Held, Rocking, Locked: The Islands of a Light Star Around an Eccentric Binary

We follow a planet-mass body around an eccentric pair of equal stars, near the 7:1 and 8:1 period ratios, by direct three-body integration for up to 60,000 orbits, to see up close why three bodies are hard. The star is held where two clocks keep time together: one angle, φ_N, comparing the pair's orbit with the star's lap, stays put while every other angle turns. Escape is one-sided: with the star's eccentricity at 0.20, 163 of 166 recorded escapes, from light and heavy stars alike, leave from the island's side nearest the pair, a door the published stability rules smooth over, and one that 41 stars started on circular orbits, as stability maps start theirs, did not find in 10⁵ of the pair's periods. An island read over 600 orbits can be slow chaos: at the pair's eccentricity 0.25, 22 of 28 inner starts lose their hold within 6,000, and the room that keeps its stars, births 7.465 to 7.505, lies below the centre the first 600 orbits point to. Mardling's (2013) resonance pendulum passes a direct test where its assumptions hold, its rate–width relation within 1.1% and its widths within about 30%, and is off by measured amounts outside them: the island's centre doubles once, at the star's eccentricity 0.348 to 0.350 rather than 0.41, and then every nearby start escapes. Inside the island the rocking locks to simple fractions of a cycle per orbit, at a fifth to within a few billionths, and odd fractions fill about three times as often as even ones. The hold drags on the star's ellipse: Mardling's coefficient, with nothing fitted, gives the drag's slope within about 6% at the star's eccentricities 0.20 to 0.26, and near 0.28 the centre's ellipse stands still, then turns backward. A heavy third star moves the islands just below the whole numbers, and the pair's short-period swing is about 1.12 times Georgakarakos's (2003) formula. At a tolerance that looks tight, the integrator's arithmetic drags every unlocked rocking rate down and fakes a slow slide of locks, and with the heavy star it shortens a slow beat by a third over 30,000 orbits; a hundredfold tighter tolerance removes both, and at a second birth a test finds the slow beat keeping pace with the pair's falling eccentricity. The method was first checked on maps with known answers, reproducing the frictionless and damped period-doubling constants and the dial between them, the area an orbit keeps per trip. Many tests began as marks made by eye on plain pictures, each written down before any number was read: a star polygon drawn on one picture is the inscribed-angle theorem, and a scatter of readings that looked chaotic is, zoomed out and read at a tight tolerance, a rotation of two angles, out to near returns 31,063 orbits apart in a run of 60,000, from which a twin nudged by 10⁻¹⁰ drifts off in a straight line, as a regular orbit's twin does; sorted blind, reads naming a relation held more often than reads naming a shape, 19 of 30 against 11 of 28. Of 490 predictions written down before their runs, 236 came out as written; every miss is kept. A preprint, without peer review and with a limited literature pass. What this paper found: Held by coupled clocks. Near seven (or eight) of the pair's periods per orbit, one angle, φ_N, stays put while every other angle turns: the pair's orbit and the star's lap keep time together. (Section 5; Figure 7.) A one-sided door. With the star's eccentricity at 0.20, 163 of 166 recorded escapes, from light and heavy stars alike, leave from the island's side nearest the pair (Figure 1); from 0.30 a second door opens on the far side. The published stability rules smooth the door over, and stars started circular do not find it. (Section 6.) Islands that are slow chaos. At the pair's eccentricity 0.25, what 600 orbits read as an island is, over 6,000, a row of rooms divided by chaotic threads, and the room that keeps its stars, births 7.465 to 7.505, lies below the centre the first 600 orbits point to, near 7.525. (Section 6; Figure 9.) Mardling's pendulum, tested piece by piece. Its rate–width relation holds within 1.1% where her assumptions hold, and her widths come within about 30% of ours. Once a chaotic layer eats the island, the relation holds instead, within about 10%, for the island and its layer together at the pair's eccentricities 0.05 to 0.20, a reading made after the fact that held when tested forward at 0.225 and missed narrowly at 0.25. (Sections 5, 6 and 10; Appendices B and C.) Locks to a few billionths. The rocking locks on simple fractions