Analyzing the rocket-equation section, I found the following statement:
The relativistic rocket equation is
Δv=ctanh(Ispclnm0m1)
where Δv is the difference in velocity, m0 is the initial mass of the probe, m1 the final mass of the replicator and the Isp/c term denotes the specific impulse of the fuel burning process. The Isp/c term can be derived from η, the proportion of fuel transformed into energy during the burning process Isp/c=√2η−η2 [24].
The fact that we can fully derive Isp from the fuel energy-transformation efficiency seemed weird to me, so I looked it up in the underlying reference and found the following quote (I slightly cleaned up the math typesetting and replaced it with equivalent symbols above, emphasis mine):
For the relativistic case, there is a maximum exhaust velocity for the reaction mass that is given by:
w=√η(2−η)c
where e is the fuel mass fraction converted into kinetic energy of the reaction mass. I was not able to improve on that derivation from a presentation point of view.
This is obviously a very similar equation, but importantly this equation specifies an upper bound on the exhaust velocity and does not say when that upper bound can be attained. Intuitively it seems to me not at all obvious that we should be able to attain that maximum exhaust velocity, since it would require the ability to perfectly direct the energy released in the fuel burning, which would naively be primarily released as heat.
The keyword that finally helped me understand the relevant rocket design is “Fission-Fragment Rocket”, which at least according to Wikipedia could indeed reach specific impulses sufficient to support the conclusions in the paper.
Ok, I think the original calculations here are still correct, if you design your rocket to directly emit the fission material at high speeds. This is a paper that proposes such a rocket design:
Dusty Plasma Based Fission Fragment Nuclear Reactor
We propose an innovative nuclear power generation system design using dusty radioactive (fissile or not) material plasma as a fuel. The fission fragments or decay products accelerated during the disintegration process to velocities of 3–5% of the speed of light are trapped and collected in a simple combination of electric and magnetic fields resulting in a highly efficient (90%), non-Carnot, DC power supply. In a conventional nuclear reactor this high kinetic energy of the fission fragments is dissipated by collisions to generate heat, which is converted to electrical power with efficiencies of no more than 50%. Alternatively, the fission fragments produced in our dusty plasma reactor can be used directly for providing thrust. The highly directional fission fragment exhaust can produce a specific impulse of one million seconds resulting in burnout velocities several thousand times those attainable today. Previous concepts suffered from impractical or inadequate methods to cool the fission fuel. In this work the heating problem is overcome by dividing the solid fuel into small dust particles and thereby increasing the surface to volume ratio of the fuel. The small size of the fuel particle allows adequate cooling to occur by the emission of thermal radiation.
Analyzing the rocket-equation section, I found the following statement:
The fact that we can fully derive Isp from the fuel energy-transformation efficiency seemed weird to me, so I looked it up in the underlying reference and found the following quote (I slightly cleaned up the math typesetting and replaced it with equivalent symbols above, emphasis mine):
This is obviously a very similar equation, but importantly this equation specifies an upper bound on the exhaust velocity and does not say when that upper bound can be attained. Intuitively it seems to me not at all obvious that we should be able to attain that maximum exhaust velocity, since it would require the ability to perfectly direct the energy released in the fuel burning, which would naively be primarily released as heat.
The keyword that finally helped me understand the relevant rocket design is “Fission-Fragment Rocket”, which at least according to Wikipedia could indeed reach specific impulses sufficient to support the conclusions in the paper.
Ok, I think the original calculations here are still correct, if you design your rocket to directly emit the fission material at high speeds. This is a paper that proposes such a rocket design:
Further comments on this rocket design:
https://forum.nasaspaceflight.com/index.php?PHPSESSID=3omp04d25qbe0qj5l5n1qfl0hp&topic=47693.20