A MISSION AND PROPULSION DESIGN ANALYSIS OF A RELATIVISTIC ELECTRON - POSITRON ANNIHILATION ROCKET

Jonathan A. WebbEmbry Riddle Aeronautical University

Prescott, Arizona 86301webbj@pr.erau.edu

ABSTRACT

Conventional liquid, solid, and electromagnetic propulsion systems are incapable of transporting large masses of payload long distances and at high velocities through stellar and interstellar space. There are many other theoretical designs which can outperform current propulsion systems, but only with minimal improvements. In order to create a craft that incorporates high thrust and a large specific impulse, a craft must utilize a fuel source that yields massive energy per unit of propellant mass. There are only three known types of energy production known that yield this type of energy. These are Nuclear Fission, Nuclear Fusion, and Matter-Antimatter Annihilation. Of these three systems of energy production, Matter-Antimatter Annihilation yields more energy than Fission and Fusion combined making it the most desirable energy source for stellar and interstellar spacetravel. This paper will describe a method of converting the electromagnetic energy created in an electron-positron annihilation into thrust to propel a spacecraft. This paper will also describe the technical problems associated with the Antimatter engine. I have titled this proposal as the Antimatter Photon Drive or APD for a simpler abbreviation

1. INTRODUCTION

In 1928 while researching the relationship of relativity and quantum mechanics a physicist, Paul Adrien Maurice Dirac theorized that positively charged electrons could possibly exist. This assumption was based on the equation

E pc mc2 2 2 2 (1.1)

Rewriting this equation in a different the form shows the equation as a positive radical.

mc E pc2 2 2 (1.2)

But due to mathematics, every equation in the radical positive form yields the same anwser when in the negative form. This changed the equation to

E E pct 2 2 (1.3)

where E t is the total energy equivalent to mc2 , (Serway, R., Moses, C., Moyer, C., (1997). Modern Physics, Second Edition. Orlando Florida, Harcourt Brace and Jovanavich),

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This means that a mass such as an electron also yields the same energy if it were a positively charged particle and not a negative particle. Dirac’s theory was proven in 1932 when Carl Anderson detected the antielectron which was later named the positron.

This theory showed that every particle in existence has an antiparticle with the exact same gravitational mass as its normal matter partner but with an opposite electrical charge and spin. The one quality of antimatter that makes it unique and invaluable in advanced space propulsion systems is the energy yielded when normal matter combines with antimatter. Whenever an electron collides with a positron, both particles annihilate each other transforming one hundred percent of the mass of both particles into energy.

1.2 Electron Positron Annihilation

The electron e positron e annihilation reaction produces its energy in a pure electromagnetic form creating two 0.511 MeV gamma rays for 99% of the annihilations and three gamma rays per 1% of the annihilations. The reaction that produces two gamma rays is written as e e . The created gamma rays are electrically neutral and are ejected from the electron-positron pairs at random angles. The overall reaction is desirable because it produces no massive particles plus the energy of the reaction can be instantaneously utilized unlike the reactants from other particle antiparticle annihilations. But unfortunately the gamma rays are neutral and can not be focused with magnetic fields.

The energy and propulsive advantage of antimatter can be shown through the charts in figure 1 and 2.

YIELDS FROM VARIOUS ENERGY SOURCES

FUELS ENERGY RELEASE J/Kg CONVERTED MASS FRACTION

CHEMICAL

LO/LH 1.35 x 107 1.25 x 10-10

Atomic Hydrogen 2.18 x 108 2.40 x 10-9

Metastable Helium 4.77 x 108 5.30 x 10-9

Nuclear Fission

U-235 8.20 x 1013 9.10 x 10-4

Nuclear Fusion

DT (0.4/0.6) 3.38 x 1014 3.75 x 10-3 CAT-DT(1.0) 3.45 x 1014 3.84 x 10-3

D He3 (0.4/0.6) 3.52 x 1014 3.90 x 10-3

pB11 (0.1/0.9) 7.32 x 1013 8.10 x 10-4

Matter Antimatter 9 x 1016 1.00

Figure 1. Energy released by various materials. weight composition corresponds to a mixture ratios for fusion fuels, i.e. (0.4/0.6) for the DT fuels. Graph from (Kammash, Terry (1995). Fusion Energy in Space Propulsion. Washington D.C.: American Institute of Aeronautics and Astronautics).

