Let's say we have a 10,000 ton ship with an18,000 kps exhaust velocity that will travel at 5% c to Alpha Centauri 3 in 86 years. Using the rocket equation, we derive the propellant mass:
E^( 15,000/18,000)= 2.3 so we need 13,000 tons of reaction mass, we'll figure out how much helium 3 and deuterium we will need later. Relativistic effects at these velocities are insignificant.
Using the kinetic energy equation K.E.=0.5( 13,000,000 kg.)(18,000,000m/s)^2= 2.1x10E21 joules This is the energy of the exhaust gases, the energy we must generate from the fusion reaction to propell the ship
Next, we'll figure ten years will be required to reach cruise velocity. Therefore: (2.1x10^21j)/(3600*24*365*10)= 6.66x10^12 joules/second for ten years or 6,659,056 megawatts of power for ten years!! A big city powerplant generates about 1000 megawatts!
Since a one megaton nuclear bomb releases 4.12x10^15 joules, this is like the energy of 1.6 kilotons of TNT every second for ten years. What an engine! Is it practical?? The ITER fusion reactor is projected to amass 10,000 tons and generate only 500 megawatts. That's only 50 watts/kg. What about a low mass laser fusion reactor that has power density similar to a nuclear fission rocket motor. From http://silver.neep.wisc.edu/~neep602/LEC25/IMAGES/fig30.GIF we find that the Pewee reactor produced 5,200 MW/cubic meter and the Phoebus 2A had a specific mass of 2.3 kg./MW. The longest lived reactor was the Nuclear Furnace. It lasted 109 minutes. So our fusion reactor might have a volume of 1280 cubic meters and amass 15,315 tons. But the ship only amasses 10,000 tons. And this thing has to run for ten years instead of 109 minutes. Also, since helium 3 yeilds 19 MW years/kg., (2.1x10E21/(3600*24*365))/19,000,000=3500 tons of He3 and 2330 tons of deuterium will also be needed. Only 0.7% of that will convert to energy and the rest will become protons and alpha particles in the plasma exhaust, so the 5830 tons of D+3He will be part of the 13,000 tons of reaction mass. The balance will be hydrogen added to the plasma for more thrust.
What if we have a fusion reactor that makes 10 kW to 100 kW/kg. as some writers have estimated? Now we have a reactor that amasses 666,000 to 66,000 tons. Still not good enough. Luckily, should we ever achieve laser fusion we might be able to build fusion rocket motors with 3500 kW to 4000 kW/ kg. See: Fusion Revisited. Now we have an engine that amasses 1900 to 1665 tons and that will work well with our 10,000 tons ship.
What if we slow down to 1%c? Then we need 1814 tons of reaction mass. Let's take 20 years to reach top speed. The rest of the trip is going to take 430 years so we can ease up on our reactor. Now we need 4.659x10E11 j/sec or a 465,924 megawatt drive. This is equal to the energy of 113 tons of TNT per second. We'll need 490 tons of He3 and 326 tons of deuterium. Actually we'll need more he3 and D because there is no such thing as 100% efficiency, so matters are even more grim. At 4000 kW/kg. we need a 116 ton engine. But can it run for 20 years? We may need supermaterials, or even new substances like ultra-dense matter. A 10,000 tons ship is about as big as a naval destroyer. The Eiffel Tower amasses 10,000 tons, so this ship could be pretty big. Maybe a dozen people could travel in a multigeneration ship, have kids at 35, die at 75 and never overpopulate the ship. When you're talking over 430 years, a destroyer doesn't seem all that big. Cryostasiss would really help, or this could be a robotic vehicle.
What if we slow down more, to 0.1%c, for interplanetary travel. We will need 168 tons of reaction mass. At this speed, 670,000 mph, we could reach Saturn in about two months. So, what can our 10,000 tons ship fusion reactor do if we take 30 days to reach cruise velocity? We can't take years to accelerate for in system travel. That would be absurd. So we're looking at 90 days to Saturn. Now we need 1.05x10E13 joules/second, the equivalent of 2.5 kilotons of TNT per second, or 10,500,000 MW. With a 4000 kW/kg. engine we get an engine mass of 2625 tons; so we have about 7375 tons of ship proper. We'd need 45 tons of He3 and 30 tons of D. That's $135 billion worth of helium 3 at $3 million/kg. Well, maybe we have robot swarms harvesting the stuff from the atmosphere of Saturn or Uranus and the price is very low. That sees plausible.
Magnetic fields from cryo-superconducting magnets could contain the super hot plasma in the engine, but the plasma will be so hot that it glimmers with X-rays and the he3 and D when it fuses will give off more radiation than the LH2 reaction mass can absorb. But what if there was such a thing as a perfect reflector at high EM radiation frequencies/energies? It could reflect all the energy into the hot plasma core and exhaust and keep all those x-rays from heating the motor. Uranium is a good reflector of soft x-rays. Maybe add a cooling system and enormous waste heat radiators, because perhaps 1% to 5% of the energy will be absorbed by the uranium. Gold, copper and silver reflect about 98%-99% visible light. Metals reflect EM radiation because the surface electrons are poorly coupled to the inner atoms. Radiation forces the electrons to oscillate and re-emit the radiation without sending energy to the inner atoms. It seems then that heavy atoms with lots of electrons and electron shells would be the best reflectors, and we find that heavy metals, like gold, are good reflectors. Uranium is used as a radiation reflector and other heavy metals like tungsten, chromium, iridium, platinum, and nickel have x-ray and gamma radiation reflecting properties.
So it looks like fusion is just fine for high speed interplanetary travel but interstellar travel will take a long time. Mag-sail beamriders are probably the best ways to achieve high speed interstellar travel, but these require massive infrastructure in the form of huge solar power collector complexes in solar orbit and particle beams of great size or artificial miniature black holes to power the particle beams. There are other ways like fusion pulse besides magnetic plasma confinement-radiation reflector- laser fusion engines to exploit fusion to make starships that travel at higher speeds.
