A relativistic rocket is any spacecraft that is travelling at a velocity close enough to light speed for relativistic effects to become significant. What "significant" means is a matter of context, but generally speaking a velocity of at least 50% of the speed of light (0.5c) is required. The time dilation factor, mass factor, and Lorentz contraction factor are 1.15 at 0.5c. Above this speed Einstein's physics are required to accurately describe motion. Below this speed, motion is reasonably well described approximately by Newtonian physics.
We define a rocket as carrying all of its reaction mass, energy, and engines with it. Thus Bussard ramjets, RAIRs[1], light sails, and maser or laser-electric vehicles are not pure rockets, though they do use reaction-drives like rockets.
Achieving relativistic velocities is difficult, requiring advanced forms of spacecraft propulsion that have not yet been adequately developed. Nuclear pulse propulsion could theoretically achieve 0.1c using current known technologies, but would still require many engineering advances to achieve this. The relativistic gamma factor ($ \gamma $) at 10% of light velocity is 1.005. Generally a rocket that achieves at least 50% of light velocity is a relativistic rocket. The time dilation factor of 1.005 which occurs at 10% of light velocity is too small to be of major significance. A 0.10c velocity interstellar rocket is thus considered to be a non-relativistic rocket because its motion is very accurately described by Newtonian physics alone.
Relativistic rockets are usually seen discussed in the context of interstellar travel, since most would require a great deal of space to accelerate up to those velocities. They are also found in some thought experiments such as the twin paradox.
The Einstein equations Edit
The relativistic factor is $ \gamma = \sqrt{1 \over 1 - v^2/c^2} $. For example:
speed | $ \gamma $ |
---|---|
0.5c | 1.15 |
0.707c | 1.414 |
0.87c | 2.0 |
0.94c | 2.93 |
0.9999c | 71 |
0.99999c | 224 |
$ \gamma $ is the amount of the relativistic mass increase, the magnitude of the time dilation factor, and the magnitude of the Lorentz contraction of spacetime in front of the interstellar spacecraft. Let $ w $ be the practical velocity of the interstellar vehicle. $ w = v\gamma/c $. with $ v $ is the velocity of the interstellar vehicle.
Newtonian physics for rockets Edit
The Newtonian version of the Tsiolkovsky rocket equation is $ v = v_e \times \ln(m_i/m_f) $, where
- ln is the natural logarithm
- $ v $ is the velocity achieved by a rocket stage at burn-out
- $ v_e $ is the exhaust velocity of the engine products (versus rocket)
- $ m_i/m_f $ is the mass fraction of the rocket
- $ m_i $ is the mass of the completely fueled rocket at ignition
- $ m_f $ is the mass of the rocket after all the fuel mass been expended and the rocket has thus reached burn-out.
The maximum exhaust velocity that a given rocket fuel combination can achieve is calculated as follows. $ v_e $ is the square root of the fraction the fuel mass M that is converted into energy times the velocity of light c.
For example the maximum rocket exhaust velocity that can be generated by a fusion fuel which converts 0.01 (1/100 th) of its mass into energy when it undergoes nuclear fusion is $ \sqrt{0.01}c = 0.11c $. If antimatter is the rocket fuel that is used then $ v_e = \sqrt{1}c = c $. If uranium or plutonium is the rocket fuel then $ v_e = \sqrt{0.001}c = 0.034c $ when the fuel undergoes nuclear fission.
Rocket math Edit
The amount of energy that can be released by converting mass into energy is $ E = mc^2 $. The kinetic energy of a mass of matter in motion using Newtonian physics is $ E_k = {m \times v^2\over 2} $. The thrust of a rocket using Newtonian physics is equal to the mass of the propellant expelled per second times the exhaust velocity of the rocket. This equation applies to all rockets except for the "photon rocket". See the nuclear photonic rocket article for details. The specific impulse of a rocket is the exhaust velocity of the rocket divided by the standard acceleration of gravity g of 9.80665 m/s².
The staging principle applied to interstellar rocket design Edit
The velocity that can achieved by a rocket at burn-out with a mass fraction of 2.72/1 is equal to the exhaust velocity of the rocket. If a rocket has a mass fraction of 5/1 it will achieve 1.6 of the exhaust velocity of the rocket. A mass fraction of 7.39/1 is required to achieve twice the exhaust velocity of a rocket. A mass fraction of 20/1 is required to achieve 3 times the exhaust velocity of the rocket. A mass fraction of 55/1 gives 4 times the exhaust velocity of a rocket, and a mass fraction of 148.4/1 gives 5 times the exhaust velocity of a rocket. These very high mass fractions may be achieved by multiple stage rockets using the principle of staging.
