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

Question

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?

Answer 1

One can certainly imagine double planet "Rocheworlds" (like the one in Robert Forward's eponymous novel) where two terrestrial worlds fill their Roche lobes until they touch.

On its own this is perfectly fine: there are indeed a fair number of asteroids and trans-Neptunian objects forming tight pairs or a merged dumbbell shaped contact binary. The planets would be tidally locked to each other, not dissipating energy through tides.

The problem is that when the rest of the solar system is taken into account tidal dissipation begins to matter. The tides will cause the planets to move a bit relative to each other, and strong tidal dissipation will dampen that motion - dragging them inwards. Hence, relatively quickly since the tidal dissipation would be strong in big and soft terrestrials, the double planet would merge.

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!

Can systems like this form? Probably. Can they remain long enough that life emerges on the planets and they become Earth-like? Hard to tell, but I suspect the answer is something like "yes, if they start sufficiently far away from each other then they will spiral in slowly enough that there is a biosphere when things get really tight." I suspect such coincidences in time and formation are rare. But the universe is large.