Hill stability

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 an adjacent pair of (coplanar) planets and their host star:

    2M
- ——————— c2h
   G2M*3

> 1 +

          m1m2
34/3 —————————————————
     m32/3(m1+m2)4/3

+ ...

(These are the most significant terms of a series.)

• M - total mass of the system.
• m₁ - mass of inner planet.
• m₂ - mass of outer planet.
• m₃ - mass of star.
• M_* - m₁m₂ + m₁m₃ + m₂m₃
• G - gravitational constant.
• c - total angular momentum of the system.
• h - total energy (kinetic energy + potential energy, the latter of which is a negative number).