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?

When one expands the two-body problem beyond the Newtonian point-particle limit, the leading-order conservative corrections are due to SRFs like general relativity, equilibrium tidal bulges, and rotational bulges.

Each SRF manifests as apsidal precession of the argument of pericenter ω, with a rate that is steeply peaked at high eccentricity (Fabrycky & Tremaine 2007; Liu et al. 2015; Hamilton & Rafikov 2021; Grishin et al. 2018; Anderson et al. 2017; Blaes et al. 2002; Veras & Ford 2010; Naoz et al. 2013):
The simplest theory that answers these questions rests on certain physical and mathematical approximations.

It assumes pure Newtonian gravity — no short-range forces (SRFs) due to general relativity or tides or rotational bulges, nor any dissipation, is included — and treats the inner binary in the ‘test particle’ approximation so the outer orbit feels no back-reaction from it.

Then the tertiary’s tidal potential is expanded in powers of the inner-to-outer semimajor axes ratio a/a2 ≪1, truncated at the leading non-trivial (quadrupole) order, and the resulting equations are double-averaged (DA) over both the inner and the outer orbit.

In this limit the dynamics preserves two integrals of motion which, in dimensionless form, can be taken as the angular momentum component perpendicular to the outer orbital plane, jz ∈[−1, 1], and the averaged Hamiltonian, or ‘Kozai constant’, CK ∈[−3/2, 1].

This setting is completely integrable (Lidov 1962; Kozai 1962) and is in fact equivalent to that of a simple pendulum (Basha et al. 2025).

Both questions (i) and (ii) can be answered precisely from just the two numbers jz and CK; the maximum eccentricity emax is mostly set by jz (with smaller |jz| giving higher emax), and the secular period PK is mostly set by CK (with smaller |CK| giving longer PK).

This Newtonian, test-particle, quadrupole, DA setting is the simple, integrable baseline against which more realistic versions of the ZLK problem are usually analyzed.

It has long been known that the answer to question (i) — the maximum eccentricity reached by the inner binary — changes somewhat when one modifies the assumptions of this simple baseline case.

Meanwhile, the answer to question (ii) — the secular period PK — was thought to be essentially robust to these changes.

The purpose of this paper is to demonstrate that, in various realistic scenarios, the (semi-)analytic answers often given to both questions (i) and (ii) may need to be revised dramatically.

In this paper we focus on the test-particle, quadrupolar ZLK problem (and on systems capable of reaching very high eccentricity, which requires |jz| ≪1).

We modify the ZLK problem in exactly two ways compared to the simplest baseline just described.

First, we include the physical effect of apsidal precession due to SRFs.

Secondly, we relax the mathematical assumption of double averaging, instead working with the SA equations.

On their own, the role of each of these modifications is well-understood, as we now overview.
| ˙ωSRF| ∝ 1 (1 −e2)n , (1) where n = 1 for GR, n ≃4.5 for equilibrium tides, and n = 1.5 for rotational bulges.

One can include SRFs in the inner orbit dynamics in ZLK calculations.

In the DA approximation, the dynamics are again integrable.

The two constants of motion are now jz, as in the baseline case described above, and D, which differs from the baseline CK only at extremely high eccentricity (see §2.1).