VARIOUS EXHAUST VELOCITIESType of Engine Propellant Content Exhaust Velocity

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(meters/sec.)Chemical Nitrogen Tetroxide/Hydrazine 2,775Chemical Oxygen/Ammonia 2,795Chemical Chlorine Tri Flouride/Hydrazine 2,883Chemical Oxygen/RP-

12,942

Chemical Oxygen/UDMH 3,040Chemical Oxygen/Hydrazine 3,069Chemical Flourine/Ammonia 3,501Chemical Flourine/Hydrazine 3,560Chemical Oxygen/Hydrogen 3,834Chemical Flourine/Hydrogen 4,021Electrothermal (Arc Jet) Ionized Hydrogen 19,613Electrostatic (ion) Cesium Electrons 196,133Electromagnetic (Plasma) Hydrogen Plasma 196,133Fission Uranium (Reactor Mixture) 11,400,000Fission Uranium to Iron 14,800,000Fusion Hydrogen to Helium 35,400,000Fusion Hydrogen to Iron 40,600,000Annihilation Matter/Antimatter 2.99x108

Figure 2. Exhaust velocity caused by various propellants.

2. THEORY OF ELECTRON - POSITRON PROPULSION

This craft will utilize the energy released form electron - positron annihilations through inelastic collisions of the ’s released from the matter - antimatter fusion. Electron and positron beams will be targeted so as to meet at the focus of a parabolic dish placed at the extreme end of a spacecraft. The particles will annihilate with each other releasing the average of two ’s per electron - positron pair.

These two high energy photons are emitted on trajectories 180o apart from each other. The photon ejected towards the craft collides with the craft’s parabolic dish transferring a component of it’s momentum in the forward direction. The photon will be recoiled of the dish transferring all of its momentum in the forward direction. This is because a parabola reflects all material perfectly that was emitted from it’s focus.

The photon ejected away from the craft will collide with the negative axis of the dish. This will initially produce a negative transfer of momentum. The photon will recoil from the dish, transferring all of it’s momentum in the forward direction of travel. The combination of the momentum transfer of photons will propel the space vehicle to relativistic velocities.

3. CONSERVATION OF MOMENTUM AND THEORETICAL MAXIMUM VELOCITIES

The method stated converts e e annihilation energy in to thrust through inelastic collisions of gamma rays. Starting from a convenient inertial frame (e.g. our solar system) the velocity profile of a space craft using an APD with perfect efficiency can be described as a succession of Lorentz boosts.

3.1 Velocity boosts using Lorentz Transformations

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Using Lorentz Transformations we can relate the velocities of the craft between the inertial frames of the craft and of an observer on earth. As the spacecraft receives continual boosts from the two rays, the craft successively jumps from one inertial frame to a new one.

Assume that the spacecraft is initially moving at velocity V with respect to our inertial frame. A small amount of dm is annihilated creating two ' ( , )s per e e pair. Because of this, the rocket loses mass while increasing its velocity to V dv . The Lorentz transformation relating the velocities between our inertial frame and the crafts is

vv Vv V c

''1 2 (3.1)

where v’ is the velocity of the spacecraft measured in the inertial frame traveling at velocity V with respect to our inertial frame. Furthermore the change in the rocket velocity dv' , in the spacecraft’s inertial frame can be related to the change in velocity observed in our inertial frame. By using eq. 2.1 the relationship between dv and dv ' is

dv vc

dv

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2 ' . (3.2)

where v is the instantaneous velocity of the spacecraft.

3.2 Conservation of Momentum

The change in velocity dv ' of the spacecraft’s inertial frame can be calculated using conservation of momentum. If a spacecraft of mass M annihilates a mass dm (e.g. an e e , pair) it will emit the two photons each with a momentum p dm c' 2 . Conservation of momentum in the spacecraft’s inertial frame can be written as

0 Mdv p' ' . (3.3)

Since the photons are ejected at various angle from the parabolic focus, the component of momentum transferred to the dish in the forward direction must be calculated. This can be shown as

cos cos

12

2 d (3.4)

For the photon ejected towards the positive axis of the dish a factor of one can be added to compensate for the hard recoil when the photon is reflected from the dish. This average is shown as

4

11

2

2

cos d (3.5)

The gamma ray ejected towards the negative axis of the parabola will also have a factor of one added to it’s component of momentum transferred in the forward direction. For this craft the dish will extend to a value of 5 6 . This average will be expressed from 2 5 6to and multiplied by two to account for the momentum transferred along the fourth quadrant of the parabolic dish.

