How is a Planet's Moment of Inertia Measured Remotely?

Question

In the July 2023 issue of Sky & Telescope there is an article titled Sights Set on Uranus. In that article the following statement is made:

Since the 1930's we've suspected that Uranus and Neptune are made mostly of ice. (Ice refers to materials that are typically liquids or gases on Earth but are frozen in the outer solar system, including water and other lightweight molecular compounds like methane and ammonia.) But that time, astronomers had measured each planet's mass, volume, and moment of inertia, a measure of how concentrated the mass is towards the planet's center.

I can easily imagine how one would estimate the mass and volume of a planet remotely in the 1930s, but how was the moment of inertia estimated? I would have thought this would have required an orbiter or a fly-by to measure the shape of the gravitational potential.

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Information below is more background information that I was able to dig up:

Wildt, R., 1939. The Constitution of the Planets. Proceedings of the American Philosophical Society, 81(2), pp.135-152.

The greatest moment of inertia, $C$, of a planet can be computed from its oblateness on the Radau-Darwin theory of the ellipticity of the> Earth.^* For a homogeneous body the dimensionless quantity, $C/Ma^2$, has the maximum value, 2/5. The values of $C/Ma^2$ given in Table I for the terrestrial planets have been taken from H. Jeffreys $\dagger$; they correspond to his models of these bodies which will be discussed in Section II. The other values have been calculated from the dynamical ellipticities of the giant planets determined by N. Lvoff.$\ddagger$ Since the deviations of the individual values from Lvoff’s mean, 0.24, are not well established at present, only this mean will be used in the further discussions; but there is evidence that the quantity $C/Ma^2$ is decidedly less for Saturn than for Jupiter.

Note: $C$ - greatest moment of inertia; $M$ - mass, unit 5.966x10²⁷ g; $a$ - equatorial radius, unit 6278.388 km

* G. H. Darwin, M.N., 60, 82, 1900; H. Jeffreys, M.N., 84, 534, 1924.
$\dagger$ H. Jeffreys, M.N., Geophys. Suppl., 4, 62, 1940.
$\ddagger$ N. Lvoff, Russ. Astr. J., 9, 68, 1932.

There three references I am currently struggling to track down.

Answer 1

I dug around quite a bit and was unable to find anyone making measurements of the type that would lead one to determine the moment of inertia of Uranus or Neptune without first making assumptions about their composition. The way to determine the moment of inertia without such assumptions is to use the gravitational quadrupole moment, $J_2$, and oblateness (see below).

Moore & Menzel (1928) report on the rotation of Neptune. They state that "it is possible to calculate the oblateness and period of rotation, first making certain assumptions as to the planet's internal constitution. Thus Jackson, assuming a constitution for Neptune similar to that of Jupiter, derives an [oblateness] of $1/65.7$..."

The same duo also reported on Uranus's rotation (Moore & Menzel 1930), but they make no mention of the oblateness.

Goody (1981) gives the most useful historical account of Uranian observations I could find. They state:

The first report of a rotation rate close to $11$ h, based on a theoretical analysis of the planet's figure, was by Berstrand (1909). At that time, there were no measurements of $J_2$ and no reliable data on the oblateness. The crucial early work was the spectrographic determination of $10.8 \pm 0.3$ h by Lowell and Slipher (1912).

Based on this I would infer that any knowledge of the moments of inertia of Uranus and Neptune relied on hydrodynamical models of the planets' interiors, particularly the Darwin-Radau Approximation, which dates back to the 19th century.

Goody references a text, Alexander (1965), as having even more historical details. If I can track that down, I'll update this.

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If you do want to determine a planet's moment of inertia remotely without assumptions, you'll need to precisely measure its gravitational field.

Any gravitational field can be expressed as a sum of spherical harmonics. This was first worked out by Laplace in the late 18th century. The terms in this sum are often referred to as the multipole moments of mass. The first $\ell=0$ term is the mass monopole, which is proportional to the object's total mass. The $\ell=1$ dipole terms vanish for the right choice of coordinates. The $\ell=2$ quadrupole terms are related to the moments of inertia of the object. The expansion continues to higher and higher terms.

As you suggest, to map out the multipoles of an object you could use geodesy. Just watch something orbit the object in question and carefully map out the motion of the orbiter. If the orbiter follows a perfect elliptical orbit, then it is moving in a purely monopolar field of spherical object. If the orbiter deviates from a perfect ellipse, then there must be higher order corrections to describe the field. The first non-zero correction to a monopole is the quadrupole, which encodes the moment of inertia.

For an axially symmetric object of mass $M$ and equatorial radius $R$ the quadrupole term is: $$J_2 = \frac{I_z - I_x}{M\,R^2},$$ where $I_z$ is the moment of inertia about the symmetric axis (i.e. rotational), and $I_x=I_y$ is the moment of inertia about an axis perpendicular to the symmetric axis (i.e. polar). This vanishes for a spherically symmetric object, because $I_z=I_x$. To fully disentangle $I_z$ from $J_2$ you'd also need the oblateness.

Luckily Uranus and Neptune have moons, so you can do geodesy without sending a satellite flyby. The earliest reports of $J_2$ and oblateness I could find are from the 1970s (e.g., Cook 1972, Trauger et al 1978). This predates the Voyager 2 flyby in 1986.

These days we can do better than $\ell=2$. Jacobson (2014) used a wide range of historical and more recent data, including observations as far back as 1912 and the Voyager 2 flyby data, to determine many parameters of the Uranian system, including multipole moments up to $\ell=6$.

With dedicated orbiters we can do even better. For example the EGM2008 model for the Earth's gravitational field (based on data from the GRACE mission) goes up to $\ell=2159$.