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I have been looking into the orbital mechanics of tidally locked worlds lately and saw some interesting things. Namely that would-be tidally locked moons/planets can have rotation periods that differ from their revolution periods, a phenomenon that has to do with orbital resonance.

Case in point mercury would have a spin-orbit resonance of about 3:2, meaning that it rotates 3 times over the time needed for 2 revolutions.

I was wondering what would be the conditions needed for a satellite to have that kind of spin-orbit resonance to it's primary body, like for example a gas giant?

And more specifically what kind of conditions would be needed to have a spin-orbit resonance of 5:2?

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It's difficult to say what the precise conditions would need to be inasmuch as we'd need to know the details of the internal structure of the satellite, but we can say for sure that in general, a higher-order spin-orbit resonances requires higher eccentricities. For instance, in the case of TRAPPIST-1e, a 3:2 resonance would require $e\gtrsim0.1$ and a 5:2 resonance would require $e\gtrsim0.3$ (Renaud et al. 2020). For the case of Mercury, simulations indicate that a 3:2 resonance would need $e\gtrsim0.00026$ and a 5:2 resonance would need $e\gtrsim0.024877$ (Correia & Laskar 2009). Mercury's eccentricity is $e\simeq0.205$, well above the limit needed for the 3:2 spin-orbit resonance it occupies.

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  • $\begingroup$ What if I told you that it was an earth-like moon with about 0.7 to 0.9 times the mass of the earth, a slightly higher density due to a greater metallic percentage (5-10% as opposed to earth's 5%), a liquid molten core surrounding a solid iron core and with an oceanic coverage of (60 to 65%)? Does this change anything? $\endgroup$ Aug 15, 2021 at 21:23
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    $\begingroup$ @JuimyTheHyena I don't know - I think you'd need to run a numerical code to solve for the motion of the moon over long timescales and see which resonance it falls into. $\endgroup$
    – HDE 226868
    Aug 15, 2021 at 23:15

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