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.
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Definition
For a planetary body with principal moments of inertia A<B<C, the moment of inertia factor is defined as
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
Ganymede has the lowest moment of inertia factor among solid bodies in the Solar System because of its fully differentiated interior, a result in part of tidal heating due to the Laplace resonance, as well as its substantial component of low density water ice. Saturn has the lowest value among the gas giants in part because it has the lowest bulk density. The Sun has by far the lowest value of all, in part because it has by far the highest central density.
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.
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.