**AAS 195th Meeting, January 2000**

*Session 58. Solar System*

Oral, Thursday, January 13, 2000, 10:00-11:30am, Regency VI
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## [58.01] On the Commensurate Motion of Pallas

*S.J. Goldstein, Jr. (U. Virginia)*

Gauss calculated in 1812 that the minor planet Pallas,
discovered by Olbers, had a sidereal mean motion exactly
18/7 times that of Jupiter. He also produced a theory for
the periodic libration of Pallas with a period of 737
Jupiter sidereal periods (Jyrs) (Werke v. 7, p. 421, p.
559).

The next example of commensurate motion for an observed
planet is the work of Charlier (Astr. Nachtrichten, no.
4094, p. 214) based on (588) Achilles, discovered by Wolf in
1906. Achilles is a Trojan, commensurate 1/1 with Jupiter.

I use a three body numerical integrator to test Gauss'
results with modern observations, taking the orbital
elements for Pallas and Jupiter from the 1997 Ephemerides of
Minor Planets. The integrator is fourth order Runge Kutta,
strictly Newtonian, and able to integrate forward and
backward for 600 Jyrs, and return to the starting point with
an accuracy of one part in 10^{8} in position and velocity.
The integrations extend 1200 Jyrs both forward and backward
from the epoch 1997 Dec. 18. They show that the mean motions
differ from the ratio 18/7 by less than one part in 6700.
Such a small difference is unlikely to arise by chance, so I
conclude that they are still synchronized.

The librations are studied by calculating the angular
heliocentric distance of the minor planet relative to
Jupiter, sampling once each 7 Jyr. They are not periodic,
but show a slow drift of 165 deg in 1200 Jyr, with a
superimposed random component. It remains to be seen if some
small changes in the 1997 orbital elements would bring
Pallas into exact agreement with the theory of Gauss.

The basic feature of exactly commensurate motion is that the
two planets are able to exchange orbital energy to maintain
the commensurability.

I thank B. G. Marsden for the references to Achilles.

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