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*To*: DotarSojat@aol.com*Subject*: Re: Physic help*From*: kgstar@most.fw.hac.com (Kelly Starks x7066 MS 10-39)*Date*: Thu, 16 May 1996 14:22:56 -0500*Cc*: 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, sl0c8@cc.usu.edu, 101765.2200@compuserve.com, MLEN3097@peachnet.edu, kgstar@most.fw.hac.com

At 2:42 PM 5/16/96, DotarSojat@aol.com wrote: >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. I realize that with fusion or anti-matter reactions you lose a little of the mass to energy. But, since in the case of fusion, the fraction lost is less than 1% of the total mass. I assumed the equation would give acceptable accuracy. Since most of the reference examples I have, use specific impulse and Exaust vel. I tried to stick with them out of familiarity. You have to admit they are convenent terms. I'm more concerned for the MAJOR differences Tim and I get when calculating fuel mass ratios. The fraction of a % of mass lost to energy conversion doesn't seem to account for more than a tiny fraction of our 100 to 1 vs 150 to 1 mass ratio disagreement. (I'm obviously just looking for rough order of magnetude numbers at this stage.) >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.)] Hum, interesting bit of history. ;) >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 Nice equations, but a little much for use on my hand calculator. ;) Thou I suppose I could fire up the free sudo-copy of Mathmatica that Apple now gives away with its operating system. If I load those equations in I should get some nice graphs. Kelly ---------------------------------------------------------------------- Kelly Starks Internet: kgstar@most.fw.hac.com Sr. Systems Engineer Magnavox Electronic Systems Company (Magnavox URL: http://www.fw.hac.com/external.html) ----------------------------------------------------------------------

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