Research: belt shapes prior art (embedded planetoids and gaps, Trojans, Oort-like edges), for ruling belt-shapes-direction

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

docs/research/2026-10-17-belt-shapes-prior-art.md

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+# Belt shapes: prior art (Eyes pass)
+
+Serves ruling: `aetheria-gravity:ruling:belt-shapes-direction` (follow-up `aetheria-gravity:follow_up:belt-shapes`).
+Session self-2026-10-17-ag5. Eyes, read-only. Facts with pointers; no conclusion or recommendation.
+Marks: (inferred) = my arithmetic or reading, not stated by the source. (gap) = looked, not found.
+Wikipedia pages were read through a summarizing fetch, so numbers are the summary's reading of the page, not verbatim; primary papers were not opened except through search snippets.
+
+## 0. Aetheria context read
+
+Operator words (ruling): a belt today is "just full belts"; she names planetoids "sweeping out an annulus", Trojans, and "Oort-like belts, ... spreading around some material at the edges of the wells". The ruling says these illustrate and do not close the space.
+
+Laws in force that bear on this (from the rulings-in-force query):
+- `belt-fills-its-slot` (Defaulted): a belt is massless, placed after siblings; slot = max(lower bound, previous sibling outer edge x q) to min(window, next sibling / q); width = min(AsteroidBeltWidth(D), slot). Production belts are slot-bound (median 11% of authored width, p90 29%).
+- `moons-within-reach-ruled`, `tail-taming-law-r3-band-loose-cusp`: children sit inside a fraction of the parent's reach (band law, reach 1/2 D), at least max(1.5 x parent Roche, ...) out, half-reach sibling spacing; what does not fit folds into the parent.
+- `binaries-orbited-from-afar`: nothing orbits on a binary's own ring ("strengthless companion on the ring" is a layout the generator must not produce); cites Holman & Wiegert 1999 (circumbinary stable beyond ~2-4 separations).
+- `asteroids-on-floater-layer`, `asteroid-tops-at-layer`, `rock-meshes-by-size`, `orbitals-above-floaters-below`: rocks sit on the floater layer, tops at layer; rock mesh ranked by size.
+- `rails-vs-field-ruled`, `tame-tails-parent-only-rails`: each orbit's period derives from its parent's pull only; rails, not n-body.
+- `well-strength-from-mass-ruled`: strength c = G m^p, p about 0.175-0.25. `core-at-roche-and-exaggeration-up`: core radius = Roche limit, a mass curve.
+
+Code read: `F:\Projects\Aetheria\Assets\Scripts\ServerShared\ZoneGenerator.cs` 557-662.
+- 558-564: belt candidates are childless planets with Mass < `BeltMassCeiling`, not binaries; each is flagged `Belt = true` with probability `BeltProbability`, at most one belt per parent (`!planet.Parent.Children.Any(p=>p.Belt)`).
+- 566-569: masses renormalized to the root's mass.
+- 576-581: `RocheMargin` 1.5, `CuspMargin` 1.5, `SiblingSpacing` .5.
+- 597: `WellMass` is 0 for a belt, so a belt contributes no pull and constrains no sibling's cusp bound.
+- 634-647: a belt is an interval inside its slot; `Distance` is the slot centre; `BeltHalfWidth = min(AsteroidBeltWidth(D), slot)/2`.
+- 625-627 / 599-603: when the outermost child overflows the window, the lightest family folds into the parent (`Fold`).
+Map (`docs/aetheria-gravity-map.md`): B53 / line 139: a belt's extent is the well tree's input ("inside L_F a node pulls as its body alone" holds for every rail round the node, and a belt is a family of rails out to its outermost asteroid); belts must keep their inner edge outside the Roche margin. Line 136: asteroid periods live on the belt (`Pose` and `BeltTargets` divide by the same float).
+(inferred) No code path read gives a belt a member body, a phase-locked cluster, or an outer-edge population; a belt is one interval per parent.
