LK problem (assuming |jz| ≪1). The columns separate the averaging level used (DA on the left, SA on the right) and the rows separate the physics included (no SRFs on top, SRFs on the bottom). Each panel sketches the inner binary eccentricity e(t) over several secular cycles. Quadrants 1, 2, and 3 represent integrable or near-integrable problems that are well-understood (semi-)analytically. The focus of this paper is the fourth quadrant — SA + SRFs — wherein adiabatic invariance is completely broken.]

2018; Hamilton & Rafikov 2019). The correction δe can be of great importance at high eccentricity if it is comparable to or greater than 1−eDA max, which is exactly the regime in which many GW-merger and hot-Jupiter calculations reside. Again, though, the secular period PK (question (ii)) does not change. Another, more slowly-acting SA effect is that the torque fluctuations can accumulate over many secular periods, producing gradual drifts of the DA constants of motion jz and CK. These drifts can be absorbed into a corrected-DA potential, the so-called Brown Hamiltonian (Brown 1936a,b,c; Soderhjelm 1975; ´Cuk & Burns 2004; Breiter & Vokrouhlick´y 2015; Luo et al. 2016; Will 2021; Tremaine 2023; Klein & Katz 2024a; Grishin 2024; Lei & Grishin 2025a,b; Gao & Lei 2025). However, this phenomenon is not relevant for the current paper. We are going to describe a distinct effect that happens on a much shorter timescale.
what physics is included (no SRFs on top, SRFs on the bottom). As just discussed, quadrants 1, 2, and 3 are well-understood territory. Quadrant 1 (DA, no SRFs) is the original, integrable ZLK problem; quadrant 2 (DA
+ SRFs) is integrable but with the high-e excursions suppressed by apsidal precession; and quadrant 3 (SA, no SRFs) is nearly-integrable, departing from the DA baseline only through small SA fluctuations δe on the timescale Pout. This paper is concerned with the fourth quadrant (SA + SRFs). One might expect the dynamics here to be a straightforward superposition of quadrants 2 and 3 — small SA fluctuations sitting on top of an SRF-suppressed DA curve. But in general this is not what happens. Instead, SA fluctuations and SRF precession interact non-trivially, driving large discrete jumps in CK and jz, and producing dynamics qualitatively different from anything in the other three quadrants. The mechanism is the following. Away from high eccentricity the SA orbital elements oscillate tightly around their ‘parent’ DA trajectories, which can be labeled by the (near-)adiabatic invariants jz and CK. But during a brief excursion in which the inner binary reaches e →1, the SRF precession rate becomes ex-
1.3. This paper: SA + SRFs The basic idea of this paper is summarized diagramatically in Figure 1. The four quadrants organize the test-particle quadrupole ZLK problem by what averaging is used (DA on the left, SA on the right) and 4 Klein & Hamilton and eccentricity e2, with a ≪a2.

We work in the testparticle limit so the only evolution of the outer orbit is in its mean anomaly.

We describe the inner orbit by the Laplace–Runge– Lenz vector e and the dimensionless angular momentum vector j = J/ � G(m0 + m1) a, satisfying j2+e2 = 1 and j · e = 0.

We choose the z-axis along the fixed outerorbit angular-momentum vector and the x-axis toward the fixed outer eccentricity vector.

The inner orbit’s longitude of ascending node Ωis measured from this x-axis.

The evolution of j and e is governed by the Milankovitch (1939) equations:
tremely large (Eq. (1)).

If the apsidal angle ω changes by an order-unity amount on the outer-orbit timescale, the DA approximation — which presumes ω varies only on the much longer secular timescale — fails.

The SA dynamics then does not oscillate back around its original parent DA trajectory after the excursion; instead, it begins to oscillate around a different DA trajectory (labeled by a different pair (CK, jz)).