Rotation-Orbit resonance

AFAIK, all the tidally locked bodies in the solar system have rotation-to-revolution ratios of either 3:2 (Mercury) or 1:1 (everything else). Mercury's 3:2 ratio is made possible by its high eccentricity (~0.2).

Is there any idea what ratios are possible, and what relationship there is between eccentricity and ratio? For example, is a 4:3 resonance possible, and, if so, at approximately what eccentricity range would it occur?

I honestly wasn't aware that there was any link between eccentricity and resonance. All of Jupiters moons are nearly in resonance, yet their orbits are pretty well circular.

AFAIK, eccentricity doesn't link with resonance, other than making resonances other than 1:1 possible. Jupiters moons are in 1:1 resonance because of their low eccentricity.

Pfhreak wrote:

Jupiters moons are in 1:1 resonance because of their low eccentricity.

Hang on, what do you mean by resonance? Usually it implies that an orbital periods is a factor of another. (or at least... sort of).

Callisto takes almost twice as long as Ganymede to orbit Jupiter.
Ganymede takes twice as long as Europa to orbit Jupiter.
Europa takes twice as long as Io to orbit Jupiter.

So we say that Europa and Io are in an 1:2 resonance.
Or we can say that Ganymede and Europa are in a 1:2 resonance.
Etcetera.

The Jovian moons are in a
1:2:4:8 resonance, for Io, Europa, Ganymede, and Callisto, respectively.

Sirius_Alpha wrote:

Hang on, what do you mean by resonance?

Rotation-to-revolution ratios in simple integer ratios — 1:1 and 3:2 been the known cases in the Solar System. Something like the orbital resonances with Jupiter that produce the Kirkwood gaps, but with the planet's rotation.

Most tidally-locked bodies in the Solar System have rotation-to-revolution resonances of 1:1 — their rotational periods and orbital periods are (excepting libration) the same. Mercury is the exception: it rotates three times for every two orbits, allegedly because of its high eccentricity. I haven't been able to find anything indicating what other rotation-to-revolution resonances are thought to be possible. Presumably, an eccentricity less than Mercury's, but still high, could produce a resonance between 1:1 and 3:2 — 4:3, for example.

Wikipedia says that Mercury's eccentricity may be responsible for keeping the 2:3 resonance stable. I wish we had more short-period planets with known rotation periods to study =(.

Sirius_Alpha wrote:

Wikipedia says that Mercury's eccentricity may be responsible for keeping the 2:3 resonance stable. I wish we had more short-period planets with known rotation periods to study =(.

The topic is discussed in arXiv:1110.2658