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\centerline{\bf EPHEMERIDES}
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\centerline{\bf OF CHAOTIC PLANETS}

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\centerline{\bf Andrea Milani}

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\centerline{\it\ddag Space Mechanics Group, University of Pisa, Italy}
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\centerline{\bf Invited review, IAU Symposium 172, Paris, July 1995}
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We have now a good understanding of the properites of the solutions of
the N--body problem over a very long time span, both in the ordered
case of conditionally periodic orbits and in the chaotic case, with
its homoclinic/heteroclinic tangle. However, when such a conceptual
framework is applied to the orbits of real solar system objects, a
paradox arises. On one hand, essentially all the orbits, when computed
within a realistic dynamical model, turn out to be chaotic. On the
other hand, most solar system objects are dynamically very old, and
indeed the orbits can be computed both analytically and numerically.
Thus celestial mechanicians spend most of their time computing
ephemerides of chaotic planets. This is of course possible over time
spans either shorter than, or comparable to, the Lyapounov time. 
Over a much longer timespan, it is not clear why such computations
could give useful results.

I will propose a possible way out of this paradox. There are chaotic
orbits for which a strong chaotic wandering occurs along only some
directions in the phase space. If the coupling among different degrees
of freedom is either weak or highly asymmetric, then a random walk in
the value of one action variable, and complete randomisation in one
angle variable, can occur together with a very slow diffusion in the
direction of the other action variables. This turns out to be the case
for the coupling of secular perturbations with mean motion resonances
of moderate to high order. Thus a random walk in the mean longitude,
which occurs for essentially all the planetary orbits, does not imply
a significant diffusion in the proper eccentricity and
inclination. For this reason the proper elements are constant --to a
very good accuracy-- for most main belt and trojan asteroids, over
timespans longer than the Lyapounov times by three orders of
magnitude. On the contrary, when chaos occurs in the secular part of
the equations of motion, large instabilities can occur for supposedly
main belt asteroids and even for the major planets.  Thus ephemerides
of chaotic planets can be reliably computed, provided only some of the
elements are described by the corresponding mean elements; this is
possible in most cases, but by no means in all cases.

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