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Hypothetically, suppose there is a situation where the Earth's moon gets neatly sliced into two equal hemispheres, and the force responsible for this slicing also creates a distance between the two halves of say, half the (average) distance from the Earth to the Moon, as it originally stood.

Do the two halves revolve around each other? Do they rotate individually? Do they each still revolve around the Earth?

EDIT: The angular momentum of each half after the division is the same as the angular momentum of the moon before the division occurred.

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Unless the force also creates an angular momentum between the two halves so that one orbits around the other, the two halves fall together again and make a really big explosion. – Peter Shor Jan 21 at 18:47
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As Peter Shor hints, this problem isn't well defined. To evolve the system, you need to specify two of (energy | position | velocity | angular momentum) for each object. You've given the initial positions, but it's not clear how fast everything is moving when you are all done rearranging things. – Chris White Jan 21 at 21:16
Noted. Will make the necessary changes. – chubbycantorset Jan 22 at 5:50
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Do you really mean that the angular momentum of each half is the same as the original, rather than the angular velocity? If it's the angular momentum then they will each have twice the angular velocity and they will either enter highly elliptical orbits or leave the Earth altogether (I think it's the latter), whereas if it's the angular velocity then they will be in orbits similar to the original one, and the answer is given by Peter Shor's comment. – Nathaniel Jan 22 at 6:00
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Also, you probably know this, but they wouldn't stay hemispherical for very long because rock isn't strong enough to prevent gravity from re-shaping them into more spherical shapes. – Nathaniel Jan 22 at 6:04
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In general two moons orbiting at the same distance is an unstable configuration, so unless you choose the initial positions and velocities of the two halves very carefully they will eventually either collide, or one or both will be ejected from orbit.

There are stable configurations for two moons e.g. the two satellites of Saturn, Janus and Epimetheus, share the same orbit, but this is a special case and is rare. Also I'm not sure if this configuration would still be stable if the bodies were as large as (half of) the Moon.

There are two points, the Lagrangian points L4 and L5, where two moons could form a stable orbit, however these points are unstable to small perturbations so moons in these positions wouldn't be stable for long.

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