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Date: Thu, 4 Apr 1996 13:26:30 -0500
From: DotarSojat@aol.com
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

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
1.169	0.76* 	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


=====================================================

Date: Thu, 4 Apr 1996 14:03:10 -0500
>To:DotarSojat@aol.com
>From:kgstar@most.fw.hac.com (Kelly Starks x7066 MS 10-39) 

Subject:Re: Optimum Interstellar Rockets
Cc: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 

>Other than complexity, would adjusting the exaust vel for optimum at
ships current vel (I.E. as the ships speed increases. Changing the exaust
velocity for the optimum for that speed) buy us anything? 

>If you look at the total amount of fuel my Explorer class needs (about
100,000,000 tons of 6Li) I'm starting to think we should at least work up
numbers for a mass conversion / anti-matter ship. Thou frankly the idea
of making and carrying a few thousand tons of antimatter bothers me a
lot. (NOT IN MY STARSYSTEM!!!!!) Ah, how far from a planet would we need
to keep the ship for safty? 

>Kelly


>At 1:26 PM 4/4/96, DotarSojat@aol.com wrote: 
>>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)]   -----



===========================================================
Date: Sun, 7 Apr 1996 16:22:56 -0400
From: DotarSojat@aol.com
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: Problems with beaming

A laser beam with wavelength lambda that has been formed by a primary
objective mirror with an aperture diameter D will di- verge in the far
field (never mind, that's where you want to work) due to diffraction. I
don't have the explicit relations at home, but I believe that the angular
width of a "diffraction- limited" beam at half maximum is the same as
that used in the Rayleigh criterion: 1.22 lambda/D.

This width is further broadened due to (1) thermal effects in the lasing
medium, by a factor called "beam quality" that is in the range 1.1-1.3,
depending on the type of laser, and (2) beam- pointing instability called
"jitter," usually about one-third the diffraction-limited beam divergence,
that enters in a root- sum-squares way. The overall angular beam width of
a contempo- rary laser "weapon" is of the order of a microradian. A factor
of ten reduction (narrowing) in beam width by the time an inter- stellar
mission would be undertaken would not be unreasonable to expect.

Regards, Rex