Double-averaging can fail to characterize the long-term evolution of Lidov–Kozai Cycles and derivation of an analytical correction

Abstract

The double-averaging (DA) approximation is widely employed as the standard technique in studying the secular evolution of the hierarchical three-body system. We show that effects stemmed from the short-timescale oscillations ignored by DA can accumulate over long timescales and lead to significant errors in the long-term evolution of the Lidov-Kozai cycles. In particular, the conditions for having an orbital flip, where the inner orbit switches between prograde and retrograde with respect to the outer orbit and the associated extremely high eccentricities during the switch, can be modified significantly. The failure of DA can arise for a relatively strong perturber where the mass of the tertiary is considerable compared to the total mass of the inner binary. This issue can be relevant for astrophysical systems such as stellar triples, planets in stellar binaries, stellar-mass binaries orbiting massive black holes and moons of the planets perturbed by the Sun. We derive analytical equations for the short-term oscillations of the inner orbit to the leading order for all inclinations, eccentricities and mass ratios. Under the test particle approximation, we derive the "corrected double-averaging" (CDA) equations by incorporating the effects of short-term oscillations into the DA. By comparing to N-body integrations, we show that the CDA equations successfully correct most of the errors of the long-term evolution under the DA approximation for a large range of initial conditions. We provide an implementation of CDA that can be directly added to codes employing DA equations.

A census of main-sequence interactions in the Multiple Star Catalogue

“…The orbit-averaging approximation can break down in cases when the timescale for secular changes is comparable to some of the orbital periods in the system. For triples, correction terms have been derived that take into account (some of) the suborbital effects that can change the long-term secular evolution (e.g., Luo et al 2016;Breiter & Vokrouhlický 2018;Lei et al 2018;Lei 2019). It is left for future work to investigate the impact of orbit-averaging corrections in higher-order systems.…”

Binary Mergers near a Supermassive Black Hole: Relativistic Effects in Triples

“…As γ approaches unity with increasing m 3 , overlapping inclination and LK resonances give rise to the widespread chaos [44], causing systems with modest I 0 to attain extreme eccentricity growth. When e max becomes sufficiently close to unity, the timescale the inner BHB spends in high-e in phase (t LK 1 − e 2 max ; [51]) becomes less than the period of the outer binary, the DA approximation breaks down, and the system enters semi-secular regime [52,53]. If it is shorter than the inner orbital period, the evolution of triples can only be resolved correctly by N-body integration.…”

The wide-binary origin of (2014) MU69-like Kuiper belt contact binaries

“…The eccentricity of the orbit becomes unbound once the fluctuation in j z , ∆j z is larger than its initial value, namely ∆j z >j z . The fluctuation is estimated analytically in 7,17 , and can be used to show that the eccentricity is unbound if…”