The Relativistic Rocket

The most reaching proposed method of propulsion, project Daedalus, can in principle send a spacecraft to at best about 10% the speed of light. It is clear that something more robust is required to explore a star system, particularly if it is more than 10 light years from our solar system. The specific impulse required to reach a significant percentage of the speed of light must be close to the speed of light divided by one-gee. The ideal case would be for s = c/g ~ 3 x 107 sec. In this case the rocket plume is composed of photons. Further, these photons must be generated by some energy source that converts half or more the initial mass of the craft into photons. This obviously requires an application of E = mc2 to convert a higher percentage of mass to energy than is currently possible with nuclear energy.

Currently a direct conversion of matter to energy requires that matter be annihilated by its interaction with antimatter. Antimatter is that elixir of science fiction fame, such as the StarTrek "antimatter pods." Antimatter is a real aspect of physics, where its prediction came from Paul Dirac with his quantum equation for spin ^h particles such as the electron. The Dirac constant h = 6.67 x 10~34 J-sec is a quantum unit of angular momentum or action. Such particles were known to be different than particles with integer h spins. Wolfgang Pauli laid down how the wave functions were antisymmetric, which meant that only one of these particles, called a fermion, could exist in a single quantum state [7.1]. A consequence of this is the selection mechanism for electrons in an atom. In each atomic shell only two electrons may exist, where they do this by having oppositely aligned spins. The difference in spin states is the quantum state difference that permits these electrons to exist in the same atomic quantum state. For this reason chemistry is possible, for otherwise all electrons would drop equally into a minimal energy state. Dirac derived a relativistic version of the quantum wave equation for the electron and found that for an electron created from a sea of virtual quanta there was a corresponding positive hole created. This positive hole was identified as the positron or anti-electron. Its quantum numbers, charge, lepton number, etc, are opposite from that of the electron. It was experimentally found by Sanderson in 1932 that a 7-ray which impacts a heavy nuclei may cause the generation of an electron and its corresponding opposite, the positron. The photon, which has no quantum numbers for the electron or positron must create both of them so there is no net creation of these quantum numbers. Thus was born the notion of antimatter, which found its way into popularizations and science fiction in the 1960s.

Every elementary particle that exist has its antiparticle, except for the photon and some other neutral bosons. There is some debate as to whether the neutrino is its own antineutrino in the form of Majorana neutrinos. A proton has its corresponding antiproton. Indeed antihydrogen atoms with a positron bound to an anti-proton have been generated. Antiparticles are today a generic feature of elementary particle physics. So the science fiction approach to pump matter and equal proportions of antimatter into some sort of chamber appears to be the simplest "first order" engineering model for some 100% conversion of mass to energy. Of course there are some obvious problems here. A proton and antiproton will produce two photons with an energy ~ 1 GeV. A method for containing these photons and directing them is an obvious complexity. An ordinary mirror simply will not work. This problem will be addressed later.

Another problem is that antimatter is not at all prevalent in the universe. Due to something called CP violations at ~ 1 TeV, far below unification energy of 1012 TeV, the universe in its expansion shortly after the big bang resulted in an excess of particles over their antiparticles. CP symmetry means that a particle state transformed by the product of two mathematical operators to have an opposite charge as well as a parity change, or change in the handedness of a wave function, is equivalent to changing the time direction of the particle CP = T-1. For more reading on quantum field theory see A. Zee's book [7.2]. This symmetry is slightly violated in the universe by a process that operated in the early universe. This resulted in a small excess of matter particles over their antiparticle complements. So there is no "wellspring" of antimatter. Antiparticle must be generated by imputing lots of energy into a process, such as the high energy collision of two protons with p + p ^ 3p + p, where p is an antiproton. Some of

Fig. 7.1. Schematic of third law of motion concept of photon rocket.

the energy of this interaction goes into the generation of identical pairs of particles and their antiparticles, where upon the antiparticles are then magnetically bottled. This is a very expensive process that must be done with high energy accelerators. Further, at the end of the day a tiny percentage of the energy input into these machines ends up converted to antimatter. This is a high energy analogue of the "hydrogen economy," where hydrogen as a fuel must be extracted from water, by electrolysis, with the input of energy from some other source. So this is an obvious difficulty with obtaining enough antimatter required for a relativistic photon rocket.