+
+## 1. Planetoid embedded in a belt that clears or sweeps a lane
+
+### 1.1 Observed cases
+| Case | Numbers | Source |
+|---|---|---|
+| Ceres, Vesta in main belt: they do not clear it | Ceres 39.2% and Vesta 10.8% of belt mass; belt total 2.39e21 kg, ~3% of the Moon; mean spacing ~965,600 km; belt "mostly empty" | https://en.wikipedia.org/wiki/Asteroid_belt |
+| Pan, Encke gap | Pan mean diameter 27.4 km (34.6 x 28.2 x 21.0); a = 133,584 km; Encke gap ~322 km; a ringlet shares Pan's orbit, taken as evidence of particles in horseshoe orbits | https://en.wikipedia.org/wiki/Pan_(moon) |
+| Pan mass and Hill radius | mass 8.6e-12 Saturn masses (~5e15 kg, inferred conversion); R_H ~19-20 km; one source states edge-to-Pan distance = 8.45 R_H | search snippet, https://arxiv.org/pdf/1211.4762 and https://en.wikipedia.org/wiki/Pan_(moon); (inferred) gap ~16 R_H wide |
+| Daphnis, Keeler gap | mean diameter 7.8 km; a = 136,505.5 km; Keeler gap ~42 km; moon raises edge waves, inner edge waves lead, outer trail | https://en.wikipedia.org/wiki/Daphnis_(moon) |
+| Shepherds | mechanism: proximity/resonant deflection; particles ahead speed up and move out, behind slow and move in. Prometheus and Pandora at F ring (only Prometheus thought to shepherd); Cordelia/Ophelia on Uranus's epsilon ring; Galatea at Neptune's ring arcs | https://en.wikipedia.org/wiki/Shepherd_moon |
+| Propeller moonlets | radial scale set by Hill radius; azimuthal extent by moonlet mass vs ring viscosity; Blériot R_H ~400-590 m; partial-gap separation ~4 R_H in simulations | https://arxiv.org/pdf/1811.05329 (via search) |
+
+(inferred) Pan's gap is 42x Pan's mean radius wide; Daphnis's gap is ~10.8x Daphnis's mean radius wide. A gap is many body radii wide and is set by Hill radius, not by body size.
+
+### 1.2 Gap-width physics
+- Hill radius: R_H = a (m / 3(M+m))^(1/3) (https://en.wikipedia.org/wiki/Hill_sphere). Scales as a x m^(1/3); for eccentric orbit evaluated at pericentre.
+- Hill-sphere stability limits quoted there: two planets need mutual separation > 2 sqrt(3) mutual Hill radii for stability; three or more planets with < ~10 mutual Hill radii are "inherently unstable".
+- Gap-opening in a ring: moonlet scattering torque (far field falls as 1/x^4 beyond ~5 R_H, Goldreich & Tremaine; near field numerical) opposed by viscous spreading (https://arxiv.org/pdf/1902.01627 and arXiv 1208.4775 via search). Gap width is therefore not a fixed multiple of R_H: it depends on ring viscosity and surface density. The Pan figure above (~8 R_H per side) is one observation.
+- Disc-planet criterion, Crida, Morbidelli & Masset 2006 (https://arxiv.org/pdf/astro-ph/0511082): 3H/(4 R_H) + 50/(q Re) <= 1, q = mass ratio, H = scale height, Re = Reynolds number; "deep gap" = ~90% of co-orbital gas removed. The two terms are the thermal limit (R_H > H) and the viscous limit (q > ~40/Re). Written for gas discs, not rings (inferred: the form transfers qualitatively to a particle disc with H = ring thickness; no source read says so).
+- Resonance-overlap (chaotic zone) width, Wisdom 1980: delta a / a ~ 1.3 mu^(2/7), mu = planet/star mass ratio; 1.6 mu^(2/7) quoted as half-width in one source (Duncan, Quinn & Tremaine 1989). Above a critical eccentricity: 1.8 e^(1/5) mu^(1/5) (Mustill & Wyatt 2012). Sources: https://arxiv.org/pdf/astro-ph/0111600, https://arxiv.org/pdf/1207.0101, https://arxiv.org/pdf/1411.1378 (all via search; coefficients disagree between half- and full-width conventions). This scaling is for a clearing planet with test particles over time, no viscosity.
+- Scalings side by side (inferred): ring-moonlet gap ~ R_H ~ m^(1/3); chaotic-zone clearing ~ mu^(2/7) a. Exponents 0.33 and 0.29 are close; prefactors differ by a viscosity-dependent factor.
+- Kirkwood gaps (resonance, not a body in the lane): 5:1 1.780 AU, 4:1 2.065, 3:1 2.502, 5:2 2.825, 7:3 2.958, 2:1 3.279 AU. Mechanism per source: overlapping nu5/nu6 secular resonances inside mean-motion resonances; orbits go chaotic and become planet-crossing within millions of years. A single-epoch snapshot shows no density dip because eccentric orbits still cross gap radii. 3:2 (3.972 AU, Hildas) and 4:3 (4.296 AU, Thule) are populated resonances. https://en.wikipedia.org/wiki/Kirkwood_gap
+
+### 1.3 Rails vs dynamics
+- Translates directly (inferred): R_H = a (m/3M)^(1/3) as a width rule needs only masses and the parent's distance; Pan/Daphnis show gap width as a multiple of R_H on a ring that is otherwise on circular rails. Kirkwood-gap positions are ratios of the parent's period (period ratio p:q gives a = a_J (q/p)^(2/3)), computable on rails.
