Now comes time to supply NASA's ice gun's barrel with 2 bar of hydrogen. Yes yes, we're probably diluting it. Hydrogen is the cost-centre here so we're pessimistically assuming the worst.
The first stretch of distance will be a rail to 1 km/s. For reasons we "will" get to on the "Saturday" post we're on a railgun. This means acceleration for the 1000 km plan: 32 m/s2. However much power the rail needs I assume it can handle the journey to the hydrogen chamber without superconductors or vacuum or anything else this expensive. So we'll knock that off the total.
Follow the usual equations. First, we need the time on the rail: 32t = 1000 m/s, 31.25 seconds. Over that time 15.625 km would be crossed. The hydrogen tube would have the rest: 984.375 km.
If the brain-smushing 100 km option is used, 96.8 m/s2 would take only 10.33 s on the rail for 5.165 km. So 95 km of hydrogen.
As to how NASA push out a real atmosphere to replace it with another one; I assume a moving membrane like in a syringe. It helps that the new atmosphere is twice the pressure. The membrane will need to be knocked out of the way either at the start ("the breech") or at the end ("the muzzle").
The cargo is supposed to be humanoid but, this must be balanced against the need to clear Earth atmo. Propose the payload is a SpaceX Dragon. That is radius 2 meters. So 22 x 95000 x π volume = 1,193,805 cubic meters. Given hydrogen density at 200 K 0.141 that - 2 bar - would be 288.9 tonnes. It'll be a bit less because NASA wanted to mix it with oxygen.
As for oxygen, the ramjet carries more of this in its own tank. NASA hoped for less than 60% wet mass; this depends on how far you want to push final velocity - somewhat (if too low, we're carrying our own propellant anyway). Even 80% worst-case compares VERY favourably against the usual Tsiolkovsky (that jerk) requirements.
The main cost as noted is hydrogen, which does not simply exist in our air. Normally 200something tonnes of H2 (and O2) would not be overly expensive but, this is Antarctica. NASA wanted the gas produced in-situ using regular wind-power, such as generated by katabasis at the Dry Valleys. NASA in the 1990s implied direct electrolysis for "Green" hydrogen (and oxygen). If an organic gas trap is found near one end or the other of the tube, they can likely be talked into "Turquoise" pyrolysis, whose energy-needs are less; oxygen would then just be condensed from the air by the usual ways.
BACKDATE 5/15
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