.

In either case, the value of CK is important because (i) its sign determines the topology of the phase space motion, separating librating cycles (CK < 0), where ω oscillates, from rotating cycles (CK > 0), where ω circulates; (ii) it changes the secular period PK, often by a large factor near separatrices; (iii) it plays the key role in determining the minimum eccentricity emin; (iv) it modulates the maximum eccentricity emax (though the key role here is played by jz).

The range of allowed CK values depends on jz.

The maximum possible range is achieved at jz = 0, namely CK ∈[−3/2, 1].

As |jz| is increased, this range gets narrower.

This means that a jump in CK beyond its allowed range must be associated with a decrease in |jz|.

We will see this occur explicitly in §3.
m2
�2
�Ω2 1sR3 1
��R1 a
2)3/2
× (1 −e2
, (10)
Gm1 m0 + m1
�a2 a
�3
εTide = 15 k2,1 m0 m1
m2
�5
�R1 a
2)3/2
× (1 −e2
. (11) Here c is the speed of light, k2,1 and R1 are the tidal Love number and radius of the inner companion, and kq,1 and Ω1s are its apsidal-motion constant and spin rate.

These coefficients share the same definition as in Liu et al. (2015).

Note this means that εGR as defined here is smaller than the ϵGR used in Hamilton & Rafikov (2021) and HR24 by a factor of 16 (see Footnote 13 of Hamilton & Rafikov 2021).

The SA approximation itself can break down in some regimes (Antonini et al. 2014).

For the examples shown in this paper, however, the qualitative findings — the jumps in CK, the secular evolution of jz, the SRFcatalyzed extreme-eccentricity behavior — are robust to both the SA approximation and the test-particle approximation.

We discuss this further (and compare with the results of direct N-body integration) in Appendix B.
2.1.

Double-averaging
3 at least for the astrophysically-relevant cases we care about, where the SRF coefficients εi are ≪1.

There are of course systems with εi ≳1 but these tend not to reach very high eccentricities anyway (Fabrycky & Tremaine 2007; Hamilton & Rafikov 2021).

If we go a step further and average ϕSA (equation (6)) over the outer orbit, we find the DA potential
ϕDA = ϕDA quad + ϕSRF, (12)

6 Klein & Hamilton
emin, eDA max, PK, etc. Both naive corrections (19) and (20) therefore fail.

Related to this, we believe the criterion (17) has been interpreted much too loosely in the past — with double averaging assumed safe when Pout/Psec is only somewhat smaller, but not much much smaller, than jDA min.

We warn against all such shortcuts.
2.2.

Validity of the DA approximation, and semi-analytic SA corrections to it We expect the DA approximation in §2.1 to be valid only when the outer orbital period Pout is much shorter than the time over which DA orbital elements evolve significantly.

The latter evolution happens most rapidly at high eccentricity, on a characteristic timescale ∼(1−
3.

NUMERICAL RESULTS
e2 max)1/2Psec (e.g., Liu & Lai 2018; Hamilton & Rafikov 2021).

Thus, formally, we expect DA theory to be valid only if
In this section we present numerical examples of nonadiabatic jump behavior and the cycle-to-cycle changes in emin, emax, PK etc. that they cause.

We show three examples, isolating one SRF at a time.

For each example we integrate equations (2) and (3) up to τ = t/Psec = 400.

We perform integrations employing either the DA potential (12) or the SA potential (6), and in each case we run both with and without SRFs.

Each example below represents an astrophysically realistic system, with definite masses, semimajor axes and eccentricities.