# Re: Physic help

```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

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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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