Extensions of Brown Hamiltonian – I. A high-accuracy model for von Zeipel–Lidov–Kozai oscillations

Triple systems with low hierarchical structure are common throughout the Universe, including examples such as high-altitude lunar satellites influenced by the Earth, planetary satellites perturbed by the Sun, and stellar binaries affected by a supermassive black hole. In these systems, non-linear perturbations are significant, making classical double-averaged models – even those incorporating the Brown Hamiltonian correction – insufficient for accurately capturing long-term dynamics. To overcome this limitation, the current study develops a high-precision dynamical model in the test-particle limit that incorporates the non-linear effects of the quadrupole-order potential arising from both the inner and outer bodies, referred to as the extended Brown Hamiltonian model. This framework specifically expresses the Hamiltonian function and the transformation between mean and osculating orbital elements in elegant, closed forms with respect to the eccentricities of the inner and outer orbits. Practical application to Jupiter’s irregular satellite shows that the long-term evolutions predicted by the extended Brown Hamiltonian model align well with the results of direct N-body simulations. The developed Hamiltonian offers a fundamental dynamical model, which is particularly well suited for describing von Zeipel–Lidov–Kozai oscillations in low-hierarchy three-body systems.