It's an interesting question. The more fuel you send up there, the longer it'll last, but also the more mass you'll be wasting by constantly having to carry it around. I suspect that it would not be practical to bring along enough nuclear fuel to make many Earth->Mars round-trips on every one of those round trips.
Land-based nuclear power plants, for what it's worth, are refueled every ~1.5 years. Nuclear-powered aircraft carriers go two decades, but they're not concerned about carrying around lots of extra mass.
Practically speaking I don't think you could refuel a nuclear-powered spacecraft in space. This is a very hard operation to do and everything involved is extremely radioactive. I don't think it could reasonably be done, either by robots or by people. Not until we have entire shipyards in space like we do on Earth, anyway. It takes 3 years to refuel a nuclear-powered aircraft carrier, that's how complicated the reactor tear-down process is (and you do need to tear the reactor down to replace the fuel).
IIRC, Naval reactors usually use Uranium enriched to much higher levels that civilian nuclear plants, both to get higher power output for smaller reactor size, and to not need refueling as often. Reactors designed to operate in space may do the same sort of thing. I wonder what other kind of optimizations and/or workarounds might be required for a real space-capable reactor.
Whatever it is, it's going to be insanely expensive. That's just guaranteed.
Honestly it's probably just better to use a larger conventional rocket. Yes in theory the nuclear engine might be more mass-efficient, but it won't be more cost-efficient, and that's really what engineering optimizes for.
This is something SpaceX is proving to be really good at that others, e.g. Robert Zubrin, just don't get. He keeps proposing alternatives to the Starship design that, while more physically efficient, would actually cost a lot more money overall to do the same job.
It certainly would be. As cool as the tech sounds, I do feel more inclined to let Elon Musk and those types see how far they can run with conventional space tech and some non-nuclear efficiency optimization work.
Being able to reliably land chemical rocket first stages for reuse probably gained us a lot more than any realistic nuclear rocket design, at least as far as getting things from Earth surface into orbit.
I'd be curious to see a side by side comparison. It might make sense to use Starship for humans and initial supplies and a nuclear thermal design for much heavier loads that would require many back and forth runs by SpaceX Starship rockets.
Or maybe the reverse is true.
Chemical rockets (trad est): 8-9 months to Mars
Chemical rockets (SpaceX est) : 6 months to Mars
NERVA nuclear thermal rockets (NASA est): 3-4 months to Mars
If compressing the time to Mars becomes the more important mission parameter due concerns over radiation, 0-gravity effects on humans or just something going wrong. Then perhaps Nuclear Thermal are the only choice for people and chemical rocket makes sense for supplies.
I'd love to see any research people have done comparing these two options and seeing which one works best.
Land-based nuclear power plants, for what it's worth, are refueled every ~1.5 years. Nuclear-powered aircraft carriers go two decades, but they're not concerned about carrying around lots of extra mass.
Practically speaking I don't think you could refuel a nuclear-powered spacecraft in space. This is a very hard operation to do and everything involved is extremely radioactive. I don't think it could reasonably be done, either by robots or by people. Not until we have entire shipyards in space like we do on Earth, anyway. It takes 3 years to refuel a nuclear-powered aircraft carrier, that's how complicated the reactor tear-down process is (and you do need to tear the reactor down to replace the fuel).