Moment of inertia factor

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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[edit]

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[edit]

Moment of inertia factor C/MR2
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[edit]

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[edit]

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[edit]

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[edit]

  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.