Turn a locked moon's pole round with its drawn node, so its axis stays on its orbit at every date the clock reaches

The IAU carries a locked moon's pole round its orbit normal on a term of the node's angle, as a
moon in a Cassini state keeps it, but at the node rate the IAU's source had: Miranda's U11 at
-2024.22 degrees a century where the JPL table its orbit is drawn from has -2030.80, Mimas's S3 at
-36505.5 against -36511.16. Over AD 1-3000 that parted Miranda's drawn axis from its drawn orbit
normal by up to 7.89 degrees (AD 9), so Uranus swung 7.6 degrees north and south on its sky every
1.41 days, and Mimas's by 2.63. Only Iapetus's pole had been put on its orbit.

lockedToOrbit now sets every periodic term whose angle turns within 5 per cent of k times the
drawn node rate (k from 1 to 9, the most the report takes, Triton's) to exactly that multiple, and
moves its constant so the angle, and the pole and W with it, are unchanged on 2025-01-01 (the
2025 pole and W of every changed moon are identical to the 1e-14 degree). Measured: the largest
offset taken is Ganymede's J5, 3.4e-2, then Rhea's 1.2e-2; the nearest term that is not a node is
6.0e-2 out (a W-only term of Miranda's), and Umbriel's W has one 1.0e-2 from ten times its node,
which the k limit leaves. Ten moons change: Deimos, Io, Europa, Ganymede, Mimas, Tethys, Rhea,
Miranda, Triton and Proteus. The Moon and Phobos, whose W carries a quadratic, are left as before,
and so is Callisto's J6, 40 per cent from its node rate.

Worst angle between the spin axis and the drawn orbit normal over AD 1-3000, every 135 days,
before and after: Miranda 7.89 -> 0.42, Mimas 2.63 -> 0.47, Rhea 0.77 -> 0.17, Triton 0.51 ->
0.15, Europa 0.33 -> 0.13, Ganymede 0.50 -> 0.16. Faces: Miranda 2.75 -> 2.39, Mimas 8.94 ->
8.89, Deimos 2.14 -> 2.08, Triton 2.88 -> 2.81; none got worse.

build.ts now checks that angle for every locked moon over the same 8114 dates as the face check,
at most 1 degree (Tethys 0.97, whose IAU pole sits 0.69 from its orbit today; Titan 0.94, whose
IAU pole is still while its node turns in 705 years), with four named ceilings: the Moon 7.1 (its
real 6.7-degree tilt, 6.98 at worst), Phobos 2 and Deimos 2 (1.81 and 1.74) and Proteus 1.2
(1.09), whose IAU poles nod with Mars's and Neptune's precessing poles while the Laplace poles
their orbits are drawn round are fixed. The obliquity check at Horizons' epoch shares the new
axisFromOrbitDeg with it. Nothing checked the axis against the orbit before: an Iapetus pole left
on its Laplace pole, 8.30 degrees off at every date, passed the ETL and the suite.

Guarded mutants, each through the solar ETL and then the suite on the data it wrote:
- node terms left at the IAU rates: the ETL fails with "Moon mimas's spin axis leans up to 2.63
  degrees ... (at most 1 expected)", and the suite with only the new test failing;
- Iapetus's pole put on its Laplace pole, W re-phased so its face today is unchanged (worst face
  16.41, under its 16.5 ceiling): the ETL fails with "Moon iapetus's spin axis leans up to 8.30
  degrees", and the suite with only the new test failing.

Also: lockedToOrbit called the Iapetus normal's circle "8.3 degrees across"; 8.3 is its radius (the
row's i = 8.298 to the Laplace plane) and it is 16.6 across. The README credited every rotation to
pck00011 without saying that a locked moon's W and node terms are re-rated to its JPL mean elements
and Iapetus's pole carried round its orbit normal; both of its lines now say so, as the body
model's doc does.

Unit suite 858/858, etl:typecheck and tsc -p tsconfig.app.json clean, solar ETL passes on the
real data.

Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>

README.md

@@ -56,7 +56,10 @@ in it is measured and what is not.
 cutting to a new scene. The Sun gets the real solar-system bodies, moving on JPL's mean orbital
 elements — Standish's for the planets, JPL SSD's satellite table for the moons, the Small-Body
 Database for Ceres, Eris, Haumea and Makemake — and turned by the IAU's rotational elements, Earth
-by the IERS Earth Rotation Angle; other
+by the IERS Earth Rotation Angle; a tidally locked moon's prime meridian (but the Moon's and
+Phobos's) turns at its JPL mean motion and its pole's terms on its node at its JPL node rate, both
+re-phased to the IAU's values on 2025-01-01, and Iapetus's pole follows its orbit normal
+(`lockedToOrbit`), so each keeps its face to its planet from AD 1 to 3000; other
 stars get their confirmed exoplanets. Orbits are drawn as ellipses and bodies are propagated
 along them by a Kepler solver to the date on the map's clock. Under them, a dashed grid marks out
 round distances in AU — 5 AU rings for the solar system, 0.01 AU rings for TRAPPIST-1 — with a
@@ -330,7 +333,7 @@ plugin's own files are kept so it can be listed from a marketplace of its own la
 ## Data credits
 
 Star catalogue: [HYG database](https://github.com/astronexus/HYG-Database) (Hipparcos, Yale
-Bright Star, Gliese) — 68 388 stars within 250 pc. Solar-system orbits: JPL approximate planetary mean elements (Standish), JPL SSD satellite mean elements and the JPL Small-Body Database; rotation: the IAU WGCCRE 2015 report via NAIF's pck00011, and for Earth the IERS Conventions 2010; physical data, and the positions the orbits are checked against: NASA/JPL Horizons. Exoplanets: NASA Exoplanet
+Bright Star, Gliese) — 68 388 stars within 250 pc. Solar-system orbits: JPL approximate planetary mean elements (Standish), JPL SSD satellite mean elements and the JPL Small-Body Database; rotation: the IAU WGCCRE 2015 report via NAIF's pck00011, with a locked moon's W and node terms re-rated to its JPL mean elements and Iapetus's pole carried round its orbit normal, and for Earth the IERS Conventions 2010; physical data, and the positions the orbits are checked against: NASA/JPL Horizons. Exoplanets: NASA Exoplanet
 Archive. Deep-sky objects: [OpenNGC](https://github.com/mattiaverga/OpenNGC). Body and skybox
 imagery: NASA/JPL/USGS public domain and Solar System Scope (CC BY 4.0) — per-file provenance
 is recorded in `src/assets/textures/README.md`.

src/app/features/galaxy-system/system-orbits-renderer.spec.ts

@@ -923,7 +923,7 @@ describe('locked moons across the clock’s window', () => {
   }
 
   it('keeps Proteus, Miranda, Mimas and Iapetus facing their planets at AD 1 and AD 3000', () => {
-    // Measured: Proteus 2.6 degrees at most over AD 1-3000, Miranda 2.8, Mimas 8.9, Iapetus 16 (9.4
+    // Measured: Proteus 2.6 degrees at most over AD 1-3000, Miranda 2.4, Mimas 8.9, Iapetus 16 (9.4
     // of it the lag of the row its orbit is drawn from). On the IAU's own W and Iapetus's straight
     // pole they were 146, 23, 49 and 87 degrees at AD 1.
     for (const jd of [1721425.5, 2816787.4]) {
@@ -934,4 +934,21 @@ describe('locked moons across the clock’s window', () => {
       expect(Math.abs(facingPlanet('iapetus'))).toBeLessThan(16.5);
     }
   });
+
+  it('keeps the axes of Miranda, Mimas and Iapetus on their drawn orbits’ normals, as a Cassini state holds them, at AD 1, today and AD 3000', () => {
+    // Measured over AD 1-3000: Miranda 0.42 degrees at most, Mimas 0.47, Iapetus 0.74. With their
+    // poles going round at the IAU's node rates, Miranda was 7.6 off at AD 1 and Mimas 2.6; with
+    // Iapetus's pole on its Laplace pole, 8.3 off at every date.
+    for (const jd of [1721425.5, 2460676.5, 2816787.4]) {
+      renderer.update(jd);
+      for (const id of ['miranda', 'mimas', 'iapetus']) {
+        const moon = renderer.members.find((member) => member.id === id)!.marker;
+        const line = moon.parent!.children.find((child) => child.name === 'orbit-line')!;
+        const axis = new THREE.Vector3(0, 1, 0).applyQuaternion(moon.quaternion);
+        const normal = new THREE.Vector3(0, 0, 1).applyQuaternion(line.quaternion);
+        // A line, not a direction: Miranda turns backwards against the IAU's pole.
+        expect((Math.acos(Math.min(1, Math.abs(axis.dot(normal)))) * 180) / Math.PI).toBeLessThan(1);
+      }
+    }
+  });
 });

