Relativistic rocket

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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 length contraction factor (all these factors equal the Lorentz factor) are 1.15 at 0.5c. Above this speed Einsteinian physics are required to describe motion. Below this speed, motion is approximately described by Newtonian physics and the Tsiolkovsky rocket equation can be used.

We define a rocket as carrying all of its reaction mass, energy, and engines with it. Bussard ramjets, RAIRs,^[1] light sails, and maser or laser-electric vehicles are not 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. The time dilation factor of 1.005 which occurs at 10% of light velocity is too small to be of major significance. A 0.1c velocity interstellar rocket is thus considered to be a non-relativistic rocket because its motion is quite 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.

Relativistic rocket equation

As with the classical rocket equation, one wants to calculate the velocity change $\Delta v$ that a rocket can achieve depending on the specific impulse $I_{{sp}}$ and the mass ratio, i. e. the ratio of starting mass $m_{0}$ and mass at the end of the acceleration phase (dry mass) $m_{1}$ . Subsequently specific impulse means the momentum produced by the exhaust of a small amount of rocket fuel divided by the mass of that small amount of rocket fuel, in an inertial reference frame where the rocket is at rest before using that small amount of fuel. Thus specific impulse is a velocity, as opposed to the common usage of the word as the ratio of momentum and weight (weight would not make much sense in this context).

Specific impulse

The specific impulse of relativistic rockets is the same as the effective exhaust velocity, despite the fact that the nonlinear relationship of velocity and momentum as well as the conversion of matter to energy have to be taken into account; the two effects cancel each other. I.e.

$I_{{sp}}=v_{e}$

Of course this is only valid if the rocket does not have an external energy source (e. g. a laser beam from a space station; in this case the momentum carried by the laser beam also has to be taken into account). If all the energy to accelerate the fuel comes from an external source (and there is no additional momentum transfer), then the relationship between effective exhaust velocity and specific impulse is as follows:

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

where $\gamma$ is the Lorentz factor.

In the case of no external energy source, the relationship between $I_{{sp}}$ and the fraction of the fuel mass $\eta$ which is converted into energy might also be of interest; assuming no losses, is

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

The inverse relation is

$I_{{sp}}=c\cdot {\sqrt {2\eta -\eta ^{2}}}.$ ^[a]

Here are some examples of fuels, the energy conversion fractions and the corresponding specific impulses (assuming no losses):

Fuel	$\eta$	$I_{{sp}}/c$
electron-positron annihilation	1	1
nuclear fusion: H to He	0.00712	0.119
nuclear fission: ²³⁵U	0.001	0.04

In actual rocket engines, there will be losses, lowering the specific impulse. In electron-positron annihilation, the gamma rays are emitted in a spherically symmetric fashion, and they almost cannot be reflected with current technology. Therefore they cannot be directed towards the rear. A simple solution would be to have a gamma ray absorber absorbing all the gamma rays moving in the forward direction, delivering part of the thrust; and letting the rest be emitted without any deflection (therefore with an angle of divergence of 180°), which cuts in half the (average) useful momentum of the gamma rays, resulting in the specific impulse being less of what it would be in the idealized case.

The formula for $\Delta v$

In order to make the calculations simpler, we assume that the acceleration is constant (in the rocket's reference frame) during the acceleration phase; however, the result is nonetheless valid if the acceleration varies, as long as $I_{{sp}}$ is constant.

In the nonrelativistic case, one knows from the (classical) Tsiolkovsky rocket equation that

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

Assuming constant acceleration $a$ , the time span $t$ during which the acceleration takes place is

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

In the relativistic case, the equation still valid if $a$ is the acceleration in the rocket's reference frame and $t$ is the rocket's proper time because at velocity 0 the relationship between force and acceleration is the same as in the classical case. Solving this equation for the ratio of initial mass to final mass gives

${\frac {m_{0}}{m_{1}}}=\exp \left[{\frac {at}{I_{{sp}}}}\right]$

with "exp" denoting the exponential function. Another related equation^[2] gives the mass ratio in terms of the end velocity $\Delta v$ relative to the rest frame (i. e. the frame of the rocket before the acceleration phase):

${\frac {m_{0}}{m_{1}}}=\left[{\frac {1+{{\frac {\Delta v}{c}}}}{1-{{\frac {\Delta v}{c}}}}}\right]^{{{\frac {c}{2I_{{sp}}}}}}$

For constant acceleration, ${\frac {\Delta v}{c}}=\tanh \left[{\frac {at}{c}}\right]$ (with a and t again measured on board the rocket),^[3] so substituting this equation into the previous one and using the identity $\tanh x={\frac {e^{{2x}}-1}{e^{{2x}}+1}}$ (see Hyperbolic function) returns the earlier equation ${\frac {m_{0}}{m_{1}}}=\exp \left[{\frac {at}{I_{{sp}}}}\right]$ .

