# Moment of inertia factor

In planetary sciences, the moment of inertia factor or normalized polar moment of inertia is an important 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]

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

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.

## Representative values

 Body Value Source Earth 0.3307 [3] Mars 0.3662 +/- 0.0017 [4] Mercury 0.346 +/- 0.014 [5] Moon 0.3929 +/- 0.0009 [6] Venus unknown

## 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.[7]

## 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.

## References

1. ^ Hubbard, William B. (1984). Planetary interiors. New York, N.Y.: Van Nostrand Reinhold. ISBN 978-0442237042.
2. ^ de Pater, Imke; Lissauer, Jack J. (2010). Planetary sciences (2nd ed.). New York: Cambridge University Press. ISBN 978-0521853712.
3. ^ 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.
4. ^ 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.
5. ^ 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.
6. ^ 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.
7. ^ Murray, Carl D.; Dermott, Stanley F. (1999). Solar system dynamics. Cambridge: Cambridge University Press. ISBN 978-0521575973.