A Performance Analysis of an Electron Positron Annihilation Rocket
Jonathan A. Webb†
Embry Riddle Aeronautical University
Prescott, AZ 86301
This paper outlines the effect of the cosine average on the performance of an antimatter
rocket using an electron positron annihilation engine. The minimum rendezvous times for
all eight planets within our solar system and earth’s moon are shown for cosine averages of
0.0001 to 2. The relationship between specific impulse, maximum burnout velocity and
thrust is shown within this paper. This paper also outlines the propellant mass
requirements for a 350 kg spacecraft launched on Hohman transfers using an electron
positron annihilation engine.
Nomenclature
= speed of light (3.00 x 108 m/s)
c
cos  = cosine average
D
e
e
g
I sp
=
=
=
=
Mi
Mf
= initial rocket mass
mf
= dry mass

m

= propellant mass flow rate
= gamma ray
= flight time
= burnout velocity


closest approach flight distance
positron
electron
acceleration due to gravity (9.81 m/s2)
= specific impulse
= final rocket mass
Introduction
In recent studies it has been shown that a spacecraft engine utilizing electron positron
annihilations offers incredible flight velocities and specific impulses(1). A spacecraft equipped
with an e  e  engine will annihilate electrons and positrons to produce gamma rays that will be
captured into a momentum transfer shield at the extreme end of a rocket system. Two cases have
been considered to capture momentum from the resultant photons as shown in Figure 1.(1)
________________
†
Student Emeritus ERAU
1
.
Figure 1. Photon absorbing and reflecting shields. Dashed line represents reflecting shield.
In the first case e  e  beams are injected into a hemispherical dish that extends
from  2    2 , and directed towards a collision at the center of the dish. At this point the
e  e  pairs annihilate producing two 511 KeV  -rays back to back. Half the resultant photons
collide with the perfectly absorbing shield, while the other half are ejected into space from the
annihilation point. The annihilation process is isotropic which produces a uniform distribution of
photon collisions across the dish.
Each collision with the shield creates a transfer of momentum to the dish and spacecraft from the
gamma rays. Each transfer of momentum to the shield boosts the spacecraft from one velocity to
a higher velocity.
Unfortunately this method absorbs all of the incident electromagnetic
radiation which causes huge temperature rises in the shield(2). Unfortunately in reality the
incident photons would cause a scattering effect inside the shield. The photons will collide with
atoms of the shield material causing a spray of electrons and lower energy photons to be ejected
at random angles. Some of these photons and electrons will be ejected at angles opposite to the
direction of motion causing some momentum to be lost thereby decreasing the specific impulse
and thrust. Currently Monte Carlo simulations are being developed to determine the efficiency
lost due to the scattering of electrons and photons.
The second configuration involves the use of a parabolic shield that extends to an arbitrary point.
In this case the shield is assumed to be a perfect reflector of the 511 KeV photons. This reflection
process creates a large increase in the momentum vector of the incident photons. This concept
yields a much larger Isp and thrust than with the previous case. Although this is by far the more
efficient of the two cases current technology lacks the ability to reflect 511 KeV photons.
Even though the reflecting shield configuration is technically the most difficult of the two it
offers the largest cosine average and Isp. The range of Isp allowed with this type of propulsion
ranges from 1000 to 30.0 x 106 seconds. As is shown the performance of this engine is highly
dependant on the cosine average obtained from the momentum transfer of the photons.
2
The efficiency at which the shield can capture the momentum from the incident photons is
measured through a cosine averaging technique. The cosine average is written mathematically as:
cos  
1

