokawa. A.. Petit. J.. Souchay. C.. Lhotka. Astronomy & Astrophysics. 565. A79. 2014. 3. 10.1051/0004-6361/201322905. 2014A&A...565A..79P .
4. Mean radius, mass, and inertia for reference Earth models. F.. Chambat. B.. Valette. Physics of the Earth and Planetary Interiors. 124. 3–4. 2001. 4. 10.1016/S0031-9201(01)00200-X. 2001PEPI..124..237C .
5. Book: Ridpath, I.. Davida. https://books.google.com/books?id=O31j9UJ3U4oC&q=115. A Dictionary of Astronomy. Oxford University Press. 2012. 115. 978-0-19-960905-5 .
6. Book: Bello . Ignacio . Britton . Jack Rolf . 1993 . Topics in Contemporary Mathematics . 5th . 512 . Lexington, Mass . D.C. Heath . 9780669289572.
7. Book: West . P. W. . 2004 . Tree and Forest Measurement . Stem diameter . 13ff . New York . Springer . 9783540403906 .