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*To*: starship-design@lists.uoregon.edu*Subject*: starship-design: Protons vs Electrons for Relativistic Electric Thrusters*From*: DotarSojat@aol.com*Date*: Tue, 4 Feb 1997 18:05:13 -0500 (EST)*Reply-To*: DotarSojat@aol.com*Sender*: owner-starship-design

Hi all Thanks to remarks by Ken Wharton (welcome Ken!) and Timothy concerning my example questions regarding relativistic electric thrusters, I have been able to distill an important question that I can start to address quantitatively. [Aside to Ken: I worked at UCRL/Berkeley before there was a UCRL/Livermore and then at UCRL/Livermore for 8 of its early years (during which it was renamed Lawrence Livermore Laboratory.) I'm glad you can provide information on the cutting edge of plasma accelerator technology. I was going to cite an "old" news item in the Scientific American, on p. 66 of the April 1987 issue, regarding plasma wake-field acceleration of electrons with a projected gradient of one billion volts per meter, a little less than the achieved levels you report. (The item also mentions laser-plasma-accelerator work then going on at UCLA.) I got the impression from the item, however, that we probably could not expect from that technology the kinds of current or the efficiencies that we must have for our thrusters.] Some selected quotes from Ken and Timothy that set the stage for the question I want to address in this note are the following: Ken wrote, on 1/27, >1) and 2)...And everything points to low mass particles being >the best. >Assuming that the final energy of the particles will be large >compared to the rest mass,...the rest mass becomes irrelevant to >the momentum... >The Big Problem, of course, is keeping the ship neutral. >Assuming we don[']t have positrons handy...we need one proton per >electron, which will severely hurt the [Isp]. [I can't type the symbols for punctuation marks that his word processor generates.] Timothy wrote, on 1/30 (quoting me), >>1. What is the best exhaust particle? Electrons, protons, >> alphas, etc.? What is the best parameter to compare them by? >Actually to determine an optimum we should first decide what we >want to optimize. ... High velocities have a low momentum:energy >ratio, but of course need a lot less mass. So you always have to >weigh between how much mass and how much energy. >Also one would want to use most of the repulsion mass that is >taken with the starship, this almost certainly means that one >needs to use ions (thus not electrons). >For the highest exhaust velocity, one should take the particle >with the highest charge:mass ratio, this would have been an >electron, so the next best thing would be a proton... Timothy wrote, on 1/30 (quoting Ken), >>3) In terms of size... Would there be an optimal length? I >>would guess no: you want the device as long as possible... >>Doubling the length will not double the mass of your entire >>ship, but it will double the amount of thrust you can get! >Previous calculations have shown that optimal exhaust velocity >depends (among other things) on the final velocity of the >starship. Just creating the highest exhaust velocity is >therefore not the main goal. So, I boil down these points to a single question: Q: What is the optimum (minimum-antimatter) performance of an antimatter-powered starship with its exhaust composed of accelerated PROTONS (with electrons dumped for charge neutralization), in comparison with that of one with its exhaust composed of accelerated ELECTRONS (with protons dumped at negligible velocity for charge neutralization)? ACCELERATED PROTONS The performance of an antimatter-powered starship with a PROTON exhaust velocity that is optimized to require a minimum mass of antimatter fuel was derived in my note of 4/4/96: "Optimum Interstellar Rockets (Minimum Antimatter Fuel)." The circumstance of minimum antimatter fuel is obtained by finding the exhaust velocity, for a given final vehicle velocity, that maximizes the propulsive energy efficiency, i.e., maximizes the conversion of exhaust kinetic energy to final vehicle kinetic energy. The 4/4 note included a derivation of the calculational procedure by which the results were obtained. Unfortunately, the analysis reported there was for only the case with 100 percent conversion of antimatter energy to exhaust kinetic energy (to be then converted to final vehicle kinetic energy by multiplying by the aforementioned maximized propulsive energy efficiency). Subsequent analysis, which I haven't yet reported, has expanded the calculation to cover antimatter conversion efficiencies less than 100 percent. The 4/4 results for acceleration of PROTONS were as follows: (with the tabulated quantities-- Uend = final proper velocity, ltyr/yr, of the starship at the end of the propulsive phase (at "burnout") Vend = final apparent velocity, ltyr/yr, ditto [The apparent velocity V is given in terms of the proper velocity U by the relation V = U/sqrt(1 + U^2) , (note: c = 1 ltyr/yr), and the inverse relation is U = V/sqrt(1 - V^2) .] optVexh = optimum apparent exhaust velocity, ltyr/yr optUexh = optimum proper exhaust velocity, ltyr/yr maxeff = maximum propulsive energy efficiency minMam/Mbo = minimum ratio of initial mass of antimatter fuel to the ship's burnout mass minMam/Mi = minimum ratio of initial mass of antimatter fuel to the ship's initial mass Mi/Mbo = mass ratio = ratio of ship's initial mass to ship's burnout mass.) Antimatter conversion efficiency = 1.0 Uend Vend optVexh optUexh maxeff minMam/Mbo minMam/Mi Mi/Mbo 0.2 0.196 0.124 0.125E+00 0.647 0.015 0.0031 4.97 0.5 0.447 0.291 0.304E+00 0.645 0.091 0.0175 5.23 1.0 0.707 0.492 0.566E+00 0.641 0.323 0.0540 5.99 2.0 0.894 0.691 0.957E+00 0.630 0.981 0.1215 8.07 3.0 0.949 0.777 0.123E+01 0.622 1.739 0.1675 10.38 4.0 0.970 0.823 0.145E+01 0.615 2.537 0.1992 12.74 5.0 0.981 0.852 0.163E+01 0.610 3.357 0.2222 15.11 [Values for antimatter conversion efficiencies less than 1.0 are available on request, as is the derivation of the calculational procedure. (Just remember that the calculational procedure described in my 4/4/96 note is an example, but is not complete.)] In the accelerated-PROTONS case, we have an optimum exhaust velocity and therefore an optimum accelerator length, as Timothy states. ACCELERATED ELECTRONS The calculational procedure described in my 4/4/96 note has been expanded to include dumping of mass at negligible velocity to bring about charge neutralization (as well as to include conversion efficiencies less than 1.0). The comparable results of the calculations for accelerated ELECTRONS (with dumping of one proton at negligible velocity for each electron for charge neutralization: "DUMP = 1836.") are as follows with the same nomenclature as above: Antimatter conversion efficiency = 1.0 Uend Vend optVexh optUexh maxeff minMam/Mbo minMam/Mi Mi/Mbo 0.2 0.196 1.000 0.184E+06 0.090 0.110 0.0900 1.22 0.5 0.447 1.000 0.478E+06 0.191 0.309 0.1908 1.62 1.0 0.707 1.000 0.931E+06 0.293 0.708 0.2927 2.42 2.0 0.894 1.000 0.163E+07 0.382 1.620 0.3817 4.24 3.0 0.949 1.000 0.217E+07 0.418 2.584 0.4186 6.17 4.0 0.970 1.000 0.257E+07 0.438 3.565 0.4382 8.14 5.0 0.981 1.000 0.289E+07 0.450 4.554 0.4503 10.11 For electrons, there is no maximum efficiency as a function of Uexh for finite values of Uexh; the efficiency increases monotonically as Uexh is increased. The maximum is replaced by an asymptote at infinite Uexh. The value of "maxeff" tabulated above is that efficiency when the increase in efficiency is 0.001 percent for an increase in Uexh of 1 percent; the tabulated value is within about 0.1 percent of the asymptote. (Note: This problem would have been almost hopelessly difficult if the parameter of optimization had been the conventional apparent exhaust velocity rather than the proper exhaust velocity; for the last line, the apparent exhaust velocity is 0.99999999999994 ltyr/yr for the stated proper exhaust velocity.) In the accelerated-ELECTRONS case, the kinetic energy efficiency stays virtually constant with increasing exhaust velocity above the tabulated value. In this case Ken is right (thrust increases directly with exhaust velocity without limit; "you want the device as long as possible"). COMPARISON Timothy cogently observes, "you always have to weigh between how much mass and how much energy." A succinct comparison between protons and electrons can be made with a table of the principal mass-related and energy-related properties of starships that would make use of the two choices of exhaust particles. The particle-accelerator energy for a specified Uexh is given by the relation particle kinetic energy = mc^2 * [sqrt(1 + Uexh^2) - 1] , where mc^2 is 938.9 MeV for protons and 0.511 MeV for electrons. For the protons' optUexh of 0.163E+01 ltyr/yr for the Uend of 5.0 ltyr/yr (achieved at a continuous acceleration of 1 g over a distance of 3.97 ltyr), the PROTON accelerator energy is about 850 MeV. For the electrons' "optUexh" of 0.289E+07 ltyr/yr for the Uend of 5.0 ltyr/yr, the ELECTRON accelerator energy is about 1,480 GeV. The thrust T is given by the relation T = iV sqrt[1 + (2mc^2/eV)] * (1 kgf/2,940 Mw) , where mc^2 is as above, i is current in amps, V is volts and eV is the accelerator energy in MeV; 1 amp*volt is 1 w. The values extracted from the Uend = 5.0 lines in the tables above or calculated from the above relations are as follows: Property Proton exhaust Electron exhaust Mass ratio 15.11 10.11 Maxeff 0.610 0.450 MinMam/Mbo 3.357 4.554 MinMam/Mi 0.2222 0.4503 Accelerator Energy (Mev) 850. 1,480,000. Thrust/amp (kgf) 0.520 500. "You pays your money and you takes your choice." I'll defer any further observations until I've seen your comments and further questions. This is only a start. Your suggestions may lead to significant changes in the approach. Rex Finke <DotarSojat@aol.com> P.S. If anyone asks for the analysis, I'll post it in another note; this note is long enough. (This analysis reproduces earlier results for the simpler conditions consistent with their less complete calculations. None of the current results seems too surprising, not even the smaller mass ratio with the electron exhaust, when one takes into account the ultra-high exhaust velocity.)

**Follow-Ups**:**starship-design: Protons vs Electrons for Relativistic Electric Thrusters***From:*Steve VanDevender <stevev@efn.org>

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