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A small moon is able to maintain a stable orbit around a planet at just under half the radius of the planet's Hill sphere. Most of Jupiter's moons are in this configuration, possibly made more stable by their retrograde orbits.

If instead we have a binary pair of planets, where each have the same mass as the planet in the former case, how can the maximum stable separation between the pair be determined? Just to clarify, I'm not interested in any moon orbiting the binary planet, I want to quantify the stability of the binary system itself.

My intuition would be that the deeper gravity well increases the range for a stable orbit, but there might be some inherent instability involved in the resulting non-spherical gravity well.

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    $\begingroup$ You might want to check out the Pluto-Charon system. IIRC, the difference in tidal forces when both bodies are aligned can vary up to 20%. Despite this, the orbits of all the system's (five!) moons occupy less than 3% of the Hill sphere. $\endgroup$
    – BMF
    Commented Apr 16 at 13:57
  • $\begingroup$ As long as planetary mass is negligible compared to star mass, mass of the planet makes almost no difference to Hill Sphere size. worldbuilding.stackexchange.com/questions/175057/… $\endgroup$
    – notovny
    Commented Apr 17 at 13:07
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    $\begingroup$ @notovny Funnily enough I had derived the √3 relationship between those periods myself just before posting this question. In practise, only orbits up to half the hill radius are stable, so √3 becomes √24 if you run the numbers (T_M ≈ 0.2T_P). This still doesn't answer whether a binary planet is more stable than a moon though, since the hill sphere doesn't make sense to apply in this case. $\endgroup$
    – Charles Rogers
    Commented Apr 17 at 14:08

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Hill spheres define the area surrounding a celestial body in which it can exert gravitational dominance over the orbit of a satellite. In the case of binary planets, where two planets revolve around a common center of mass (barycenter), the concept of Hill spheres becomes more intricate due to the gravitational effects of both planets. Each planet in a binary system possesses its own Hill sphere, which is determined by its mass, the mass of its companion, and their separation distance. The dimensions of each Hill sphere are influenced by these factors: closer planets have smaller, overlapping Hill spheres, while farther planets have larger, distinct Hill spheres.

Satellites can maintain stable orbits within the Hill sphere of each planet if they are sufficiently distant from the gravitational influence of the other planet. When the Hill spheres overlap, a combined region of stability can emerge around the barycenter. Moreover, binary planets possess Lagrange points where the gravitational forces and orbital motion balance each other, providing relatively stable positions for satellites. Consequently, Hill spheres in binary planet systems create intricate yet predictable areas for stable satellite orbits, which are influenced by the masses of the planets and the distance between them.

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