1 22

52

cos d (3.6)

Equations (3.5) and (3.6) are then added together, giving a value of 2 318. as the total momentum transferred along a parabola extending from 5 6 5 6 to .

Taking this average into account we can write eq.3.3 as:

0 2 318 Mdv dm c' .(3.7)

Solving for dv' , we find that dv dm M c' .2 318 . Using the relationship dm dM,we can rewrite dv' as

dv dMM

c 2 318. . (3.8)

By combining equations 3.2 and 3.8 we can write the change in the spacecraft’s velocity (measured in our inertial frame) as a function of the mass expended by the spacecraft.

dv vc

dMM

c

1 2 3182

2 . . (3.9)

This differential equation can be solved by separation of variables.

dvv c

c dMM

v

M

M

i

f

12 318

2 20 . . (3.10)

Integrating both sides we obtain

ln lnc vc v

MM

f

i

2 (3.11)

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A running parameter can then be defined that relates initial and final masses of the rocket M Mf i 1 , where 0 1 Substituting this into eq. (2.8), we get

v c xx

11

(3.12)

where

x MM

i

f

2 318 2 31811

. .

(3.13)

The relationship between velocity as a fraction of the speed of light (beta) and the fraction of the spacecraft used as propellant (alpha) can be graphed as,

Although this type of performance is very tantalizing, it is also very difficult to obtain. In order to obtain this maximum value of velocity, the propulsion system must transform one hundred percent of the energy released from the matter - antimatter fusion into thrust. The process of transforming that much energy into thrust, although theoretically is not impossible, is very difficult. It is also important to note that an antimatter propulsion system using only one thousandth of the energy created within the annihilation can still offer significant increases in rocket performance. The amount of the energy utilized for thrust is dependent upon the manner in which the propulsion system combines particle antiparticle pairs and how the resultant particles are captured and focused.

4. ENGINEERING AND MECHANICS OF THE SPACECRAFT

Figure 3. A graph of a perfect APD relating fraction of the speed of light (beta), and fraction of the spacecraft used as propellant (alpha).

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As stated earlier, the velocity that this type of craft can obtain is proportional to the momentum transferred from ’s to the spacecraft. The efficiency at which the spacecraft’s mechanical parts can do this will determine the craft’s actual velocity profile. The mechanical systems most related to the APD’s efficiency are as follows. The method of confining the propellant to storage tanks, the particle beam delivery system, the method of annihilation, and the method of reflecting ’s and electrons.

4.1 Propellant storage

The storage of antimatter propellant is much more complex than that of conventional chemical propellants. In order to keep the electron - positron pairs from annihilating in storage they must be stored separately. Unfortunately this contributes to two problems.

The first is that the similar charges of the particles repel each other. This repulsive force makes it very difficult to store the propellant in a concentrated area. The second problem is caused by the energetic nature of the positrons. If the positrons were to escape storage, they would annihilate with electrons from whatever object they collided with.

These problems lead to the use of magnetic fields to contain the propellant. In this type of storage, the magnetic field acts as the wall of the propellant tank, ending the problem of positrons annihilating with the electrons in the wall of the storage tank. Unfortunately using an electromagnetic field is in this manner is complicated, and no current system is capable of confining kilogram quantities of antimatter.

Fortunately extensive research is being done in this area by Dr. Gerald Smith of Pennsylvania State University, for storing nanogram quantities of antiprotons using Penning traps. A Penning trap uses electromagnetic fields and lasers to confine charged particles and plasma’s into small areas.

In this type of a magnetic bottle, electromagnetic fields are used to confine propellant. The lasers are used to decrease the kinetic energies of the particles within the trap. The photons within the lasers are used to collide with the oscillating electrons and positrons, at the moment when the velocity vectors of the photons and particles are anti-parallel. This collision decreases the momentum of the particles within the penning trap, making them easier to store. For the benefits of this projects it will be assumed that an augmented extrapolation of a penning trap can be built.

4.2 Propellant Annihilation

Although it would seem logical to design the particle delivery beams next, the intended velocity of propellant entering the parabolic dish must be know first.