In spite of these problems we will press on with the basic concept of the photon rocket. The issue of high energy confinement or directing energy will be discussed later in this chapter. The problem of obtaining antimatter will be discussed in the Chapter 9. We assume here that these problems have been solved. By the first law of relativistic motion the acceleration of this rocket must be measured in an inertial reference frame. The time as measured by observers on the inertial reference frame is denoted by t, called the coordinate time, and the time as measured by a clock on the accelerating rocket is the proper time and is denoted by T, previously called s. The acceleration of the rocket is denoted by g. The distance the spacecraft has travelled at any time is d, the velocity is v and 7=1/— (v/c)2.

The four momentum of a body in flat spacetime, such as a rocket, is

The four-momentum has the spatial momentum and the energy. The energy the rocket before it starts to accelerates has the initial energy

where M is the fuel mass and m is the payload mass. Once the fuel, presumably matter plus anti-matter, is used it is converted to photon energy plus the final energy of the system

where Eph is the energy of the photons generated. Conservation of energy tells us that Ei = Ef and so

Similarly there is a conservation of momentum. Before accelerating the total spatial momentum is zero in the Earth frame,

After the fuel is converted to energy, the final spatial momentum is that of the ship plus that of the photons directed in the opposite direction

By conservation of momentum Pf = Pi, and so

Eliminating Eph from these two conservation equations gives

and the fuel to payload ratio is then

Now appealing to the equations for an accelerated reference frame with

this ratio is then

So in order to accelerate to the velocity v under a constant acceleration g this is the required fuel/payload mass ratio. Now as a practical matter rockets are efficient for this ratio = 10 or so. The specific impulse is s = c/g = 3 x 107 sec, and so s ln(M/m + 1)= T = 7.2 x 107 sec . (7.12)

So the rocket can accelerate for about two years at a g = 10m/sec2. Put this into the equation for the gamma factor and velocity

which is pretty fast. For a low gamma rocket a ratio M/m = 2 will accelerate at g = 10m/s2 for a proper time s log(M/m +1) = T ~ 3.3 x 107 sec , (7.14)

which is approximately 1.05 year to reach a velocity v = .76c at one gee. The Lorentz factor is 7 = 1.5 that gives the coordinate time on Earth t = (c/g)sinh(gT/c) or t = 3.88 x 107 sec or 1.23 yr, which is longer than the proper time on the craft. A velocity v ~ 0.8c is sufficient for sending a probe to a star within a radius of 50 ly.

Of course this assume a 100% efficient motor! Thermodynamics and engineering ugliness creeps into the issue. Yet it is apparent that if we could generate antimatter by some means, quantum black holes, magnetic monopoles or ... that a relativistic probe could be sent to some of the nearby stars. Also the use of rocket stages will increase this efficiency some, yet the practical limit is about 3 or maybe 4 stages. Also this does not assume the rocket decelerates to its destination. For that consider half the M/m for the midway trip and then compute the same for the return trip. For a probe only a one way trip is considered.

