# How big/powerful would your fusion reactor need to be to power the engines of a 32km long starship?

Let's assume the reactor takes up about 4 vertical kilometers and has the same radius as the ship (4km). This gives you a volume of about 2.01e11 m^3, which given about 3-5% of the reactor's estimated mass means your available reaction mass (presumably mostly hydrogen) at any given time would be a few gigatons. This is feasible if you harvest asteroids and comets in interstellar space and could probably last you decades, if not centuries without having to make stops to harvest more fuel (if I'm wrong, please, PLEASE correct me on this). The ship itself weighs around 5.5e18 kg, and I'd like it to be able to pull 1g for an extended period of time (long enough to reach moderate fractions of light speed, up to .1c).

Does my ship's reactor need to be bigger? How much wattage could you get out of a generator this size with this much reaction mass? How long would it last? I really don't know how fusion power compares to fission and fossil fuels in terms of efficiency and energy production, so I could really use some help.

• The thing that can be calculated more or less from first principles is the amount of fuel you need (heaps). Calculating the size of the engine itself is much trickier because that is dependent upon rules of thumb for engine size per output in a kind of engine that has no real world equivalent. You probably won't go far wrong in estimating that at least 90-95% of the total mass would need to be fuel and engines if you are going to want to accelerate at 1 g for as long as it takes to reach 0.1c, slow down at the same rate, and cruise for a long time on non-propulsion power. – ohwilleke Oct 27 '16 at 0:43
• Are there many interstellar asteroids/comets? I have the distinct feeling that there aren't many at all. This affects the feasibility of your long-term plans. – user6511 Oct 27 '16 at 5:34
• "This is feasible if you harvest asteroids and comets in interstellar space and could probably last you decades, if not centuries without having to make stops to harvest more fuel". As user6511 notes, this is probably not correct. In addition to the shortage of interstellar asteroids and comets, the other problem is that collecting them while cruising along at 0.1c is not a viable proposition. – ohwilleke Oct 27 '16 at 5:36
• The time to accelerate to 0.1 c at 1 g is about five weeks. That's not an exact figure, but it's close enough to grasp what's involved timewise for acceleration and deceleration. A quick calculation gives me 3,061,224,5 seconds, say, three million seconds or roughly 36 days. OK, slightly more than five weeks, but close. – a4android Oct 27 '16 at 8:30

This is an awful lot of numbers considering there are no net energy generating fusion reactors to compare with. How are we supposed to know the net energy density of a system that doesn't exist? How are we supposed to guess what volume is needed for support equipment?

The one question I can answer is how much energy you will get out of a gigaton $\left(1\times10^{12}\text{kg}\right)$ of hydrogen. Even that depends on the type of hydrogen involved, but I'll assume D-D fusion, otherwise you will need a partitioned storage for D-T. D-D fusion releases 2.7 MeV per fusion (averaged between two possible reactions), and D has an atomic mass of 2 g/mol, so: $$1\times10^{12} \text{kg} \cdot 6.02\times10^{23} \frac{\text{atom}}{\text{mol}}\cdot \frac{1 \text{ mol}}{.002\text{kg}} \cdot \frac{2.7 \text{ MeV}}{2 \text{ atom}} \cdot \frac{1.6\times10^{-13}\text{ J}}{1 \text{ MeV}}= 6.5\times10^{25} \text{J}$$

That will at least give you an order of magnitude of the energy you are producing, though I ignored ancillary fusion reactions, you might use different hydrogen molecules, etc.

You can expect power of engine to be 10 MW/m3 per volume of plasma inside that engine (maybe more, the engine is big and that helps for faster burning) - in your case (it might not work that way for your 4 x 8 km engine but) probably 2e+18 W power generated by that engine.

Density of plasma isn't high, 1.4e20 particles/m3, total mass of plasma inside the chamber is about 93 tonne.

With thermonuclear engine you are limited to ISP ~3'000'000 sec (This is theoretical value, so no improvement is possible). To achieve a speed of 0.1 c, reaction mass should be ~2/3 of the ship's mass (only acceleration, or 9/10 with deceleration (both figures are theoretical limits, so no improvement is possible)).

