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the best heat exchangers i can think of would be to use a water to water, ie have the radiators submerged in water, have this 'bath' water take have values that regulate the relese of water to the vacum of space, the relesed water would evaporate and absorb energy (heat) from the tank. this is the same method of cooling that was used in the spacesuits used by appollo astronuts.
I was actually hoping compressed or liquified air would do the trick... since we'd be using a lot of it and throwing it away through the exhaust, anyway (how much extra exhaust scavenging do you think we can get when venting to vacuum? ). The main worry is how cold is cold enough? We'd have too much cooling unless we regulated the amount of liquid air getting into the heat exchangers.
the oscilation motion shoudl be engineered out with the use of dashpots (shock absorbers)
Hmmm... add a few weeks of testing to the schedule... Branson can!
there is no sound in space (not strickly true but it is in this case)
George Lucas didn't get rich from making silent movies... I think artistic license can be used in the filming of the attempt. Or better yet... use microphones mounted on the road surface to pick up the vibrations of the car as it passes.