Fleming, H.J., and D.P. Hamilton 1999. On the origin of the Trojan asteroids: Effects of Jupiter's mass accretion and radial migration. Icarus, submitted.

We present analytic and numerical results which illustrate the effects of Jupiter's accretion of nebular gas and subsequent radial migration on its Trojan companions. We approximate the system by the planar circular restricted three-body problem, assume small Trojan libration amplitudes, and employ an adiabatic invariant calculation which shows that Jupiter's thirty-fold growth from a $10 M_\oplus$ core to its present mass causes the libration amplitudes of Trojan asteroids to decrease to $\sim 40\%$ of their original size. The calculation also shows that Jupiter's predicted inward radial migration of $\sim 1$ AU has comparatively little effect on the Trojans, causing an increase in their libration amplitudes of only $\sim 4\%$. In each case, the area enclosed by small tadpole orbits, if made dimensionless by using Jupiter's semi-major axis, is approximately conserved. Similar adiabatic invariant calculations for inclined and eccentric Trojans show that Jupiter's mass growth leaves the asteroids' eccentricities and inclinations essentially unchanged, while the planet's inward radial migration causes a $\sim 4\%$ increase in both of these quantities.

Numerical integrations confirm and extend these analytic results. We demonstrate that our predictions remain valid for Trojans with small libration amplitudes even when the asteroids have low, but nonzero, eccentricities and inclinations and/or Jupiter has an eccentricity similar to its present value. The integrations also show that Trojans with large libration amplitudes, including horseshoe orbits, are even more strongly affected by Jupiter's mass growth and radial migration than simple scaling from our analytic results would suggest. Further, the numerical runs demonstrate that Jupiter's predicted mass growth is sufficient to cause the capture of asteroids initially on horseshoe orbits into stable tadpole orbits. Thus, if Jupiter captured most of its Trojan companions before or while it accreted gas, as seems probable, then Jupiter's growth played a significant role in stabilizing Trojan objects by systematically driving them to lower libration amplitudes.

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