By applying the Lorentz transformation on the acceleration, one can calculate the end velocity $\Delta v$ as a function of the rocket frame acceleration and the rest frame time $t'$ ; the result is

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

The time in the rest frame relates to the proper time by the following equation:

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

Substituting the proper time from the Tsiolkovsky equation and substituting the resulting rest frame time in the expression for $\Delta v$ , one gets the desired formula:

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

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

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

Since rapidities, contrary to velocities, are additive, they are useful for computing the total $\Delta v$ of a multistage rocket.

Matter-antimatter annihilation rockets

It is clear on the basis of the above calculations that a relativistic rocket would likely need to be a rocket that is fueled by antimatter. Other antimatter rockets in addition to the photon rocket that can provide a 0.6c specific impulse (studied for basic hydrogen-antihydrogen annihilation, no ionization, no recycling of the radiation^[4]) needed for interstellar space flight include the "beam core" pion rocket. In a pion rocket, antimatter is stored inside electromagnetic bottles in the form of frozen antihydrogen. Antihydrogen, like regular hydrogen, is diamagnetic which allows it to be electromagnetically levitated when refrigerated. Temperature control of the storage volume is used to determine the rate of vaporization of the frozen antihydrogen, up to a few grams per second (amounting to several petawatts of power when annihilated with equal amounts of matter). It is then ionized into antiprotons which can be electromagnetically accelerated into the reaction chamber. The positrons are usually discarded since their annihilation only produces harmful gamma rays with negligible effect on thrust. However, non-relativistic rockets may exclusively rely on these gamma rays for propulsion.^[5] This process is necessary because un-neutralized antiprotons repel one another, limiting the number that may be stored with current technology to less than a trillion.^[6]

Design notes on a pion rocket

The pion rocket has been studied independently by Robert Frisbee^[7] and Ulrich Walter, with similar results. Pions, short for pi-mesons, are produced by proton-antiproton annihilation. The antihydrogen or the antiprotons extracted from it will be mixed with a mass of regular protons pumped inside the magnetic confinement nozzle of a pion rocket engine, usually as part of hydrogen atoms. The resulting charged pions will have a velocity of 0.94c (i.e. $\beta$ = 0.94), and a Lorentz factor $\gamma$ of 2.93 which extends their lifespan enough to travel 2.6 meters through the nozzle before decaying into muons. 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. These can't be reflected by any known material at the energies involved, although they can undergo Compton scattering. They can be absorbed efficiently a shield of tungsten placed between the pion rocket engine reaction volume and the crew modules and various electromagnets to protect them from the gamma rays. The consequent heating of the shield will cause it to radiate visible light, which could then be collimated to increase the rocket's specific impulse.^[4] The remaining heat will also require the shield to be refrigerated.^[7] The charged pions would travel in helical spirals around the axial electromagnetic field lines inside the nozzle and in this way the charged pions could be collimated into an exhaust jet that is moving at 0.94c. In realistic matter/antimatter reactions, this jet only represents a fraction of the reaction's mass-energy : over 60% of it is lost as gamma-rays, collimation is not perfect, and some pions are not reflected backwards by the nozzle. Thus, the effective exhaust velocity for the entire reaction drops to just 0.58c.^[4] Alternative propulsion schemes include physical confinement of hydrogen atoms in an antiproton and pion-transparent beryllium reaction chamber with collimation of the reaction products achieved with a single external electromagnet; see Project Valkyrie.

