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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