Sensitive Dependence on Initial Conditions

Physics World has a somewhat puzzling news article about the solar system:

Physicists have known for some time that the motions of Pluto and the inner planets are chaotic. This means that a small external force on a planet could, over time, cause a major change in the position of the planet within its orbit. Although no planets are likely to collide or be ejected from the Solar System anytime soon, the chaos means that the orbits of these planets cannot be forecasted with any long-term reliability.

Whether the orbits of the gas giants are chaotic, however, is less certain — some computer simulations have found chaos while others have not. One cause of this confusion could be “numerical artefacts” — spurious errors that accumulate, for example, when a computer rounds-off the results of successive calculations.

“What’s puzzling about that?” you say. Well, that’s not the puzzling bit. The puzzling bit is here a little further down:

But according to Hayes the discrepancy is actually a result of the uncertainty in astronomers’ knowledge of the current conditions of the planets, which are used as input to the simulations. Although from observations we know the orbital positions of the gas giants to an accuracy of a few parts in 10 million, even that tiny amount of uncertainty can make the difference between regularity and chaos in a system, he says.

One of the hallmarks of chaos the phrase in the title: sensitive dependence on initial conditions. A tiny change in the initial values of a system lead to a huge change in the position some time down the road.

So, given that, doesn’t it seem like if you ask “Is this orbit chaotic?” and the answer is “I don’t know unless I know the position to better than a part in 107,” that really ought to translate to a “yes?”

(I know, I know, there’s a very formal definition of chaos, in terms of how quickly a two trajectories diverge from one another, and it’s possible to have islands of regularity within great seas of chaos on the phase diagram of a system. Still, I did a double-take when I first saw that description…)


  1. #1 Upstate NY
    September 26, 2007

    You mean for “practical purposes” the trajectory of Jupiter is chaotic? He’s propagating the trajectory over 200 million years. Not practical anyway.

    It isn’t easy to numerically determine “chaotic” vs “regular” in general, especially over those long times. “Chaotic” doesn’t necessarily mean it will leave the solar system — it could stay close to its stable periodic orbit for a long time. No one likes calculating Lyapunov exponents.

    “Chaotic” just sounds good in a headline. Will this help him get funding? Keep your eye on the ball.

  2. #2 CS
    September 26, 2007

    There is a difference between talking about orbits and positions. The paper is most likely referring to slow variables like eccentricity and the angular momentum vector. Position is poor variable to use for determining if the system is chaotic because it is rapidly oscillating. Suggesting the system is chaotic just because we cannot predict the position in 200 million years is a little unfair.

    Think of it this way: suppose there is only one planet, of negligible mass, in the solar system and it is in a circular orbit. Now suppose you know the Sun’s mass to only one part in a million. That corresponds to an uncertainty of one part in 1000 in the orbital period. Now suppose you wanted to know where that planet will be in exactly 1000 years (assume the orbital period is approximately 1 year). During that time, the planet could very well have gone through 999 or 1001 orbits. 999.5 orbits puts the planet in a hugely different position than 1000.0 orbits. The uncertainty in the position is essentially 100%: it has a non-trivial likelihood of being in any position along the orbit. However, that does not mean the orbit is chaotic. The orbit is actually still a circle. (The total angle advanced around the angular momentum vector, which increases by 2 pi for each orbit, is actually a more appropriate variable to use here: the uncertainty in that variable is only 1/1000, but would exactly specify the position.)

    In some systems, instabilities can grow exponentially (vary input x by fraction f and parameter y will vary by f*exp(t/t0), where t is time and t0 is the time scale of the instability). In others, instabilities are dampened and do not grow exponentially (variation in y is proportional to variation in x). Some orbital parameters of the inner planets fall into the first case. However, the question is whether or not the the same applies to the larger outer planets. Some simulations show they do, some don’t. Hayes is saying that, given the uncertainties in input parameters, we do not yet know which case we are going to end up in.

    Really, the question is: if I want to know orbital parameters at some point in the future to an accuracy of O(10^-5), do I need to know my inputs to O(10^-5) or O(10^-100). Hayes response is: “don’t ask that question at 10^-5. Ask it at 10^-7.”

    (I oversimplified some of the above, but the point remains.)

  3. #3 Luke
    September 27, 2007


    In a related preprint, Hayes says that about 70% of the initial conditions he tried resulted in chaotic orbits for the outer planets. If these are all equally valid, the outer solar system is probably chaotic. Eventually (hundreds of years from now?) we’ll have good enough initial conditions that we’ll know whether it’s chaotic or not.

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