Moment of Inertia Factor
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Moment of Inertia Factor

In planetary sciences, the moment of inertia factor or normalized polar moment of inertia is a dimensionless quantity that characterizes the radial distribution of mass inside a planet or satellite.

## Definition

For a planetary body with principal moments of inertia A<B<C, the moment of inertia factor is defined as[1][2]

${\displaystyle {\frac {C}{MR^{2}}}}$,

where C is the polar moment of inertia of the body, M is the mass of the body, and R is the mean radius of the body. For a sphere with uniform density, C/MR2 = 0.4. For a differentiated planet or satellite, where there is an increase of density with depth, C/MR2 < 0.4. The quantity is a useful indicator of the presence and extent of a planetary core, because a greater departure from the uniform-density value of 0.4 conveys a greater degree of concentration of dense materials towards the center.

## Solar System values

The Sun has by far the lowest moment of inertia factor value among Solar System bodies; it has by far the highest central density (,[3][note 1] compared to ~13 for Earth[4][5]) and a relatively low average density (1.41 g/cm3 versus 5.5 for Earth). Saturn has the lowest value among the gas giants in part because it has the lowest bulk density .[6]Ganymede has the lowest moment of inertia factor among solid bodies in the Solar System because of its fully differentiated interior,[7][8] a result in part of tidal heating due to the Laplace resonance,[9] as well as its substantial component of low density water ice. Callisto is similar in size and bulk composition to Ganymede, but is less differentiated.[7][8] The Moon is thought to have a small core, but its interior is relatively homogenous.[10][11]

Body Value Source Notes
00 Sun 0070 0.070 [3] Not measured
01 Mercury 0346 0.346 ± 0.014 [12]
02 Venus 0338unknown[note 2]
03 Earth 0331 0.3307 [14]
04 Moon 0393 0.3929 ± 0.0009 [15]
05 Mars 0366 0.3662 ± 0.0017 [16]
06 Ceres 0370 0.37[note 3] [17] Not measured (Darwin-Radau relation)
07 Jupiter 0254 0.254 [18] Not measured (approximate solution to Clairaut's equation)
08 Io 0378 0.37824 ± 0.00022 [19] Not measured (Darwin-Radau relation)
09 Europa 0346 0.346 ± 0.005 [19] Not measured (Darwin-Radau relation)
10 Ganymede 0311 0.3115 ± 0.0028 [19] Not measured (Darwin-Radau relation)
11 Callisto 0355 0.3549 ± 0.0042 [19] Not measured (Darwin-Radau relation)
12 Saturn 0210 0.210 [18] Not measured (approximate solution to Clairaut's equation)
13 Titan 0341 0.3414 ± 0.0005 [20] Not measured (Darwin-Radau relation)
14 Uranus 0230 0.23 [18] Not measured (approximate solution to Clairaut's equation)
15 Neptune 0230 0.23 [18] Not measured (approximate solution to Clairaut's equation)

## Measurement

The polar moment of inertia is traditionally determined by combining measurements of spin quantities (spin precession rate or obliquity) and gravity quantities (coefficients in a spherical harmonics representation of the gravity field).

## Approximation

For bodies in hydrostatic equilibrium, the Darwin-Radau relation can provide estimates of the moment of inertia factor on the basis of shape, spin, and gravity quantities.[21]

## Role in interior models

The moment of inertia factor provides an important constraint for models representing the interior structure of a planet or satellite. At a minimum, acceptable models of the density profile must match the volumetric mass density and moment of inertia factor of the body.

## Gallery of internal structure models

 The Sun (C/MR2 = 0.070) Saturn (C/MR2 = 0.210) Ganymede (C/MR2 = 0.3115)
 Earth (C/MR2 = 0.3307) Callisto (C/MR2 = 0.3549) The Moon (C/MR2 = 0.3929)

## Notes

1. ^ A star's central density tends to increase over the course of its lifetime.
2. ^ Based on a theoretical model of Venus's interior, its mean moment of inertia has been predicted to be 0.338.[13] Because this model makes assumptions about the interior that are not verified by observations, the prediction is of limited value.
3. ^ The value of 0.37 given for Ceres is the mean moment of inertia, which is thought to better represent its interior structure than the polar moment of inertia (0.39), due to its high polar flattening.[17]

