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Is the Solar System stable?

You can see this Wikipedia page.
In May 2015 I was at the conference of Cedric Villani at Sharif university of technology with this title: "Of planets, stars and eternity (stabilization and long-time behavior in classical celestial mechanics)" , at the end of this conference one of the students asked him this question and he laughed strangely(!) with no convincing answer!
Edit: The purpose of "long-time" is timescale more than Lyapunov time, hence billions of years.

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    $\begingroup$ Depends on your notion of stable. I'm basing many of my plans on the assumption that civilization will be around long enough to support my children and their children. Gerhard "Knows It's A Risky Assumption" Paseman, 2015.07.03 $\endgroup$ – Gerhard Paseman Jul 4 '15 at 0:47
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    $\begingroup$ If you give a somewhat mathematically rigorous definition of the stability, partial (numerical) answers can be found in J. Laskar's papers. $\endgroup$ – Piyush Grover Jul 4 '15 at 1:03
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    $\begingroup$ Although it's been about 10 years since Pluto was demoted from the list of planets, it still looks strange to me to see "8 planets" instead of 9. Perhaps billions of years from now there will be 15 planets after the opposing astronomers make another definition of what a planet is that lets in Pluto and its cousins. A relevant question is whether the definition of a planet will remain stable. $\endgroup$ – KConrad Jul 4 '15 at 1:21
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    $\begingroup$ Not only is the solar system unstable, but try getting a plumber on weekends. $\endgroup$ – Gerry Myerson Jul 4 '15 at 4:56
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    $\begingroup$ Wittyness aside, there are some real questions here before one can begin to answer. For example: On what timescale do you want stability? The solar system might be stable for the next million years or so but unstable if you look billions of years into the future. Consider for example that the number of planets has decreased over the last few billion years through collisions. Who is to say that earth won't collide with mars in the future for example? $\endgroup$ – Johannes Hahn Jul 4 '15 at 15:18
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Due to chaotic behaviour of the Solar System, it is not possible to precisely predict the evolution of the Solar System over 5 Gyr and the question of its long-term stability can only be answered in a statistical sense. For example, in http://www.nature.com/nature/journal/v459/n7248/full/nature08096.html (Existence of collisional trajectories of Mercury, Mars and Venus with the Earth, by J. Laskar and M. Gastineau) 2501 orbits with different initial conditions all consistent with our present knowledge of the parameters of the Solar System were traced out in computer simulations up to 5 Gyr. The main finding of the paper is that one percent of the solutions lead to a large enough increase in Mercury's eccentricity to allow its collisions with Venus or the Sun.

Probably the most surprising result of the paper (see also http://arxiv.org/abs/1209.5996) is that in a pure Newtonian world the probability of collisions within 5 Gyr grows to 60 percent and therefore general relativity is crucial for long-term stability of the inner solar system.

Many questions remain, however, about reliability of the present day consensus that the odds for the catastrophic destabilization of the inner planets are in the order of a few percent. I do not know if the effects of galactic tidal perturbations or possible perturbations from passing stars are taken into account. Also different numerical algorithms lead to statistically different results (see, for example, http://arxiv.org/abs/1506.07602).

Some interesting historical background of solar system stability studies can be found in http://arxiv.org/abs/1411.4930 (Michel Henon and the Stability of the Solar System, by Jacques Laskar).

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This paper (Batyrin and Laughlin, 2008) seems to indicate that we are doomed.

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    $\begingroup$ "Doomed" in what sense? Gerhard "Igor's Not From Around Here?" Paseman, 2015.07.03 $\endgroup$ – Gerhard Paseman Jul 4 '15 at 3:09
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    $\begingroup$ @GerhardPaseman Planets leaving, other planets crashing into the Sun. O, the humanity! $\endgroup$ – Igor Rivin Jul 4 '15 at 3:27
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    $\begingroup$ Seems to me (from skimming the paper) that Earth and the gas giants had orbits stable enough to last until the sun burns out. Of course, if you're planning to move to Mars long term, you might reconsider. Gerhard "It's Not Like Bradbury Said" Paseman, 2015.07.03 $\endgroup$ – Gerhard Paseman Jul 4 '15 at 3:47
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    $\begingroup$ First, they came for Mars... $\endgroup$ – Igor Rivin Jul 4 '15 at 3:52
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    $\begingroup$ I'm not moving anywhere! It was honour to play with you guys! $\endgroup$ – მამუკა ჯიბლაძე Jul 4 '15 at 4:34
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in a conference in Paris, Jacques Féjoz said (and i quote from memory) that the big planets seem to be stable, while the small ones chaotic. if i remember well, it was based on numerical evidence, intuition and the known results on the stability of the planar many-body problem...

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The work of Wisdom is relevant (for example, see http://rspa.royalsocietypublishing.org/content/413/1844/109.short) but not conclusive. Numerical work suggests overall stability over very long time periods.

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