+- Assumes dynamics: the Wisdom width and Crida criterion are statements about what happens over time to test particles; on rails they would be authored widths. Viscous gap-refilling has no rail analogue.
+- Mismatch with Aetheria's law (inferred): Aetheria's reach and Roche come from c = G m^p with p 0.175-0.25, not m^(1/3), so a Hill-radius-derived width would be a different curve of mass than the field's.
+
+## 2. Trojans and co-orbitals
+
+### 2.1 Geometry and dynamics
+- Tadpole orbits librate about L4/L5 (60 deg ahead/behind); horseshoe orbits wrap L3 to L4 to L5; the tadpole-horseshoe separatrix sits near libration angle ~24 deg (https://arxiv.org/pdf/1311.6750). Horseshoe orbits need libration beyond ~312 deg full excursion (i.e. tadpole within ~156 deg of the Lagrange point angle) per https://arxiv.org/pdf/2402.15168; Janus-Epimetheus are horseshoe partners, Polydeuces a tadpole at Dione's L5 (https://en.wikipedia.org/wiki/Horseshoe_orbit).
+- Stability: L4/L5 linearly stable for mass ratio mu below the Routh/Gascheau value 0.03852 (https://arxiv.org/pdf/2510.13479, https://ar5iv.labs.arxiv.org/html/2007.13814). Horseshoe-existence upper limit disagrees by source (~4e-4, ~9.5e-4 to 1.2e-3, "Saturn-mass").
+- Width scaling (via search, https://arxiv.org/pdf/1509.02276 and astro-ph/0302135): tadpole boundary delta a = (2 sqrt2 / sqrt3) sqrt(mu), in units of the planet's a, as quoted; horseshoe width scales ~mu^(1/3) (Hill-sphere separatrix) in one source and mu^(1/2) in a generalization; smaller mu allows larger libration amplitude.
+- Pan's ringlet is a horseshoe population inside a gap (Section 1.1): a co-orbital structure and a gap in one body.
+
+### 2.2 Real populations
+- Jupiter Trojans (Wikipedia, Feb 2026 count): 9,738 at L4 (Greek), 5,931 at L5 (Trojan); L4/L5 about 1.6 (inferred from counts; ScienceAlert via search also says ~1.6x). Estimated true counts: L4 ~160-240k >2 km, ~600k >1 km; L5 comparable. Revised 2008-9 estimate 6,300 and 3,400 above 2 km (may be 2x high). https://en.wikipedia.org/wiki/Jupiter_trojan
+- Same page: libration amplitude 0.6 to 88 deg, mean ~33 deg; libration period ~150 yr; each swarm ~26 deg along the orbit; inclinations to ~40 deg; a = 5.05-5.35 AU (mean 5.2 +/- 0.15); swarm width ~0.6 AU (two Hill radii); along-orbit extent ~2.5 AU; up to 17% unstable over the Solar System's age; collisional fragments below ~40 km; largest Hektor ~200 km.
+- Asymmetry: observational bias is proposed (L5 lies toward the Galactic centre, crowded sky) and, separately, dynamical origins (fast outward Jupiter migration yields more stable L4 orbits, Li et al. via https://www.sciencealert.com/the-mysterious-asymmetry-of-jupiters-asteroids-may-finally-be-explained; capture during chaotic giant-planet evolution, Vokrouhlicky 2024 https://www.boulder.swri.edu/~bottke/Reprints/Vokrouhlick%C3%BD_2024_AJ_167_138_Debiased_Trojans.pdf). The sources do not settle the cause. Neptune Trojans: 27 at L4 and 4 at L5 on an undated mirror, first L5 found 2008 (https://en.wikipedia.org/wiki/Neptune_trojan, via search).
+- Hildas, 3:2 with Jupiter: triangle ~1 AU sides, apices 20-40% wider, apices align with Jupiter's L5, L4, L3; a 3.7-4.2 AU (mean 3.97); e < 0.3; >7,000 members; 5.0-5.5 yr near aphelion at apices vs 2.5-3.0 yr on sides; period ~7.9 yr. https://en.wikipedia.org/wiki/Hilda_asteroid
+- Cluster extent vs mass ratio (inferred): Jupiter mu is ~1e-3, so R_H/a = (mu/3)^(1/3) ~ 0.07 and R_H ~ 0.36 AU at 5.2 AU, consistent with the page's swarm width of ~2 R_H (0.6 AU). The observed Jupiter scale is therefore ~2 R_H radial width and ~26 deg (swarm) / ~33 deg (mean libration amplitude) along the orbit. Section 2.1 gives the sources' scalings with mu for other mass ratios.
