The magnetic field acts on a fusion reaction 300m away from the hull. It acts like a spring; it requires no energy input to absorb the kinetic energy of the charged fusion reaction products and transmit it to the spaceship. Using figures from the cited paper, thrust efficiency is 75%.
Thanks to this arrangement, only 0.0127% of the unwanted energies from the fusion reaction are intercepted and absorbed as waste heat by the heatshield.
Some of the heat can be converted into electricity using to power the laser igniting the fusion reactions. The generator can be of the superconducting magnetohydrodynamic type, and the laser could be cryogenically cooled. This makes them both extremely efficient. The electrical power that needs to be generated to run the laser can be very small if the fusion gain is extreme (small ignition, big fusion output).
Putting these percentages so far together, 0.00216% of the fusion reaction energy ends up as heat in the heatshield.
Using that percentage, it is now evident that we have a very large 'multiplier' to play with. For every watt that the heatshield can survive, 46,300 watts of fusion output can be produced.
A heatshield absorbing 2.09 GW of heat means that its Epstein drive can have an output of 96.8 TW. About 2.2kg of fuel is consumed per second.
83% of that fusion power is in the form of useful charged particles, and the magnetic field turns 75% of those into thrust. So, 41.5% of the fusion power becomes thrust power; which is 60.25 TW.
The effective exhaust velocity of a Deuterium-Helium3 reaction can be as high as 8.9% of the speed of light. This assumes 100% burnup of the fusion fuel. Because we are using an excess of Helium-3, this might be reduced to 6.3% of the speed of light.
With this exhaust velocity, we get a thrust of 6.37 MegaNewtons.
An empty 250 ton Rocinante would accelerate at 2.6g with this thrust.
We know it can accelerate harder than that but it cannot handle any more fusion power. So, it must increase its thrust by injecting water alongside fuel into its exhaust.
There is a linear relationship between exhaust velocity and thrust at the same power level, but a square relationship between thrust and mass flow.
Halving the exhaust velocity doubles the thrust but quadruples the mass flow rate. The Rocinante can have a 'cruise' mode where only fuel is consumed to maximize exhaust velocity, and a 'boost' mode where more and more water can be added to the exhaust to increase thrust.
It is useful to know this, as we must now work out just how much fuel (Deuterium and Helium-3) and extra propellant (water) it needs 1800km/s is done in the 'boost' mode, and then 2200km/s in the 'cruise' mode, for a total of 4000km/s. How much fuel and propellant does it need?
As with any rocket equation calculation, we need to work backwards.
Mass ratio = e^(DeltaV/Exhaust Velocity)
An exhaust velocity of 6.3% of the speed of light and a deltaV requirement of 2200km/s means a mass ratio of 1.123.
The 250 ton Rocinante needs to first be filled with 30.75 tons of fusion fuel. A 1:2 mix of Deuterium (205kg/m^3) and Helium-3 (59kg/m^3; it won't freeze) has an average density of 107.6kg/m^3, so this amount of fuel occupies 285m^3. It represents about 4.9% of the spaceship's 12x12x40 m internal volume.
And now the 'boost' mode. 5g of acceleration while the spaceships gets lighter as propellant is being expended means that thrust decreases and exhaust velocity increases gradually over the course of the engine burn. The propellant load can to be solved iteratively... on a spreadsheet.
Using 0.25 ton steps for water loaded onto the Rocinante, it can be worked out that an initial mass of 352 tons is required. This represents an additional 57 tons of water and 17.25 tons of fuel.
The full load is therefore 57 tons of water in 57m^3, and 48 tons of fusion fuel in 446m^3. Together, they fill up 8.7% of the Rocinante's internal volume.
The thrust level during the acceleration to 1800km/s varies between 13.77MN and 17.27MN. It takes just over 10 hours to use up all the water.
Boosting to 12g would require that this thrust be increased further, between 33.05MN and 41.44MN. However, it could only be sustained for 106 minutes, until 751km/s is reached.
