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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. $\endgroup$ Apr 26, 2016 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 $\endgroup$ Apr 26, 2016 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, 2019 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? $\endgroup$ Apr 26, 2016 at 12:37
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Transit of the probe at (near-)relativistic speed doesn't require much more than SR. The need for GR is less about speed and more about spacetime curvature. GR would have more relevance for propagating the planetary orbits (remember that historic observations of such were how the need for it was first suggested), but there is not going to be much more spacetime curvature in the Alpha Centauri system than our own Solar System and hence the need to take account of GR is about the same (if not less, if you're going to Proxima in particular) as for that and the same techniques there could be used which is mostly to just take a few low-order correction terms on the Newtonian gravitational potential, I believe.

The relevant parameter when considering chaos is the Lyapunov time which, for our Solar System is around 5-10 Ma (150 - 300 Ts) and I doubt that for another star system this will be off by orders of magnitude. This describes the strength of error growth and is exponential (one Lyapunov time is - I believe - one $e$-folding of the error in the initial conditions), so at 20 years we're talking 0.0004% of one Lyapunov time, chaotic effects from neglect of GR will be negligible.

For the craft itself, weak curvature and relativistic speed, SR together with Newton-Maxwell gravity (more commonly called by the cringey name of "gravitoelectromagnetism") emanating from the star and planets under the data given from the above would be (more than?) sufficient, I'd think.

ADD: That said, what I wonder if their concern is in relation to more is regarding getting precise enough measurements of the relevant inputs for the planetary motions: the planetary and stellar masses (or better, "gravitational parameters" i.e. mass times $G$, or $GM$, since that is the most feasible to measure to high precision), distances, and velocities. Errors in these will have a much greater effect on the ephemeris than will errors from simplifying assumptions in the relevant physics.

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I would assume that using a slight variant of the restricted three-body problem would give a reasonable solution. Basically, the probe is the only truly relativistic object in the system, and its mass is sufficiently small that we could treat it as a "test particle". This means that the following technique would (I suspect) give a reasonable approximation:

  1. Simulate the motions of the stars & exoplanets. Newtonian & post-Newtonian methods that work in the Solar System should be sufficiently accurate for this part of the problem. The post-Newtonian (relativistic) methods that one might employ here are, of course, a limit of GR.
  2. Solve for the Newtonian & post-Newtonian metric given the known positions and velocities of the stars & exoplanets. GR also comes in here. The post-Newtonian piece would effectively be an expansion of the metric in a power series in $v/c$, where $v$ is a velocity scale; the leading-order term in this expansion will simply yield the Newtonian potential. Both the CM velocities of the bodies and their internal velocities (i.e., thermal motions & rotational motions) would be relevant here, though if one of these velocity scales is large compared to the other, it might be possible to neglect the smaller motions.
  3. Solve for the motion of the probe in the resulting metric, treating the probe as a relativistic test particle.

It should not be necessary to employ the full power of numerical GR to solve this problem, simply because to a very good approximation the metric will be nearly flat.

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