The Hamiltonian for von Zeipel–Lidov–Kozai oscillations

Abstract

ABSTRACTThe Hamiltonian used in classical analyses of von Zeipel–Lidov–Kozai or ZLK oscillations in hierarchical triple systems is based on the quadrupole potential from a distant body on a fixed orbit, averaged over the orbits of both the inner and the outer bodies (‘double averaging’). This approximation can be misleading, because the corresponding Hamiltonian conserves the component of angular momentum of the inner binary normal to the orbit of the outer binary, thereby restricting the volume of phase space that the system can access. This defect is usually remedied by including the effects of the octopole potential, or by allowing the outer orbit to respond to variations in the inner orbit. However, in a wide variety of astrophysical systems, non-linear perturbations are comparable to or greater than these effects. The long-term effects of non-linear perturbations are described by an additional Hamiltonian, which we call Brown’s Hamiltonian. At least three different forms of Brown’s Hamiltonian are found in the literature; we show that all three are related by a gauge freedom, although one is much simpler than the others. We argue that investigations of ZLK oscillations in triple systems should include Brown’s Hamiltonian.