We know that the Sun gets so big billions of years later and the Earth will get destroyed.
What will the orbital period of Jupiter (Sun-Jupiter system) and its moon Europa (Jupiter-Europa system) be when the habitable zone reaches Europa? Also, what will the side-real rotation period be for the two objects?
Is there a way to calculate them?
When the Sun becomes a red giant there will be some mass loss, making Jupiter's orbit wider and slower. The extent is fairly small during the initial expansion, but about 50% of the mass at the end. Simulations suggest Jupiter does end up with about twice the current semimajor axis, and hence up to 4 times the period it has now ($P\propto \sqrt{a^3/M_\odot}$).
When the habitable zone passes the Jupiter region depends a bit on stellar model, but it is in about 5 gigayears. This is a long time compared to the slow evolution of the orbits.
Currently the inner Galilean moons are in a 1:2:4 orbital resonance (the Laplace resonance). However, there is a lot of tidal dissipation in Io, making the system slowly shift. (Lari, Sailenfest & Fenucci 2020) model the future evolution of the moons. During the first billion years the periods go up a small amount as the moons become a bit more distant but retain their resonance. However, after 1.4 Gyr, Ganymede approaches the 2:1 mean-motion resonance with Callisto and chaotic effects show up. At this point various scenarios of resonance can play out, but practically for this question the effects are minor: the moons keep on slowly expanding their orbits.
So when Europa becomes warm, it will likely have an orbit maybe 11 Juiter radii out rather than 10, with a period of about 100 hours rather than 85. Not too diifferent.
