If you wanted to reach Alpha Centauri in 40 years, one way to do it would be to boost a spacecraft up to 10 percent of lightspeed as quickly as possible and then let it coast to destination. Or you could do something entirely different: push your payload at constant acceleration halfway to Centauri, turn it around at the halfway point, and perform a uniform deceleration that gets you to into Centauri space with zero speed. To achieve the latter — no small feat, needless to say — requires a constant acceleration of 0.0105*g*.

That number comes from the work of Italian physicist and mathematician Claudio Maccone, whose new paper “Radioactive Decay to Propel Relativistic Interstellar Probes Along a Rectilinear Hyperbolic Motion (Rindler Spacetime)” discusses a novel way to design an interstellar probe. Maccone’s study of constant acceleration (using what special relativity calls ‘hyperbolic motion’) shows that it could provide an ideal mission profile if we can find a way to propel a probe through processes of radioactive decay.

Hyperbolic motion has a fascinating history, first mentioned by Hermann Minkowski in 1908 in the same lecture where he introduced the idea of spacetime. Maccone notes a relatively unheralded 1930 book by Robert Esnault-Pelterie called *L’Astronautique* as using special relativity to examine relativistic interstellar flight, and also gives a nod to the German scientist Eugen Sänger, who suggested a design for a photon rocket in the 1950s (Sänger used electron/positron annihilation to produce gamma rays as exhaust, though how such an exhaust stream would be directed seems an insuperable difficulty).

But it was Carl Sagan whose work with hyperbolic motion fired the imagination of many of today’s scientists. In his 1963 paper “Direct Contact among Galactic Civilizations by Relativistic Spaceflight,” (*Planetary and Space Science* 11, pp. 485–98), Sagan drew from Robert W. Bussard the idea that an interstellar ramjet could accelerate continuously at 1 g. He then worked out the effects of relativistic time dilation as the ramjet continues to close on lightspeed.

Remarkably, with a constant acceleration of 1 g, a crew would reach the galactic center in a ship’s time of 20 years, even as 32,000 years passed on Earth. Even the Andromeda Galaxy would be within reach, 25 years away by ship’s time, while fully two million years passed on Earth. If all this sounds familiar, it may be because it stirs memories of Poul Anderson’s wonderful novel *Tau Zero* (New York: Doubleday, 1970), in which a runaway starship does something similar.

**Image**: M31, the Andromeda Galaxy. A constant acceleration of 1 g, as Carl Sagan showed, would allow a crew to reach it within a human lifetime, although millions of years would pass back on Earth. Credit: Adam Block/NOAO/AURA/NSF.

Maccone goes on to discuss the work of Jakob Ackeret (1898-1981), who in 1946 published the rocket equation for special relativity, an extension of the Newtonian rocket equation worked out originally by Konstantin Tsiolkovsky in 1903. Quoting Maccone: “…if one knows the probe’s velocity profile, then the Ackeret equation yields its mass decrease in proper time; or, the other way round, if the law by which the mass decreases in proper time is known, the velocity profile in proper time is obtained.”

For hyperbolic motion, the mass of the propellant decreases exponentially. The key point: “…this exponential decrease of the propellant mass formally has just the same equation as the radioactive decay equation. The way thus is paved to exploit the radioactive decay as a propulsion system to achieve the uniformly accelerated mission profile of the hyperbolic motion.” The relationship between the constant acceleration of hyperbolic motion, the radioactive decay constant and the exhaust speed of the decaying radioactive material can be used to select the appropriate radioactive material as a propellant to achieve the needed constant acceleration for a given target star.

Maccone’s paper “Radioactive Decay to Propel Relativistic Interstellar Probes Along a Rectilinear Hyperbolic Motion (Rindler Spacetime)” appeared in *Acta Astronautica* 57 (June, 2005), pp. 59-64.

{ 7 comments }

Hi Paul

Any pointers if we wanted to quantify non-hyperbolic motion? For example if thrust was constant? I’ve wracked my brains over this one and trawlled the Web but I’ve found zippo.

Adam, I don’t have anything to suggest other than that I’ll run that question past Claudio Maccone to see if he will comment.

Hi Paul

Yes please! That’d be fantastic. The main question is the derivation of distance. It’s pretty straightforward integration for non-relativistic motion for a constant thrust rocket (acceleration increases as mass decreases), and determining the speed and proper-time elapsed are trivial problems, but the coordinate time and rest-observer distance has me stumped.

It might not be a closed form integral, but even knowing that is a help – like when I tried to solve an elliptical integral, only to discover there was no closed form. Knowing that would’ve saved a few neurons.

Adam

Adam, these are questions I can’t answer either, but I’ll put the question to Dr. Maccone, who should be back in Italy by now.

Adam, I have a quick response from Dr. Maccone. Let me give it to you verbatim:

“Dear Reader of the “Centauri Dreams” web site,

I am not aware of any solved-in-closed-form motions other than the

hyperbolic motion.

But let me congratulate you about your attempt to find a solution for the

constant-thrust case.

May I then suggest that:

1) (if you have not done it already) you download (for free and legally)

Maxima from the web site:

http://maxima.sourceforge.net/

2) you re-cast in Maxima your attempted constant-thrust solution.

3) just enjoy seeing how Maxima solves any integral of rational functions

and – I believe but did not try yet – also elliptic integrals if one

pre-loads the elliptic functions package into the system… Perhaps maxima

will find the solution you are seeking !

With neurons saved !”

Hi Paul

Excellent! Thanks for that!

Adam

Hi Paul and Adam;

An idea suddenly popped into my mind last night as a means for providing energy for interstellar manned spacecraft. Basically, the idea involves the production of two rotors, or multiple systems of two rotors, wherein the rotors are separated by a distance on the order of 10 EXP -9 meters to 10 EXP -15 meters. The rotors would be composed of precisely manufactured and configured materials wherein some unknown means would be developed to selectively alternately switch on and switch off degeneracy pressure between the inner surfaces of the rotors in such a manner that the rotors would repel each other in such a manner that they would start to rotate in a manner analogous to that of an electric motor.

For stronger degeneracy interaction, the rotors might be made of fermionic matter of some types that approach the density or surpass the density of the atomic nucleus. Perhaps some sort of method of producing solid neutronium rotors that are stable against radioactive decay would enable repulsive pressures similar to the neutron degeneracy pressures that hold up neutron stars from collapse. The caveat being the production of stable forms of neutronium since neutronium is stable in neutron stars only because of the extreme gravity and pressure of neutron stars which causes the neutronium to be continually regenerated.

Another option might be the production of quarkonium rotors for even closer contact, perhaps on the order of 10 EXP -17 meters to 10 EXP -18 meters for even greater repulsive forces. Quarkonium made of strange matter composed of roughly equal numbers of up, down, and strange quarks might be possible for far distance future materials engineers.

Perhaps the main problem that would need to be overcome is how to selectively turn on and off degeneracy pressure which is a result of particles being fermions or those having an odd multiple of half a unit of quantum spin which is a fundamental property of the respective fermions. Perhaps the particles could somehow be arranged such that adjacent particles exist in oppositely aligned spins (-, +, -, +, …) wherein the spins or particles on opposing rotor surfaces would temporarily be somehow induced to have the relative spin configuration of (+,+,+, ….) and/or (-.-.-, …). The big question is how to do this on a macroscopic scale for long enough temporal durations in order to derive useful energy from the process.

Thanks;

Jim

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