Take two copies of the same three bodies and shift the starting position of one by a part in ten thousand.
For a while after starting, the two lie on top of each other. There is no telling which is which.
Then they begin to separate. And in the end the paths are nothing alike: one body flies off in one copy while it stays put in the other.
Two bodies do not behave this way. Shift the start slightly and the path shifts slightly. What happens with three is that the manner of the shifting is itself different.
The gap grows by multiplication, not addition. In every fixed span of time, it doubles.
Plot it against time with each step up the axis a factor of ten and the growth becomes a straight line. The slope says how fast it opens.
The reciprocal of that slope is the Lyapunov time: how long it takes for the gap to grow by a factor of , about .
The shorter the Lyapunov time, the less can be foreseen. And that time is a property of the system, unchanged by any refinement of measurement.
An awkward consequence follows.
If the gap doubles and doubles, then running it backwards, measuring the start ten times more finely buys only one more step of foresight.
A hundred times finer buys two, a thousand times three. Pile on orders of magnitude and what comes back is only addition.
Improving a measurement ten thousandfold is an enormous undertaking. It extends the horizon by four Lyapunov times. That is what being unable to see far ahead amounts to.
The Lyapunov time of the solar system is estimated at about million years, a figure Laskar obtained numerically in .
The present positions of the planets are known as precisely as they can be measured. Even so, the uncertainty grows tenfold every million years.
A hundred million years out, there is no saying where along its orbit a planet will be. Whether the Earth is on the near side of the Sun or the far side becomes unanswerable.
The orbits themselves survive. That the orbit of the Earth will still be an ellipse of much the same size can be said. What cannot be said is where on that ellipse it will be.
This is easily confused, so let it be said plainly: none of this is randomness.
From the same start, the same path follows every time, exactly. The equations contain no hesitation and no jitter, and the next instant is settled uniquely.
Settled, and still not foretellable. The start cannot be known with perfect precision, and that alone puts the far future out of reach. This is exactly what Poincare saw in .
Weather forecasts run out at about two weeks for the same reason. The next chapter turns to what this opening actually does in the solar system.