The Verification Venue · a resonance, integrated until it answers

The Gap That Carves Itself

Every interactive orrery shows a mean-motion resonance as a quiet island where orbits lock and stay, and the received account of the 3:1 says the island is really a machine: it pumps the eccentricity of what it traps until nothing is left, and the gap is carved by chaos rather than by anything static. That is a claim about a computation nobody watching an orrery can afford. This page runs it: a compiled symplectic engine sweeps 1,028 particles across the 3:1 for three million steps each, 3.08 billion particle-steps, on every core you have. In this model, one perturber on a circular orbit at mass ratio 0.001, the measured answer is no. Zero of 1,024 particles crossed the escape threshold in 4,774 perturber orbits, and the chaos profile across the resonance is flat. A particle planted where the model really is chaotic escapes in the same run, which is what makes that zero a measurement rather than a broken detector.

Drag the slider to place a test particle inside or outside the resonance, press launch, and watch sixty-four clones integrate in front of you. The resonant clones librate in a tidy band, their neighbours circulate, everything looks calm, and from this picture the resonance looks like a place that holds things. The interesting question is whether that picture survives 4,774 perturber orbits for each of 1,028 particles, where the orrery above gives 637 orbits to 64 of them, and the honest way to find out is to run it rather than to assert it. The full sweep below starts its particles at e₀ = 0.2, an eccentricity at which the resonance is wide and any pumping should be at its fastest inside the run, and integrates every one of them for 30,000 time units. What comes back is a resonance that is plainly there and plainly not carving: read the escape fractions and the Lyapunov profile below, and then read the two planted controls that say what the same instruments do when there is something to see.

Where to watch · core of the resonance

Presets move the watched clone. They do not change the model: the same mass ratio, step and threshold govern every clone on this page.

The mechanism, named. The critical argument of the 3:1 commensurability is σ = λ − 3λ′ + 2ϖ: the particle's longitude minus three times the perturber's longitude plus twice the longitude of its perihelion. In the frame corotating with the perturber, where λ = θ + t and λ′ = t identically, this reduces exactly to σ = θ + 2ϖ, and that is the form integrated and displayed on this page. (An earlier revision displayed 2ϖ − 3θ, which differs from σ by −4θ and therefore winds four times per orbit for every particle, resonant or not; it carried no libration information.) Deep in a 3:1 resonance σ librates, swinging inside a bounded band instead of winding around the circle. Circulation versus libration is a winding number, so it is unchanged if mean longitudes are used instead of true ones; this page uses the true-longitude form because it needs no Kepler equation. Watch the σ strip under the traces: it is how you can tell a trapped clone from a passing one. Measured offline on the shipped reference over 1,000 time units, σ turns 0.7 and 0.9 times for the two particles planted at a_res, 2.8 times for the sweep particle nearest to it, and 73 times for one at the low edge of the sweep: a hundredfold separation in the same instrument. The resonance is unmistakably there. What this page then measures is that being trapped in it, in this model, does not take you anywhere.

Everything below is a stated parameter of this model, chosen for it, not a measurement of the solar system. No figure from the observational literature appears anywhere on this page.

Four numbers that moved, and why each moved. The diagnostics used to be read off the wrong state. Because the two half-kicks where neighbouring steps meet are fused into one full kick, the pair sitting in the registers between steps is the position at time t with the momentum at t + h/2, and every recorded eccentricity and Jacobi error was computed from that mismatched pair. It cost an order: the recorded Jacobi error was O(h) instead of the map's O(h²), measured 30 to 130 times larger than the map actually delivers, and every recorded eccentricity carried a bias of about 0.02. Undoing the half kick before each diagnostic fixed all four numbers at once. The drift ceiling goes back to 0.005, from the 0.1 it had been restated to: the 1e-2 delivery that justified 0.1 was the unsynchronised reading, and the real delivery over the full run is 7.2e-4. The pointwise cross-method horizon goes back to 5,000 steps from 2,000, because the comparison that was failing at 5,000 was comparing a leapfrog sample held for a hundred steps against a Runge-Kutta sample taken at the step, offset by one sample; sampled properly, the worst disagreement over 5,000 steps is 1.9e-3. The circular control moved from a = 0.72 to a = 0.36: at 0.72 a circular start reaches e ≈ 0.020, which two independent integrators agree on, so it is dynamics rather than artefact, but a control that pumps twice its own ceiling is not a control. At 0.36 the same measurement gives 0.0025. And the planted resonant starts are now asked for max e ≥ 0.23 rather than 0.25, because 0.25 was only ever reached through the 0.02 bias; read at the synchronised state the same two orbits deliver 0.2388 and 0.2356, and Runge-Kutta puts the second at 0.2355.