Notes

This formula can be derived from the fact that a spaceship's total rest energy before using an infinitesimal amount of fuel ( $dm_{{fuel}}$ ) has to be equal to the sum of the total relativistic energies of the accelerated fuel and the accelerated spaceship:
$m_{{ship}}\ c^{2}={\frac {dm_{e}\ c^{2}}{{\sqrt {1-{\frac {v_{e}^{2}}{c^{2}}}}}}}+{\frac {(m_{{ship}}-dm_{{fuel}})\ c^{2}}{{\sqrt {1-{\frac {(dv_{{ship}})^{2}}{c^{2}}}}}}}$

We can get rid of the infinitesimal quantity in the denominator of the right hand side of the sum on the right hand side by calculating the Taylor series (in $dv_{{ship}}$ ) and ignoring everything except constant and linear terms. Because only the square of $dv_{{ship}}$ occurs to begin with, there will be no linear term in the Taylor expansion, and the denominator indeed becomes 1:

$m_{{ship}}\ c^{2}={\frac {dm_{e}\ c^{2}}{{\sqrt {1-{\frac {v_{e}^{2}}{c^{2}}}}}}}+m_{{ship}}\ c^{2}-dm_{{fuel}}\ c^{2}$

Now the ship's rest energy cancels on both sides of the equation and we can rearrange the equation:

$dm_{{fuel}}\ c^{2}={\frac {dm_{e}\ c^{2}}{{\sqrt {1-{\frac {v_{e}^{2}}{c^{2}}}}}}}$

(I)

Due to the definition of $\eta$ , the following equation holds:

$dm_{e}=(1-\eta )\ dm_{{fuel}}$

(II)

Substituting $dm_{{fuel}}$ in terms of $dm_{e}$ and $\eta$ in equation (I) and dividing by $c^{2}$ and $dm_{e}$ yields

${\frac {1}{1-\eta }}={\frac {1}{{\sqrt {1-{\frac {v_{e}^{2}}{c^{2}}}}}}}$

Solving this equation for $v_{e}$ :

$v_{e}=c\ {\sqrt {2\eta -\eta ^{2}}}$

(III)

From the basic laws of relativity, the infinitesimal relativistic momentum of the exhaust is

$dp_{e}={\frac {dm_{e}\ v_{e}}{{\sqrt {1-{\frac {v_{e}^{2}}{c^{2}}}}}}}$

Substituting (II):

$dp_{e}={\frac {dm_{{fuel}}\ (1-\eta )\ v_{e}}{{\sqrt {1-{\frac {v_{e}^{2}}{c^{2}}}}}}}$

The specific impulse $I_{{sp}}$ is, by definition, this infinitesimal momentum divided by the infinitesimal mass of the fuel:

$I_{{sp}}={\frac {dp_{e}}{dm_{{fuel}}}}={\frac {(1-\eta )\ v_{e}}{{\sqrt {1-{\frac {v_{e}^{2}}{c^{2}}}}}}}$

Substituting (III) and simplifying results in the final formula:
$I_{{sp}}=c\ {\sqrt {2\eta -\eta ^{2}}}$

Sources

The star flight handbook, Matloff & Mallove, 1989
Mirror matter: pioneering antimatter physics, Dr. Robert L Forward, 1986

References

↑ http://web.archive.org/web/20060323032407/http://scientium.com/diagon_alley/commentary/bowden_essays/sotm/starship/rockets.htm
↑ Forward, Robert L. "A Transparent Derivation of the Relativistic Rocket Equation" (see the right side of equation 15 on the last page, with R as the ratio of initial to final mass and w as the specific impulse)
↑ http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html
↑ 4.0 4.1 4.2 http://arxiv.org/PS_cache/arxiv/pdf/0910/0910.1965v1.pdf
↑ http://thefutureofthings.com/articles/33/new-antimatter-engine-design.html
↑ http://science.nasa.gov/science-news/science-at-nasa/1999/prop12apr99_1/
↑ 7.0 7.1 http://www.relativitycalculator.com/images/relativistic_photon_rocket/ANTIMATTER_ROCKET_FOR_INTERSTELLAR_MISSIONS.pdf

External links

Physics FAQs: The Relativistic Rocket
Javascript that calculates the Relativistic Rocket Equation
Spacetime Physics: Introduction to Special Relativity (1992). W. H. Freeman, ISBN 0-7167-2327-1
The Relativistic Photon Rocket

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