I would like a earth-moon system like Pluto-Charon (but with earth instead of Pluto) where both will be tidally locked to each other within something like a billion years. As the moon would probably lock to earth before the opposite how big the moon has to be?
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$\begingroup$ According to a theory proposed by Arpita Roy, Jason T. Wright, et. al., the Earth's moon tidally locked in ~100 days, considerably less than a billion years. $\endgroup$– BillDOeAug 10, 2018 at 20:05
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$\begingroup$ semanticscholar.org/paper/…. They post a formula for calculating the time it takes to tidally lock a satellite. $\endgroup$– BillDOeAug 10, 2018 at 20:12
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1 Answer
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There's a lot of variables for the timing, but according to the formula from Wikipedia, the time ratio is inversely proportional to the square of the mass.
If our Moon will tidally lock the Earth to it in 50 billion years, a moon 7 times the mass should tidally lock the Earth, everything else being equal, about 7^2 or 49 times faster. About 1 billion years.
There's numerous variables, such as how large the oceans are on your planet, the density, the radius, how long the surface stays molten after the impact, the initial rotation rate and initial distance. (that one is proportional to the 6th power).
If somebody wants to give a better and more numerical explanation, please feel free, but this should get you started.
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1$\begingroup$ Tidal lock also depends on asymmetric density of the planet/moon in question - the more asymmetry, the easier to lock. $\endgroup$ Aug 10, 2018 at 13:22