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SRF Catalysis of Extreme Orbital Evolution 17
APPENDIX
A.

INITIAL CONDITIONS AND NUMERICAL INTEGRATION
masses and the two semimajor axes—whereupon the dimensionless set (A1) maps onto dimensional initial conditions.

In the next subsection we list such dimensional realizations for each example.

In §3 we quoted the initial conditions of each example as standard orbital elements (e, ω, i, Ω, and so on).

That is not how we actually set up or run the integrations.

In practice we work entirely in dimensionless variables: we fix each system by a small set of dimensionless numbers, build the initial j and e vectors from that set, integrate the dimensionless equations of motion, and only afterwards—when we want a concrete astrophysical system—convert to physical units.

Up to an overall timescale, the dynamics of any system we study is fixed by the nine dimensionless numbers
A.1.

Dimensional realizations of the numerical examples The three numerical examples of §3 are specified by their dimensionless parameter sets (Appendix A); the dynamics depend only on these dimensionless numbers, so each parameter set corresponds to a whole family of dimensional systems related by scaling, all evolving identically as functions of τ = t/Psec.

For reproducibility, Table 1 gives one explicit, high-precision dimensional realization of each example — the masses, semimajor axes, eccentricities, body parameters and orientation angles one would supply to a secular or N-body integrator.

The inner orbit starts at its DA eccentricity minimum, e = emin(jz, CK), with ω = 0 (rotating branch, CK > 0; Appendix A); for these parameter sets (CK = 10−6) this is e = √CK = 1.0 × 10−3, and the round value e = 0 quoted in the main text is shorthand for this small but nonzero minimum.

For the GR example (§3.1) we list three realizations of the same dimensionless parameter set, all orbiting an m2 = 107 M⊙SMBH on an e2 = 0.8 orbit: a binary neutron star (NS+NS), a 10 + 10 M⊙BBH, and the 30 + 30 M⊙BBH shown in the main text.

For the remaining two examples (tides and rotation, §§3.2-3.3) we show only one realization.
jz, CK, Ω, e2, εGR, εRot, εTide, Pout/Psec, M2. (A1) All systems that share this set evolve identically as functions of τ = t/Psec.

We specify these numbers first, and everything else follows from them: the pair (jz, CK) labels the underlying no-SRF DA quadrupole cycle; e2 and Ωorient the inner orbit relative to the outer apse; (εGR, εRot, εTide) set the SRF strengths through Eqs. (9)–(11); and Pout/Psec and the initial outer mean anomaly M2 fix the phase of the outer orbit, which single-averaging retains.8 From this set we build the initial state in three steps.

First, from (jz, CK) we compute the minimum eccentricity emin of the no-SRF DA quadrupole cycle (Antognini 2015; Basha et al. 2025; Kinoshita & Nakai 2007; Vashkov’yak 1999) and start every integration there, e = emin, so that each run begins at a well-defined phase of its cycle.

Second, we fix the apsidal branch from the sign of CK: we take ω = π/2 for librating cycles (CK < 0) and ω = 0 for rotating cycles (CK > 0).

Third, we orient the orbit in the outer frame with Ω.

These three choices determine the initial j and e.