It is interesting to note a spacecraft that accelerates at one-gee will reach velocities very close to the speed of light relative to the Earth in a very short time. The following chart illustrates this

It is apparent that a clock on the continually accelerating craft will tick at a rate far slower than a clock on Earth and the craft can reach enormous distances in a time frame much smaller than time on Earth. Low gamma craft cut off at 7 < 2 for this may be the practical limit for any probe designed to send information back to Earth within a time frame reasonable on Earth. Higher gammas do not produce significantly reduced time frames for a return message. For much higher gammas the time rate on Earth races far ahead of that on the craft. Most nation states, civilizations or empires have a life that is measured in a few centuries. A craft that is sent to higher gammas to some very distant destination will transmit its data to an increasing number of future generations here on Earth. To engage in such a program would require that humanity settle its affairs according to some long term project of sustainability over this time period. There is no historical precedent for this. In other words for higher gamma rockets it is likely that by the time the mission results are sent back in a signal back to Earth, civilization may well exist in some other paradigm, such as a dark age, where nobody is here to receive the message. Curiously the seeds of a dark age may already be germinating in our age as seen with resource depletion issues and the current rise in the social interest and power of religious beliefs.

It may be possible to send crews on such higher gamma craft, but the information they garner may be held only by them. For exceedingly high gammas this is almost certainly the case. If such craft could be arranged the crew could reach space beyond our galaxy with 11 years of their proper time, and reach the observed boundary of the entire universe within about 25 years of ship time. Yet what ever knowledge they obtain is theirs, for by this time conditions on Earth will have evolved beyond anything predictable, and in the latter case Earth will have been toasted by death of the nova stage of the sun. The universe when it comes to such space travel is similar to a black hole: Those who wish to travel to such heights will exit permanently the society they leave behind. Of course such a space mission could only be prepared by some exceedingly wealth class of "lunarians," who have technologies far beyond those considered here, which are in turn technologies beyond today's. Such craft involve ideas such as the Bussard ramjet that can consume vast amounts of material coming in front of the craft. Such architectures are outside any consideration that could ever be executed by any reasonable plenipotentiary structure.

It is most likely that a photon rocket will have an acceleration less than one-gee [7.3]. For a proper time T the value of 7 is 7 = cosh(gT/c) and for T = (c/g)a cosh(Y) then it takes a proper time T = 3.95 x 108 sec or 12.5 years for g =1 m/sec2 to reach 7 = 2, and T = 1.98 x 108 sec or 6.3 yr for g = 2 m/sec2 to reach the same velocity. The coordinate time as measured here on Earth is t = (c/g) sinh(gT/c) are then 5.2 x 108 sec or 16 years for g =1 m/sec2 and about 8 years for g = 2 m/sec2. The distance travelled may be shown to be d = (c2/g)(cosh(gT/c) - 1),

which is 9 x 1016 m or about 9.5 light years for g =1 m/sec2 and about half that for g = 2 m/sec2. A computer run of various accelerated missions that reach y = 2 are seen in equation 7.16 below. A spacecraft accelerated to Y = 2 sent on a mission 20 light years distant a 1 m/s2 craft will take ~ 33 years of coordinate time to reach the destination, while a 10 m/sec2 craft will take 21.3 years to reach the destination. This shows a rocket with a higher acceleration does not produce great savings over a lower accelerated rocket. For very high accelerations to y = 2 the limiting time is obviously 20 years. High gamma rocket produces less and less mission time as measured on Earth compared to low gammas.

g (m/s2)

T (yr)

t (yr)

d(iy)

1.0

12.54

16.49

9.525

2.0

6.271

8.248

4.762

3.0

4.181

5.499

3.175

4.0

3.136

4.124

2.381

5.0

2.509

3.299

1.905

6.0

2.090

2.749

1.587

7.0

1.792

2.356

1.360

8.0

1.567

2.062

1.191

9.0

1.394

1.833

1.058

10.