If for example we were to stack three 5/1 mass fraction thermonuclear rocket stages together to make an interstellar launch vehicle, and the exhaust velocity were 0.11c: then theoretically the third stage of the rocket could give its payload a rapidity of 0.52c at burn-out. To convert between rapidity, which is the integrated acceleration & time profile of vehicle measured from a co-moving observer, and speed as seen by an inertial observer, the basic equation is v = c.tanh(R/c) ,where v is speed, c is the speed of light, and R is the rapidity. Thus the 3 stage fusion rocket with a total mass ratio of 125 (5 x 5 x 5) and exhaust velocity of 0.11c is doing 0.486c at burn-out.
The actual burn-out velocity for the ideal multiple stage thermonuclear rocket mentioned above would be 0.486c. However this is not the average trip-speed as the vehicle's acceleration isn't taken into account. For constant high acceleration the distance travelled during acceleration is v/2 times the acceleration time, in the non-relativistic approximation. Taking relativity into account the acceleration distance is s = c^2/a*($ \gamma $-1) where $ \gamma $ is the Lorentz mass-dilation factor discussed above. If the intent of the rocket designer is to actually fly to Alpha Centauri A & B and Proxima Centauri on an undecelerated interstellar trajectory, this would take perhaps 11.6 years plus the acceleration time.
If you want to decelerate at the end of your interstellar journey to actually orbit Alpha Centauri, then using the same mass-ratio as above your maximum interstellar cruising velocity would be perhaps 0.26c, and it would take 16.9 years plus the acceleration and deceleration periods to actually get there. On a round trip trajectory however, the maximum cruising velocity would be about 0.13c, and it would take 33.8 years to reach the Alpha Centauri system plus both acceleration and deceleration time. Realistic staged rockets are probably constant thrust, rather than constant acceleration, and this complicates the computations markedly. In the non-relativistic approximation the distance accelerated at constant thrust is $ x = u.t[1 - ln(r)/(r-1)] $ , where u is the exhaust velocity, t the acceleration time and r the mass-ratio. Counter-intuitively this means that the highest attainable burn-out speed isn't necessarily faster, as the total acceleration time increases with the increasing mass-ratio, but the average speed while accelerating approaches the exhaust velocity. In this case an optimal maximum speed can be computed for a given constant thrust, total travel distance, and exhaust velocity.
Matter-antimatter annhilation rocket design Edit
It is clear on the basis of the above calculations that a relativistic rocket should be a rocket that is fueled by antimatter. The only known antimatter rocket other than the photon rocket that can provide 0.5c needed for interstellar space flight is the "beam core" pion rocket. In a pion rocket antimatter is stored inside superconducting electromagnetic bottles in the form of electromagnetically levitated frozen antihydrogen. Laser beams vaporize and ionize the antihydrogen a rate of a few grams per second. The antihydrogen ions flow through a superconducting steel electromagnetic vacuum pipe. The electromagnetic field is generated by liquid helium cooled coils.
The pion rocket has a superconducting nozzle with electromagnets of 10 teslas or more. Antihydrogen and regular hydrogen are diamagnetic which allows them to be electromagnetically levitated.
The physics of anti-proton–proton annhilation rocket propulsion Edit
Antiprotons will be released at a maximum rate of a few grams per second. Pions, also known as pi-mesons, are produced by proton-antiproton annihilation. The antiprotons will be mixed with an exactly equal mass of regular protons in the form of liquid hydrogen pumped inside the nozzle of a pion rocket engine. All of these pions have a velocity of 0.94c when produced by proton-antiproton annihilation. The pions will also have a relativistic time dilation factor $ \gamma $ of 2.93 which extends their lifespan enough to travel 10 to 1000 meters through the nozzle, before decaying into gamma rays. Sixty percent of the pions will have either a negative, or a positive electric charge. Forty percent of the pions will be neutral. The neutral pions will decay immediately into gamma rays. Gamma rays can be reflected by some materials such as beryllium. They can also be absorbed and blocked by other materials such as lead, steel, gold and uranium. The pion rocket driven starship will need a radiation shield placed between the pion rocket engine nozzle and the crew modules to protect them from the gamma rays. A collimator made of a gamma ray reflector such as beryllium metal is wrapped around the pion rocket nozzle.
