# Last edited on 2014-05-31 17:10:04 by stolfilocal
# NOT POSTED

[quote author=Richy_T link=topic=178336.msg7058713#msg7058713 date=1401561500]
[quote author=MaxwellsDemon link=topic=178336.msg7056266#msg7056266 date=1401550868]
But to run a simulation you need to be able to solve the 3-body equations... Unless you're running some sort of simplified heuristic simulation, in which case it could never be absolutely deterministic anyway.
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Absolutely not. A simulation would typically rely on the basic laws of physics, not complex solutions for particular situations.
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Yes, there are methods for solving (in the jargon, "integrating") differential equaitions that give solutions as accurate as any closed formula with relatively moidest computing effort; probably less than it takes to mine a bitcoin block[*].  "Runge-Kutta" is the first name to look up in that field.  However, as the simulated time ength increases, you need to keep more digits and use a smaller time step, to get the required accuracy at the end.  

In the 1980s someone at MIT built a tabletop computer just to simulate the motion of the planets of the solar system.  He ran it for several million years in the future and claimed that, while the orbits of other planets were stable on that time scale, the orbit of Pluto was chaotic.

That simulation in particular was probably bogus.  The guy omitted Mercury, Venus, Earth, Mars as being "insignificant", as well as all the satellites, comets, and asteroids.  He also ignored the flattening of the Sun and planets at the poles (which changes slightly their gravitational field), the Ooort cloud, and of course all the new "planets" that were uknown then.  The impact of those omitted details may indeed have been negligible, but, IIRC, he did not provide an estimate of the error.

I suppose that the simulation has been redone many times since then, with  more accurate models and integration algorithms.  The numerical errors (from rounding and approximation) MAY grow exponentially, but not necessarily.  If a planetary system is stable (non-chaotic), the numerical integration can probably be coded so that it is stable too.  (There are systems of 3 or more bodies  that evolve in a complicated but periodic way, and therefore will never eject any body, at least in theory; I don't know stable they are against external perturbations.)

If the system is chaotic, the main obstacle in such simulations is not the numerical errors, but the uncertainty in the data.  The mass, position, speed, and shape of most planets is known only with a few decimal digits of accuracy.   Indeed their mass is often estimated from perturbations that they cause on the motion of other nearby bodies, over the few years in which they have been observed[**].  That lack of data probably makes it pointless to simulate  the motion of those bodies, and of those that are are affected by them, for more than a few tens of orbits.

As interesting as simulating the future is running the simulation backwards.  A fringe author called Velikowski got famous decades ago for his claim that mythical tales of gods and events in heavens were distorted records of major changes in the Solar System, such as Venus moving from Jupiter (or near it) to its present place, Mars doing a very close pass to Earth, etc.  For a while Carl Sagan was quite obsessed with the guy (who had no scientific education IIRC) and spent much time ridiculing his theories, on the argument that planets cannot just move around like that.  But it is not entirely clear that his claims are completely absurd, since the same "chaotic limit" that prevents accurate simulations in the far future also prevents us from knowing where the planets were in the distant past.  

[*][**] Words added to ensure that this post is is not off-topic.