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I recently read about the Breakthrough Initiative to launch "StarShot", a nano-probe that is designed to travel to Alpha Centauri at $0.2c$. One of the challenges to be solved involves the precise determination of the orbital ephemerides of any exoplanets in the system.

My question is related to how we might go about simulating not only the orbital ephemerides of the planets but also the probe itself, given that it is travelling at relativistic speeds. In essence, what do we need to consider to simulate the entire physical system?

Presumably, the usual method for speeds $v \ll c$ is to consider a Newtonian n-body approach and simply integrate the equations of motion with the gravitational inverse square law, using a numerical scheme such as Runge-Kutta or, something more sophisticated, like a symplectic integrator?

To me, this seems appropriate only for the planets themselves and not the probe. Given that speeds near $c$ are occurring it should be necessary to include at least a special relativistic approach. However, after more consideration, given that gravitation is involved (even without strong-field regions such as black holes) it would seem to be necessary to involve general relativity.

If the latter is the case, does that mean we would need to bring in the heavy machinery from numerical relativity, such as a foliating curved space-time and solving the Einstein field equations numerically, with a matter distribution of planets as the initial condition?

To me this seems to be "using a sledgehammer to crack a nut" when all we really need is the trajectory of the probe from the Alpha Centauri reference frame and the positions of the planets from the probe reference frame.

In addition, does anybody know of any available software (open source or otherwise) that might help with this simulation?

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  • $\begingroup$ I am not fully aware of field equations applications in n body motion, but when we propagate any asteroid or comet within solar system for a long time frame, we do use relativistic corrections to inverse square law. You can check them out in Seidelmann's 'Explanatory supplement to the Astronomical Almanac' I believe something similar would be done for the probe trajectory. $\endgroup$ – Astroynamicist Apr 26 '16 at 11:26
  • $\begingroup$ I've just had a look through the book you mentioned on Google Books and have come across some references due to Kiloner. This discussion on Relativistic Celestial Mechanics is also interesting: scholarpedia.org/article/Relativistic_Celestial_Mechanics $\endgroup$ – Michael Halls-Moore Apr 26 '16 at 11:57
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    $\begingroup$ $v=0.2c$ is just barely relativistic: $\gamma\approx 1.0206$. Also, the probe has a tiny mass compared to the stars & planets of the two systems, and most of its trajectory is in rather flat space. So Newtonian physics with minor relativistic corrections should be adequate. $\endgroup$ – PM 2Ring Feb 16 at 12:35
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"Presumably, the usual method for speeds v≪c is to consider a Newtonian n-body approach and simply integrate the equations of motion with the gravitational inverse square law, using a numerical scheme such as Runge-Kutta or, something more sophisticated, like a symplectic integrator?"

It might work for non-relativistic but for more than two bodies it is known that the equations of motion under gravity are chaotic.

General relativity equations can be solved but the solution may make sense only "locally" in weak-field limit. Small errors can result in huge deviations over galactic scales.

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  • $\begingroup$ You're right - the error accumulation will be a problem in the general case. I do wonder though, over the time-frame that the StarShot probe is travelling (20 years in the Sun/Solar System frame), whether the errors in the Alpha Centauri ephemerides will be significant? $\endgroup$ – Michael Halls-Moore Apr 26 '16 at 12:37

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