### Hill stability

(planetary orbits which will remain in the same order)

**Hill stability** is the stability of a planetary system
such that the planetary
orbits will remain as is, i.e., the innermost orbit
will remain innermost. It does not exclude the
possibility of the outermost planet being ejected.

A sufficient-but-not-necessary
criterion is the following inequality regarding
adjacent (coplanar) planets and the star:

2M
- ——————— c^{2}h
G^{2}M_{*}^{3}
> 1 +
m_{1}m_{2}
3^{4/3} —————————————————
m_{3}^{2/3}(m_{1}+m_{2})^{4/3}
+ ...

- M - total mass of the system
- m
_{1} - mass of inner planet
- m
_{2} - mass of outer planet
- m
_{3} - mass of star
- M
_{*} - m_{1}m_{2} + m_{1}m_{3} + m_{2}m_{3}
- G - gravitation constant
- c - total angular momentum of the system
- h - the energy

(*orbits,celestial mechanics*)
http://faculty.washington.edu/rkb9/publications/bg08a.pdf

https://arxiv.org/abs/1210.0321

index