tokens: (the second average). This means that the short-term effects in the time-scale of the outer binary’s period are incorporated into the double-averaged Hamiltonian, but the short-term effects in the time-scale of the inner binary’s period are neglected. Thus, the application conditions require that the elements of the test particle are slow variables in the time-scale of the inner binary’s orbital period (i.e. the secular frequencies of the elements need to be much smaller than the mean motion of the inner binary). It is evident that the modified approach broadens the application range compared to the classical double averaging. As for the computational efforts about the classical and modified double-averaged models, we report in Table 1 the time required by the numerical integration of different-order models over a time span of 1.5 × 10 4 yr for the Sun-perturbed Uranus–Caliban system (the initial conditions and time span are the same as the ones of Fig. 6 ). It is observed that, compared to the classical approach, the modified version requires additional computation time. Therefore, when the application conditions of the classical approach are satisfied, the classical averaged model is preferable because in this case the periodic corrections considered in the modified approach become negligible but the computation time increases. Table 1. Computation time for the numerical integration of different-order averaged models about the Sun-perturbed Uranus–Caliban system. The initial conditions and time span are the same as the ones of Fig. 6 . The computation time about the full model over the same period is 9.720 s. In some physical systems, such as Jupiter’s irregular satellite systems, the strong perturbation coming from the disturbing body (the Sun) could make the elements of test particles no longer slow variables in the time-scale of the inner binary’s orbital period. In this case, the short-term effects appearing in the first average could accumulate and produce significant errors to the secular evolution. To overcome this problem, the short-term effects would need to be incorporated also in the first average (over the inner binary’s orbital period). 6 CONCLUSIONS In this work, we developed a modified double-averaging approach, which was used to formulate a more accurate Hamiltonian in comparison to the secular Hamiltonian formulated by the classical double averaging. To test the modified approach, the resulting double-averaged Hamiltonian was applied to a variety of hierarchical triple systems, covering planetary, and stellar scales. In this approach, the disturbing function appearing in the Hamiltonian is averaged twice in order to eliminate the short-term oscillations induced from the orbital motions of inner and outer binaries. When the perturbation coming from the disturbing body is relatively strong, the short-term effects neglected in the classical double averaging may accumulate over long time-scales. This leads to the
ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O