Let's assume for a moment the N-prize folks change their minds and require an orbit unsupported by anything except orbital velocity. How might this be achieved by means of the air tower? Rail gun?
10 grams is 154 grains, which is slightly larger than the rounds I carry in my 9mm. Those achieve a velocity of approximately 300 m/s. With an electromagnetic rail gun the size of a car one might
be able to propel a 10 gram bullet to the 100 km orbital velocityof 7,847 m/s. Just so people have a better feel for the mass, .45 ACP ammo falls squarely in the N-Prize weight range.
On the other hand, is there any requirement for there to be a payload? What if a balloon hauled a large, but gossamer-thin solar sail of 15 grams to, say, 50 km (164,000 ft). Would that be high enough for solar wind to accelerate it into space? Or is that still far too deep in the Earth's atmosphere for a solar sail to work? I suspect the latter, so what might be the absolute minimum altitude at which a solar sail with no payload but itself could be blown further away from the Earth's atmosphere?
Short of a balloon/rocket/X approach, where X is a solar sail or some sort of high-tech, super-lightweight railgun, I see no way of remaining in budget while achieving 9 orbits. And if it's any sort of gun, the rocket will have to take it to at least 100 km. Perhaps a balloon/rocket/railgun/solar sail, where the sail would work for final orbital insertion. With a good enough railrun we might be able to eliminate the rocket stage and just do a balloon/railgun/solar sail. I don't think an ion drive would withstand the massive g's produced by a railgun, and there remains of the issue of how to control the solar sail once it's deployed above the atmosphere.
Truly a gnarly problem!
First, is it even possible, or like the much smaller car dealership versions, will it simply flap all over the place? If it can be made to be reasonably stable, would it be enough to lift a small payload, such as packaged gossamer sail? I don't think deploying the sail would be difficult - simply package it in a rocket-spun hocky puck then deploy. If it's circular it'll just fan out and ill be gyro-stabilized to mainain both shape and orientation. How long would it take the thinnest of sails to accelerate at 100 km? If it's too long, then we'll have to haul up a small rocket to at least start it on it's way. We might discover the sail acceleration is so slight that a rocket would have to be used to get it, say, 90% of the way there, at which point why not use it to take us all the way there?