+
+### 2.3 Rails vs dynamics
+- Translates directly (inferred): a Trojan on a rail is the parent's rail with phase offset +/-60 deg and the same period; a tadpole libration is a small superposed oscillation of phase and radius (period ~150 yr = ~12.7 Jupiter years, from the page) that a rail model can fake with a phase wobble or omit. The same-radius, same-period constraint is exactly what a rail system gives for free.
+- Assumes dynamics: libration amplitude distribution, the tadpole/horseshoe separatrix, and the 17% leak are effects of the three-body problem; on rails they are authored.
+- Aetheria constraint cross-reference (inferred): the planet's Trojan zone sits on the planet's own orbital radius, 60 deg away. Under `binaries-orbited-from-afar` a layout "on the same ring as a pair of bodies" was ruled out; the Trojan layout is a different case (same ring as one body, 60 deg apart) and the read of ZoneGenerator shows no check for it either way. Pull of the host planet near L4/L5 would also be inside the planet's reach only if reach exceeds the 60-deg chord (= radius of the orbit, an equilateral triangle) (inferred: chord to L4 equals the orbit radius D, while band reach is 1/2 D, so L4/L5 lie beyond the planet's own reach in the current law).
+
+## 3. Oort-like and scattered-disk material at a well's edge
+
+- Oort cloud: theorized ~2,000 to 200,000 AU; outer spherical region ~20,000-50,000 AU radius, inner torus ~2,000-20,000 AU (Hills cloud, Hills 1981, tens to hundreds of times as many nuclei as the outer cloud); outer edge set by the Sun's Hill sphere / Galactic tide truncation ~100,000-200,000 AU; isotropy attributed to giant-planet scattering followed by stellar passes, molecular clouds and Galactic tide randomizing inclinations; outer cloud possibly trillions of objects >1 km but only ~5 Earth masses total (Halley proxy). https://en.wikipedia.org/wiki/Oort_cloud
+- Sun's Hill sphere: unstable outer boundary 230,000 AU (1.1 pc; 3.6 ly), per https://en.wikipedia.org/wiki/Hill_sphere. Other stable-fraction numbers (1/2, 2/3 of R_H) were not on that page (gap).
+- Scattered disc: perihelion > 30 AU (about 30-35 closest), a > 50 AU out past 100 AU (one scheme to 2000 AU), e up to ~0.8, i up to ~40 deg; dynamically unstable under Neptune scattering, feeding centaurs/JFCs and outward to the Oort cloud. Detached objects (Sedna) have perihelion beyond Neptune's reach, Brown places Sedna in the inner Oort cloud. Kuiper belt ~30-50 AU: classical objects untouched by Neptune, resonant objects locked. https://en.wikipedia.org/wiki/Scattered_disc
+- (gap) Kuiper cliff at ~48 AU (2:1 resonance edge): not found on the pages read.
+- Distribution shapes, the three zones in order of distance from the star (inferred): flat-ish plane belt, thick plane-biased scattered disc (inclination to 40 deg, high e), isotropic sphere. Inclination spread grows with distance; edge is a tidal truncation, not a hard wall.
+- Rails vs dynamics: the extent rule (Hill sphere of the host in the larger potential) is a static radius, rail-compatible; the isotropy is an outcome of long stochastic evolution, authored as a distribution on rails (inferred). Aetheria is a flat 2D zone with an orbital plane and layers (rulings on layers), so "isotropic" has no direct meaning there (inferred); the radial edge rule and the sparse mass are what translate. Aetheria's own edge law: `dark-beyond-populated-area` (light falls off outside the populated area, soft not hard).
+
+## 4. Implementations
+
+Evidence quality here is low: most sources were forum posts or search snippets; several primary sources would not load (Starsector javadoc returned HTTP 402; fandom wiki and docs.rs accrete source not readable).
