My question is simple:

Is the Solar system stable?

You can see this Wikipedia page.
Edit: Sorry, because i think my question is more about mathematics and classical mechanics of planets in billion years scale, than astronomy. But i think this is our fortuity that Earth didn't doomed in past billion years ago and it's possible that we will have less than 8 planet's before Sun will be ruin.

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    $\begingroup$ Could you clarify a bit? It seems like the Wikipedia page you have linked answers your question. $\endgroup$ – Mitch Goshorn Jul 4 '15 at 0:41
  • $\begingroup$ My notion of stability is if we ignore the decreasing of Sun mass (because of nuclear reactions in the center of sun) and we assume only the mechanics of the planets by Newtone's law of gravitation whit present situations and speeds and masses, then billion years later we have 8 planets in Solar System (specially Earth!) or not $\endgroup$ – Ab_Sh Jul 4 '15 at 1:09
  • $\begingroup$ Normally, I would agree that the question is not sufficiently clear. However, since there is an answer that the user is happy with, somebody obviously understood what was asked. @2000 - if you wouldn't mind, could you please edit the question to add in the details you added in your comment. $\endgroup$ – Donald.McLean Jul 4 '15 at 4:29
  • $\begingroup$ Your question is not simple. On what timescale? $\endgroup$ – ProfRob Jul 4 '15 at 6:57
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    $\begingroup$ @WayfaringStranger I don't think this work specifically considers the stability of the solar system does it? Doesn't it just calculate a timescale for the big rip and then assume the solar system (which of course must exist in an entirely different configuration in 22 billion years in any case) is disrupted shortly before that. $\endgroup$ – ProfRob Jul 4 '15 at 23:16

The wikipedia page you linked to tells you that the solar system is gravitationally "chaotic", in part because the mass of the sun is not fixed over time.

But even more simply than that, focusing just on the gravity (ignoring loss of stellar mass, etc.), the solar system is an N-body problem. We have 8 planets, a sun, and millions of asteroids, comets, and who knows how many individual particles gravitationally bound to our sun (plus ones that aren't and are just passing through the neighborhood, so to speak). When you have more than 2 bodies, the solutions to the N-body problem are unstable. What this means is that, say we describe the N-body problem with data $D$ (the "initial conditions", or a perfect description of the state of the system at some specific point in time). With a given complete data set the (Newtonian) gravitational evolution of the system is completely determined (but so difficult to do we can only approximate it). What instability means here is that if we have some other data set $D'$ that is only a little bit different from $D$, then the differences between the evolution from $D$ and $D'$ will become exponentially large over long enough time scales. So what may seem like minor differences now will result in radically different looking solar systems in the long run.

Since all of our observations can never give exact values, but only a range of values, there is necessarily a bit of uncertainty in what the exact gravitational state of our solar system is. We have very poor data on the exact asteroid and comet content of our solar system, and even planetary data has significant error margins. All of this means that there are lots of justifiable picks for the data $D$, each differing by a small amount from each. But due to the instability, eventually these data will produce radically different futures from each other. Currently we can only predict the solar system's evolution up to a few million years or so (the exact value stated can vary wildly depending on how you opt to define and compute the Lyapunov time). After that the evolutionary tracks become so disparate we can't really say we're predicting anything other than "it'll definitely do something".

One way or another, it is currently impossible for us to make any clear assertions about what the solar system will look like on a timescale of billions of years. Maybe all 8 planets will still be there; maybe their orbits will be very similar, but maybe they'll have much different orbits; maybe several planets will have been ejected from the solar system. At best we can observe a few things that lead certain objects to be most likely to undergo significant alteration. For example, Jupiter and Mercury appear to have a certain orbital resonance right now which could ultimately lead Mercury to undergo a significant orbit change. This may ultimately cause it to collide with another planet, or the sun, or be ejected from the solar system entirely. But maybe it won't. It's hard to say.

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    $\begingroup$ I think this is incomplete, in the sense that while one cannot make a definitive prediction, one can certainly make probabilistic predictions (and people have done so) on timescales considerably longer than a few million years. It's like saying we can't predict the weather. $\endgroup$ – ProfRob Jul 4 '15 at 7:01
  • $\begingroup$ @RobJeffries A fair point. I kind of glossed that over in my mind. I'll see if I can put together some details in that regard. $\endgroup$ – zibadawa timmy Jul 5 '15 at 3:52

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