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    Long-term orbit predictability: was [Nav-l] Delta-T
    From: George Huxtable
    Date: 2004 Dec 5, 11:32 +0000

    I wrote, on 2 Dec.-
    >> Not everything in astronomy is deterministic. Even in a system, such as the
    >> solar system, in which more than two bodies have a mutual gravitational
    >> interaction, there's a limit to the long-term predictability of it all.
    >> This is the famous "three-body problem".
    After a spot of initial misunderstanding between me and Alex, I wrote-
    For the purposes of our own period on the Earth, and for hundreds, perhaps
    a few thousands, of years in the future and the past, solar system motions
    are indeed predictable to adequate accuracy, based on our own recent
    observations of the system dynamics. Alex and I agree about that, when he
    >This is not a problem for modern astronomy.
    >If we are talking of "predictions" (forward or backward)
    >for the period of few thousand years, this will cause no problems
    >(in principle)
    except that I would describe it as sufficient predicability "in practice",
    rather than "in principle"
    But the further you get from the present-day, error in these predictions
    increases. It's not, simply, that we haven't measured the present positions
    and velocities accurately enough, though that plays a part. But you can get
    to a point when infinitesimal changes in those initial conditions lead to a
    divergence in the predicted results. It's comparable to the "beating of a
    butterfly's wing" effect that causes meteorology to be so unpredictable;
    but over a MUCH longer timescale.
    Where Alex says-
    >The "long term" prediction is indeed impossible, (because the
    >many-body problem of celestial mechanics has no simple solution)...
    We agree: that's the topic I was writing about, and said so.
    but then he goes on to say-
    >...but "long term" in this sentence refers to billions years.
    Here we disagree, and strongly. Unpredictable divergences show up over a
    MUCH shorter period, tens or perhaps hundreds of thousands of years
    (depending on what accuracy is being called for).
    >For all "practical purposes" such prediction can be done.
    No argument over that.
    Since then, I've unearthed some copies of past papers from my archives on
    that topic, solar system stability, in which I was taking an interest at
    the time.
    A paper by Sussman and Wisdom, from MIT, "Numerical evidence that the
    motion of Pluto is chaotic", in Science, July 88, claimed to show that if
    we took two possible Plutos, at infinitessimally different positions at the
    present day,  then about 20 million years from now the differences between
    their positions would be increasing exponentially (by a factor of e) every
    20 million years. After a few such time-intervals, it would become
    impossible to predict future positions.
    But Pluto is very much at the weird-end of the solar systam, performing a
    complex dance interlocking with Neptune's orbit so they don't collide, It's
    also of no interest to navigators.
    Of more direct interest, then, is a paper by Laskar, from BIH, Paris, "A
    numerical experiment on the chaotic behaviour of the solar system" in
    Nature 338 (March 1989), which was backed by other workers from Cornell
    reported in that same issue. He concluded that predictability of the orbits
    of the inner planets of the solar system, including the Earth, is lost
    "within a few tens of millions of years".
    But I haven't been following the topic since those days, and that work may
    have been discredited, withdrawn, or greatly modified over the intervening
    years. If so, I would be interested to learn. Perhaps Frank knows.
    Otherwise, it implies that the motions of the planets will literally become
    unpredictable over some finite period of time of tens of millions of years
    (well beyond any timescale that will directly concern us!).
    Such chaos doesn't necessarily impliy that major catastrophes, such as
    collisions or ejections of planets from the system, will result.
    Frank Reed has quoted a review which stated -
    "In the inner solar system,  overlapping secular resonances have been
    identified as the possible source of  chaos. For example, Mercury, in 10^12
    years, may suffer a close encounter with  Venus or plunge into the Sun. In
    the outer solar system, three-body resonances  have been identified as a
    source of chaos, but on an even longer time scale of  10^9 times the age of
    the solar system. On the human time scale, the planets do  follow
    their orbits in a stately procession, and we can predict their  trajectories
    for hundreds of thousands of years. That is because the mavericks,  with shorter
    instability times, have long since been ejected. The solar system  is not
    stable; it is just old!"
    (from Ann. Rev. Astro., Sep 2001)
    I see no serious divergence between what I have reported and what that
    review claims.
    "we can predict their  trajectories for hundreds of thousands of years." Agreed.
    The review seems to identify chaos with catastrophe, in predicting-
    "Mercury, in 10^12 years, may suffer a close encounter with  Venus or
    plunge into the Sun". Which may indeed be the case. But the orbits become
    unpredictable, long, long, before there's enough energy been transferred to
    give rise to such a dramatic event. We were discussing unpredictability,
    not catastrophe.
    Earlier, Frank had written-
    "But to George's point, the Solar System has some moderate chaotic behavior
    on long time scales, and that implies that the computational cost increases
    exponentially. That makes it deterministic but in a useless manner."
    Which I suggest is no more than word-play. If an infinitesimal
    initial-difference has an exponential blow-up as a consequence, then those
    end-results eventually become indeterminate. By spending more on
    "computational cost" you might follow the process a bit further, but that's
    all. And you need to know the initial conditions more and more precisely,
    before you start.
    contact George Huxtable by email at george@huxtable.u-net.com, by phone at
    01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
    Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

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