One idea to get to ramjet speeds might be to let gravity do the work.
First of all I get it: we wanted to accelerate at 32 m/s2 or even 96.8. Let's say we have dug out a little railway before the main tunnel. The main problem for Antarctica: it's not a vacuum-tube. Terminal-velocity applies.
This is root[2mg / ρAC]. A Dragon is A=4π. g=9.81 m/s2 down but we'll have this on an angle, so multiply that by cos or sin something. ρ=1.766 kg/m3 at 200K. C, the drag-coëff, is low for a ramjet say 0.1. So root[27.77 m cosφ / π]. (PS. Chromium doesn't do MathML.)
NASA's minimum mass was 100000 kg because why not. Let's just slide the thing down 45° so, "cos or sin something" is 1/√2. √625148 = 790 m/s.
We can tweak this or that: pile on more mass, tilt the ramp more downward, do something about the drag. The tweak which would help the most would be to lower ρ - to suck out more air beforehand. Um. Microwave the ice, let it re-freeze? Collapse and repeat for next time?
With the railgun it's simpler. The Mach number is a bit lower at Antarctic 200 K, but not so much lower; 1 km/s should be Mach 3, within Naval parameters (before they killed the project).
Next Question: whether it cost more to suck out the air (which up here isn't being replaced) or to run that railgun open to the surface. Keeping in mind a sight more tunneling is required, toward 1 km/s, if the acceleration thereto be 6.94 m/s2 (on that 45°) versus 32 m/s2.
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