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. Since a
moment of inertia has dimensions of mass times length squared, the moment of inertia factor is the coefficient that multiplies these.
where C is the first
principal moment of inertia of the body, M is the
mass of the body, and R is the mean
radius of the body.[1][2] For a
sphere with uniform density, .[note 1][note 2] For a
differentiated planet or satellite, where there is an increase of density with depth, . 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 2/5 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 (162 g/cm3,[3][note 3] 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 (0.687 g/cm3).[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 not part of the orbital resonance and is less differentiated.[7][8] The
Moon is thought to have a small core, but its interior is otherwise relatively homogenous.[10][11]
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 and/or
obliquity) with
gravity quantities (coefficients of a
spherical harmonic representation of the gravity field). These
geodetic data usually require an orbiting
spacecraft to collect.
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.
^For a sphere with uniform density we can calculate the moment of inertia and the mass by integrating over disks from the "south pole" to the "north pole". Using a density of 1, a disk of radius r has a moment of inertia of
whereas the mass is
Letting and integrating over we get:
This gives .
^For
several other examples (in which the rotation axis is the axis of symmetry if not otherwise specified), a solid cone has a factor of 3/10; a uniform thin rod (rotating about its center perpendicularly to its axis, so R is length/2) has a factor of 1/3; a hollow cone or a solid cylinder has a factor of 1/2; a hollow sphere has factor of 2/3; a hollow open-ended cylinder has a factor of 1.
^A star's central density tends to increase over the
course of its lifetime, aside from during brief core nuclear fusion ignition events like the
helium flash.
^The value given for Ceres is the mean moment of inertia, which is thought to better represent its interior structure than the polar moment of inertia, due to its high polar flattening.[17]
^
abSohl, 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.
^Konopliv, Alex S.; Asmar, Sami W.; Folkner, William M.; Karatekin, Özgür; Nunes, Daniel C.; Smrekar, Suzanne E.; Yoder, Charles F.; Zuber, Maria T. (January 2011). "Mars high resolution gravity fields from MRO, Mars seasonal gravity, and other dynamical parameters". Icarus. 211 (1): 401–428.
Bibcode:
2011Icar..211..401K.
doi:
10.1016/j.icarus.2010.10.004.
^
abMao, X.; McKinnon, W. B. (2018). "Faster paleospin and deep-seated uncompensated mass as possible explanations for Ceres' present-day shape and gravity". Icarus. 299: 430–442.
Bibcode:
2018Icar..299..430M.
doi:
10.1016/j.icarus.2017.08.033.
^Fortney, J.J.; Helled, R.; Nettlemann, N.; Stevenson, D.J.; Marley, M.S.; Hubbard, W.B.; Iess, L. (6 December 2018).
"The Interior of Saturn". In Baines, K.H.; Flasar, F.M.; Krupp, N.; Stallard, T. (eds.). Saturn in the 21st Century. Cambridge University Press. pp. 44–68.
ISBN978-1-108-68393-7.