The mechanics of electron-positron annihilation described in (section 1.2) are simple and will always produce the similar results. The problem with efficiency has nothing to do with the mechanics of annihilation, but with forcing all of the propellant to

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annihilate. Because electrons and positrons are such small point particles, they can pass within hundredths of a centimeter and not annihilate. Thus if the beams are not aligned perfectly, which would be very hard if not impossible to do, not all the propellant will annihilate. Therefore a new approach must be taken.

To gain maximum efficiency, the two beams are steered using electric fields. Instead of sending the propellant through the delivery beams at high velocities, the propellant will be sent at low velocities. This will allow the electronegativity of the electrons and positrons to capture each other, forming atoms of positronium.

Positronium is a unique type of atom. It has no neutrons or protons, only electrons and positrons. The electron and positron orbit each other around the center of mass of the system. Since each particle has the exact same mass, the center of mass is directly in the center of the atom. From the second the positronium atom is formed the orbits of the electrons and positrons begin to decay. This decay lasts for 1.25x10-10s, ending when the electron-positron pair annihilate.

Since the particles will be traveling at such low velocities, the kinetic energies of the particles of propellant will not be enough to overcome the force of attraction between the electrons and positrons. This allows for particles of propellant to be offset by small distances and still be captured by each other to annihilate. This makes it possible for beams to be installed with tolerances for error.

4.3 Particle delivery system

The method of delivering electrons and positrons to the focus of the parabolic dish are less complex than that of storage. Physicists have been confining antimatter and normal matter into beams for decades at laboratories such as Fermi and Cern Laboratories.

The beams for this craft must interface properly with the propellant confinement chambers to ensure that no propellant is lost when the particles leave the magnetic storage fields. The propellant must also not be accelerated beyond a certain velocity. This low velocity will decrease the mass of magnets aboard the ship, and will help to ensure that the electrons and positrons capture each other to form positronium.

Both electron and positron beams will utilize quadrapole magnets to confine the propellant beams. The magnets will be of opposite polarity to confine the opposite charges of both propellants.

4.4 Momentum transfer form gamma rays and electrons

As described earlier, the heart of this system relies in the collisions and hard recoil of 0.511 MeV photons over a parabolic surface. The collision with the dish causes a component of momentum to be transferred to the spacecraft, in the direction of travel. This momentum transfer amounts to 200 % of the total momentum of the photon.

This is difficult because the energy of the photons created in electron-positron annihilations do not allow for collisions on hard surfaces. The wavelength of the photon

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can be treated as the cross section of a particle. The wavelength of these photons is so small that the photons will travel though the crystal lattice spacing of any known structure and diffract of the individual atoms of the structure.

The wavelength of a 511 KeV gamma ray, such as would be created in a Electron Positron Annihilation, can be expressed by De Broglie’s matter wave equation of wavelength,

hceV

(4.1)

where h is Planks Constant in eV.s. By using this equation it is determined that the wavelength of a 511 KeV gamma ray is 0.00243 nm. Unfortunately This is much smaller than the lattice spacing of most practical solids.

Therefore in order to reflect the photons, the energy of the photons must be reduced. This must be done in a manner that allows for the energy lost by the photons to be absorbed by the spacecraft.

Upon research in this area it was discovered that electrons at rest with respect to the spacecraft’s inertial frame have a wavelength equal to that of 0.511 MeV gamma rays. This was found using Comptons wavelength equation.

' cos 0 1hm ce

(4.2)

from (Serway, R., Moses, C., Moyer, C., (1997). Modern Physics, Second Edition. Orlando Florida, Harcourt Brace and Jovanavich).

This similarity between the wavelength of the electrons and photons, allows collisions to occur with great frequency between the photons and electrons. The collisions between the ’s and the electrons will transfer energy from the photons to the electrons. In theory if a shower of ’s were sent through a free electron gas cloud, the electrons would absorb energy from the photons. When the photons exited the opposite side of the cloud, their energy would have been decreased and their wavelengths would have increased, allowing the photons to collide with the lattice structure of the parabolic dish. After collision the photons will be reflected in a hard recoil

The energy transferred to the electrons would be in the form of kinetic energy. This would cause the electrons to move in the direction of impact, out of the cloud and into the dish, like the photons. Unlike the photons the electrons have a mass which would cause degradation of the shield. In order to counter this a magnetic field in the form of a z-pinch focused at the apex of the dish will capture the electrons, absorb their impact, and reflect them in the same manner as the photons. The summation of the momentum transferred to the dish from initial impacts and hard recoil of both the photons and electrons will cause forward movement of the spacecraft.

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