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I am not trained in applied mathematics and am asking as a layperson.

It seems that the motions of the planets can be reliably predicted for thousands of years. Yet, if we consider a relatively simple system, the compound pendulum for example, its motion is highly unpredictable.

Chaotic pendulum - guess when it will stop flipping

The pendulum is highly constrained by its pivot points, therefore its components do not escape from the system. However, with a solar system, there is no rigid attachment and it seems intuitive that planets could randomly fly off into space as a result of chaotic orbits.

Question

Is our solar system inherently stable or unstable?

If unstable then how did it hang together long enough to form in the first place?

If stable, how can stability and predictability be consistent with many bodies moving relatively to one another?

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The solar system is chaotic, but it is also stable!

The fixed and linkages between the bars of a double pendulum allow for very rapid energy transfer between the arms. This makes the chaotic motion develop rapidly.

The interactions between planets are gravitational and much much weaker, moreover, the planets are heavier and it takes a lot more energy to deflect them. This means that in the medium to long-term it is possible to predict exactly where a planet will be.

In the very long-term the system is chaotic, but the gravitational linkage between planets is not strong enough to actually cause one of them to deviate very far from its orbit, unless a resonance develops. So it is chaotic in the sense that we won't know which side of the sun Mars will be on. But it is stable in the sense that we know that Mars will continue to be the planet between Earth and Jupiter (but perhaps slightly closer to Earth or slighty closer to Jupiter)

There is only one potential resonance: it is possible that in the very distant future, Mercury could develop a resonance with Jupiter, which could (in 3 to 4 billions years) eject it from the solar system.

So it is the very rigidity of the double pendulum that makes chaos develop so quickly. Whereas the looseness of gravity means the motion of the planets can be predicted far into the future.

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  • $\begingroup$ Can we say that stability in such systems evolves? I.e. early in the life of a solar system, the planets that are most likely to be thrown out, are thrown out. And that what remains is the planets that are least likely to be ejected? Or must the chaos develop over the very long term? $\endgroup$ Mar 6 at 21:03
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    $\begingroup$ The early solar system was complicated by the lingering presence of a protoplanetary disc, and a complex set of resonances that developed and broke see en.wikipedia.org/wiki/Grand_tack_hypothesis It is likely that any system of more than one star and one planet will be chaotic, in the mathematical sense, but chaos doesn't imply ejections. $\endgroup$
    – James K
    Mar 7 at 9:56
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Is our solar system inherently stable or unstable?

Yes. This is not an either / or question. There are multiple conjectures regarding instabilities in both the early and late solar system:

Why are solar systems stable and not chaotic?

Not all star systems are stable. Astronomers have discovered multiple star systems that appear to have been racked with instabilities. They have discovered hot Jupiters, exoplanets whose orbits about a star are highly inclined with respect to one another, and exoplanets that appear to be out of order (our solar system has terrestrial planets, then gas giants, then ice giants; that is not always the case). But astronomers have also discovered star systems that appear to be nice and orderly, like our solar system.

There appear to be stabilizing as well as destabilizing influences in the formation of a star system. The stabilizing influences apparently dominated in our solar system. One argument in favor of the rarity of intelligent life in the universe is that intelligent life can only form in star systems that exhibit long-term stability. After all, it took bout three billion years on our Earth for life to evolve beyond simple single cellular organisms, about another billion years for complex life to begin to explode, and about another half a billion years for sapient life to arise.

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