Islands of Stability Orbits
Around the Sun, stable orbits sit in islands. Jupiter clears some lanes of the asteroid belt and shelters crowds in others, and the same cause does both.
An asteroid's year depends on its distance from the Sun. Where that year is a simple fraction of Jupiter's, say one third of it, the asteroid meets Jupiter at the same points of its orbit over and over, and the repeated pulls add up instead of averaging out. These places are called resonances. Some of them empty the belt. Others hold asteroids in place for billions of years. In the physics of orbits, the stable regions have been called islands of stability for decades: islands of regular motion in a chaotic sea.
Which sense of island. The belt below is a chart-island, as on the nuclear page: regions of a chart whose occupants have outlasted their neighbors. The map further down shows the original dynamical islands, regions of regular motion in a system without friction. The theory page says this second picture carries over to its own islands only as chaotic saddles and long transients. The test on this page was built to check where one proposition stops applying because of that. It ran twice, and neither of its two runs counts: the first did not measure what it was built to measure, and the corrected run's own control failed.
The belt, measured
Every numbered asteroid, counted by its distance from the Sun. The scale is logarithmic, so that the small groups beyond the main belt show. Dashed lines mark Jupiter's resonances: red where the belt is emptier than just beside it, green where it is fuller.
| Resonance | Distance (au) | Within 0.01 au | Same width beside | Ratio | Reading |
|---|---|---|---|---|---|
| 4:1 | 2.06 | 52 | 374.5 | 0.14 | gap |
| 3:1 | 2.50 | 583 | 16,829.0 | 0.03 | gap |
| 5:2 | 2.82 | 2,512 | 15,003.0 | 0.17 | gap |
| 7:3 | 2.96 | 5,509 | 10,409.0 | 0.53 | gap |
| 2:1 (the Hecuba gap) | 3.28 | 158 | 3,495.0 | 0.05 | gap |
| 3:2 (the Hildas) | 3.97 | 1,210 | 201.0 | 6.0 | cluster |
| 4:3 (the Thule group) | 4.29 | 7 | 1.5 | 4.7 | cluster |
| 1:1 (the Trojans) | 5.20 | 1,506 | 1,026.0 | 1.5 | cluster |
Of 895,910 numbered asteroids, the deepest gap is at the 3:1 resonance: 583 asteroids within 0.01 au of it, against 16,829 in the same width beside it. The Trojans spread over about 0.3 au around Jupiter's own distance: 10,590 asteroids there, against 14.5 on average in equal widths either side. The database classes 10,633 of them as Jupiter Trojans, and 3,821 asteroids sit in the Hilda group at the 3:2 resonance.
Why the same cause cuts both ways: in a gap, the repeated pulls make the orbit chaotic, its elongation wanders until it crosses the orbit of Mars, and the asteroid is eventually removed. Wisdom showed this for the 3:1 gap in 1982 and 1983. In the islands, the resonance keeps the repeated meetings away from Jupiter: the Hildas are at the far point of their orbits whenever Jupiter passes, and the Trojans stay sixty degrees ahead of it or behind it.
Islands in a chaotic sea
The standard map is the textbook model of motion near a resonance: at each step a kick of strength K changes a momentum, and the momentum moves an angle. Click anywhere in the square to start an orbit and watch where it goes. Near the middle, orbits trace closed curves: that is the island. Elsewhere they scatter through the chaotic sea. Raise K and the sea floods the islands. Lower it and the islands grow and merge.
Click in the square to start an orbit.
Chaotic orbits that come close to an island tend to stick to it: its edge is layered with smaller islands, and an orbit can linger among them for a very long time before it drifts off. In the asteroid belt that shows up as asteroids that sit near a resonance's edge far longer than chance would suggest.
What the test asked
Does a long stay near an island predict a long remaining stay?
Proposition 3 says that under a steady stream of disturbances, a stay's age is no evidence of how long it has left. Orbits without friction are the case where that should fail: a stay near an island that has already lasted a long time tends to last much longer, because the orbit is stuck in the island's layered edge. The test followed 10,000 orbits of the map for a million steps, once undisturbed and once with a small random kick at every step. Only the disturbed arm meets proposition 3's condition, so only it can count for or against the proposition. The undisturbed arm shows whether the condition is needed. Theory predicted both answers, so this was a check of the proposition's scope rather than a discovery.
