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 (), which is twice the mean radius, is also used. For a non-spherical object, the mean radius (denoted or ) 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.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/3/33/Ellipsoide.svg/300px-Ellipsoide.svg.png)
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
edit2D
editThe area of a circle of radius R is . Given the area of an non-circular object A, one can calculate its mean radius by setting
or alternatively
For example, a square of side length L has an area of . Setting that area to be equal that of a circle imply that
Similarly, an ellipse with semi-major axis and semi-minor axis has mean radius .
For a circle, where , this simplifies to .
3D
editThe volume of a sphere of radius R is . Given the volume of an non-spherical object V, one can calculate its mean radius by setting
or alternatively
For example, a cube of side length L has a volume of . Setting that volume to be equal that of a sphere imply that
Similarly, a tri-axial ellipsoid with axes , and has mean radius .[1] The formula for a rotational ellipsoid is the special case where .
Likewise, an oblate spheroid or rotational ellipsoid with axes and has a mean radius of .[4]
For a sphere, where , this simplifies to .
Examples
edit- For planet Earth, which can be approximated as an oblate spheroid with radii 6378.1 km and 6356.8 km, the mean radius is . The equatorial and polar radii of a planet are often denoted and , 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 .[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
editReferences
edit- ^ 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.
- ^ Bello, Ignacio; Britton, Jack Rolf (1993). Topics in Contemporary Mathematics (5th ed.). Lexington, Mass: D.C. Heath. p. 512. ISBN 9780669289572.
- ^ West, P. W. (2004). "Stem diameter". Tree and Forest Measurement. New York: Springer. pp. 13ff. ISBN 9783540403906.