# How much fuel is needed for a constant-acceleration rocket?

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.)

• This may be well answered by "Why are rockets so big?"/ Sep 19, 2014 at 16:46
• Are constant-acceleration trips around the solar system at all feasible - yes. You're on one right now. :P Sep 19, 2014 at 18:12
• @dmckee: not as far as I can see. I know how to calculate the delta-V for a rocket with a constant exhaust velocity and mass fraction. But I don't know the mass fraction or delta-V here, and I'm not even concerned with delta-V; I'm just trying to figure out if I can maintain a certain acceleration for a certain amount of time. Sep 19, 2014 at 19:49
• In practice one would keep up acceleration on the order of 1e-3g for months and years (or decades for interstellar flights) with electric rocket propulsion. ISP of 100,000 is possible with nuclear fragment propulsion: en.wikipedia.org/wiki/Fission-fragment_rocket Sep 19, 2014 at 19:51

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.

Literal tons. Tons and Tons of fuel. Even with an antimatter reaction, constant acceleration would take literal tons of material.

http://nathangeffen.webfactional.com/spacetravel/spacetravel.php

This calculator should give you the exact maths dependent on fuel conversion. Hydrogen fusion for instance is, 0.007%, which is widely efficient compared to all our current forms of power generation, but still not enough for practical 1g acceleration.

For example, if you want to travel for 1 lightyear in 1 ton ship, using fusion, you'd need a minimum of 864 tons of fuel. Even with a anti-matter drive, you'd need 6 tons of fuel, 6 times the mass of the ship.