Contributed Poster Session, Wednesday, October 14, 1998, 2:00-3:40pm, Hall of Ideas

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*Heather J. Fleming, Douglas P. Hamilton (U. Maryland)*

The growth of Jupiter from a protoplanetary embryo and its subsequent radial migration have long been considered as potential mechanisms for the capture of near-Jupiter objects into librating tadpole orbits, and for the subsequent evolution of these objects (Yoder, 1979, {\it Icarus}). We have shown previously with numerical simulations that the effects of Jupiter's mass increase dominate the effects of its radial migration and cause a significant decrease in the libration amplitudes of Trojan asteroids. Other numerical studies have demonstrated that planetesimals could also be captured by a growing proto-Jupiter (Mazari {\it et al.}, 1998, {\it Icarus}).

Here we present an adiabatic invariant analysis which
determines the effects of Jupiter's growth and migration on
Trojan objects. If Jupiter and the Trojan are on circular
coplanar orbits and the Trojan libration amplitude is small,
the motion of the Trojan can be described by the Hamiltonian
H = {1 \over 2}a_{J}^{2} \dot \phi^{2} + {1 \over 2} \omega^{2}
a_{J}^{2} \phi^{2} where a_{J} = Jupiter's semi-major axis,
\omega = frequency of Trojan libration, and \phi =
difference between the Trojan and Jovian mean longitudes.
For adiabatic changes in this system, the action J = \int p
dq = \pi \sqrt{27 G \over 4} A^{2} m_{J}^{1/2} a_{J}^{1/2} is
conserved, where A is the Trojan libration amplitude. This
gives us the relation between initial (i) and final (f)
quantities: {A_{f} \over {A_{i}}} = ({ m_{Jf} \over
{m_{Ji}}}) ^{-1/4} ({ a_{Jf} \over
{a_{Ji}}})^{-1/4}. We also numerically model the
three-body Sun-Jupiter-Trojan system as Jupiter grows from
~10 M_{\}oplus to its current mass. These simulations
confirm our analytic results for growth timescales \gtrsim
10^{4} years and for libration amplitudes \lesssim
40^{\}circ. For larger orbits, the decrease in libration
amplitude is steeper, indicating that Jupiter's growth is
even more effective at capturing and stabilizing Trojan
objects than is suggested by the analytic formula. Our
analytic result remains an excellent approximation for
Trojans with small eccentricities or inclinations (e
\lesssim 0.1, i \lesssim 0.1 radians). Furthermore, the
eccentricities and inclinations of the Trojans are
essentially unaltered by Jupiter's growth. Thus, the current
distribution of Trojan eccentricities and inclinations may
contain information about their primordial values.

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