# What is the tightest orbit binary planets can orbit each other?

If there were two Earth-like planets in a tight orbit around each other, how close could they be to each other without colliding? How quickly would they have to orbit to be stable? Would they be habitable? Could something similar to a Roche lobe form from their atmospheres?

The orbital speed calculation looks a bit messy. Basically, the mass of the touching planets (that we need for the speed) is their density times the volume of their Roche lobe, which lacks analytical expression. However, one can approximate them fairly well as spheres of radius (see also Eggleton) $$r=0.38 R$$ where $$R$$ is their orbital distance. So we get $$M\approx 0.2298 \rho R^3.$$ Kepler's law gives us the period $$T^2 = \left(\frac{4\pi^2}{2GM}\right)R^3$$ (note the $$2GM$$ term rather than $$GM$$ - this is for a double pair), or $$T = \sqrt{\frac{2\pi^2}{ 0.2298 G \rho}}.$$ For Earth-density worlds I get $$T=4.2512$$ hours. Note that this is independent of the size of the system!