Mean radius (astronomy)

{{Short description|A measure for the size of planets and other Solar System objects}}

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{{Redirect|Mean diameter (astronomy)|other uses|Diameter (disambiguation)}}

{{Redirect|Dimensions (astronomy)|other uses|Dimension (disambiguation)}}

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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.{{cite journal|url=https://www.aanda.org/articles/aa/pdf/2011/04/aa15811-10.pdf|title=Distorted, nonspherical transiting planets: impact on the transit depth and on the radius determination|first1=J.|last1=Leconte|first2=D.|last2=Lai|first3=G.|last3=Chabrier|journal=Astronomy & Astrophysics|volume=528|issue=A41|year=2011|pages=9|doi=10.1051/0004-6361/201015811|arxiv=1101.2813 |bibcode=2011A&A...528A..41L }} 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.{{cite journal|url=https://perso.math.u-pem.fr/pajor.alain/recherche/docs/Mil-Paj-isot.pdf|title=Isotropic position and inertia ellipsoids and zonoids of the unit ball and normed n-dimensional Space|first1=V. D.|last1=Milman|first2=A.|last2=Pajor|pages=65–66|journal=Geometric Aspects of Functional Analysis: Israel Seminar|year=1987–88|publisher=Springer|location=Berlin, Heidelberg}} In astronomy, the dimensions of an object are defined as the principal axes of that special ellipsoid.{{cite journal|url=https://www.aanda.org/articles/aa/pdf/2014/05/aa22905-13.pdf|title=High precision model of precession and nutation of the asteroids (1) Ceres, (4) Vesta, (433) Eros, (2867) Steins, and (25143) Itokawa|first1=A.|last1=Petit|first2=J.|last2=Souchay|first3=C.|last3=Lhotka|journal=Astronomy & Astrophysics|volume=565|issue=A79|year=2014|pages=3|doi=10.1051/0004-6361/201322905|bibcode=2014A&A...565A..79P }}

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 $R=(a^{2} \cdot c )^{1/3}$ .

A tri-axial ellipsoid with axes $a$ , $b$ and $c$ has mean radius $R=(a \cdot b \cdot c)^{1/3}$ . 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.{{Cite book|title=The Stability of Rotating Liquid Masses|last=Lyttleton|first=R.|year=1953|url=https://books.google.com/books?id=aBO9vOoIsA8C|publisher=Cambridge University Press|isbn=9781107615588}}

Examples

For planet Earth, which can be approximated as an oblate spheroid with radii {{val|6378.1|u=km}} and {{val|6356.8|u=km}}, the mean radius is $R=\left((6378.1~\text{km})^{2}\cdot6356.8~\text{km}\right)^{1/3}=6371.0~\text{km}$ . The equatorial and polar radii of a planet are often denoted $r_{e}$ and $r_{p}$ , respectively.{{cite journal|url=http://frederic.chambat.free.fr/geophy/inertie_pepi01/chambat_valette_publie01_with_errata.pdf.pdf|title=Mean radius, mass, and inertia for reference Earth models|first1=F.|last1=Chambat|first2=B.|last2=Valette|journal=Physics of the Earth and Planetary Interiors|volume=124|issue=3–4|year=2001|page=4|doi=10.1016/S0031-9201(01)00200-X|bibcode=2001PEPI..124..237C }}
The asteroid 511 Davida, which is close in shape to a tri-axial ellipsoid with dimensions {{val|360|×|294|×|254|u=km}}, has a mean diameter of $D=(360~\text{km}\cdot294~\text{km}\cdot254~\text{km})^{1/3}=300\text{ km}$ .{{cite book|url=https://books.google.com/books?id=O31j9UJ3U4oC|title=A Dictionary of Astronomy|first=I.|last=Ridpath|publisher=Oxford University Press|year=2012|pages=115|isbn=978-0-19-960905-5 }}
Assuming it is in hydrostatic equilibrium, the dwarf planet Haumea has dimensions 2,100 × 1,680 × 1,074 km,{{cite journal | title = Haumea's Shape, Composition, and Internal Structure | last1 = Dunham | first1=E. T. | last2 = Desch | first2=S. J. | last3 = Probst | first3= L. | date = April 2019 | journal = The Astrophysical Journal | volume = 877 | issue = 1 | page = 11 | doi = 10.3847/1538-4357/ab13b3 | bibcode = 2019ApJ...877...41D | arxiv = 1904.00522 | s2cid = 90262114 | doi-access = free}} resulting in a mean diameter of $D=\left(2100~\text{km}\cdot1680~\text{km}\cdot1074~\text{km}\right)^{1/3}=1559~\text{km}$ . 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.{{cite journal | title = Photometric Observations Constraining the Size, Shape, and Albedo of 2003 EL₆₁, a Rapidly Rotating, Pluto-Sized Object in the Kuiper Belt | journal = Astrophysical Journal | volume = 639 | issue = 2 | pages = 1238–1251 | doi = 10.1086/499575 | bibcode = 2006ApJ...639.1238R | arxiv = astro-ph/0509401 | last1 = Rabinowitz |first1 = D. L. | date = 2006 | last2 = Barkume | first2 = K. | last3 = Brown | first3 = M. E. | last4 = Roe | first4 = H. | last5 = Schwartz | first5 = M. | last6 = Tourtellotte | first6 = S. | last7 = Trujillo | first7 = C. | s2cid = 11484750}}

References

Category:Radii

Category:Units of measurement in astronomy

Mean radius (astronomy)

Calculation

Examples

See also

References