# Mass to fuel ratio of a hypothetical spaceship propelled using a helium-3 reactor

Hey there. I’m trying to get a good idea of said mass to fuel ratio. Is the energy that can be extracted from Helium-3 in a nuclear reactor much higher than that of the oil-based fuels with the same mass that we use on Earth? Assuming that the ship has to be able to accelerate at 1.5g for 1 year and ignoring all other energy needs of the crew. Since we’re talking about a ratio here I guess we don’t need the ship’s mass but if you do it is 10,000,000 kg. I am not saying that it is constantly accelerating, it needs to have enough fuel to be able to do so, though.

How much mass in helium-3 is needed to do that assuming 100% conversion of energy into propulsion?

With my limited understanding I have made this calculation in the case of hydrogen atom antimatter annihilation, please tell me if I am correct and if not please say what I did wrong or provide the correct calculation.

Thanks a lot for helping.

You need to look up the mass of the reaction pieces -

The energy produced is the difference between the mass of 2 x He3 and 1 x He + P; that tiny amount of mass converted to energy is recovered.

The logical question is “then what?”? The practical trick is converting energy output into a usable form.

If you’re using a fusion reactor to create heat, and then using that heat to power a rocket motor, you are still limited by the amount of reaction mass you have on board. Very soon you’re going to run out of reaction mass even though your fusion reactor could keep running for decades.

I don’t have time to check the OP’s math in detail, but several years ago, I did math to figure out fusion energy yields for spaceship propulsion for ships like Star Trek’s Enterprise or Star Wars Star Destroyers. My numbers showed that they could easily fuse tons of hydrogen per second and that the biggest bottleneck in believability wasn’t the fuel/energy requirements, but how they deal with all that waste heat in a vacuum. (After all, waste heat is going to be there no matter how energy dense your fuel source is.)

Not necessarily. In a sci-fi scenario where He3 fusion or antimatter annhilation is even a possibility, we’re going to have options other than rockets. Ion engines get a lot more energy out of reaction mass than a rocket motor does, and an ion engine could use the waste products of the fusion reaction as the ions in the reaction mass, getting double-use out of them. There’s also the possibility of using some kind of magnetic scoop to pick up interstellar gas and accelerate that with the ion engine so that you don’t have to carry the reaction mass with you. (In fact, since most of what you scoop up is Hydrogen, some people theorize propulsion in which the scoop provides both the nuclear fuel and the ions. This is probably not practical, though.)

I think the relativistic rocket page on Wiki will be of use. Specifically, for the reaction 2 [sup]3[/sup]He -> [sup]4[/sup]He + 2 p, we have eta = 0.002471 and so a specific impulse of about 0.0496c.

Beyond that, you’ll have to be a little more specific. Do you mean 1 year of ship-board time, or 1 year of Earth time? Because they’re not the same thing, due to relativity. And you do have to take relativity into account; if you use the naive Newtonian form v = a*t, you find that after 1 year of acceleration at 1.5 g you’ll be moving at 155% the speed of light, which is impossible. (Your hydrogen-antihydrogen calculation is wrong for the same reason, I’m sorry to say.)

http://www.5596.org/cgi-bin/thrusters.php

other calculators can be found here
http://www.projectrho.com/public_html/rocket/calculators.php

Although ion engines are not thermal rockets (that is, they do not heat a fluid inside of a chamber and use directed release of the resulting pressure to change the momentum of the vehicle) they still use rocket propulsion (i.e. they carry the propellant mass) and obey the rocket equation. Although ion engines have a high exhaust velocity and corresponding improvement in propellant mass efficiency (typically measured as specific impulse) they tend to be massively energy inefficient and transfer the bulk of energy put into them into heat.

Scooping gas out of the interstellar medium–in essence, making a interstellar jet engine–does address the problem of having to carry propellant but comes with its own set of challenges insofar as you have to be going fast enough to collect sufficient propellant to get useful thrust (~0.01 to 0.02 c) and will result in drag on your collector which limits the upper end of speed (around 0.12c for a Bussard-type ramjet), so you cannot just accelerate indefinitely.

