# Re: Physic help

```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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