2
 1  cos   d
(1)
1
The maximum theoretical cosine average that can be obtained is a factor of 2. The engines
burnout velocity and specific impulse both increase as the cosine average approaches 2, this is
shown in fig.2 and 3.
Performance Equations
In the case of an electron positron annihilation engine the thrust, specific impulse and cosine
average are all related. The manner in which they are related is best shown in the specific
impulse and thrust equations shown below and in figure 2.
F
I sp 
cos 
2
m c 
(2)
cos  c
(3)
2g
As is seen in figure 2 very high specific impulses can be achieved with low cosine averages.
Figure 2. Isp Vs. <cos  >
It should be noted in figure 2 that a cosine average of only 0.0001 representing an engine
propulsive efficiency of only 0.005% still yields an I sp of 1500 seconds. This is a larger specific
impulse then even Nuclear Thermal Rockets can offer.
Using equation 2 in conjunction with a 50 mg/s propellant mass flow rate, the thrust vs. specific
impulse and cosine average can be graphed and is shown in figure 3. Figure 3 shows that thrusts
on the order of kilo-newtons require cosine averages of 0.2 or higher. Cosine averages lower than
0.2 would probably be best used for low mass spacecraft applications due to the lower achievable
thrust. It should be noted that higher thrusts can be achieved with larger propellant mass flow
rates.
3
Figure 3. Thrust Vs. <cos  > and Isp, dm/dt = 50 mg/s
Velocity Profile
The velocity profile for an APD derived by Smith and Webb(1) is written as:
1 x 
v  c

1 x 
(4)
where
M
x i
M
 f




cos 
 1 


1  
cos 
(5)
by setting the mass fraction equal to 100, representing 100% of the spacecrafts total mass is used
as fuel the maximum flight velocity can be graphed as a function of cos  . The graphs for
maximum burnout velocity as a function of cosine average and I sp and are shown in figure 4
Figure 4. Burnout velocity Vs. <cos  >.
Figure 4 shows a dramatic increase in Isp and burnout velocity as a function of cosine average. As
expected the maximum burnout velocity approaches the speed of light as the cosine average
approaches 2. In theory spacecraft with cosine averages of 0.30 or greater can reach significant
fractions of the speed of light.
Interplanetary Mission Analysis
4
To best show the use of an antimatter engine, minimum one way rendezvous times to all eight
planets within the solar system were calculated and graphed as a function of cosine average and
Isp using an all propulsive trajectory code. It was assumed that all spacecraft started from the
exact same position and velocity vector of earth (i.e. a C3 = 0 orbit) and assumed the exact same
orbit in reference to its target planet.
All calculations were made assuming launch occurred when the planetary alignment brought the
two bodies (earth and the target planet) to their closest approaches. The time needed for plane
changes and orbit insertion were not considered within this analysis and are negligible compared
to the total trip time.
The equation used to describe the one way time of flight for the various spacecraft is written
below(3):

