Hill stability

Hill stability of a planetary system means the planetary orbits will remain with the innermost orbit remaining innermost, and with no pair of the planets change switches places in the order of the planets from innermost to outermost. However Hill stability does allow the possibility of an 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).