That is just a horribly written article, and not strictly relevant to the question of the o.p.

To answer the o.p., the question requires more than just the theoretical power output of the D-[SUP]3[/SUP]He or [SUP]3[/SUP]He-[SUP]3[/SUP]He reaction; we would actually need to know details about how the system converts the output of the fusion reaction into pressure (thermalization) or voltage potential (induction). There are almost certain to be enormous losses and resultant waste heat, which is a significant problem for a spacecraft with limited radiative surface and internal low temperature reservoirs. (Even for low powered satellites and space probes that we currently operate which do not have a high temperature propulsion system, internal heating is a significant problem and limitation.) The greatest mass efficiency comes from obtaining the highest exhaust velocity from the expelled propellant. The greatest energy efficiency comes from converting as much of the imparted energy as possible into momentum (directed energy) and minimizing heat (randomized energy). This is the reason chemical rockets have large nozzles at the end; so that the expended gas (plume) can expand and cool, imparting a small but useful fraction of energy to the nozzle and thus to the rocket.

Given even very high temperatures (~10[SUP]8[SUP] K) for a thermal rocket using hydrogen as the propellant, the I[SUB]sp[/SUB] comes out to be around 10,000 to 20,000 seconds, which gives a payload to propellant mass ratio in the millions or billions to accelerate to ~0.01c. An I[SUB]sp[/SUB] of around 80,000 s, giving a mass ratio of 45:1, is about the threshold to be able to attain this speed, although if you wanted to decelerate the ratio is then squared, giving a final ratio of greater than 2000:1. So, even nuclear fusion doesn’t provide temperatures sufficient to attain sufficient exhaust velocity and specific impulse for interstellar transit in a human lifetime.

I didn’t go through the calculations done by the o.p. in detail, but I will note that the energy from fusion is not direct conversion of the mass of the hydrogen (or helium in the case of [SUP]3[SUP]He) but the nuclear binding energy that is released in the fusion process.

Regarding [SUP]3[SUP]He fusion, this reaction offers about two orders of magnitude less specific power output than the D-T reaction at a triple product condition that is 200 times less than the D-T reaction (lower is worse); basically, that means that the practical power output for a given amount of fusion fuel is lower by a factor of roughly 40*10[SUP]3[/SUP]. Given that we cannot even achieve over unity D-T fusion at this point, the excitement of using [SUP]3[SUP]He as a fusion fuel, which stems entirely from the almost aneutronic products, is somewhat premature to say the least.

If you could somehow contrive to accelerate at 1.5 * g (14.7 m/s[SUP]2[SUP]) for one year, you would achieve a speed of about 0.84c. At a specific impulse of 20,000 the mass ratio is somewhere around 10[SUP]153[/SUP], which is greater than the number of atoms in the universe. Again, specific impulse has to be up on the order of 100,000 seconds before mass ratios for rocket propulsion to become even remotely reasonable to achieve relativistic speeds.

Stranger

An Isp of 1 million seconds would be enough to get to 0.1c without an incredibly high mass ratio;
m0 = 1 [sup]29979245 / (9800000)[/sup]= 21.309

none of the fusion alternatives (not even D-T or D-He3) are powerful enough for that. Maybe something using antimatter might be good enough, but there are problems with waste heat, neutrinos and gamma rays in all the designs imagined so far.

I think that any interstellar mission will need to be propelled at least in part by some kind of beamed propulsion, such as Jordin Kare’s Sailbeam
http://www.niac.usra.edu/files/library/meetings/fellows/oct01/597Kare.pdf
Kare also suggests the use of a magnetic sail as a brake to provite propellantless deceleration - but this only works at high speeds. Once you get to lower velocities you will still need a rocket.

So you start your deceleration by using a ramscoop to fill up your hydrogen tanks, and then once your speed drops enough for the drag to not be enough any more, you burn off the hydrogen you just collected to finish stopping.

An excellent idea; thanks.