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

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

The volume of a sphere of radius R is ${\frac {4}{3}}\pi R^{3}$ . Given the volume of an non-spherical object V, one can calculate its mean radius by setting

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

or alternatively

R_{\text{mean}}={\sqrt[{3}]{\frac {3V}{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 that

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

Similarly, a tri-axial ellipsoid with axes $a$ , $b$ and $c$ has mean radius $R_{\text{mean}}={\sqrt[{3}]{a\cdot b\cdot c}}$ .^[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 $R_{\text{mean}}={\sqrt[{3}]{a^{2}\cdot c}}$ .^[4]

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

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 $R={\sqrt[{3}]{6378.1^{2}\cdot 6356.8}}=6371.0{\text{ km}}$ . 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 360 km × 294 km × 254 km, has a mean diameter of $D={\sqrt[{3}]{360\cdot 294\cdot 254}}=300{\text{ km}}$ .^[5]

References

^ ^a ^b Leconte, J.; Lai, D.; Chabrier, G. (2011). "Distorted, nonspherical transiting planets: impact on the transit depth and on the radius determination" (PDF). Astronomy & Astrophysics. 528 (A41): 9. arXiv:1101.2813. Bibcode:2011A&A...528A..41L. doi:10.1051/0004-6361/201015811.
^ Milman, V. D.; Pajor, A. (1987–88). "Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space" (PDF). Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics. Vol. 1376. Berlin, Heidelberg: Springer. pp. 65–66. doi:10.1007/BFb0090049. ISBN 978-3-540-51303-2.
^ Petit, A.; Souchay, J.; Lhotka, C. (2014). "High precision model of precession and nutation of the asteroids (1) Ceres, (4) Vesta, (433) Eros, (2867) Steins, and (25143) Itokawa" (PDF). Astronomy & Astrophysics. 565 (A79): 3. Bibcode:2014A&A...565A..79P. doi:10.1051/0004-6361/201322905.
^ ^a ^b Chambat, F.; Valette, B. (2001). "Mean radius, mass, and inertia for reference Earth models" (PDF). Physics of the Earth and Planetary Interiors. 124 (3–4): 4. Bibcode:2001PEPI..124..237C. doi:10.1016/S0031-9201(01)00200-X.
^ Ridpath, I. (2012). "Davida". A Dictionary of Astronomy. Oxford University Press. p. 115. ISBN 978-0-19-960905-5.

[Leconte-1] Leconte, J.; Lai, D.; Chabrier, G. (2011). "Distorted, nonspherical transiting planets: impact on the transit depth and on the radius determination" (PDF). Astronomy & Astrophysics. 528 (A41): 9. arXiv:1101.2813. Bibcode:2011A&A...528A..41L. doi:10.1051/0004-6361/201015811.

[2] Milman, V. D.; Pajor, A. (1987–88). "Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space" (PDF). Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics. Vol. 1376. Berlin, Heidelberg: Springer. pp. 65–66. doi:10.1007/BFb0090049. ISBN 978-3-540-51303-2.

[3] Petit, A.; Souchay, J.; Lhotka, C. (2014). "High precision model of precession and nutation of the asteroids (1) Ceres, (4) Vesta, (433) Eros, (2867) Steins, and (25143) Itokawa" (PDF). Astronomy & Astrophysics. 565 (A79): 3. Bibcode:2014A&A...565A..79P. doi:10.1051/0004-6361/201322905.

[Chambat-4] Chambat, F.; Valette, B. (2001). "Mean radius, mass, and inertia for reference Earth models" (PDF). Physics of the Earth and Planetary Interiors. 124 (3–4): 4. Bibcode:2001PEPI..124..237C. doi:10.1016/S0031-9201(01)00200-X.

[Dictionary-5] Ridpath, I. (2012). "Davida". A Dictionary of Astronomy. Oxford University Press. p. 115. ISBN 978-0-19-960905-5.

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