Can a planet be in spin-orbit resonance with its parent and in resonance with other planets at the same time?

Question

Mercury is tidally locked in a 3:2 spin-orbit resonance with the sun. Other resonances like 1:1, 2:1, 5:2 can occur as well. Many objects, like the moons of Jupiter, are locked in mean-motion resonance, meaning that each object orbits an integer amount of orbits for each integer amount of orbits its neighbors complete. This means that the system is very stable and will return its initial state eventually.

I was wondering if a member of a resonance chain could be locked into a higher order, non 1:1 spin-orbit resonance with it parent, while still being in mean-moton resonance with the other planets?

Specifically, I was wondering if this roughly drafted system is realistic?

• Valhalla A: early F-type star
• Valhalla b: super-earth, lava world with sodium atmosphere
• Valhalla c: super-earth, lava world with frozen nightside
• Valhalla d: ice-giant, helium dominated atmosphere due to proximity to the sun
• Valhalla e: super-earth, carbon-planet that formed due to a variation in the composition of the protoplanetary disc
• Valhalla f: gas-giant in the habitable zone, almost 9 $Mj$

Valhalla b to e are really tightly grouped close to the star in mean-motion resonance. All of them are locked in a 1:1 spin-orbit resonance, except for Valhalla e, which is in a Mercury like 3:2 spin-orbit resonance. I was wondering if Valhalla f, could explain this state of the inner system. It has herded the planets together so close to the star and interacts with Valhalla e so that it is more eccentric than the other planets. Is that a believable and stable setup?

The structure of the system is supposed to be similar to the innermost system of Kepler 90.

Answer 1

The answer to your first question - Can a planet be in spin-orbit resonance with its parent and in resonance with other planets at the same time? - is yes.

It is not uncommon for planets or moons to be in orbital resonance with each other. The Jovian moons Ganymede and Europa are in 4:1 and 2:1 resonance relative to Io, and Pluto and Neptune are in 3:2 orbital resonance. Some extrasolar planets also display orbital resonance.

Now, if two planets in orbital resonance both are in spin-orbit resonance with their parent sun, they will also be in resonance with each other. Say that two planets are in 2:1 orbital resonance, and the innermost planet is in 3:2 (1.5:1) spin-orbit resonance, while the outermost are in 3:1 spin-orbit resonance, their rotations will be in 1:1 resonance with each other: 2*1.5:3 Every time the inner planet completes 2 orbits, it rotates 3 times - and in the same time, the outer planet completes one orbit, rotating 3 times. Hence, their rotational periods are the same.

However, I don't think that the system you describe is possible. F-stars are 1-1.4 solar masses, and from the mass–luminosity relation, we hence get that their luminosity is at most 3.8 times that of our Sun. This would make the habitable zone reach out about twice as far as in our solar system (since light and heat diminishes with the square of the distance). Most estimates of the outer edge of our habitable zone are within 2 astronomical units (AU), which would make it 4 AU for your system. I doubt that a superjupiter of 9 Jovian masses in such an orbit would allow stable planets except very close to the sun. Jupiter's presence, at 5.2 AU from our sun, has prevented a planet forming where the Asteroid Belt is. Mars orbits at 1.5 AU and could probably not be much further out (perhaps a maximum of 2 AU). At a rough estimate, I doubt any inner planet in your system could be stable farther out than around ½ AU, which would make it blistering hot, given the higher lumonosity of the sun. A planet at this distance would receive about three times the heat that Mercury gets. If the star's luminosity is less, then the outer edge of the habitable zone will also be less, bringing your superjupiter even closer to the sun, possibly making any stable inner orbit impossible.

One caveat to this is that planets in orbital resonance will make closer orbits more stable. However, five stable planets within your superjupiter is stretching things quite a bit.