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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 108 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.