1.255

1.650

These results illustrate the length contraction and time dilation of relativity. An observer watching a rocket accelerate away will find that the time measured on the rocket's clock is slowing down. From the perspective of an observer on the rocket the time required to travel a distance is smaller than what her compatriots on Earth observe. Similarly, the distance between two points along the direction of motion is reduced according to the astronaut. This length contraction means that the time required to traverse the distance between two points is reduced. For a y = 2 relativistic rocket this distance is reduced by which means that the corresponding proper time required to travel that distance is also halved. From the perspective of an observer on Earth there is no change in the distance between two points, but the rocketship is observed to be shortened by half and its clock is seen to run at half speed. Figure 7.2 illustrates how an observer on a

Fig. 7.2. The length contraction of a lattice of points seen by a relativistic observer.

rocket with 7 = 2 will see a lattice of points in space. This illustrates the "clocks and rods" view of relativity. An observer who is accelerating will see the distance between these lattice points continue to shrink. The distance between lattice points asymptotically approaches zero as the velocity v ^ c.

A minor consideration needs to be addressed as well. The rocket travelling through space at a 7 = 2 had best not run into a small particle. Even a microgram mass particle that runs into the spacecraft will have a devastating impact. The energy released would be ^ mc2, which would be 9 x 107 J, which is equivalent to about 40/6s of explosives. Even ordinary hydrogen in interstellar space would cause damage, as it would become a radiation flux on the ship. Thus the craft would be best made needle shaped. A radiation field in front of the ship might be used to ionize the hydrogen and a magnetic field used to deflect the charged particles. Potentially some form of vaporizing beam needs to be employed in front as well to eliminate threats due to larger particles such as dust.

This is the basic theory for the relativistic photon rocket. It is apparent that there are some technical problems here. The first big one is how are photons with energies nearly a billion times the energy of optical photons to be harnessed or channelled into a thrust? A large flux of these photons will destroy any material they run into. Obviously something must absorb these photons or scatter them into lower energy photons. If the matterantimatter interactions involve electrons and positron an obvious approach would be to Compton scatter these photons. Here the photons scatter off of charged particles to yield some of their energy to this particle and so the photon has a lower energy. This might in a rough way be extended to proton-antiproton interactions as well, even though this ignores processes such as the production of n mesons.

The four momentum of a photon is hcv(c, n), where h is Planck's unit of action in quantum mechanics, v is the frequency of the photon, and n is the direction the photon propagates in space. The four momentum of a particle of mass m is (E, mv). The momentum of the photon and particle is conserved in the scattering process. The energy of the photon and particle changes by hv + E = hv' + E', (7.18)

and similarly the photon and particle momentum change by hcvn + mv = hcv'n' + mv' . (7.19)

For simplicity the initial momentum of the charged particle is considered to be zero mv = 0. By squaring the momentum of the electron after the scattering with n • n' = cos(0) the change in the wavelength of the photon is shown to be h

The term h/mc is the Compton wavelength of the particle, which for the electron is 3.86 x 10"11 cm. The wavelength of radiation produced by e — e+ annihilations is ~ 10"10 cm, and so an ionized gas with electrons will multiply scatter these photons. If these electrons are dense enough multiple scattering will increase their wavelength by a factor of around 103. Similarly a gas of nucleons will scatter ~ GeV photons to the ~ 10 MeV range in energy, which is considerably more modest.

A possible way to contain the energy of an matter-antimatter interaction is to consider the following reaction

In this way a substantial amount of the energy of the interaction is carried off by the tritium ion. If we assume that the initial four momenta of the 2He4 and p are considered to have zero spatial momentum then

for M the mass of the initial He + p and m the mass of the tritium nucleon. It is easy to show that which gives a y = 1.13 (v = .47c) and the tritium has a kinetic energy K = (7 — 1)mc2 or .4mpc2. This is 8% conversion of the initial mass energy into the kinetic energy of a charged nuclei. There are still photons with an energy of 1.61 GeV, which are very high energy.

Similarly consider the reaction

Much of the kinetic energy is transmitted to the electron. For the electron Y ~ 1000 and the energy of the gamma ray photon is ~ 1.5 GeV. This is a 25% conversion of the mass-energy into the kinetic energy of a charged particle. So there is still considerable energy not given to the photon, where it is evident that only for y for m/M ^ 0 do we still get half the energy output in a photon with a 1.0 GeV energy. There is a further problem that a neutron and an antiproton have no electrical attraction for each other and so do not readily combine.

