archive: SETI Re: [ASTRO] FWD from Bill Larson: "Is the solar system

SETI Re: [ASTRO] FWD from Bill Larson: "Is the solar system

Larry Klaes ( lklaes@bbn.com )
Mon, 03 May 1999 15:42:00 -0400

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>Date: Mon, 03 May 1999 19:21:18 +0200
>From: Zlatko Papic <zlatko@beotel.yu>
>Organization: Home
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>Subject: Re: [ASTRO] FWD from Bill Larson: "Is the solar system chaotic?"
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>Reply-To: Zlatko Papic <zlatko@beotel.yu>
>
>> Is the solar system chaotic?
>
>Yes. You can take a look at Laskar's "A numerical example of the chaotic
>behaviour of the Solar system"(Nature,vol 338, 16th March 1989), but the
>first steps were made by Jack Wisdom in early 80ies.
>In that particular article, Laskar achieved an accuracy to the second
>order in the planetary masses and to the fifth order in eccentrcity and
>inclination. He included the corrections from general realtivity and the
>Moon. The estimated LCE(Maximum Lyapunov Exponent) he obtained was
>suprisingly high(~1/5Myr^-1). The Lyapunov Characteristic Exponent is
>used in dynamical astronomy to measure the extent of the chaos in the
>system. If you take two particles with the same initial conditions, and
>let them evolve thorugh the time, they can either remain close or their
>trajectories can diverge aat a very slow pace-this is a stable system.
>The other possibility for them is when the ditance between them
>increases linearly with time-this is the quasiperiodic motion. And when
>the distance between them increase exponentially-you have a chaos. And
>the Lyaounov exponent is the measure of that chaos, mathematically, you
>can express it as:
>
> Gamma= lim (1/t)*ln(d/d0),
> t->infty
>
>where t is the time, d and d0 the distances between the particles. the
>higher the Lyapunov exponent, the stronger the chaos. For any particluar
>trajectory of the n-dimensional system there can be n distinct Lyapunov
>exponents, depending on the phase-space direction from the reference
>trajectory to the test trajectory. In Hamiltonian systems, the exponents
>are paired: for each non-negative there is a non-positive with the same
>magnitude. For chaotic trajectories, the largest Lyapunov exponent is
>positive; for quasiperiodic trajectories, all of the Lyapunov exponents
>are zero.
>
>However, the stability of the Solar system has always been a task which
>was aimed to achieve. But all the attempts to prove it have failed so
>far. Arnold, for instance, has shown that a large proportion of possible
>Solar systems are quasiperiodic if the masses, and orbital inclinations
>and eccentricities of the planets are sufficiently small. The actual
>Solar system however, does not meet the conditions to fulfil the proof.
>Certainly, its great age suggests a high level of stability, but the
>nature of the long-term motion still remains the puzzle.
>
>For your information, you may take a look at the proof of instability
>for Pluto, Science vol 241, "Numerical evidence that the motion of Pluto
>is chaotic". The chaos chaos in the motion of asteroids is a different
>and not less interesting story, but if you find it interesting, contact
>me off the list because it may not be so exciting to other list-members.
>
>Regards,
>
>Zlatko
>
>
>