Effects of the Radial Migration of Jupiter on the Trojan Asteroids
Heather. J. Cohen, Douglas. P. Hamilton (U. Maryland)
There is considerable debate about the origin of the Trojan asteroids which librate about the L4 and L5 Lagrangian equilibrium points near Jupiter. In order to extract information about the early population of Jupiter Trojans from current observations, we must understand how the evolution of the Solar System has affected their orbits. Current theories of solar system formation predict that Jupiter migrated inward by a few tenths of an AU as it expelled planetesimals out into the Oort cloud. We examine how this radial migration of Jupiter has affected the orbits of its librating companions.
We use a fourth-order Runge-Kutta integrator with adaptive stepsize to calculate the motion of a three-body system consisting of the Sun, Jupiter, and a massless asteroid. Jupiter is initially placed on a circular orbit. We then add a small drag force acting on Jupiter to produce slow inward or outward migration. We start the Trojan asteroid on a low eccentricity orbit ( ) which looks like a smooth tadpole when viewed in the reference frame rotating with Jupiter's mean motion. We find that the asteroid's amplitude of libration, , increases by a small amount for inward migration and decreases by a similar amount for outward migration. For a change in Jupiter's semi-major axis of , we find a change in of . These results are independent of the drag rate for drag rates slower than . The change in appears to be independent of the choice of equilibrium point, L4 or L5, and the fractional change in libration amplitude, , appears to be independent or at most a very weak function of the size of the initial tadpole orbit. Finally, the smooth tadpole orbits remain smooth as they grow or shrink, suggesting that the eccentricities of the initially circular asteroid orbits do not change drastically.
Although our results indicate that Jupiter's inward migration acted to destabilize the orbits of its Trojan asteroids, the effect was probably too small to be significant. Corresponding changes in libration amplitude may be significant for the present Trojan companions or hypothetical past Trojan companions of other bodies, such as Neptune, our Moon, and the Saturnian satellites, which have undergone substantially more radial migration than Jupiter.