I don't know...Ted Taylor was the one chosen to design the bombs. Based on his history that part would have been a lead-pipe cinch. The rest of it--well, as long as they avoided the giant-spring-looks-like-a-sperm interstellar version there was no Unobtanium in the design stuff I've seen. Now the test system NASA demanded before man-rating it? THAT would scare the shit out of anyone.
I'm sure they could explain the launch as an H-bomb test, perhaps of an exotic kind. Perhaps launch it over the Pacific during the day so that fewer people will witness the strangely regular flashes of light as it climbs to orbit.
The ship supposedly has politics. If it has politics, rather than vicious arguments about risk management and consumables accounting and people taking showers that use more than 4 litres of water, it presumably has margin. Which means it's freaking vast.
So it's quite possible that it's got a crew of ten thousand or so, or more; they're building manufacturing capability for new technology as they get instructions transmitted to them. Probably also cursing their local indium shortage.
If was going to try to make that work as narrative I'd be talking about how someone came up with direct conversion, matter to energy, but the minimum scale is enormous and the exhaust is the sort of thing you'd normally associate with the spin poles of a black hole. No one can figure out how to use it on Earth, Cold Warriors determined to preserve the species managed to launch this one ship which was just everything and everyone packed really tightly, they grabbed a big hunk of nickel-iron for radiation shielding and something with volatiles to stack behind that and are pushing the whole big mass while they make stuff they're going to need at the destination. (That's still some millions of tonnes in launch mass, it's completely crazy as a thing to do, but it's something like an explanation.)
Of course then I have to explain why the exhaust isn't making the amateur astronomy press twice a week, but that's a fairly small conspiracy as these things go. (Or it's so far into the gamma you can't see it without an orbiting telescope?)
...and then the big door opens and they see that painted on the outside is "VAULT 76", and then an old tape recorder starts up and a deep voice intones "War...war never changes."
But if it's a hoaxed ship, the self-contained environmental system doesn't have to be self contained. Just rig up some black-box technobabble water and air treatment system that amazingly never breaks down and only has to have the filters replaced once a year. They just don't see that the pipes go into the black box and make a right turn to the water treatment plant and air vents on the surface.
We can say the drive flare emerges in a near-perfect collimated beam, so it's hard to spot from Earth...and when the Russians took a second look at the energy output and realized the Giant Death Laser it made they promptly shat a brick.
I notice the starship is supposed to have launched in 1963. Give the Russians a few months to freak out and organize a response...the project must have been insanely classified, only known to a few American politicians at the very top...obviously the Russian protest was lodged on December 22nd, 1963.
Actually, dragging in an Area 51 or Roswell reference might make it less implausible. "Yeah, we've got a star drive. It'll move fifty million tons through space and get it to another star in a few decades. Thing is, we've only got the one..."
If it's got a life support system for hundreds of people, it's likely to have a truly vast number of pipes and valves and boxes and tubes and tanks... Stuff that doesn't break down can go un-messed-with and unnoticed pretty much forever.
The best place to hide a needle is in a stack of needles.
I strongly suspect that some mixture of wanting to make certain that the drive didn't blow up the ship or massively irradiate the crew with worry about the political implications of detonating dozens of nukes in orbit killed Project Orion. There's a non-zero chance that using Project Orion for a mission to Mars could have get of a nuclear war, especially after the SALT II treaty (1979), which prohibited orbital nukes.
I was thinking more "Gosh, these humans are likely to blow up their planet and go extinct in the wild. If we want to preserve a viable population, we better act now. But let's put them in an environment that makes sense to them, so they don't freak."
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