this post was submitted on 08 Jul 2023
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Worldbuilding
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You might be mixing up a few things, since there's a bit here that doesn't make sense.
Gravity has very little to do with orbital or rotational period, and a lot to do with the mass of the planets. (You are right the planet would be slightly more spherical, though.)
This may also depend heavily on atmospheric composition, density, and eccentricity of orbit more than just the binary arrangement.
One other topic I didn't see brought up: Are the planets and their orbit arranged planar to the star, such that they will regularly eclipse each other? Or are they tilted so that they'll never actually intersect?