With a total mass ship and fuel - 5.5e18 kg , based on ISP above and power of the engine, acceleration will be 2.4e-8 m/s2

For 1 g acceleration with given power and mass of ship + fuel will be

$v_{exhaust}=\frac{2\cdot Power_{reactor}}{m_{ship}\cdot a}=0.074m/s$

• density of produced energy in reactor chamber potentially may be higher that 10MW/m3, hard to tell, may be 3-4 orders of magnitude, and to get 1g with ISP = 3'000'000 sec it have to be 8-9 orders of magnitude higher, then 10 MW/m3. It may or may not happen, with given size, higher plasma density and higher magnetic field may help.

For 1 g acceleration your reactor have to generate about 8e+26W .

Efficiency depend on reactor size, reaction type - but as usual bigger size is better, fusion reactions are better if neutrons aren't produced, for example, He3+D - expect something 70-80% (size of reactor - bigger is better in this case too) - so I would say efficiency may be pretty high, and in that case I would take it as 1.0, because calculating it makes no sense in that case and it should be 0.8 and above for that case.

I may be a bit off with my calculations, but it should look something like this.

• Given how a modern chemical rocket can get you something more akin to 300-400 seconds of specific impulse, I'm not sure 3,000,000 seconds Isp is really much of a limitation in practice. I think someone here on Worldbuilding coined it: Her Isp is measured in weeks (in this case, just under five weeks). – a CVn Oct 27 '16 at 8:36
• @MichaelKjörling proton 18MeV is about 6'000'000sec, Helium 18MeV - is about 3'000'000 sec - I'm too lazy to calculate average ISP, I was told exhaust are about 0.1c. Looking at those numbers it looks like reasonable approximation. He3+D is about 18MeV per fusion - if I'm correct one of highest if not most highest per nucleon. So upper limit for ISP for thermonuclear reaction can be calculated, based on energy they produce and mass of particles. Or you about something else? – MolbOrg Oct 27 '16 at 12:25

Since you are willing to imagine nuclear fusion is an already developed science, then you pretty well have to imagine that a few pre-requisite developments have been made as well. Inside a star, gravity helps greatly to contain a plasma. An artificially induced fusion reaction depends on either inertial confinement or magnetic confinement of plasma, with our best efforts so far heavily favoring magnetic confinement via electromagnets made with superconductors. If you have adequate field strength superconductive magnets to contain a plasma then you also have the means to build a Bussard ramjet or RAIR to collect the low density hydrogen gas latent in space. After you have accelerated to .1 C, you are able to harvest fuel as you travel accelerating indefinitely. So making a roughly reasonable description of size and mass for the ship depends a great deal more on your parameters than anything else. 1 How large is the crew complement? Unmanned ships may be quite a bit smaller as the crew has to be separated from the radiation at both the ramjet end and the engine end. 2 What is the purpose of the ship? A freighter may need to be quite a bit larger to prevent radiation damage to cargo. A Ramjet is not very likely going to be a warship in terms of ship to ship combat unless it is huge. The drive exhaust would be a powerful weapon, but aiming it at a target at relativistic velocities is impractical. So a warship is probably a ship that carries personnel and equipment to fight at the destination. Or the ship itself is the weapon as its intent is to render a planet uninhabitable with the drive radiation upon arriving. 3 What is the distance the ship will travel? A ramjet will only collect fuel while the ramjet is operating in front of the ship. So it must collect enough fuel to operate the fusion engine to decelerate to a stop, before turning off the ramjet and using thrust to point the engine opposite to the established velocity vector. Research the ramjet or read this ISBN 0-471-61912-4 for better descriptions of details to make calculations of size and mass.

In the war-hammer 40K universe a Emperor class ship can have up to 3 million men as crew, with up to 20k people maintaining the engine bays etc and be anything up to 35km long, and 15km wide (The height varies from ship to ship, as every chapter is different.), during one crusade a emperor class ship was seen to be fitted with dual engine bays to power 6 clusters of engines that where 1k in radius each. Thats a really large reactor, although in War-Hammer they do have machine spirits lol.That being said your reactor would be fine, it should generate enough power (Compared to other ships of similar size in other universe)

• Warhammer 40K is heavy in hand weaving and you don't provide any scale for the Emperor battleship (couldn't find those my self), so -1. Try to at least find some scale comparison between Rogue Trader or BattleFleet Gothic. – MakorDal Oct 27 '16 at 11:13
• an emperor class battleship has know to be anywhere up to 35km in length, and 15km in width – Ryan white Oct 27 '16 at 11:44
• Give the details and sources in your post, to give at least a fictional equivalent. – MakorDal Oct 27 '16 at 11:47