# Can you explain the pattern of Hill sphere sizes of the objects of the Solar system?

I found this image on calculations of Hill sphere for planets/dwarf planets of the Solar system.

I found it interesting that variation of Hill sphere is intuitive for the first five planets, as the variation is similar to variation of mass/radius of those planets. Mercury has the smallest Hill sphere, Venus/Earth/Mars quite similar, and a giant leap from Mars to Jupiter.

But, Saturn has its Hill sphere bigger than Jupiter, even it is smaller than Jupiter. and this anomaly continues to Uranus and Neptune: They have progressively larger Hill spheres.

And Hill spheres of Pluto and Eris are quite larger than Mercury, Venus, Earth, and Mars.

This was quite surprising for me. Could someone explain why this—for the lack of a better word—anomalies are there?

Hill sphere is the region of space around a satellite where the satellite wins the gravitational tug-of-war with its primary.

If the mass of the primary object is $M$, mass of the satellite is $m$, semi-major axis of satellite is $a$, and eccentricity of the orbit of the satellite is $e$, then the radius $r$ of the Hill sphere for satellite is given by:

$$r \approx a (1-e) \sqrt[3]{\frac{m}{3 M}}$$

Note that this formula does not take into account the other objects in the vicinity.

The anomaly pointed out in the question is not really an anomaly. The contributing factor for the surprise values is the semi-major axis of the planets ($a$).

Take Jupiter and Saturn for example: Saturn has only around $30\%$ of the mass of Jupiter, and if the two gas giants had the same semi-major axis, this mass reduction will make Saturn's Hill sphere around $68\%$ than that of Jupiter. But Saturn is around $84\%$ farther from Sun than Jupiter. This is just enough to make Saturn's Hill sphere slightly larger than that of Jupiter.

Thinking along the same lines, we can also explain why Uranus, Neptune, Pluto, and Eris have surprisingly large Hill spheres.

The definition of Hill sphere is the region where the given object's gravity is dominant. In this area the object's gravity pulls more strongly than anything else; and everything else combined.

The primary competition for a planet is the sun. The further you get from the sun then the weaker its gravity is. This means it's easier for Neptune's gravity to exceed the sun's than it is for Jupiter's gravity. And it so happens that the masses are just right that Neptune has the larger Hill sphere.

If you stuck Jupiter further out in the solar system, then its Hill sphere would increase as a result.

• Just to be clear: Your answer is that it's because the planets are further out from the sun? Mine is that they're further away from each other. Both are (as far as I can see) valid. – HDE 226868 Sep 13 '14 at 23:34
• @HDE226868 The other planets do affect things, but not as much as the Sun. If a planet was a greater influence than the sun then we'd start orbiting the planet. But the planets don't do that. But they will complicate things. Stable orbits tend to be well within the Hill sphere due to radiation pressures and additional gravity sources perturbing orbits closer to the boundary. – zibadawa timmy Sep 13 '14 at 23:38
• But the moons of, say, Neptune, orbit Neptune, and not the Sun. – HDE 226868 Sep 13 '14 at 23:39
• @HDE226868 Yes, as they are in Neptune's Hill sphere, which is where Neptune gravitationally dominates. Those moons have their own Hill sphere, and their primary competition is Neptune (or whatever body they orbit). – zibadawa timmy Sep 13 '14 at 23:41
• And so my point is that the reason that Neptune's Hill sphere is so large is that there is no other primary competitor to Neptune. In fact, if you want to get into Hill-sphere-sception, we shouldn't be comparing the Sun's influence to that of the planets, because the Sun's Hill Sphere (relative to other stars) is the entire solar system. – HDE 226868 Sep 13 '14 at 23:43

I take it there isn't exactly a coincidence between you finding my question regarding geostationary orbits and you asking this question about Hill Spheres? :-)

The chart you found does seem counter-intuitive at first. But consider this chart:

(I wish I could add it here, but I can't manage to add it as an image, just as a link).

There's another pattern, and it has to do with distances from the Sun, and from nearby bodies. As you go out into the outer solar system, planets start to get farther apart. For example, Uranus is twice as far away as Saturn, separated by 10 AU at their closest, and Neptune is, at the closest position to Neptune, 10 AU away. That means that each planet is separated by a large margin from other planets, and Uranus and Neptune have virtually nothing else to contend with in the outer solar system, because they are so far away from any other massive bodies that could take over (i.e. Jupiter and/or Saturn).

Pluto, Ceres, and Eris are interesting cases. As far as I can tell, they have large Hill spheres because they are the largest of a collection of similar bodies. Ceres dominates the asteroid belt, and Pluto is so big it was once (in what now seems like ancient times) considered to be a planet. Eris, too, is quite large.

The only anomaly here is, actually, Pluto - and that's only for a [relatively] short portion of a time. It comes closer towards the Sun than Neptune for a portion of its orbit, which would seem to indicate that Neptune shortens Pluto's Hill sphere, but in reality, the two are rarely near where their orbits intersect at the same time.