Relativistic rocket

A relativistic rocket is a hypothetical rocket- propelled spacecraft whose airspeed is so close to the speed of light that significant relativistic effects occur. When one speaks of significant relativistic effects, it depends on the context, but one can roughly say that the vehicle moves at least at half the speed of light (0.5 c). At 0.5 c the Lorentz factor γ (gamma), and with it time dilation , mass factor and length contraction all have the value 1.15. At these and higher speeds, relativistic physics are required to describe the movement. For slower rockets, Newtonian physics and Ziolkowski's basic rocket equation are sufficiently good approximations.

General

A missile is defined as having all of its reaction mass, energy and engines with it. Vehicles with buzzard ramjet , sun sails , maser or laser- electric propulsion are therefore not rockets.

In order to achieve relativistic speeds, advanced space propulsion methods are necessary that are not yet sufficiently developed today. With a nuclear pulse drive , using technology known today, 0.1 c could theoretically be achieved, but this would also require numerous technical developments. The Lorentz factor γ at 0.1 c is 1.005. A time dilation of 1.005 that occurs at 0.1c is too small to have any meaningful effect. An interstellar rocket that moves at 0.1c is therefore to be regarded as non-relativistic, and its movement can be described with almost sufficient accuracy using Newtonian physics.

As a rule, relativistic rockets are discussed in connection with interstellar space travel , because they would usually need long distances to accelerate to such high speeds. They also appear in thought experiments such as the twin paradox.

Relativistic basic rocket equation

As with the classical basic rocket equation , the aim here is also to calculate the speed increase Δv that a rocket can achieve if the specific momentum and the ratio between launch mass m ₀ and empty mass m _{1 are} given. The specific momentum has the dimension of a speed and indicates the momentum that is transferred to the rocket by the ejection of a certain amount of fuel, divided by the mass of this amount of fuel. ${\ displaystyle I_ {sp}}$

Specific impulse

The specific momentum of relativistic rockets is equal to the effective outflow velocity, although the non-linear relationship between velocity and momentum and the conversion of mass into energy must be taken into account. The two effects cancel each other out. The following applies:

{\ displaystyle I_ {sp} = v_ {e}}

Of course, this only applies if the rocket has no external energy source (for example a laser beam from a space station. In this case, the pulse transported by the laser beam would also have to be included in the calculation). If all the energy to accelerate the fuel comes from an external source without any additional momentum being transmitted at the same time, the relationship between the effective exhaust velocity and the specific momentum is as follows:

{\ displaystyle I_ {sp} = {\ frac {v_ {e}} {\ sqrt {1 - {\ frac {v_ {e} ^ {2}} {c ^ {2}}}}}} = \ gamma _ {e} \ v_ {e},}

(where is the Lorentz factor ). ${\ displaystyle \ gamma}$

If there is no external energy source, then the relationship between and the proportion of the fuel mass that becomes energy is also of interest . Assuming there are no losses is ${\ displaystyle I_ {sp}}$ ${\ displaystyle \ eta}$

{\ displaystyle \ eta = 1 - {\ sqrt {1 - {\ frac {I_ {sp} ^ {2}} {c ^ {2}}}}} = 1 - {\ frac {1} {\ gamma _ {sp}}}.}

The inverse relation is:

{\ displaystyle I_ {sp} = c \ cdot {\ sqrt {2 \ eta - \ eta ^ {2}}}.}

In this table, the components converted to energy and the corresponding specific impulses are shown for some fuels, based on the speed of light (losses not taken into account, unless otherwise stated):

fuel	${\ displaystyle \ eta}$	${\ displaystyle I_ {sp} / c}$
Electron - positron - pair annihilation	1	1
Proton - antiproton - pair annihilation , only with charged pions	0.56	0.60
Electron - positron - annihilation single hemispheric absorption of gamma radiation	1	0.25
Electron - positron - pair annihilation with hemispherical Compton scattering	1	> 0.25
Nuclear fusion : H to He	0.00712	0.119
Nuclear fission : ²³⁵U	0.001	0.04

Delta v

To simplify the calculations, we assume that the acceleration in the reference system of the rocket (the relativistic self- acceleration ) is constant during the acceleration phase . However, the result also applies to variable acceleration as long as is constant. ${\ displaystyle I_ {sp}}$

In the non-relativistic case, the (classical) Ziolkowski rocket equation gives that

{\ displaystyle \ Delta v = I_ {sp} \ ln {\ frac {m_ {0}} {m_ {1}}}}

Assuming constant acceleration , the duration of the acceleration phase is: ${\ displaystyle a}$

