ABSTRACT Hierarchical triples are promising environments for producing exotica such as black hole mergers and hot Jupiters, because of the von Zeipel-Lidov-Kozai (ZLK) effect, whereby a distant tertiary can torque an inner binary to high eccentricity over secular timescales. In the double-averaged (DA) approximation to ZLK, this eccentricity excitation is suppressed by apsidal precession due to ‘short-range forces’ (SRFs) like relativity and tidal/rotational bulges. Here we show that, when the DA approximation is relaxed, SRFs often catalyze, rather than suppress, extreme eccentricity behavior. This occurs because SRFs can drive large, discrete jumps in the binary’s effective ‘adiabatic invariants’ during high-eccentricity episodes. These nonadiabatic jumps can dramatically alter the maximum/minimum eccentricity and secular period of astrophysically relevant triples, including some for which SRFs were previously thought irrelevant. Even the angular momentum component jz evolves secularly — to our knowledge, this is the first time such evolution has been demonstrated from a quadrupole-order, three-body mechanism. In short, binaries may explore much more of phase space than is implied by any (semi-)analytic ZLK theory of which we are aware. We demonstrate this at the test-particle quadrupole level; in a companion paper we show how even more-extreme behavior occurs when the jumps are combined with octupolar ZLK evolution.
1. INTRODUCTION
Petrovich 2015; Anderson et al. 2016; Mu˜noz et al. 2016; Owen & Lai 2018; Vick et al. 2019; Teyssandier et al. 2019; O’Connor et al. 2021; Stephan et al. 2021; Angelo et al. 2022; Stephan et al. 2024; Weldon et al. 2025), binary black hole (BBH) mergers (Miller & Hamilton 2002; Blaes et al. 2002; Wen 2003; Antonini & Perets 2012; Naoz et al. 2013; Antonini et al. 2016; Stephan et al. 2016; Liu & Lai 2017; Hoang et al. 2018; Grishin et al. 2018; Liu & Lai 2018; Hamilton & Rafikov 2019; Liu & Lai 2019; Martinez et al. 2020), neutron-star gravitational-wave sources (Feng et al. 2026), blue stragglers (Perets & Fabrycky 2009; Naoz & Fabrycky 2014; Antonini et al. 2016), tidal disruption events (Ivanov et al. 2005; Melchor et al. 2024, 2025), Type Ia supernovae (Thompson 2011; Katz & Dong 2012), and white-dwarf binaries, cataclysmic variables (CVs), and low-mass X-ray binaries (LMXBs) (Shariat et al. 2023, 2025a,b). Whatever the astrophysical context, a theory of ZLK dynamics needs to answer two basic questions:
Hierarchical triple systems are widely studied as potential origin sites of exotic astrophysical objects because of the famous von Zeipel-Lidov-Kozai (ZLK) oscillations (von Zeipel 1910; Lidov 1962; Kozai 1962)1. Precisely, an inclined tertiary companion can periodically torque an inner binary to very high eccentricity — i.e., to very small pericenter distance — on secular timescales. At small pericenter distances, interactions between the binary components can occur that would not have been possible in the absence of the ZLK effect, such as dissipation through gravitational wave emission or tides, or even head-on collisions. This has led many authors to identify ZLK oscillations as the culprit behind the formation of (at least some) hot Jupiters and the eccentricity distribution of cold Jupiters (Naoz et al. 2011, 2012;
Corresponding author: Ygal Y. Klein ygalklein@gmail.com chamilton@princeton.edu
(i) what is the maximum possible eccentricity emax that the inner binary can reach? and
1 For a historical overview, see Ito & Ohtsuka (2019).

2 Klein & Hamilton (ii) how long does it take to get there, i.e., what is the secular period (or ‘Kozai period’) PK?