## References

1. ^ Hubbard, William B. (1984). Planetary interiors. New York, N.Y.: Van Nostrand Reinhold. ISBN 978-0442237042. OCLC 10147326.
2. ^ de Pater, Imke; Lissauer, Jack J. (2015). Planetary sciences (2nd updated ed.). New York: Cambridge University Press. ISBN 978-0521853712. OCLC 903194732.
3. ^ a b Williams, D. R. "Sun Fact Sheet". Planetary Fact Sheets. NASA. Retrieved .
4. ^ Robertson, Eugene C. (26 July 2001). "The Interior of the Earth". USGS. Retrieved 2007.
5. ^ Hazlett, James S.; Monroe, Reed; Wicander, Richard (2006). Physical geology : exploring the earth (6. ed.). Belmont: Thomson. p. 346. ISBN 9780495011484.
6. ^ Williams, David R. (7 September 2006). "Saturn Fact Sheet". NASA. Archived from the original on 9 April 2014. Retrieved 2007.
7. ^ a b Showman, Adam P.; Malhotra, Renu (1999-10-01). "The Galilean Satellites" (PDF). Science. 286 (5437): 77-84. doi:10.1126/science.286.5437.77. PMID 10506564.
8. ^ a b Sohl, F.; Spohn, T; Breuer, D.; Nagel, K. (2002). "Implications from Galileo Observations on the Interior Structure and Chemistry of the Galilean Satellites". Icarus. 157 (1): 104-119. Bibcode:2002Icar..157..104S. doi:10.1006/icar.2002.6828.
9. ^ Showman, Adam P.; Stevenson, David J.; Malhotra, Renu (1997). "Coupled Orbital and Thermal Evolution of Ganymede" (PDF). Icarus. 129 (2): 367-383. Bibcode:1997Icar..129..367S. doi:10.1006/icar.1997.5778.
10. ^ Brown, D.; Anderson, J. (6 January 2011). "NASA Research Team Reveals Moon Has Earth-Like Core". NASA. NASA.
11. ^ Weber, R. C.; Lin, P.-Y.; Garnero, E. J.; Williams, Q.; Lognonne, P. (2011-01-21). "Seismic Detection of the Lunar Core" (PDF). Science. 331 (6015): 309-312. Bibcode:2011Sci...331..309W. doi:10.1126/science.1199375. PMID 21212323.
12. ^ Margot, Jean-Luc; Peale, Stanton J.; Solomon, Sean C.; Hauck, Steven A.; Ghigo, Frank D.; Jurgens, Raymond F.; Yseboodt, Marie; Giorgini, Jon D.; Padovan, Sebastiano; Campbell, Donald B. (2012). "Mercury's moment of inertia from spin and gravity data". Journal of Geophysical Research: Planets. 117 (E12): E00L09-. Bibcode:2012JGRE..117.0L09M. doi:10.1029/2012JE004161. ISSN 0148-0227.
13. ^ Aitta, A. (April 2012). "Venus' internal structure, temperature and core composition". Icarus. 218 (2): 967-974. Bibcode:2012Icar..218..967A. doi:10.1016/j.icarus.2012.01.007.
14. ^ Williams, James G. (1994). "Contributions to the Earth's obliquity rate, precession, and nutation". The Astronomical Journal. 108: 711. Bibcode:1994AJ....108..711W. doi:10.1086/117108. ISSN 0004-6256.
15. ^ Williams, James G.; Newhall, XX; Dickey, Jean O. (1996). "Lunar moments, tides, orientation, and coordinate frames". Planetary and Space Science. 44 (10): 1077-1080. Bibcode:1996P&SS...44.1077W. doi:10.1016/0032-0633(95)00154-9. ISSN 0032-0633.
16. ^ Folkner, W. M.; et al. (1997). "Interior Structure and Seasonal Mass Redistribution of Mars from Radio Tracking of Mars Pathfinder". Science. 278 (5344): 1749-1752. Bibcode:1997Sci...278.1749F. doi:10.1126/science.278.5344.1749. ISSN 0036-8075.
17. ^ a b Park, R. S.; Konopliv, A. S.; Bills, B. G.; Rambaux, N.; Castillo-Rogez, J. C.; Raymond, C. A.; Vaughan, A. T.; Ermakov, A. I.; Zuber, M. T.; Fu, R. R.; Toplis, M. J.; Russell, C. T.; Nathues, A.; Preusker, F. (2016-08-03). "A partially differentiated interior for (1) Ceres deduced from its gravity field and shape". Nature. 537: 515-517. Bibcode:2016Natur.537..515P. doi:10.1038/nature18955. PMID 27487219.
18. ^ a b c d Yoder, C. (1995). Ahrens, T., ed. Astrometric and Geodetic Properties of Earth and the Solar System. Washington, DC: AGU. ISBN 0-87590-851-9. OCLC 703657999.
19. ^ a b c d Schubert, G.; Anderson, J. D.; Spohn, T.; McKinnon, W. B. (2004). "Interior composition, structure and dynamics of the Galilean satellites". In Bagenal, F.; Dowling, T. E.; McKinnon, W. B. Jupiter : the planet, satellites, and magnetosphere. New York: Cambridge University Press. pp. 281-306. ISBN 978-0521035453. OCLC 54081598.
20. ^ Iess, L.; Rappaport, N. J.; Jacobson, R. A.; Racioppa, P.; Stevenson, D. J.; Tortora, P.; Armstrong, J. W.; Asmar, S. W. (2010-03-12). "Gravity Field, Shape, and Moment of Inertia of Titan". Science. 327 (5971): 1367-1369. Bibcode:2010Sci...327.1367I. doi:10.1126/science.1182583. PMID 20223984.
21. ^ Murray, Carl D.; Dermott, Stanley F. (13 February 2000). Solar System Dynamics. Cambridge: Cambridge University Press. ISBN 978-1139936156. OCLC 40857034.