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[21P.12] A Mechanism for the Capture and Stabilization of Trojan Asteroids

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_J² \dot \phi² + {1 \over 2} \omega² a_J² \phi² 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} A² 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 10⁴ 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.