Over on Youtube I've been having fun with a troll who claims that rockets push on air and therefore cannot work in a vacuum. (I say 'troll' because I can't believe that he really thinks this way, though I suppose it's possible.)
I pointed out that achieving a stable, closed orbit around the earth requires rockets that work in vacuum because, in 2-body orbital mechanics, a satellite always returns to the point in space where it performed its last maneuver. Newton's Cannonball shows this very clearly. If that point is in the atmosphere, then the satellite will burn up after a single orbit.
But it got me thinking; is it possible to achieve a closed and reasonably stable earth orbit using 3-body mechanics? Say you could achieve TLI velocity with a cannon on the earth (let's ignore atmospheric drag and accuracy issues.) You put your projectile into an Apollo-like free return trajectory, but with the return perigee above the atmosphere. What would the resulting trajectory be, an earth escape trajectory or a closed, highly elliptical orbit?
I haven't simulated this because I don't have the tools, but my intuition tells me that it would be the latter. I say that because a TLI creates a highly elliptical but still closed orbit, and passing in front of the moon in its orbit in a free-return trajectory will only lose orbital angular momentum to the moon.
But it might not even be possible to achieve a free-return lunar trajectory with a single impulse within the atmosphere. The Apollo trajectory had a perigee just outside the atmosphere (at its parking orbit altitude) and you can't achieve exactly the same trajectory with a cannon. But it might not be necessary, either.
Another possibility is to aim behind the moon in its orbit, something like the final trajectories of the Apollo 8 and 10-12 S-IVB's. This usually achieves earth escape and a solar orbit, but Apollo 12's S-IVB was known to enter a high earth orbit at least for a time.
Any ideas?
