Moment of inertia factor

Template:Short description 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.

Definition

For a planetary body with principal moments of inertia $A<B<C$, the moment of inertia factor is defined as

$\frac{C}{M{R}^2}$,

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. For a sphere with uniform density, $C/M{R}^2=2/5$.Template:RefnTemplate:Refn For a differentiated planet or satellite, where there is an increase of density with depth, $C/M{R}^2<2/5$. 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 (Template:Val,Template:Refn compared to ~13 for Earth) 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 (Template:Val). 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.

Body	Value	Source	Notes
Template:Hs Sun	Template:Hs 0.070		Not measured
Template:Hs Mercury	Template:Hs 0.346 ± 0.014
Template:Hs Venus	Template:Hs 0.337 ± 0.024
Template:Hs Earth	Template:Hs 0.3307
Template:Hs Moon	Template:Hs 0.3929 ± 0.0009
Template:Hs Mars	Template:Hs 0.3644 ± 0.0005
Template:Hs Ceres	Template:Hs 0.36 ± 0.15Template:Refn		Not measured (range reflects different assumptions for original spin rate)
Template:Hs Jupiter	Template:Hs 0.2756 ± 0.0006		Not measured (two-layer model calculations constrained by Juno gravity data)
Template:Hs Io	Template:Hs 0.37824 ± 0.00022		Not measured (Darwin-Radau relation)
Template:Hs Europa	Template:Hs 0.346 ± 0.005		Not measured (Darwin-Radau relation)
Template:Hs Ganymede	Template:Hs 0.3115 ± 0.0028		Not measured (Darwin-Radau relation)
Template:Hs Callisto	Template:Hs 0.3549 ± 0.0042		Not measured (Darwin-Radau relation)
Template:Hs Saturn	Template:Hs 0.22		Not measured (Darwin-Radau relation)
Template:Hs Enceladus	Template:Hs 0.3305 ± 0.0025		Not measured (Darwin-Radau relation)
Template:Hs Rhea	Template:Hs 0.3911 ± 0.0045		Not measured (Darwin-Radau relation)
Template:Hs Titan	Template:Hs 0.341		Not measured (Darwin-Radau relation)
Template:Hs Uranus	Template:Hs 0.23		Not measured (approximate solution to Clairaut's equation)
Template:Hs Neptune	Template:Hs 0.23		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.

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

• 
  The Sun (C/MR² = 0.070)
• 
  Saturn (C/MR² = 0.22)
• 
  Ganymede (C/MR² = 0.3115)
• 
  Earth (C/MR² = 0.3307)
• 
  Callisto (C/MR² = 0.3549)
• 
  The Moon (C/MR² = 0.3929)