Mean radius (astronomy)

A sphere (top), rotational ellipsoid (left) and tri-axial ellipsoid (right)

The 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] In astronomy, the dimensions of an object are defined as the principal axes of that special ellipsoid.^[3]

Calculation

The dimensions of a minor planet can be uni-, bi- or tri-axial, depending on what kind of ellipsoid is used to model it. Given the dimensions of an irregularly shaped object, one can calculate its mean radius:

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

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

For a sphere, which is uni-axial ( $a=b=c$ ), this simplifies to $R=a$ .

Planets and dwarf planets are nearly spherical if they are not rotating. A rotating object that is massive enough to be in hydrostatic equilibrium will be close in shape to an ellipsoid, with the details depending on the rate of the rotation. At moderate rates, it will assume the form of either a bi-axial (Maclaurin) or tri-axial (Jacobi) ellipsoid. At faster rotations, non-ellipsoidal shapes can be expected, but these are not stable.^[5]

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 $R=\left((6378.1~{\text{km}})^{2}\cdot 6356.8~{\text{km}}\right)^{1/3}=6371.0~{\text{km}}$ . 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 tri-axial ellipsoid with dimensions 360 km × 294 km × 254 km, has a mean diameter of $D=(360~{\text{km}}\cdot 294~{\text{km}}\cdot 254~{\text{km}})^{1/3}=300{\text{ km}}$ .^[6]
Assuming it is in hydrostatic equilibrium, the dwarf planet Haumea has dimensions 2,100 × 1,680 × 1,074 km,^[7] resulting in a mean diameter of $D=\left(2100~{\text{km}}\cdot 1680~{\text{km}}\cdot 1074~{\text{km}}\right)^{1/3}=1559~{\text{km}}$ . The rotational physics of deformable bodies predicts that over as little as a hundred days, a body rotating as rapidly as Haumea will have been distorted into the equilibrium form of a tri-axial ellipsoid.^[8]

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 and normed n-dimensional Space" (PDF). Geometric Aspects of Functional Analysis: Israel Seminar (GAFA). Berlin, Heidelberg: Springer: 65–66.
^ 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.
^ Lyttleton, R. (1953). The Stability of Rotating Liquid Masses. Cambridge University Press. ISBN 9781107615588.
^ Ridpath, I. (2012). A Dictionary of Astronomy. Oxford University Press. p. 115. ISBN 978-0-19-960905-5.
^ Dunham, E. T.; Desch, S. J.; Probst, L. (April 2019). "Haumea's Shape, Composition, and Internal Structure". The Astrophysical Journal. 877 (1): 11. arXiv:1904.00522. Bibcode:2019ApJ...877...41D. doi:10.3847/1538-4357/ab13b3. S2CID 90262114.
^ Rabinowitz, D. L.; Barkume, K.; Brown, M. E.; Roe, H.; Schwartz, M.; Tourtellotte, S.; Trujillo, C. (2006). "Photometric Observations Constraining the Size, Shape, and Albedo of 2003 EL₆₁, a Rapidly Rotating, Pluto-Sized Object in the Kuiper Belt". Astrophysical Journal. 639 (2): 1238–1251. arXiv:astro-ph/0509401. Bibcode:2006ApJ...639.1238R. doi:10.1086/499575. S2CID 11484750.

[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 and normed n-dimensional Space" (PDF). Geometric Aspects of Functional Analysis: Israel Seminar (GAFA). Berlin, Heidelberg: Springer: 65–66.

[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.

[5] Lyttleton, R. (1953). The Stability of Rotating Liquid Masses. Cambridge University Press. ISBN 9781107615588.

[Dictionary-6] Ridpath, I. (2012). A Dictionary of Astronomy. Oxford University Press. p. 115. ISBN 978-0-19-960905-5.

[Dunham2019-7] Dunham, E. T.; Desch, S. J.; Probst, L. (April 2019). "Haumea's Shape, Composition, and Internal Structure". The Astrophysical Journal. 877 (1): 11. arXiv:1904.00522. Bibcode:2019ApJ...877...41D. doi:10.3847/1538-4357/ab13b3. S2CID 90262114.

[Rabinowitz2005-8] Rabinowitz, D. L.; Barkume, K.; Brown, M. E.; Roe, H.; Schwartz, M.; Tourtellotte, S.; Trujillo, C. (2006). "Photometric Observations Constraining the Size, Shape, and Albedo of 2003 EL₆₁, a Rapidly Rotating, Pluto-Sized Object in the Kuiper Belt". Astrophysical Journal. 639 (2): 1238–1251. arXiv:astro-ph/0509401. Bibcode:2006ApJ...639.1238R. doi:10.1086/499575. S2CID 11484750.

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Calculation

Examples

See also

References