This collimator will reflect the gamma rays into a beam at its focus, and then reflect the beam of gamma rays into space. Some additional thrust will be obtained from the radiation pressure of the reflected gamma rays on the gamma ray collimator when they are reflected by it. The charged pions travel in helical spirals around the axial electromagnetic field lines inside the nozzle. In this way the charged pions are collimated into an exhaust jet that is moving at 0.94c. This exhaust velocity is approximately 282,000,000 m/s. if we expelled 1 kg of these pions per second the pion engine would have a thrust of 282,000,000 newtons. This is enough force to accelerate a mass of 28,200,000 kilograms upward at 10 m/s² on the surface of the earth. This is a mass of 28,200 metric tonnes.
Calculating actual thrust, and acceleration Edit
Newton's second law may be used to calculate the acceleration given an interstellar rocket per second by a pion rocket motor. If the pion rocket engine driven interstellar rocket has a mass of 3000 tonnes and it expels 1 gram of charged pions per second at 0.94c then the thrust will be $ V_e \times M \times \gamma $ = 282,000,000 m/s × 0.001 × 3.34 = 952 kN approximately. Newton's second law is F = ma. Solving for "a" gives 952,000 N/3,000,000 kg = 0.31 m/s². To accelerate at 1g, that is, Earth gravity, we would need to expel about 334 g/s of charged pions.
The nozzle design Edit
A pion rocket can operate with a nozzle length of just 1 meter. However, it is appropriate to extract all of the momentum of all the pions before they decay if this can be achieved. To achieve this the rocket nozzle might be a kilometer in length. There is a distinct tradeoff between the efficiency of thruster, and the mass of the rocket engine. A very long nozzle provides some radiation protection to the rest of the vehicle from the ultra-high energy gamma ray radiation produced by the decay of the pions.
The construction, operation, and assembly Edit
The pion rocket propelled starship must be assembled and fueled elsewhere in the solar system very far from the earth, because of the great radiation danger that would be created if the antimatter should accidentally escape confinement from the magnetic bottles, and create a very powerful gamma ray explosion. The starship may be up a kilometer in length and 100 meters in diameter. It may have a mass of 3000 t. It would have a rotating crew habitat with a massive gamma ray radiation shield behind it. An alternative design has been proposed by Charles Pellegrino & James Powell and called "Valkyrie". Instead of the drive system pushing the crew section, it pulls the crew section on very long cables, maximising the radiation protection. The interstellar vehicle, "Venturestar", featured in the recent hit-movie "Avatar" is a modified "Valkyrie" design that also uses a large laser-sail when leaving and arriving in Earth's solar system, but brakes into and accelerates away from the Alpha Centauri system using a matter/antimatter drive, thus minimising the round-trip propellant required.
Shielding pion rocket craft during their operation in space Edit
There would be a massive metallic interstellar dust, erosion, and micro-meteorite shield on the top of the habitat. There may also be a magnetic cosmic ray shield generated around it. There would be an array of high energy lasers, particle beams and some missiles mounted on the starship. These would be used to destroy objects in the path of the starship.
The equipment sensors, and other equipment needed during interstellar space flight Edit
There will be many very high frequency microwave radar transmitter-receivers for detecting objects around the starship. There will be a variety of other instruments also mounted on the starship. Communications will be optical signals using laser beams. There will be a renewable life support system in the crew habitat modules. There will be shuttles and probes propelled by antimatter rockets based in hangars on the starship. The mass fraction of a 1 stage pion rocket driven starship will be 4/1 or 5/1. For a 3000 t 4/1 mass fraction starship, then we will need 2250 t of fuel; 1125 t of it will be liquid hydrogen, and 1125 t of it will be antihydrogen ice in the magnetic storage bottles discussed previously.
The mass-constraints for a 3000 metric tonne mass pion rocket Edit
The mass of the fuel tanks and the pion rocket engine can not exceed 450 t, and the maximum mass of the crew habitat, and any other vehicles based on it will be limited to 300 t. This pion rocket will be capable of accelerating up to 0.5c two times before all the fuel is used. Maximum round trip velocity for a one stage vehicle is is thus limited to 1/4th of light velocity. One way decelerated velocity is limited to 0.5c. One way undecelerated cruising velocity will be about 0.9c to 0.99c for a one stage vehicle. A minimum of a two-stage (4/1) mass fraction pion rocket is required to achieve a round trip cruising velocity of 0.5c. If 4 stages were used a round trip cruising velocity of up to 0.87c is theoretically possible. These figures show that the best starship designs are based on a non-rocket systems such as light sails, interstellar ramjets, interstellar rairs, or laser-maser electric propulsion interstellar vehicles. This is because all of rocketry is limited by the rocket equation, and also limited by the maximum amount of energy that can be released from the rocket fuel.
Sources Edit
- The star flight handbook, Matloff & Mallove, 1989
- Mirror matter: pioneering antimatter physics, Dr. Robert L Forward, 1986