My son and I have been discussing the concept of a constant-acceleration rocket, as described here and here. We're willing to assume advanced technology such as a fusion rocket, which, according to some sources, could someday provide a specific impulse $I_{sp}$ in the neighborhood of 100,000 s. We're trying to figure out if constant-acceleration trips around the solar system are at all feasible with this sort of technology.
So our question is this: assuming such an $I_{sp}$, what's the ratio of fuel to payload we would need to keep up an acceleration of 1G for a few days? What if we settled for 0.5G?
(Note that the second reference provides some handy equations and examples for a rocket converting mass to radiation with 100% efficiency, but I don't understand how to generalize that to a more realistic exhaust velocity.)
Best Answer
Not a full answer, but too long for a comment. Maybe this can lead you the right way.
I'm going to ignore the changing mass for a moment and assume a ship with a fuel fraction of $0.5$. If we could get enough thrust from the engines to give a $1g$ thrust to the fuel, we could give a $0.5g$ thrust to the ship.
If you have an engine with $I_{sp} = x$, and a thrust $F$ can burn for $x$ time with a quantity of fuel that has a weight of $F$.
This means engines at that power would run for $100000s$, or just over a day.
Now the nice part is things get better from there. You'd either maintain thrust and increase acceleration to $1g$ as the fuel is exhausted in a day, or you'd throttle down to maintain $0.5g$ and the fuel would last longer. Given that, I assume there's a nice log equation to show the exact relationship between fuel fraction and burn time.