Theory of the Trojan asteroids

Abstract

In the previously published Parts I and II of the paper, the author has constructed a formal long-periodic solution for the case of 1∶1 resonance in the restricted problem of three bodies to 0(m ^3/2), wherem is the small mass parameter of the system. The time-dependencet(λ, α,m), where λ is the mean synodic longitude and α is related to the Jacobi constant, has been expressed by ahyperelliptic integral. It is shown here that with the approximationm=0 in the integrand, the functiont(λ, α, 0) can be expanded in a series involving standardelliptic functions. Then the problem of inversion can be formally solved, yielding the function λ(t, α, 0).

Similarly, the normalized period τ(α,m) of the motion can be approximated by theHagihara hyperelliptic integral τ(α, 0), corresponding tom=0. This integral is also expanded into elliptic functions. Asymptotic forms for τ(α, 0) are derived for α∼0 and for α∼1, corresponding to the extreme members of thetadpole branch of the family of orbits.