+
+| System | Rails or n-body | How these shapes are placed | Pointer |
+|---|---|---|---|
+| Elite Dangerous Stellar Forge | static, generated once | Newsletter #26: systems formed from first principles; matter accretes in protoplanetary discs over simulated time guided by chemistry and angular momentum; #36: stellar encounters may remove outer planets and capture others. Ring systems of planets and brown dwarfs reported unrealistically large (player analysis of Stellar Forge). Belt "clusters" are static; a player proposal wants them dynamic. Max 512 bodies per system (secondhand). (gap) the placement rule for belt clusters and rings was not found | https://mwomercs.com/forums/topic/146675-elite-dangerous/page__view__findpost__p__3654624, https://forums.frontier.co.uk/goto/post?id=5341120, https://forums.frontier.co.uk/goto/post?id=7453534 |
+| Space Engine | Keplerian orbits (inferred from the forum), no n-body in results | Asteroid belts for imported systems made by a community Python tool (JackDole's Asteroid Belt Creator) from parameters; a user calls procedural moon orbits unrealistic. (gap) no source for Trojans, Kuiper belt or Oort cloud generation | https://forum.spaceengine.org/viewtopic.php%3Ft%3D262, https://forum.spaceengine.org/viewtopic.php%3Ft%3D604 |
+| Starsector | rails | `StarSystemGenerator.addOrbitingEntities()` adds planets plus rings and asteroid fields; the OLD age setting yields mostly rings and debris, YOUNG a higher planet-to-debris ratio; entities take orbit angle, radius, days; hand-made systems place belts by authored radius. (gap) method signatures and Trojan handling not retrievable | https://starsector.wiki.gg/wiki/Star_System_Tutorial |
+| Stellaris | no orbit simulation | belts need a fixed radius in generation scripts; off-centre belts and belts in binary systems unsupported (modder reports) | https://admin-forum.paradoxplaza.com/forum/threads/offcentered-asteroid-belts.1113769/latest, Steam Synth Collective changelog |
+| EVE Online | static | each system's belts appear to be fixed per system; two geometries: ~50 km semicircle or ~70 km sphere; player observation only | https://wiki.eveuniversity.org/Asteroid |
+| Oolite | placed by authored coordinates | asteroid fields at an authored radius with no check against planet orbits; vanilla clumps "don't resemble real belts"; an add-on places asteroids at Lagrange points | https://bb.oolite.space/viewtopic.php?p=14995 |
+| Dole / Accrete / StarGen and ports | static output | Dole 1969 RAND "Formation of Planetary Systems by Aggregation"; Rust port `accrete`: parameters dust_density_coeff A, dust-to-gas k, b critical mass for gas giants, post_accretion_intensity; outputs planets plus a moons and rings stage. (gap) dust-band sweeping and effect limits not read; StarGen source (gryphel.com) not opened | https://docs.rs/crate/accrete/latest, https://www.gryphel.com/c/sw/astro/stargen |
+| Universe Sandbox | n-body | Trojan made by placing a ring and converting a body to a Trojan, per a community guide citing preconditions (primary >= 25x secondary; secondary >= 10x body) | https://gamepretty.com/universe-sandbox-how-to-make-a-trojan-2021/ |
+| Kerbal / Principia | stock KSP: patched conics on rails; Principia: n-body | a forum thread reports Principia kicking Bop and Vall out via Tylo; no source on Principia Trojans; stock KSP Lagrange points not simulated (forum title) | https://github.com/mockingbirdnest/Principia, https://forum.kerbalspaceprogram.com/topic/205593-lagrange-points-not-simulated |
+| Celestia | (gap) nothing found | | |
+| Endless Space | (gap) not searched (budget) | | |
+
+Failure modes found in sources:
+- Elite ring systems too large (player analysis).
+- Stellaris off-centre belts silently dropped when added as a child of a star; no belts in binaries.
+- Oolite belts placed by coordinates unrelated to planetary orbits.
+- Principia: moons ejected by neighbours (an n-body failure, not a placement one).
+- Nothing found in these sources about placement inside another body's Roche or Hill sphere (gap): the only Roche/Hill rules found are the physics ones above.
+
+What decides where clusters may sit, per source: authored radius (Oolite, Stellaris, Starsector hand-made, EVE); accretion outcome (Dole/Stellar Forge); resonance/Hill-radius criteria are in the literature only (Sections 1-3). No implementation found ties belt or cluster location to a Hill-radius or resonance rule (inferred from absence in what was read, not shown).
+
+## 5. Gaps in this pass
+Space Engine, Celestia, Endless Space, Stellar Forge placement rules, StarGen source, Starsector API signatures, the Kuiper cliff, Lindblad-resonance gap-opening criterion for rings (Lissauer-Shu-Cuzzi 1981, Goldreich-Tremaine 1980 not opened), exact horseshoe and tadpole prefactors (sources disagree). Budget was 40 calls.