Recently I did the math to figure out how large a polywell fusion reactor would have to be to support the power requirements of a warp bubble, assuming the power requirements are always equal to those of the Ent-D. This is, of course, an invalid assumption but the Ent-D energy graph is all I have go on, so I'm sticking to this assumption.
Turns out a 14 gigawatt polywell fusion reactor would have a magnetic grid on the order of three meters in diameter. Around that would then go the power grid and containment shell, making the whole thing on the order of 6 meters in diameter. This is larger than would fit in Cochrane's Pheonix. Indeed, all the fusion reactors I can calculate for simply would not scale to fit the physical dimensions of Pheonix while still putting out 14GW of power, which is the power requirements to overcome the Warp 1 hump for Ent-D.
So that got me thinking. First off, there's more than enough power in the 3-6kg of plutonium found in a modern nuclear weapon to supply one to two hours of continuous 14GW, if all of it were fissioned, and that's at about 25% efficiency. (Six kilograms is enough, by itself, for a warhead but I don't know how much less can be gotten away with by using fusion tampers and U238 reflectors, etc. So I don't know how much plutonium Cochrane would have found in the ICBM. If it's a MERV --which is likely-- there shouldn't be a problem.) However, actually releasing the energy locked up in the plutonium is a major problem. Current methods release it all at once, and that's no good for obvious reasons. (Ka-BOOM!!)
So, I did the math and you need about 1/5 gram of Pu239 to burn per second to get 14GW. This requires approximately 2^67 plutonium atoms to fission and, thus, approximately 2^67 neutrons to make them fission.
Now, plutonium averages more than two neutrons released per fission --about 2.5, depending on circumstances. Assuming, on average, 0.5 neutrons per reaction are lost and 2 neutrons are absorbed to create more fissions, it would take (SURPRISE!!) 2^67 generations of neutron doubling to burn the needed plutonium.
In case you were wondering, 2^67 is a rather large number of atoms and yet it's nowhere near enough for critical mass. In fact, it's not even a mol of atoms, 2^79.
But there's a trick that can, theoretically, be done. If you hit a plutonium nucleus with an antiproton, the plutonium explodes in a shower of small fragments, including neutrons. As it turns out, you can probably produce from 2^4 (16) to 2^5 (32) neutrons in one interaction.
(Plutonium239 has 93 protons and 145 neutrons. One proton will be destroyed. If the remaining protons are separate exactly into two nuclei, there will be two new palladium atoms of 46 protons each. The heaviest stable palladium atom has 64 neutrons, leaving 17 from the original plutonium that are not welcome in either of the new atoms. However, the amount of energy released by a proton-antiproton event should be enough to disintegrate the nucleus to small chunks, potentially releasing 52 neutrons.)
For ease of calculation, let's assume the smaller number (2^4 = 16) is the average number of neutrons released per AM/Pu239 reaction. If you can get 23 generations of neutron duplication within the plutonium after the initial antimatter barrage, then you need 2^40 antiprotons to get the 2^67 total fissions you need. That many antiprotons is about 1.8 trillionths of a gram (1.8 picograms).
Ok. Let's add some inefficiencies. Let's say 90% of the neutrons from the antiproton barrage are lost, 60% of the plutonium doesn't fission and 60% of the energy is lost before making it to the warp coils. That would mean the Pheonix would use up a gram of plutonium and a tenth a nanogram of antiprotons would be used up every second to excite the coils with 14GW.
Again, 1 gram of Plutonium and 0.1 nanogram of antiprotons every second...
In the movie, the Phoenix's engines are active for less than a minute. The exact amount of time depends on how you count the seconds between cuts. But, if we put an absolute maximum of 60 seconds travel time out and another 60 seconds back, and further assume 14GW is needed throughout this time, that will give an absolute maximum to the amount of plutonium and antimatter needed for this scenario: 120 grams Pu and 12 nanograms of antiprotons.
Now, we can assume there's at least one kilogram of usable plutonium in the missile Phoenix was made from. But what about the antiprotons? Well, that's a problem. It's estimated there are about 160 nanograms of antiprotons in the Van Allen belt. Could collecting them be so efficient that Cochrane could get 7.5% of them? I suppose it's possible. But getting it down from orbit could be difficult depending on the condition of space infrastructure at the time. It seems likely it was pretty bad since he used an ICBM to launch his prototype.
Still, maybe before their war with the Eastern Coalition, the United States had a small supply of antiprotons and Cochrane had access to what was left over... I don't know.
[...Hrm... There is the distinct possibility that some nuclear weapons of the mid 21st century are fusion/antimatter hybrids, where chemically imploded fusion fuel converges onto a central container of antimatter. The resulting M/AM explosion would strike the implosion shockwave so hard that the fusion fuel would fuse completely, releasing several orders of magnitude more energy than the antimatter/matter explosion by itself. If these weapons are extent throughout the post-apocalyptic US, there might be enough in these weapons to use for the Pheonix. Or, maybe not. It depends on... all kinds of things actually... Anyway, continuing...]
