The Planar Motion of a Trojan Asteroid

Abstract

The planar motion of an asteroid in an orbit close to that of Jupiter is considered within the framework of the restricted circular three-body problem. The solution is derived asymptotically to second order using the two-variable expansion procedure for the case of small Jupiter-sun mass ratio. It is shown that the solution derived under the assumption of small Jupiter-sun mass ratio becomes singular for orbits that approach Jupiter. The nature of the singularity is exhibited as a guide for future work valid for this case. The results are given in explicit form for the coordinates as functions of time including both short and long periodic terms. Finally, the present solution is specialized to the family of long periodic Trojan orbits about the sun-Jupiter libration points.