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Notes -
I made the point about exhaust velocities mostly wrt fission engines using thermal gas as a propellant. If you have a fission reactor as an energy source, getting a heating your propellant to a much higher temperature than your fuel elements seems challenging.
The length of your brachistochrone would depend on your available acceleration. If you have unlimited thrust, the fastest path is a straight line (relativity aside). If your thrust is very limited, I would expect that you will spend a lot of time orbiting the Earth while prograding until you escape it eventually, and then you will spend a lot of time circling the sun until your intercept.
However, I agree that 10mm/s^2 is still a usable amount of thrust within the solar system. The Dawn spacecraft got around with much weaker engines, but it definitely increased the transition time.
I am more skeptical about fission fragment engines. Sure, the exhaust velocity -- a few percent of c -- is amazing. But for every fragment which escapes and generates thrust, another one (or three) will hit your spacecraft. Because energy scales with v^2, if you want a decent thrust, that will mean an ungodly amount of energy. 1kN times 0.01c is something like a few Gigawatts of thermal power, similar to what a large commercial nuclear reactor might have. Cooling this away in space would be challenging. And if you add a reaction gas to get more momentum per energy (at lower exhaust velocities), you still have to confine that gas magnetically, which also adds overhead.
Right. So you run open-cycle. You mix your fuel with your propellant, stick it into your nozzle, and burn it at plasma temperatures (remember, while terrestrial nuclear reactors burn up their fuel over the course of years, this is not actually required; a nuclear bomb burns its fuel to reasonable completion inside a microsecond). You will need cooling systems for the nozzle, of course, but the temperature gradients are all in the right direction.
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