However, there's another problem: the design of the reactor. How do you get a gram of Plutonium and a tenth gram of antiprotons to interact correctly every time?
Here's what I'm thinking: free nucleon lasers.... Let me explain.
Free electron lasers take electrons, accelerate them in a linear accelerator and then make them go past a set of magnets whose poles alternate. The electrons wiggle and emit lased light in the process. But here's the important part: a stream of electrons, after being oscillated like that, stop being a stream and conglomerate into chunks, the size and separation depends on the frequency and amplitude of the oscillations.
You can do the same thing with nucleons. The laser light that is emitted is in the x-ray range but as long as the path of the x-rays is away from the cockpit, you can engineer around them and there shouldn't be a problem. So, you have two nucleon accelerators, one for Pu239 and one for antiprotons. The soul reason for the tracks is to conglomerate and focus the nucleons into small clumps of, say, 1 milligram for Pu and 1 picogram for the antiprotons. Then you split the two streams into four and merge a stream set of Pu/AM in each nacelle, next to their warp coils at a frequency of 1 kilohertz and that will give 14GW of power to the warp coils --7GW per nacelle.
You could even have a generator that produces electrical power that uses a pair of these clumps at 10 hertz, making 140 megawatts of power for the ship's other systems.
Of course, if you could increase the efficiencies even a little bit, that would mean much less antimatter/plutonium would be used. With 100% efficiency, you would only need 0.2g of plutonium to fission per second for the 14GW we need. If Cochrane could get 30 generations of neutron doubling after the antimatter bombardment, then he'd only need 15 femtograms (15E-15) of antimatter (2^33 antiprotons) per second --while still assuming 16 neutrons per Pu/AM reaction-- to get the 2^67 fissions we're looking for. Again, this is at 100% efficiency.
Furthermore, the assumption of 14GW throughout the flight is ludicrous. Far more likely, usage would peak at 14GW, and even that's an unlikely worst-case scenario based on Ent-D power usage --warp fields should scale in power usage depending on the size of the vessel and the materials of the warp coils.
Still, I think it should be possible to find a way to strongly confine clumps of plutonium nuclei and smack them with clumps of antiprotons, producing both the power and the particles needed to charge Phoenix's warp coils.
Now, I see yet another problem with this design. I've got little explosions going off inside the nacelles at 14 mega joules each. That's the equivalent of 3.3 kg of tnt, each. A second's worth (500 pulses) of these pulses, together, is equivalent to 1.6 tonnes of tnt in each necelle. Admittedly, that's spread out over their length. Still, that's a lot of force to be contained!
Thoughts? What did I get wrong?
Turns out a 14 gigawatt polywell fusion reactor would have a magnetic grid on the order of three meters in diameter. Around that would then go the power grid and containment shell, making the whole thing on the order of 6 meters in diameter. This is larger than would fit in Cochrane's Pheonix. Indeed, all the fusion reactors I can calculate for simply would not scale to fit the physical dimensions of Pheonix while still putting out 14GW of power, which is the power requirements to overcome the Warp 1 hump for Ent-D.
So that got me thinking. First off, there's more than enough power in the 3-6kg of plutonium found in a modern nuclear weapon to supply one to two hours of continuous 14GW, if all of it were fissioned, and that's at about 25% efficiency. (Six kilograms is enough, by itself, for a warhead but I don't know how much less can be gotten away with by using fusion tampers and U238 reflectors, etc. So I don't know how much plutonium Cochrane would have found in the ICBM. If it's a MERV --which is likely-- there shouldn't be a problem.) However, actually releasing the energy locked up in the plutonium is a major problem. Current methods release it all at once, and that's no good for obvious reasons. (Ka-BOOM!!)
So, I did the math and you need about 1/5 gram of Pu239 to burn per second to get 14GW. This requires approximately 2^67 plutonium atoms to fission and, thus, approximately 2^67 neutrons to make them fission.
Now, plutonium averages more than two neutrons released per fission --about 2.5, depending on circumstances. Assuming, on average, 0.5 neutrons per reaction are lost and 2 neutrons are absorbed to create more fissions, it would take (SURPRISE!!) 2^67 generations of neutron doubling to burn the needed plutonium.
In case you were wondering, 2^67 is a rather large number of atoms and yet it's nowhere near enough for critical mass. In fact, it's not even a mol of atoms, 2^79.
But there's a trick that can, theoretically, be done. If you hit a plutonium nucleus with an antiproton, the plutonium explodes in a shower of small fragments, including neutrons. As it turns out, you can probably produce from 2^4 (16) to 2^5 (32) neutrons in one interaction.
(Plutonium239 has 93 protons and 145 neutrons. One proton will be destroyed. If the remaining protons are separate exactly into two nuclei, there will be two new palladium atoms of 46 protons each. The heaviest stable palladium atom has 64 neutrons, leaving 17 from the original plutonium that are not welcome in either of the new atoms. However, the amount of energy released by a proton-antiproton event should be enough to disintegrate the nucleus to small chunks, potentially releasing 52 neutrons.)
