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Optimum Interstellar Rockets
- To: T.L.G.vanderLinden@student.utwente.nl, kgstar@most.fw.hac.com, stevev@efn.org, jim@bogie2.bio.purdue.edu, KellySt@aol.com, zkulpa@zmit1.ippt.gov.pl, hous0042@maroon.tc.umn.edu, rddesign@wolfenet.com, David@interworld.com, lparker@destin.gulfnet.com
- Subject: Optimum Interstellar Rockets
- From: DotarSojat@aol.com
- Date: Thu, 4 Apr 1996 13:26:30 -0500
MEMORANDUM
TO: LIT/SSD Discussion Group
FROM: Rex Finke
SUBJECT: Optimum Interstellar Rockets (Minimum Antimatter Fuel)
INTRODUCTION
Timothy van der Linden points out in his calc.txt that there is
an optimum ratio of exhaust velocity to final rocket velocity
relativistically (as I had calculated earlier for non-relativistic
velocities --undocumented). The existence of this optimum indic-
ates that there is a minimum in the amount of antimatter fuel
required to accelerate a starship to any given final mission
velocity.
This memo provides the numbers that show how the ratio of the min-
imum antimatter mass to initial starship mass varies with the de-
sired mission velocity at the end of the first, acceleration burn.
ANALYSIS
We define the following operative quantities:
V = "apparent" velocity = starmap distance/Earth time, in ltyr/yr
U = "proper" velocity = starmap distance/starship time, in ltyr/yr
Vend, Uend are the velocities at the end of the acceleration burn
(at "burnout")
Vexh, Uexh are the exhaust velocities
g = relativistic energy factor "gamma" = 1/sqrt(1 - V^2)
U = g V
V = U/sqrt(1 + U^2)
gend = gamma for Vend
gexh = gamma for Vexh
M = starship mass (= Mi initially; = Mbo at burnout)
r = starship mass ratio = Mi/Mbo
Ma = annihilation mass used during acceleration burn for rela-
tivistic rocket = twice the mass of antimatter = 2 Mam
Mp = mass of propellant used during acceleration burn for non-
relativistic and relativistic rockets
The propulsive energy efficiency (let's call it eff) is the ratio
of the final vehicle kinetic energy to the total exhaust kinetic
energy.
Non-relativistically-
final vehicle energy = (1/2) Mbo Vend^2
total exhaust energy = (1/2) Mp Vexh^2
eff = (Mbo/Mp)(Vend/Vexh)^2
Now from the rocket equation
Mi/Mbo [= (Mbo + Mp)/Mbo] = exp(Vend/Vexh)
we get
Mp/Mbo = exp(Vend/Vexh) - 1
If we set x = Vend/Vexh to simplify, we get for the energy effic-
iency the expression
eff = x^2/(exp(x) - 1)
This has a maximum value 0.648 for x = 1.59.
So, if the burnout velocity of a non-relativistic rocket is 1.59
times its exhaust velocity, the energy efficiency is a maximum of
64.8 percent. I.e., the final vehicle energy can be no greater
than 64.8 percent of the exhaust energy. This limitation is not
an important consideration for a non-relativistic rocket because
energy is subordinate to mass.
Relativistically-
final vehicle kinetic energy = Mbo (gend - 1) c^2
total exhaust kinetic energy = Mp (gexh - 1) c^2 = Ma c^2
(no energy losses)
which gives Mp = Ma/(gexh - 1)
but relativistically Mp = Mi - Mbo - Ma
Ma/(gexh - 1) = Mi - Mbo - Ma
Ma = (Mi - Mbo)(gexh - 1)/gexh
so the energy efficiency, which is the ratio of the final vehicle
kinetic energy to the total exhaust kinetic energy, is
eff = Mbo (gend - 1) c^2/(Ma c^2)
= Mbo (gend - 1) gexh/[(Mi - Mbo)(gexh - 1)]
= (gend - 1) gexh/[(Mi/Mbo - 1)(gexh - 1)]
= (gend - 1) gexh/[(r - 1)(gexh - 1)]
The relativistic rocket equation, in its "velocity-parameter"
form, is
theta = Vexh ln r
and the definition of the velocity parameter is
tanh(theta) = Vend
or sinh(theta) = Uend
Note: asinh(Uend) = ln [Uend + sqrt(Uend^2 + 1)]
so r = exp[asinh(Uend)/Vexh]
With this relation we have all of the parameters to calculate
eff = (gend - 1) gexh/[(r - 1)(gexh - 1)]
The expression for eff is evaluated using a Fortran computer pro-
gram, OPTVEXH, a description and a copy of which are given in the
Appendix.
RESULTS
The results of the calculations of the optimum Vexh, the maximum
energy efficiency and the minimum ratios of antimatter mass to
burn-out mass and to initial mass are given in the table below
for ascending values of the mission final proper velocity Uend.
Included in the table are values of Vend, to illustrate the degree
of saturation of apparent velocity, and of the optimum Uexh, to
give a value (not otherwise meaningful) to which to relate the
Uend, in order to examine the behavior of the ratio.
