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starship-design: Protons vs Electrons for Relativistic Electric Thrusters

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)?


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

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


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").


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.)