Result of the corrected test, r2
The corrected test ran once, on 30 September 2026, after its rules were deposited as 10.5281/zenodo.23071492. Its random seed comes from that identifier.
| Arm | Role | R, with its interval | Stays counted | Call under the rules |
|---|---|---|---|---|
| Undisturbed, stays counted from entry | the control | 0.31 (0.31 to 0.31) | 157,412,984 | unresolved |
| Disturbed, a kick of 0.1 at every step, each stay from the center | the test | 0.99 (0.98 to 0.99) | 28,417,946 | memoryless |
The control did not show the memory theory predicts, so the instrument failed its own check. In the undisturbed arm half of all stays ended within 2 steps, and all but one in a hundred within 30: the two ages the rules compare fall among quick passes through the neighborhood, before any sticking shows. The disturbed arm's stays were memoryless, but a test whose control fails counts neither way. Verdict on proposition 3: void. Under the rules the test is void, and no verdict on proposition 3 is entered.
The first run, r1, recorded as invalid
The test ran once, on 30 September 2026, after its rules were deposited as 10.5281/zenodo.23066015. Its random seed comes from that identifier.
| Arm | R, with its interval | Stays counted | Call under the rules |
|---|---|---|---|
| Undisturbed | 0.67 (0.66 to 0.68) | 1,240,547,674 | unresolved |
| Disturbed, a kick of 0.1 at every step | 81.42 (81.29 to 81.54) | 1,120,798,688 | memory |
R compares the time a stay has left once it reaches an age only one stay in a hundred reaches with the time left at the median age. Memoryless stays give 1, and stays that stick give much more. Under the deposited rules, memory in the disturbed arm kills proposition 3 in this model, and that is the registered verdict.
Recorded as invalid. After the run, the test turned out not to measure what it was built to measure. Its neighborhood, a square drawn around the island, does not contain the island: the island's own orbits swing outside the square on every lap. So the stays it counted are fragments of laps, not stays near the island. In the undisturbed arm, none of 1,240,547,674 stays lasted more than 4 steps, where theory expects stays of thousands of steps at an island's edge. The author has recorded the run as invalid: its verdict counts neither for nor against proposition 3, and its numbers stay here as they came out. A corrected test, r2, followed. Its neighborhood is measured from the island's full edge, its undisturbed arm must first show the stickiness theory predicts before any verdict counts, and every disturbed stay starts at the island's center, so that all have the same depth. It was deposited and run once. Its result is above.
Cost line and fragility line
Cost line. Nothing is spent to keep an asteroid in a resonance: orbits without friction keep their energy. As with nuclei, the island is paid for once, by where the asteroid ended up.
Fragility line. The Trojan islands exist only because Jupiter is light compared with the Sun: Jupiter's share of their combined mass would have to be below about 0.0385 for the points they circle to be stable, and it is about 0.001. The islands also leak slowly: pulls from the other planets let some Trojans drift out over the age of the Solar System.
How the propositions meet the orbits
| Proposition | On the orbits | Status |
|---|---|---|
| P1 Sparsity | Not tested. The main belt is full except at its gaps, and the islands beyond it are sparse. | not applied |
| P2 The evidence rule | Illustrated. An orbit stuck to an island's edge looks regular for a long time before it leaves: proposition 2's sticky boundary, in the system where the phrase comes from. | illustration |
| P3 Exit statistics | Tested for its scope: whether a stay's age predicts its remaining length, with and without a steady disturbance. It ran twice: the first run was invalid and the corrected run void, so neither counts. | no verdict |
| P4 Cost | Does not transfer: nothing is spent to hold an orbit, as with nuclei. | does not transfer |
| P5, P6 | Framing propositions, stated for organizations. | not applied |
Source and method
Data: NASA/JPL Small-Body Database (SBDB) Query API, query https://ssd-api.jpl.nasa.gov/sbdb_query.api?fields=spkid,a,e,i,class&sb-kind=a&sb-ns=n, retrieved 2026-09-29, 36,954,812 bytes, SHA-256 1de4bfccccfb39a32895f353bb404c69021c42cf3f7b6273b4443af6f8ba1536. Data: NASA/JPL Small-Body Database, Solar System Dynamics Group, Jet Propulsion Laboratory. The elements are osculating (the orbit at the database's epoch), not proper elements, which blurs the gaps and groups slightly.
Resonances: an asteroid that makes p orbits while Jupiter makes q sits at Jupiter's distance times (q/p)2/3, by Kepler's third law, with Jupiter's mean distance 5.2029 au (Standish, JPL). At each resonance the count within 0.01 au is compared with the mean of two windows of the same width, 0.05 au either side. The Trojans spread wider, so their island is also measured over 0.3 au against equal widths 0.6 au either side. The map: p′ = p + K sin θ, θ′ = θ + p′, both modulo 2π (Chirikov, Physics Reports 52, 1979). On the 3:1 gap: Wisdom, Astronomical Journal 87, 1982, and Icarus 56, 1983. The frozen test (r1) is deposited, with its code, as 10.5281/zenodo.23066015.