### 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?

8 years ago

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.