Co-Orbital Motion with Slowly Varying Parameters

Abstract

We consider the dynamics of a test particle co-orbital with a satellite of mass m _s which revolves around a planet of mass M ₀ ≫ m _s with a mean motion n _s and semi-major axis a _s. We study the long term evolution of the particle motion under slow variations of (1) the mass of the primary, M ₀, (2) the mass of the satellite, m _s and (3) the specific angular momentum of the satellite J _s. The particle is not restricted to small harmonic oscillations near L ₄ or L ₅, and may have any libration amplitude on tadpole or horseshoe orbits. In a first step, no torque is applied to the particle, so that its motion is described by a Hamiltonian with slowly varying parameters. We show that the torque applied to the satellite, as measured by ∈_s = j_s/(n _s J _s) induces an distortion of the phase space which is entirely described by an asymmetry coefficient α = ∈_s/μ, where μ = m _s/M. The adiabatic invariance of action implies furthermore that the long term evolution of the particle co-orbital motion depends only on the variation of m _s a _s with time. Applying a constant torque to the particle, as measured by ∈_s = j_s/(n _s J _p) is then merely equivalent to replacing α = ∈_s/μ by α = (∈_s − ∈_p)/μ. However, if the torque acting on the particle exhibits a radial gradient, then the action is no more conserved and the evolution of the particle orbit is no more controlled by m _s a _s only. We show that even mild torque gradients can dominate the orbital evolution of the particle, and eventually decide whether the latter will be pulled towards the stable equilibrium points L ₄ or L ₅, or driven away from them. Finally, we show that when the co-orbital bodies are two satellites with comparable masses m ₁ and m ₂, we can reduce the problem to that of a test particle co-orbital with a satellite of mass m ₁ + m ₂. This new problem has then parameters varying at rates which are combinations, with appropriate coefficients, of the changes suffered by each satellite.