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. Since a moment of inertia must have dimensions of mass times length squared, the moment of inertia factor is the coefficient that multiplies these.

Definition

For a planetary body with principal moments of inertia, 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 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 has a moment of inertia of
whereas the mass is
Letting and integrating over we get:
This gives.
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 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 and a relatively low average density. Saturn has the lowest value among the gas giants in part because it has the lowest bulk density. 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. Callisto is similar in size and bulk composition to Ganymede, but is not part of the orbital resonance and is less differentiated. The Moon is thought to have a small core, but its interior is otherwise relatively homogenous.
BodyValueSourceNotes
Sun 0.070Not measured
Mercury 0.346 ± 0.014
Venusunknown
Earth 0.3307
Moon 0.3929 ± 0.0009
Mars 0.3662 ± 0.0017
Ceres 0.36 ± 0.15Not measured
Jupiter 0.2756 ± 0.0006Not measured
Io 0.37824 ± 0.00022Not measured
Europa 0.346 ± 0.005Not measured
Ganymede 0.3115 ± 0.0028Not measured
Callisto 0.3549 ± 0.0042Not measured
Saturn 0.22Not measured
Enceladus 0.3305 ± 0.0025Not measured
Rhea 0.3911 ± 0.0045Not measured
Titan 0.341Not measured
Uranus 0.23Not measured
Neptune 0.23Not measured

Measurement

The polar moment of inertia is traditionally determined by combining measurements of spin quantities with gravity quantities. These geodetic data usually require an orbiting spacecraft to collect.

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.

Gallery of internal structure models