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*To*: T.L.G.vanderLinden@student.utwente.nl, KellySt@aol.com, stevev@efn.org, jim@bogie2.bio.purdue.edu, zkulpa@zmit1.ippt.gov.pl, hous0042@maroon.tc.umn.edu, rddesign@wolfenet.com, David@interworld.com, lparker@destin.gulfnet.com*Subject*: Re: Physic help*From*: DotarSojat@aol.com*Date*: Thu, 16 May 1996 14:42:25 -0400*cc*: sl0c8@cc.usu.edu, 101765.2200@compuserve.com, MLEN3097@peachnet.edu, kgstar@most.fw.hac.com

To Kelly Beware of using chemical-rocket parameters for fusion/antimatter rockets. The parameter to be most wary of is the specific im- pulse, "Isp." Isp is the rocket-engine thrust per unit mass flow rate (out of the nozzle). We are interested in an equation that gives the velocity incre- ment, "delta-V," of the rocket stage in terms of the ratio of the initial mass of the stage to the final mass, the "mass ratio". The equation is called the "rocket equation" in the West and the "Tsiolkovsky equation" in Russia (after the first person to derive it). The depletion in mass (initial mass minus final mass) for a chemical rocket stage is solely propellant. For a fusion/antimatter rocket, however, the mass depletion is the sum of the propellant mass (out of the "nozzle," or accelerator) and the mass converted to energy in the fusion or matter/antimatter- annihilation reaction. For a chemical rocket stage, the parameter that links the delta-V with the function of the mass ratio (the natural log) is simply the exhaust velocity, Vexh, i.e., delta-V = Vexh ln(mass ratio) . [Isp was invented, I believe, to allow English-system (foot-pound- second) engineers to talk about rocket performance with metric- system (meter-kilogram-second) engineers by "non-dimensionalizing" the exhaust velocity. This was done by dividing it by the cons- tant gc, i.e., Vexh/gc = Isp . This reduces the units to "seconds", which are common to both sys- tems. The constant gc is only incidentally equal in value to the standard acceleration of gravity; it is properly referred to as the conversion factor from mass to force units, either 32.17405 lbmass-ft/(sec^2-lbforce) or 9.80665 kgmass-m/(sec^2-kgforce). (To be totally correct, the units of Isp should be lbforce-sec/ lbmass or kgforce-sec/kgmass, but the ratios lbforce/lbmass and kgforce/kgmass are usually just left out of both Isp and gc.)] For a fusion/antimatter rocket, additional parameters that must be included in the rocket equation are the ratio of the fusion/anti- matter mass to the mass converted to reaction energy (Timothy's "f") and the efficiency of conversion of reaction energy to ex- haust kinetic energy (let's call it "eta"). Also, the velocity increment must be put into relativistic terms. The relativistic fusion/antimatter rocket equation becomes much more complicated than the chemical rocket equation, i.e., the increment in apparent velocity is given by delta-V/c = tanh[(gexh eta/[(eta) + f(gexh - 1)]) * (Vexh/c) ln(mass ratio)] where gexh = sqrt[1 - (Vexh/c)^2] or the increment in proper velocity is given by delta-U/c = sinh[(gexh eta/[(eta) + f(gexh - 1)]) * (Vexh/c) ln(mass ratio)] (Note: for fusion, i.e., f greater than 1, the (eta) term does not appear because some of the fusion reaction products can be used as propellant.) In the case where f = 1 (matter/antimatter annihilation), the rel- ativistic rocket equation becomes delta-U/c = sinh[(gexh eta/[eta + gexh - 1] * (Vexh/c) ln(mass ratio)] In the simplistic antimatter-rocket case where eta = 1 (100 per- cent conversion of reaction energy to exhaust kinetic energy), the relativistic rocket equation reduces to delta-U/c = sinh[(Vexh/c) ln(mass ratio)] in a form somewhat similar to the chemical rocket equation. (Note that all velocities are non-dimensionalized now by dividing them by c.) I hope this properly expresses the shortcomings of using chemical- rocket performance relations for fusion/antimatter rockets. Rex

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