For ease of calculation, let's assume the smaller number (2^4 = 16) is the average number of neutrons released per AM/Pu239 reaction. If you can get 23 generations of neutron duplication within the plutonium after the initial antimatter barrage, then you need 2^40 antiprotons to get the 2^67 total fissions you need. That many antiprotons is about 1.8 trillionths of a gram (1.8 picograms).
Ok. Let's add some inefficiencies. Let's say 90% of the neutrons from the antiproton barrage are lost, 60% of the plutonium doesn't fission and 60% of the energy is lost before making it to the warp coils. That would mean the Pheonix would use up a gram of plutonium and a tenth a nanogram of antiprotons would be used up every second to excite the coils with 14GW.
Again, 1 gram of Plutonium and 0.1 nanogram of antiprotons every second...
In the movie, the Phoenix's engines are active for less than a minute. The exact amount of time depends on how you count the seconds between cuts. But, if we put an absolute maximum of 60 seconds travel time out and another 60 seconds back, and further assume 14GW is needed throughout this time, that will give an absolute maximum to the amount of plutonium and antimatter needed for this scenario: 120 grams Pu and 12 nanograms of antiprotons.
Now, we can assume there's at least one kilogram of usable plutonium in the missile Phoenix was made from. But what about the antiprotons? Well, that's a problem. It's estimated there are about 160 nanograms of antiprotons in the Van Allen belt. Could collecting them be so efficient that Cochrane could get 7.5% of them? I suppose it's possible. But getting it down from orbit could be difficult depending on the condition of space infrastructure at the time. It seems likely it was pretty bad since he used an ICBM to launch his prototype.
Still, maybe before their war with the Eastern Coalition, the United States had a small supply of antiprotons and Cochrane had access to what was left over... I don't know.
[...Hrm... There is the distinct possibility that some nuclear weapons of the mid 21st century are fusion/antimatter hybrids, where chemically imploded fusion fuel converges onto a central container of antimatter. The resulting M/AM explosion would strike the implosion shockwave so hard that the fusion fuel would fuse completely, releasing several orders of magnitude more energy than the antimatter/matter explosion by itself. If these weapons are extent throughout the post-apocalyptic US, there might be enough in these weapons to use for the Pheonix. Or, maybe not. It depends on... all kinds of things actually... Anyway, continuing...]
However, there's another problem: the design of the reactor. How do you get a gram of Plutonium and a tenth gram of antiprotons to interact correctly every time?
Here's what I'm thinking: free nucleon lasers.... Let me explain.
Free electron lasers take electrons, accelerate them in a linear accelerator and then make them go past a set of magnets whose poles alternate. The electrons wiggle and emit lased light in the process. But here's the important part: a stream of electrons, after being oscillated like that, stop being a stream and conglomerate into chunks, the size and separation depends on the frequency and amplitude of the oscillations.
You can do the same thing with nucleons. The laser light that is emitted is in the x-ray range but as long as the path of the x-rays is away from the cockpit, you can engineer around them and there shouldn't be a problem. So, you have two nucleon accelerators, one for Pu239 and one for antiprotons. The soul reason for the tracks is to conglomerate and focus the nucleons into small clumps of, say, 1 milligram for Pu and 1 picogram for the antiprotons. Then you split the two streams into four and merge a stream set of Pu/AM in each nacelle, next to their warp coils at a frequency of 1 kilohertz and that will give 14GW of power to the warp coils --7GW per nacelle.
You could even have a generator that produces electrical power that uses a pair of these clumps at 10 hertz, making 140 megawatts of power for the ship's other systems.
Of course, if you could increase the efficiencies even a little bit, that would mean much less antimatter/plutonium would be used. With 100% efficiency, you would only need 0.2g of plutonium to fission per second for the 14GW we need. If Cochrane could get 30 generations of neutron doubling after the antimatter bombardment, then he'd only need 15 femtograms (15E-15) of antimatter (2^33 antiprotons) per second --while still assuming 16 neutrons per Pu/AM reaction-- to get the 2^67 fissions we're looking for. Again, this is at 100% efficiency.
Furthermore, the assumption of 14GW throughout the flight is ludicrous. Far more likely, usage would peak at 14GW, and even that's an unlikely worst-case scenario based on Ent-D power usage --warp fields should scale in power usage depending on the size of the vessel and the materials of the warp coils.
Still, I think it should be possible to find a way to strongly confine clumps of plutonium nuclei and smack them with clumps of antiprotons, producing both the power and the particles needed to charge Phoenix's warp coils.
Now, I see yet another problem with this design. I've got little explosions going off inside the nacelles at 14 mega joules each. That's the equivalent of 3.3 kg of tnt, each. A second's worth (500 pulses) of these pulses, together, is equivalent to 1.6 tonnes of tnt in each necelle. Admittedly, that's spread out over their length. Still, that's a lot of force to be contained!
Thoughts? What did I get wrong?
