What constraints are there on the orbits of moons in a Laplace resonance?
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I'm interested in giving a conworld moons orbiting in a Laplace resonance, like the inner Galilean moons of Jupiter.
Here are the constraints I believe hold for moons in a Laplace resonance (orbiting prograde, because I'm not sure if retrograde works):
- The semi-major axes must be within some fraction of the planet's Hill radius. (0.6, I think? Also, I could believe that the resonance would stabilize a farther-out orbit, but I didn't know, so I didn't rely on that idea.)
- The sidereal orbital periods must be nearly in a 1:2:4 ratio
- The deficit is made up by the rate of precession of the periapses of the inner moons, which fixes their annular period, but not that of the outer moon. (I think it would be constrained in a de Sitter resonance, which is similar but not the same.)
- Despite the importance of the apsidal precession, the orbits have low eccentricity.
It would be helpful to know if I'm wrong about any of that, but mostly I wanted to establish the stuff I am sort of confident about, as a prelude to the stuff that I'm not at all sure of.
When it came to the rate of nodal precession, I found some formulas that gave nonsense values for the moon, so I assume that the nodal month for a single moon has to be taken on faith from the worldbuilding. However, in the context of a Laplace resonance:
- Is there any relation between the moons in terms of longitude of the ascending nodes?
- In terms of the rate of precession of the longitude of the ascending nodes?
- In terms of the orbital inclinations?
- In terms of the librations of the orbital inclinations?
- Can I actually ignore the eccentricity as "small", and, if not, are there any resonance couplings I should know about?
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