Dm f
2D
2
I sp g
F
(6)
Due to the fact that the lower Isp engines produce smaller thrust, the calculations were separated
into two categories. The lower thrust engines are assumed to propell an unmanned satellite of
5000 kg dry mass. The higher Isp engines are assumed to pushing a manned spacecraft of 50000
kg dry mass. The propellant mass flow rate assumed in the calculations for all transfer times is
50 mg/s.
Figure 5. Interplanetary transfer times for low Isp, low cosine average S/C of 5000 kg.
5
Figure 6. Propellant mass fraction for interplanetary transfers, low Isp, low cosine average with S/C
of 5000 kg.
Figure 7. Interplanetary transfer times for high Isp, high cosine averages with a S/C of 50000 kg.
Figure 8. Propellant mass fractions for high Isp, high cosine averages for a S/C of 50000 kg.
6
Figure 9. Earth-Lunar Transfer Time Vs. <cos()> and Propellant Mass.
Figures 5 through 9 show very rapid transfer times to all eight planets using very little propellant
mass even for engines with low cosine averages. Figure 5 and 6 shows that engines with low
cosine averages should only be used to propel low mass spacecraft due to the low thrust to weight
ratio. Even with a lower thrust, spacecraft of 5000 kg or less can be delivered to the inner
planets within 9 weeks or the outer planets within 12 months.
Engines with cosine averages larger than 0.1 are well suited to drive high mass spacecraft as
shown in the figures 7 an 8. Spacecraft can be delivered to any point in the solar system within a
few days to weeks with a propellant mass fraction of less than 6 percent. A spacecraft using an
engine with a cosine average of 1.9 can deliver a 10 metric ton spacecraft to the moon in less than
7 hours using 1.11 kg of propellant
The above graphs are assuming direct trajectory calculations and does not take into account the
thrust to weight ratio. In each of the above figures the time of flight will increase above the
calculated times for lower cosine averages. The flight times will deviate from the graphed times
at cosine averages below 0.3 for the figure 7 trajectories and 0.0025 for the figure 5 trajectories.
This is because the acceleration of each spacecraft will not be enough to compete with the
acceleration due to the suns gravity. This will not allow the spacecraft to fly a direct trajectory
and will more than likely require the spacecraft to either spiral out of the solar gravity towards its
desired burnout velocity or fly a propulsive conic trajectory.
Figures 5 through 9 show the maximum performance capabilities of an antimatter engine for
interplanetary propulsion applications. The above figures show the fastest transit times and the
most propellant consuming trajectories. This engine could also be used on conventional Hohman
transfers with extreme savings in propellant mass. The propellant requirements for both the
absorbing shield at a cosine average of 2/ and the reflecting shield with a cosine average of
1.985 are shown in figure 10 compared to that of a chemical engine with a specific impulse of
450 seconds.
All Hohman transfers assume the spacecraft is in an initial earth parking orbit of 400 km with an
eccentricity of 0.60. The plane change for each transfer is assumed to occur at a true anomaly of
90 degrees. It is also assumed that the spacecraft enters into a 0.95 eccentricity orbit with a
perigee altitude of 300 km around the target planet.
7
Hohman Flight Parameters for all Eight Planets
Flight Time/Propellant Mass
400
350
300
250
Absorbing Mp gm
200
Reflecting Np gm
150
Chemical Mp kg
100
Flight Time years
50
0
Mercury
Venus
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
Planets
Figure 9. Hohman transfer parameters for an absorbing/reflecting APD and chemical engine.
As is seen in fig. 9 both forms of electron positron annihilation engines offer extreme savings in
propellant mass over chemical engines. In every interplanetary transfer the chemical engine at
450 seconds of specific impulse requires hundreds of kilograms of propellant where the
antimatter concepts only need a few grams of propellant. It should also be noted that 450 seconds
of specific impulse is currently the highest Isp achievable with any type of chemical propellants
namely with the use of liquid hydrogen and liquid oxygen.
Conclusion
The performance of an antimatter engine relying on electron positron annihilations is highly
dependant on the cosine average. This is a performance characteristic highly dependant on
advances in the engineering materials field. In order to obtain high cosine averages a reflecting
shield is needed. Unfortunately no shield is currently capable of reflecting 0.511 MeV photons.
High specific impulse engines can still be obtained with an absorbing shield concept.
Unfortunately the thrust would have to be limited to incredibly low levels in order to prevent the
shield from melting under the extreme conditions(2).
The results of this study show that even at low cosine averages and specific impulses an
antimatter engine offers performance well above that of any other known propulsion systems and
can allow for extremely rapid interplanetary travel. Due to the potential of this engine further
study is not only warranted but is needed to push the envelope of space exploration. In order for
an antimatter engine to become a reality advances in the storage and production of antimatter are
required as are advances in reflecting shield materials.
Bibliography
1
Smith, D., Webb, J., (2001) The Antimatter Propulsion Drive A Relativistic Propulsion System,
AIAA Paper 2001-3231.
2
Webb, Jonathan, (2002) Thermal Analysis of a Tungsten Radiation Shield for Beamed Core
Antimatter Rocketry, Embry Riddle Aeronautical University
3
Kammash, Terry, (1995) Fusion Energy in Space Propulsion, Washington D.C., American
Institute of Aeronautics and Astronautics
4
Gaidos, G., Lewis, R.A., Meyer, K., Schmidt, T., Smith, G.A. (1998), AIMStar: Antimatter
Initiated Microfusion for Pre-cursor Interstellar Missions. Conference on Applications of
Thermophysics on Microgravity and Breakthrough Propulsion Physics. STAIF-99 Albuquerque,
NM. January 31-February 4, 1999
5
Chakabarti, S., Dundore, B., Gaidos, G., Lewis, R.A., Smith, G.A. (1998). AntiprotonCatalyzed Microfission/fusion Propulsion Systems for Exploration of the Outer Solar System and
8
Beyond. The 34th Joint Propulsion Conference, AIAA Paper 98-3589, Cleveland, OH. July 1315, 1998.
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