of a cycle per orbit, the core of the lock at a fifth to within 3.5 × 10⁻⁹ over 6,000 orbits, and odd fractions fill about three times as often as even ones. (Section 8; Figures 11 and 12.) The hold drags the ellipse, then turns it back. At the island's centre the star's ellipse turns at two-thirds of its free rate. Across the island the drag follows the hold's grip, and Mardling's coefficient, taken at each line's mean eccentricity with nothing fitted, gives its slope over 600 orbits within about 6% at the star's eccentricities 0.20 to 0.26. Near 0.28 the centre's ellipse stands still, and above it turns backward. (Section 9; Figure 13.) One doubling, then escape. The island's centre doubles once, between the star's eccentricities 0.348 and 0.350, where the pendulum would only at 0.41; then every nearby start escapes. (Section 10; Figure 14.) A heavy third star. Its islands sit just below the whole numbers instead of above, and the pair's short-period swing is about 1.12 times Georgakarakos's (2003) formula, at half the period ratio he tested. (Section 11; Figures 15 and 16.) The arithmetic can fake resonance physics. At relative tolerance 10⁻⁹ every unlocked rocking rate falls by about 2.8 × 10⁻⁸ cycles per orbit with each orbit, on the line measured most closely, enough to fake a slide of locks; a hundredfold tighter tolerance shrinks the fall to 0.0093 of itself. With the heavy star the same drift shortens a slow beat by a third over 30,000 orbits, and at a second birth a test finds the slow beat keeping pace with the pair's falling eccentricity. Ten times tighter again, over 60,000 orbits, the creep is about a hundred times smaller. At looser tolerances it moves the orbit's two rates in the proportion a nudge in birth does, leaving the eccentricity's second line almost still, though it is no nudge in birth: it drains the energy where a nudge outward adds to it, and it keeps its sign where the drifts of the energy and of the angular momentum both change theirs. Made again on a second machine, the long runs gave the same verdict in every test registered before they were sent there. (Section 3; Appendix A.5.) The bench and the guesses. On maps with known answers the method reproduced the frictionless and damped doubling constants, 8.721082 and 4.669197, and found the dial between them (Section 4; Figure 6). Of 490 predictions written before their runs, 236 came out as written, 211 did not and 43 could not be read. (Section 3.) How the eye found it. Several results began as the author's marks on plain pictures, each written down as a test before anything was measured: a clockwork star polygon that is the inscribed-angle theorem, dead zones that runs he never drew on avoid, and the beat hidden in the centre's rocking. Sorted blind, picture-reads that name a relation held 19 times in 30, those that name a shape 11 in 28. (Section 3; Figure 4.) The kinds of thing measured here are known: mean-motion resonance islands, their chaotic edges, secondary resonances, slices whose highest point resembles a vanishing twist, and doubling centres (Section 14). What this paper adds, as far as our literature passes found, is their measurement in one circumbinary setting by direct integration, over runs of up to 60,000 orbits, with every number tied to a run that reproduces on a laptop. It tests Mardling's pendulum in the test-particle limit, where it holds and where it misses. It finds a one-sided door at a resonance beyond the stability limit, which the published stability rules smooth over and which a star started circular, as the stability maps start theirs, does not find; and slow chaos that 600-orbit maps read as islands. It measures the rocking's lock at a fifth to a few billionths of a cycle per orbit, which fractions' locks fill most, the drag's size from Mardling's coefficient with nothing fitted and where, over longer records, it stops being one number; a single doubling followed by escape; and the heavy star's tide as Georgakarakos's short-period response. And it puts a size on the drift that a tolerance which looks tight fakes in resonance frequencies. Each section opens with what it found, with what is known beside it. This is a preprint, not peer reviewed, with a limited literature pass. It was built with the assistance of Claude (Anthropic) in algorithm and program development and in drafting; the declaration page says how each cited work was read and how Claude was used. CC BY 4.0. It cites the author's field guide for its way of working and the author's Crossover Atlas for the circle-valued counting it relies on.