Antimatter-Catalyzed Micro-Fission/Fusion: Please check my math
I've already found at least one mistake, so I'm trying to spell this out in full for other people to verify.
Obviously a lot of this is going to depend on what values we use for things like the cost of antimatter, and will ignore a lot of actual construction, labor, transportation, etc., but I'm just trying to get a rough estimate.
Details on the engine are taken from Atomic Rockets, which lists the fuel pellets as 3 grams of nuclear fuel (molar ratio of 9:1 of Deuterium:Uranium 235) coated with a spherical shell of 200 grams of lead. The lead shell is to convert the high energy radiation into a form more suited to be absorbed by the propellant.
Each pellet produces 302 gigajoules of energy (about 72 tons of TNT) and are fired off at a rate of 1 Hz (one per second).
The pellet explodes when it is struck by a beam containing about 1×10^11 antiprotons.
Pellet Composition and Fuel Costs
200 grams of lead 2.7 grams of Deuterium = 10.64 USD 0.3 grams of U235 = 24.08 USD (erring high and guessing 90% enrichment to be safe)
1×10^11 antiprotons (injected separately in a beam, but listed here for cost-per-pellet)
Antiproton mass = 1.67×10^−27 kg So for each pellet: (1×10^11) x (1.67×10^−27) = 1.67x10^-16 kg or 1.67x10^-13 grams of antiprotons
Atomic Rockets here quotes Centauri Dreams saying that antimatter currently costs about 100 trillion USD per gram. So (1.67x10^-13) x (1x10^14) = 16.7
16.70 USD of antiprotons per fuel pellet.
Total fuel cost per pellet (ignoring lead and silicon carbide propellants, and ignoring manufacturing, etc.) = 51.42 USD
The original ICAN-II ACMF design called for a total of 65 nanograms of antiprotons, but let's call it 100 nanograms here to make things easier.
At 100 trillion USD per gram of antiprotons, that should be 100,000 USD per nanogram, or 10 million USD for 100 nanograms to fuel the whole ship.
Increasing the number of pellets to match takes us from 453,000 to 696,923 pellets. And at a cost of 51.42 USD each that's about 35.84 million USD total.
Rocket Performance
That seems... not bad at all for a rocket, considering how most of the open-cycle nuclear-thermal rockets are costing anywhere from 50,000 to 1 million USD per second to run.
(Cost per pellet here is incorrect, based on Atomic Rockets' figure of 7x10^14 antiprotons per nanogram, rather than doing my own calculation from the mass of an antiproton. Unless I'm mistaken, which I may be)
An ACMF engine is cheaper BY FAR in cost per second than any fission rocket; even the most crazy high-performance NSWR costs more than 90x as much.
On top of that, the performance an ACMF rocket seems to offer is pretty fantastic. Atomic Rockets lists an exhaust velocity of 132,435 m/s, and thrust of 180kN, with an engine Thrust/Weight Ratio of 0.68.
Plugging those into the Delta-V and Acceleration nomograms from the same site:
That's astonishingly good performance.
Around 92 km/s of dV with a propellant mass ratio of 2! Even if only 20% of your mass is propellant you're still getting around 30 km/s!
And you can meet the 0.05 m/s^2 minimum acceleration requirement with a ship massing 3,500 tonnes, off of just the one engine! (Including Deuterium, U235, and lead in each fuel pellet, plus the mass of the ablative silicon carbide shell, I come up with a total propellant mass of 698.5 tonnes)
So we're looking at lower thrust than solid-core and gas-core nuclear-thermal rockets, but better exhaust velocity than even the best possible gas-core option, and still with decent thrust. (for right now we're ignoring the Pulsed Solid-Core NTR, since I don't know enough to evaluate its claims of exhaust velocities ranging from 135km/s to 13,500km/s, 4.5% of c!)
How Much Kaboom?
According to Atomic Rockets' Boom Table, 100 nanograms of antimatter should be about the equivalent of 4.3kg of TNT.
So a breach in a ship's antiproton containment will kill the engine, but not catastrophically explode the ship.
Antimatter Production - Energy Requirements
Atomic Rockets says that at current inefficiency of 0.00000002 efficiency it will take 0.0075 Joules to make 1 antiproton, or 4.483×10^24 Joules (4.48 yottaJoules) to make one kilogram of antiprotons.
But let's say we want a fleet of 100 ACMF ships, each loaded with 100 nanograms of antiprotons. That's 10 micrograms, which is 10^-5 grams or 10^-8 kilograms.
(4.483x10^24) / 10^8 = 4.483x10^16
So making 10 micrograms of antiprotons would take 44.83 petajoules (4.483x10^16 joules)
A Watt is 1 Joule / Second, so 4.483x10^16 joules divided by seconds in a year (3.15576x10^7), gets you 1.42x10^9 watts:
1.42 gigawatts continuously to fuel 100 ships each year (My previous number of 14.2 GW was off, so please check me on all this)
That’s… so easy.
That’s one larger-than-average fission power plant. There’s more than 20 solar farms that can do that (though not round-the-clock). There are at least 39 hydroelectric dams that produce more than double that number
That’s 7,174,231 m^2 of current 14.5% efficient solar panels at Earth’s distance (a circle 3.02km across) Or 1,074,443 m^2 (a circle 1.17km across) at Mercury. (14.5% efficiency means you need enough panels to receive 9.8GW of sunlight)
Using that original figure of 100 Trillion USD per gram of antiprotons, 10 micrograms is 0.00001 grams for the entire fleet, at a cost of 1 Billion USD. The cost of one current American Arleigh Burke-class destroyer is 2.2 Billion USD. So this seems like a very modest cost to fuel an entire 100-strong space fleet. Even including the U235 and Deuterium it's 3.584 Billion USD.
And referring back to the Boom Table, the whole antimatter factory is only producing the antiproton equivalent of 430kg of TNT, which is less than the warhead of one Tomahawk cruise missile.
Final Request
Please check my math, or point out anything else you think I may be missing.
In terms of rocket engines, this seems maybe too good to be true.
If the numbers I'm basing this on are correct, and I haven't made too much of an error in my math, it seems like by using a tiny amount of fission and fusion fuel, and a microscopic amount of antimatter, you can get performance that is significantly cheaper AND significantly better than even the most hypothetical fission rockets, for a cost so modest that any space military with the means could hardly pass it up.
For a space setting without abundant and easy fusion power (fusion power plants still don't exist, or have an Alpha/Specific Power that makes them unusable onboard a spacecraft due to their mass), this seems like a very appealing option. It does still use up scarce fissile fuels, so you may still try to move all regular traffic by other means (and you still need to solve the space access problem through non-rocket means to be viable, so you'd likely keep using that same elevator/tether/laser/mass driver solution for interplanetary transfers anyway) but for spacecraft demanding independent high-performance, such as a military not wanting to rely on civilian transportation networks, ACMF may be the way to do it.
