Moment of inertia factor

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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.

Definition

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.

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 1] 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 less differentiated.[7][8] The Moon is thought to have a small core, but its interior is relatively homogenous.[10][11]

Body Value Source Notes
Sun 0.070 [3] Not measured
Mercury 0.346 ± 0.014 [12]
Venus unknown[note 2]
Earth 0.3307 [14]
Moon 0.3929 ± 0.0009 [15]
Mars 0.3662 ± 0.0017 [16]
Ceres 0.37[note 3] [17] Not measured (Darwin-Radau relation)
Jupiter 0.254 [18] Not measured (approximate solution to Clairaut's equation)
Io 0.37824 ± 0.00022 [19] Not measured (Darwin-Radau relation)
Europa 0.346 ± 0.005 [19] Not measured (Darwin-Radau relation)
Ganymede 0.3115 ± 0.0028 [19] Not measured (Darwin-Radau relation)
Callisto 0.3549 ± 0.0042 [19] Not measured (Darwin-Radau relation)
Saturn 0.210 [18] Not measured (approximate solution to Clairaut's equation)
Titan 0.3414 ± 0.0005 [20] Not measured (Darwin-Radau relation)
Uranus 0.23 [18] Not measured (approximate solution to Clairaut's equation)
Neptune 0.23 [18] 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 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.[21]

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 = 0.210)
Ganymede (C/MR2 = 0.3115)
Earth (C/MR2 = 0.3307)
File:Callisto diagram.svg
Callisto (C/MR2 = 0.3549)
The Moon (C/MR2 = 0.3929)

Notes

  1. ↑ A star's central density tends to increase over the course of its lifetime.
  2. ↑ Based on a theoretical model of Venus's interior, its mean moment of inertia has been predicted to be 0.338.[13] Because this model makes assumptions about the interior that are not verified by observations, the prediction is of limited value.
  3. ↑ The value of 0.37 given for Ceres is the mean moment of inertia, which is thought to better represent its interior structure than the polar moment of inertia (0.39), due to its high polar flattening.[17]

References

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  21. ↑ Lua error in package.lua at line 80: module 'strict' not found.
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