The extreme Uend of 5 ltyr/yr represents the final velocity reach-
ed at a continuous acceleration, a, of one g (1.0324 ltyr/yr^2)
over a distance of 3.97 ltyr. (The acceleration distance
s = [sqrt(1 + U^2) - 1]/a .) Only for destinations beyond about
8 ltyr or accelerations greater than one g need one consider Uends
greater than 5 ltyr/yr.
(Note: these calculations assume no energy losses in converting
annihilation energy to exhaust kinetic energy. The correction for
energy losses would be to divide the minMam values by the conver-
sion efficiency.)
Uend Vend optVexh optUexh maxeff Uend/optUexh minMam/Mbo minMam/Mi
----non-relativistic---- 0.648 1.59 --- ---
0.2 0.196 0.124 0.125 0.647 1.60 0.0153 0.0029
0.5 0.447 0.291 0.304 0.645 1.64 0.0914 0.0174
1.0 0.707 0.492 0.566 0.640 1.77 0.323 0.0535
1.1690.76* 0.541 0.643 0.639 1.82 0.422 0.0666
2.0 0.894 0.691 0.957 0.630 2.09 0.981 0.1211
3.0 0.949 0.777 1.235 0.622 2.43 1.739 0.1689
4.0 0.970 0.823 1.450 0.616 2.76 2.537 0.1972
5.0 0.981 0.852 1.625 0.611 3.08 3.357 0.2210
--------
*Timothy's selection
OBSERVATIONS
The minimized amount of antimatter is a small fraction of the
starship's initial mass, less than 25 percent for mission proper
velocities as high as 5 light-years/year for 100 percent conver-
sion efficiency.
The maximum energy efficiency decreases slowly as mission proper
velocity is increased, but remains over 60 percent up to a mission
proper velocity of 5 light-years/year.
The ratio of the mission proper velocity to the optimum exhaust
proper velocity increases fairly slowly at first from 1.59 at low
velocities to almost double at a mission proper velocity of 5
light-years/year.
The implications of the values of the optimum exhaust velocity
need to be examined, in terms of their conversion to MeV for
exhaust particles.
------------------------------------------------------------------
APPENDIX. Program OPTVEXH
For an input value of final proper velocity Uend, the program cal-
culates the Vend, the gend and the theta. Then for values of Vexh
increasing from 0.01 in increments of 0.01, the program calculates
the eff until a maximum is passed. The optimum Vexh is calculated
by fitting a second-degree curve to the three points that include
the maximum. The value of the maximum is simply taken to be the
value preceding the drop. The ratios of initial antimatter mass
to Mbo and Mi are derived from expressions above-
Ma/Mbo = (r - 1)(gexh - 1)/gexh
eff = (gend - 1) gexh/[(r - 1)(gexh - 1)]
= (gend - 1)/(Ma/Mbo)
Ma/Mbo = (gend - 1)/eff
Mam = (1/2) Ma
Mam/Mbo = (gend - 1)/(2 eff)
Mam/Mi = (Mam/Mbo)(Mbo/Mi)
= (Mam/Mbo)/r
= (gend - 1)/(2 eff r)
C PROGRAM OPTVEXH 4/2/96
101 FORMAT(2X, 21H Final Proper Vel = ?)
102 FORMAT(2X, 15H Opt Exh Vel = , F6.4, 18H Max Energy Eff = ,
& F6.4, 17H Antimatter/Mi = , F6.4)
103 FORMAT(2X, 8H VEXH = , F4.2, 7H EFF = , F6.4)
2 CONTINUE
WRITE(*,101)
READ(*,*) UEND !final proper velocity, ltyr/yr
IF(UEND .EQ. 0.) GO TO 99
VEND = UEND/SQRT(1. + UEND*UEND)
GAMEND = 1./SQRT(1. - VEND*VEND)
THETA = LOG(UEND + SQRT(UEND*UEND + 1.)) !asinh
VEXH = 0.01
VEXHN = VEXH
1 CONTINUE
VEXHNN = VEXHN
VEXHN = VEXH
VEXH = VEXH + 0.01
RN = R
R = 1.01
IF(VEXH .GT. .05) R = EXP(THETA/VEXH)
GAMEX = 1./SQRT(1. - VEXH*VEXH)
EFFNN = EFFN
EFFN = EFF
EFF = (GAMEND - 1.) * GAMEX/((R - 1.) * (GAMEX - 1.))
C WRITE(*,103) VEXH, EFF
IF(EFF .LT. EFFN .AND. VEXH .GT. 0.1) THEN
Y1 = EFF
Y2 = EFFN
Y3 = EFFNN
X1 = VEXH
X2 = VEXHN
X3 = VEXHNN
A = ((Y1-Y2)*(X2-X3)-(Y2-Y3)*(X1-X2))/
& ((X1*X1-X2*X2)*(X2-X3)-(X2*X2-X3*X3)*(X1-X2))
B = ((Y1-Y2) - A*(X1*X1-X2*X2))/(X1-X2)
OPTVEXH = -B/(2.*A)
AMRATIO = (GAMEND - 1.)/(2.*EFFN*RN)
WRITE(*,102) OPTVEXH, EFFN, AMRATIO
GO TO 2
END IF
GO TO 1
99 STOP
END