{\ displaystyle t = {\ frac {I_ {sp}} {a}} \ ln {\ frac {m_ {0}} {m_ {1}}}}

The equation also applies in the relativistic case, when the acceleration is in the reference system of the rocket and the on-board time, because at time 0 the relationship between force and acceleration is the same as in the classic case. ${\ displaystyle a}$ ${\ displaystyle t}$

If you apply the Lorentz transformation to the acceleration, you can calculate the final velocity relative to the reference system at rest (i.e. to the reference system of the rocket before the acceleration phase) as a function of the intrinsic acceleration in the reference system moving with it and the time in the reference system at rest. The result is ${\ displaystyle \ Delta v}$ ${\ displaystyle t '}$

{\ displaystyle \ Delta v = {\ frac {a \ cdot t '} {\ sqrt {1 + {\ frac {(a \ cdot t') ^ {2}} {c ^ {2}}}}}} }

The related form of movement is also called relativistic hyperbolic movement . The time in the stationary reference system is related to the on-board time according to the following equation:

{\ displaystyle t '= {\ frac {c} {a}} \ sinh \ left ({\ frac {a \ cdot t} {c}} \ right)}

Inserting the on-board time into Ziolkowski's equation and inserting the resulting time in the resting system into the expression for , one obtains the formula: ${\ displaystyle \ Delta v}$

{\ displaystyle \ Delta v = c \ cdot \ tanh \ left ({\ frac {I_ {sp}} {c}} \ ln {\ frac {m_ {0}} {m_ {1}}} \ right)}

The formula for the corresponding rapidity (the hyperbolic areatangent of the speed divided by the speed of light) is simpler:

{\ displaystyle \ Delta r = {\ frac {I_ {sp}} {c}} \ ln {\ frac {m_ {0}} {m_ {1}}}}

Because rapidities, as opposed to relativistic velocities, can simply be added, they are useful to calculate the total value of for multi-stage rockets. ${\ displaystyle \ Delta v}$

Rocket propulsion through matter-antimatter annihilation

From the above calculations, it can be seen that a relativistic missile will likely need to be propelled by antimatter. In addition to the photon rocket, the “beam core” pion rocket is a conceivable type of antimatter rocket with which the speed of 0.5 c required for interstellar flights can be achieved. In a pion rocket, antimatter is stored in the form of frozen anti-hydrogen in superconducting magnetic bottles . (Anti-hydrogen and normal hydrogen are diamagnetic and can therefore be kept in suspension by magnetic fields.) Laser vaporises and ionizes the anti-hydrogen at a rate of a few grams per second. The pion drive could require a superconducting nozzle with magnets of 10 Tesla or more.

For the construction of a pion rocket

Robert Frisbee and Ulrich Walter examined the pion rocket independently and with similar results. Pions (also referred to as pi-mesons) are in the pair annihilation of protons with anti-protons generated. In the magnetic "combustion chamber" of a pion rocket engine, the antiprotons, in the form of frozen anti-hydrogen, are to be brought together with exactly the same amount of normal protons. The resulting charged pions have a velocity of 0.94 c (ie β = 0.94) and a Lorentz factor γ of 2.93, which extends their lifespan to such an extent that they move 2.6 meters through the nozzle before they disintegrate into muons . 60% of the pions are either negatively or positively charged and 40% are electrically neutral. The neutral pions immediately decay into gamma rays. Gamma radiation of this energy cannot be reflected with any known material, but it is subject to Compton scattering . A shield made of tungsten could effectively shield the crew and anti-hydrogen tanks from gamma radiation. The charged pions move in helical lines around the axial electromagnetic field lines in the nozzle and can thus be bundled into a beam that theoretically exits at 0.94 c. If 1 kg of pions were ejected per second, the pion engine would have a thrust of 282 meganewtons, but in real matter-antimatter reactions, 78% of the fuel's mass energy is lost as gamma radiation and the effective jet speed therefore drops to only 0.33 c. Diamagnetism could be used to store anti-hydrogen ice in a superconducting magnetic vacuum bottle. Their temperature would have to be kept below 1 K to prevent anti-hydrogen from subliming and causing annihilation on the vessel walls .

literature

Robert L. Forward: Mirror matter - pioneering antimatter physics. Wiley, New York 1986, ISBN 0-471-62812-3 .
Eugene F. Mallove, et al .: The Starflight Handbook. Wiley, New York 1989, ISBN 0-471-61912-4 .
Marc G. Millis, Eric W. Davis: Frontiers of propulsion science. American Inst. Of Aeronautics and Astronautics, Reston 2009, ISBN 978-1-56347-956-4 , Relativistic Limits of Spaceflight, pp. 455-470.
Ulrich Walter: Relativistic rocket and space flight. Acta Astronautica, Vol. 59, Issue 6, Sept. 2006, pp. 453-461.