# Moment of inertia factor of Venus and Pluto

Wikipedia's Moment of inertia factor begins:

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.

The article lists values for 18 solar-system bodies including the Ceres and many moons.

But Venus and Pluto are conspicuously absent from this table.

What are the RI factor values for Venus and Pluto? Is there any source of reasonable calculation done for these two?

C/MR2

According to this note on the page Venus has a range of 0.327–0.342. Pluto is not mentioned anywhere.

• Interesting question! I've never heard of moment of inertia factor before. I've made some edits to the question to quote some of the referenced material so that readers don't necessarily have to leave this site just to understand the question. Feel free to edit further. – uhoh Jan 22 at 16:41
• This is NOT my field, but searching through my New Horizons papers, it looks like you need either a lot of assumptions or a degree-2 gravity model for the moment of inertia. Since New Horizons was a flyby mission, no degree-2 model could be constructed, so you can't calculate it for Pluto. For Venus, per the note, it looks like we need more close spacecraft flybys/orbits. Note that while that table lists 18 values, only FOUR of them are measured, the rest are modeled. If no one more knowledgeable writes something, I'll try to expand this into a real answer. – Stuart Robbins Jan 23 at 9:11
• @StuartRobbins I think that’s exactly right - I’d encourage you to write up an answer. I’ll note that we can be almost certain that the number for Pluto will be not too far below 0.4 (or 2/5, the value you get for a uniform-density sphere), since it’s quite unlikely that Pluto would be strongly differentiated. But of course it would be great to have a measurement of that! – Eric Jensen Jan 23 at 15:07

Measuring the moment of inertia of a planetary body is not straightforward and simple, but instead it requires a lot of data about how the body rotates, models for how mass is distributed across the body, and the gravity field of the body (these are related things, i.e. the gravity field can tell you how mass is distributed). We have so many satellites in orbit of Earth and so much ground research that we know this well enough to measure the moment of inertia of our own planet. The same is reasonably true for the Moon, and more recently for Mars.

The MESSENGER spacecraft had a highly elliptical orbit around Mercury for several years and so was also able to provide data to reasonably estimate Mercury's moment of inertia, too, but to far less precision: Based on the Wikipedia article from the question, the Moon's moment of inertia has a relative uncertainty of ±0.2%, while Mercury's is ±4% (for completeness, Mars is ±0.1%). Mars' is better known, despite having had fewer orbiting spacecraft, because it has two moons that help.

Reading through the paper on Mars' moment of inertia gives some idea of everything you have to do to really get at this value: Love number, second-degree gravity field, correct things for atmospheric drag (including tides within the atmosphere), account for orbital changes in the moons, map the precession rate, measure the solar torque since Mars is oblate, look at all of Mars' orbital parameters (eccentricity, obliquity, mean motion, spin rate, precession) ... it's a lot. Once they had all that, with their associated uncertainties, they could calculate the moment of inertia. But, they suggested this could then be modified based on mass asymmetries across Mars, such as the giant Tharsis volcanic region providing a load on that side of the planet, which could tweak one of the terms that goes into the calculation and change the moment of inertia from 0.3644 to 0.3652.

I didn't provide links to what all the terms mean in the above paragraph (someone can edit if they wish), the real point is that we need to know a lot to actually measure the moment of inertia of a planetary body, and we simply don't have those for most bodies in the solar system. The Wikipedia article that's linked in the question does list values for 17 (not 18) bodies, noting that Venus is unknown (hence 17 bodies), but then the other 13 are listed as "not measured" but rather calculated through other means.

For example, several list the Darwin-Radau equation method, which relates the body's shape and spin to moment of inertia. Since we can relatively easily see those, we can guess at the moment of inertia via those means, but it makes a lot of assumptions about the interior structure. Since the interior structure of Venus is likely more complicated than something like Io, whomever wrote the Wikipedia article chose not to list them in the table, though two references are given with values of 0.327–0.342.

The internal mass distribution of Pluto is also quite complicated given what we know about Sputnik basin and the glacier within it. We also don't even have a full shape model for Pluto or Charon because of the lighting geometry during the New Horizons flyby in 2015, where south of about 45° was in permanent night for this part of Pluto's orbit (same with Charon). Without even the shape, you can't use the Darwin-Radau equation to estimate the moment of inertia, though based on limb profiles no flattening was detected so I suppose some estimates could be made. But, like the Venus estimates, how accurate those would be is debatable might not be at all useful.

Keep in mind that the moment of inertia for a perfectly uniform, solid sphere is 0.4, so a planetary body is going to be less than that (unless it's perfectly uniform and solid). Values less than that indicate the mass is more concentrated toward the center, but even with Earth's large metallic core, the value is still 0.3307, which is not hugely different from 0.4 when you're trying to precisely measure all of these different values that go into the calculation. So, if you calculate something like 0.35±0.05 for Pluto (to make up numbers) based on your limited data, is that really useful? Maybe. Or, maybe not.

• This is a beautiful answer! I think it was my edit to the OP's question that added the "18". – uhoh Jan 23 at 23:24
• Thanks for the efforts and time in detailing the explanation. Is there any reliable RI factor value for Venus for practical use? – Majoris Jan 24 at 4:17
• @Majoris As I noted in my comment on your question, this really is not my field, so I have very little to go on to say if there's any reliable moment of inertia value for Venus. I see no reason why the values given in the "Note 4" on the Wikipedia page aren't reasonable (they make sense, bracketing Earth's value), so I would go with those unless someone has a reason to not. – Stuart Robbins Jan 24 at 19:25