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)).

Thus the system has ‘jumped’ from one DA cycle to another in a single high-e episode, and this new cycle may have a totally different minimum/maximum eccentricity and secular period to the old one.

As a result, the answers to questions (i) and (ii) above vary substantially from one secular cycle to the next.

These nonadiabatic jumps were first reported by Hamilton & Rafikov (2024) (hereafter HR24), who studied the test-particle quadrupole ZLK problem with GR precession.

They were also identified in the simulations of Rasskazov & Rafikov (2024).

These papers called the effect relativistic phase space diffusion, but we now think this was a poor choice of name.

The jumps are neither diffusive — a single jump can shift CK by O(1) in one secular cycle — nor necessarily driven by GR, since any SRF whose apsidal precession rate diverges at high eccentricity through Eq. (1), equilibrium tides and rotational bulges included, will drive jumps of the same kind.2 They also occur under a much wider (and more realistic) range of conditions than were appreciated by HR24, and they do cause jumps in jz, whereas HR24 believed they did not.

It is therefore worthwhile to revisit this problem and some of its implications in light of our new understanding.

The rest of this paper is organized as follows.

In §2 we set up the dynamical framework of our problem, introduce the SA and DA equations, and discuss the validity criteria for both of them.

In §3 we provide numerical examples, with various SRFs included, in which nonadiabatic jump behavior drives radical changes in emax, PK and associated quantities.

In §4 we discuss our results in the context of earlier literature and outline future extensions.

We summarize in §5.
dj dτ = j × ∇jϕ + e × ∇eϕ, (2) de dτ = j × ∇eϕ + e × ∇jϕ, (3)
where τ ≡t/Psec is the time normalized by a characteristic secular timescale �a2 a
Psec = 1
�3 (1 −e2
2)3/2 Pin
2π m0 + m1
m2
� G(m0 + m1)a
=
, (4) Φ0
where Pin is the inner orbital period, and
Φ0 ≡ Gm2a2
(5)
a3 2(1 −e2 2)3/2 ,
measures the strength of the tertiary’s perturbation upon the binary, and ϕ ≡Φ/Φ0 is the dimensionless perturbing potential (i.e., the true perturbing potential Φ normalized by Φ0).

In general ϕ includes contributions from the tertiary’s tidal potential, as well as any SRF.

For this paper we expand the tidal potential in the small parameter a/a2 ≪1 and keep only the quadrupolar term.

We now make the SA approximation, i.e., we average ϕ over the inner orbit while fixing the outer body’s instantaneous position r2 = r2ˆr2.

The full dimensionless SA perturbing potential is then
2.

DYNAMICAL FRAMEWORK We consider an inner binary with component masses m0 and m1 and semimajor axis a, perturbed by a distant mass m2 on an outer orbit with semimajor axis a2
ϕSA = ϕSA quad + ϕSRF, (6)
where
2 This conservative, single-averaging effect is distinct from the relativistic “precession resonance” of Kuntz (2022), in which gravitational-wave radiation shrinks the inner orbit until its apsidal precession rate is commensurate with the outer orbit.

That mechanism requires both dissipation and strong apsidal precession, whereas our jumps invoke neither.