So consider this process as something that produces a gas of photons with some energy with some reaction mass particles. With the 2He4 + p ^ T + y process there would then exist a gas of photons and tritium nuclei. Under multiple scatter of the nuclei by these photons there will exist a thermodynamic equilibrium. In other words the energy of the photons will become distributed equally amongst the tritium nucleons. This is the equipartition theorem of statistical mechanics. So the tritium nuclei with a mass-energy equivalent of 3 GeV will come to equilibrium with an energy of 1.61 GeV. This means that these nucleons have an energy that is 54% of their rest mass. The temperature of this gas of tritium nucleons would be around ~ 1012 K. The average velocity of the outward plume would then be v ~ 0.58c. Some process must permit the existence of this plasma, so that the actual rocket material does not come in contact with energies and temperatures at this level. Obviously this plasma must be magnetically contained. The problems here are quite evidently manifold.

From here the pure photon rocket has been modified into a rocket with a high velocity plume of massive particles. A rocket with a four momentum P = (mc2, 0) that shoots out some 5m of particles with a velocity u will then have the four momentum P' = (5j(m — 5m)c2, 5(yv)(m — 5m)c), Y = Y(v) due to the plume four momentum 5P = (y'5mc2, — y'5uc), with y' = y'(u), so that mc

The momentum part gives the differential expression

m where it is easier to convert this to an integral in dy with f m /

with the result

The gammas and mass ratios of interest are displayed in the chart below

These gammas and mass ratios are comparable to pure photon rocket. The accelerations, distances and times of flight for the pure photon rocket derived above hold for this modified version as well.

A spacecraft that annihilates matter is commonly thought to use antimatter, as indicated above. This antimatter is then contained in "antimatter pods," to use the term from Star Trek, which are essentially tanks or bottles. Clearly this antimatter must not touch the walls of any container, for if it does an explosion will result. In the case of antiprotons, which are negatively charged particles, a magnetic field can bottle these particles as the moving charged particles spiral around the magnetic field lines. If the magnetic field pinches off at the ends of the tanks this acts as a sort of mirror. However, magnetic fields do not pinch off completely, so the ends will be leaky. Further for 7 = 2 it is clear that the mass ratio is larger than 2. This requires bottling up lots of antiprotons. The mutual electrostatic repulsion between these particles will be enormous and impossible to hold. Thus a spacecraft that consists of ~ 1/4 antimatter will have to contain it in a neutral form, such as anti-hydrogen — an antiproton with a positron around it. However, this is neutrally charged and a magnetic bottle will not work. It will be hard to bottle up anti-hydrogen so that it does not touch the walls of a container.

In quantum field theory there is a conserved quantity called the baryon number B. A proton has a baryon number B = 1 and an antiproton has B = —1. Thus a process p + p ^ 27 will conserve a net zero baryon number. Baryon number, which is based on more fundamental quantum numbers associated with quarks, appears to be a strictly conserved, and no experiment has observed any violation of B. However, black holes will violate baryon number, and some theoretical physics with Grand Unified Theories (GUTs) indicate possible violations of B as well. So it is possible that with GUT physics or with quantum black holes that matter might be directly converted to energy by violating B. These prospects will be discussed in Chapter 9 on the technical requirements for starcraft.

The next approach for star travel to be discussed here is the photon sail. This approach does not suffer from some of the technical difficulties of the relativistic rocket. So it would appear that the photon sail is more likely to first travel to a star. However, the photon sail requires delicate construction in space on a colossal scale. This poses difficulties for the photon sail that might be just as daunting as those with the relativistic rocket. If the technical difficulties discussed above can be solved the relativistic rocket would be much more modest in size, maybe not any larger than a standard launch vehicle.

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