Summary

The secular motion of Pallas has been studied by numerically integrating the Lagrange planetary equations for 700 000 yr. Two terms with periods of approximately 15 000 yr and 190 000 yr dominate the secular elements. The former corresponds to a perturbation with argument 2ω and the latter to a perturbation with argument |$\omega -{\omega }_\text{J}$|⁠, where |$\omega\,\text{and}\,{\omega }_\text{J}$| are the arguments of perihelion of Pallas and Jupiter respectively, defined with Jupiter's plane as the reference plane. In the 15 000-yr periodic term, which has the greater amplitude, the maximum value of the eccentricity and the minimum value of i, the inclination, occur when |$\omega =90^{\circ}\,\text{and}\,270^{\circ} $| and the minimum value of e and maximum value of i when |$\omega =0^{\circ}\,\text{and}\,180^{\circ}$|⁠. This decreases the possibility of close approaches to Mars and Jupiter and increases the stability of the orbit. The inclination of Pallas at its minimum values over the 15 000-yr period shows remarkably little change at |$28^{\circ}.2\pm0^{\circ}.3$|⁠. Pallas is found to have a slow retrograde motion for the longitude of perihelion and this is unusual for asteroids. It circulates with a period of 544 600 yr.

A numerical integration over 100 000 yr showed that the perturbations arising from the 18:7 commensurability with Jupiter are not significant in the secular motion of Pallas.

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