The solar system has held its present shape for billion years. All eight planets go the same way round, on very nearly the same plane.
What about the future? Will it look the same in a billion years, or in ten?
Neither the three laws of Kepler nor the inverse square says anything about it. Both concern two bodies and leave out the pull of the planets on one another.
The question has been with celestial mechanics from the beginning, and it is not fully settled now.
At the end of the eighteenth century Laplace and Lagrange gave an answer.
Expanding the perturbations term by term produces two kinds of term: those that grow without limit as time passes, and those that merely oscillate. If any of the first kind survives, a planet eventually departs.
The two showed that the growing terms cancel against one another and vanish, leaving only the oscillating ones. So the orbits merely swing within fixed bounds, and the solar system is stable forever.
That conclusion stood for a century. Laplace was the man who declared that the motion of the heavens needed no divine hand, and this calculation was what his confidence rested on.
In Poincare re-examined the calculation.
The perturbation series does grow more accurate as terms are added, at first. So it looks convergent. Past a certain term, though, each further term makes it worse, and in the end it diverges.
Such a series is called asymptotic. Truncated early it is an excellent approximation; summed to infinity it does not exist. The proof of Laplace had treated a non-convergent series as convergent.
That work changed how the three-body problem was seen. The word chaos did not yet exist, but this is where the opening out of small initial differences was first recognised.
If the series will not serve, the only route left is stepping. Computers finally made that possible.
In Laskar ran integrations over billion years, from slightly different starts. Each run opens out differently, so a single run means nothing; the result is a proportion across many.
In most of them the eight planets keep circling much as they do now. In about one run in a hundred, the eccentricity of Mercury swings far. Above its orbit can cross that of Venus.
From there: Mercury falls into the Sun, or collides with Venus, or is thrown out and disturbs the inner planets on its way. In none of those does the present arrangement survive.
For the next billion years nothing much moves. Every calculation agrees on that, so it can be said with confidence.
Beyond that the runs fan out. As the previous article showed, the uncertainty grows tenfold every five million years, so no single run can tell you about five billion.
Something arrives before that worry matters, however. In about billion years the Sun swells and swallows the inner planets. By then, whether the orbits are stable is no longer a question anyone has.
We began with the three laws of Kepler and have arrived here. Two bodies can be solved, three cannot, and stepping still yields values. And to the largest question there is, no answer has yet been found.