Official art by Ryan Dening.
In 'cruise' mode and with no water loaded, the Rocinante would have 3320km/s of deltaV and can cross the distance between Earth and Saturn in 10 to 12 days at any time of the year.
At 12g, it can sprint out to a distance of 21.2 thousand kilometres in about 10 minutes, and 0.76 million kilometres in an hour.
Scaling
This proposed design can be easily scaled to adjust for different figures for mass, acceleration and deltaV.
The variable will be the ignition point distance from the spaceship and therefore the magnetic field strength of the coils in the 'nozzle'. A stronger field allows for fusion products to be redirected from further away, so that an even smaller portion of the harmful energies are intercepted.
If we assumed a ten times greater density for the Rocinante, for example, we would have an empty mass of 2500 tons. To adjust for this while maintaining the same performance, we would simply state that the fusion reaction is ignited 10^0.5: 3.16 times further away, or 948m. The 'multiplier' mentioned earlier jumps from 46,300 to 461,300, just over 10 times better than before. In other words, the fusion output can be increased 10 times and all the performance falls back in line with what was calculated so far.
Consequences
Beyond what we've seen on the show or read from the books, there could be some interesting consequences to having this sort of design.
Visually, for example, the rear end of spaceships would glow white hot. They cannot come close to each other while under full power, as then they'd expose each other's flanks to intense heat from the fusion reactions.
You might have noticed from a previous diagram that a portion of the fusion plasma travels all the way up the magnetic fields without being redirected. This could be the reason why we see 'gas' in the 'nozzle'; it is simply the leaking plasma hitting a physical structure and being compression heated up to visible temperatures.
A failure of the magnetic fields would immediately subject the heatshield to 5x its expected heat intensity. This would quickly raise the temperature by a factor 2.23, so it would turn from solid to explosively expanding gas. Not exactly a 'failure of the magnetic bottle', but a similarly devastating result.
On the other hand, the magnetic field passively provides shielding against most of the radiation that can affect space travellers. If it is strong enough to repel fusion protons, then it could easily deal with solar wind protons and other charged particles, as found in the radiation belts around Earth or Jupiter. This could be a reason why we don't see thick blankets of radiation shielding all around the hull.
Our proposed engine design is pulsed in nature. We want smooth acceleration, so we want as many small pulses in such quick succession that the spaceship feels a near-continuous push. This can be achieved with as few as 10 pulses per second, or hundreds if possible.
However, even at 10 pulses per second, you need to shoot your fusion fuel from the fuel stores to the ignition point 300 meters away at a velocity of 3km/s. This is can be accomplished by a railgun, and it is incidentally a good fraction of the projectile velocities used in combat.
Could the Belters in the Expanse simple have pointed their fuel injection railguns in the opposite direction to equip themselves with their first weapons?
Similarly, an intense laser is needed to ignite the fuel quickly enough to achieve an extreme fusion gain. Doing so from 300m away requires a short wavelength and a focusing mirror… which are also the components needed to weaponize a laser. If a laser can blast a fuel hard enough to cause it to ignite, then it could do the same to pieces of enemy spaceships, and all that is needed to extend the range is a bigger mirror.
There is also a claim made where the Rocinante's fuel reserves are 'enough for 30 years'. This cannot mean propulsion. Even at a paltry 0.1g of acceleration in 'cruise' mode, the Rocinante can consume all of its fuel in just 40 days. Add in a lot of drifting through space without acceleration, and we're still looking at perhaps a year of propulsion. It is much more likely that this claim refers to running the spaceship; keeping the lights on, the life support running and the computers working. That sort of electrical demand is easily met by the energy content of fusion fuels.
Finally, keep in mind that the propulsion technology described here is not specific to the setting of the Expanse. It respects physics and you can introduce it to any setting where real physics apply. In other words, it is a ready-made and scalable solution for having rapid travel around the Solar System without much worry about propellant, radiators, radiation shielding and other such problems!