src/app/shared/models/body.model.ts

@@ -105,8 +105,9 @@ export interface BodyRecord {
   /**
    * Where the body's pole points and which way its prime meridian faces at any date, from the IAU
    * WGCCRE 2015 report (Archinal et al. 2018) as NAIF's `pck00011.tpc` carries it, but that a locked
-   * moon's W turns at its drawn orbit's rate and Iapetus's pole goes round with its orbit's, so they
+   * moon's W, and its pole's terms on its node, turn at its drawn orbit's rates and Iapetus's pole
-   * keep their faces to their planets over the clock's AD 1 to 3000 (see `lockedToOrbit` in the
+   * goes round with its orbit's, so they keep their faces to their planets, and their poles round
+   * their orbits', over the clock's AD 1 to 3000 (see `lockedToOrbit` in the
    * ETL). Where present it alone sets how the body is drawn, and the ETL checks the period and
    * obliquity above against it. Absent where the report gives none: Hyperion tumbles, and Nereid,
    * Eris, Haumea and Makemake have no model.

src/assets/data/bodies.json

@@ -821,9 +821,8 @@
       "terms": [
         {
           "angleDeg": [
-            121.46893664,
+            121.49945940952146,
-            660.22803474,
+            660.1059470044942
-            0
           ],
           "ra": 3.09217726,
           "dec": 1.83936004,
@@ -831,9 +830,8 @@
         },
         {
           "angleDeg": [
-            231.05028581,
+            231.2716139683453,
-            660.9912354,
+            660.1059470044942
-            0
           ],
           "ra": 0.22980637,
           "dec": 0.1432532,
@@ -841,9 +839,8 @@
         },
         {
           "angleDeg": [
-            251.37314025,
+            251.44553184217244,
-            1320.50145245,
+            1320.2118940089883
-            0
           ],
           "ra": 0.06418655,
           "dec": 0.01911409,
@@ -918,8 +915,8 @@
       "terms": [
         {
           "angleDeg": [
-            283.9,
+            283.6369874084692,
-            4850.7
+            4851.752021563342
           ],
           "ra": 0.094,
           "dec": 0.04,
@@ -983,8 +980,8 @@
       "terms": [
         {
           "angleDeg": [
-            355.8,
+            355.4537739825443,
-            1191.3
+            1192.6848661542538
           ],
           "ra": 1.086,
           "dec": 0.468,
@@ -1066,8 +1063,8 @@
         },
         {
           "angleDeg": [
-            119.9,
+            117.57926275587673,
-            262.1
+            271.38269483015966
           ],
           "ra": 0.431,
           "dec": 0.186,
@@ -1211,8 +1208,8 @@
       "terms": [
         {
           "angleDeg": [
-            177.4,
+            178.81408536763,
-            -36505.5
+            -36511.15618661257
           ],
           "ra": 13.56,
           "dec": -1.53,
@@ -1327,8 +1324,8 @@
       "terms": [
         {
           "angleDeg": [
-            300,
+            300.02841306210905,
-            -7225.9
+            -7226.013649136892
           ],
           "ra": 9.66,
           "dec": -1.09,
@@ -1437,8 +1434,8 @@
       "terms": [
         {
           "angleDeg": [
-            345.2,
+            342.2970571615749,
-            -1016.3
+            -1004.6885465505693
           ],
           "ra": 3.1,
           "dec": -0.35,
@@ -1703,8 +1700,8 @@
       "terms": [
         {
           "angleDeg": [
-            102.23,
+            103.87516350424971,
-            -2024.22
+            -2030.8004738534437
           ],
           "ra": 4.41,
           "dec": 4.25,
@@ -1721,8 +1718,8 @@
         },
         {
           "angleDeg": [
-            204.46,
+            207.75032700849943,
-            -4048.44
+            -4061.6009477068874
           ],
           "ra": -0.04,
           "dec": -0.02,
@@ -2028,8 +2025,8 @@
       "terms": [
         {
           "angleDeg": [
-            177.85,
+            177.83706224270992,
-            52.316
+            52.367749612333185
           ],
           "ra": -32.35,
           "dec": 22.55,
@@ -2037,8 +2034,8 @@
         },
         {
           "angleDeg": [
-            355.7,
+            355.67412448541984,
-            104.632
+            104.73549922466637
           ],
           "ra": -6.28,
           "dec": 2.1,
@@ -2046,8 +2043,8 @@
         },
         {
           "angleDeg": [
-            533.55,
+            533.5111867281297,
-            156.948
+            157.10324883699957
           ],
           "ra": -2.08,
           "dec": 0.55,
@@ -2055,8 +2052,8 @@
         },
         {
           "angleDeg": [
-            711.4,
+            711.3482489708397,
-            209.264
+            209.47099844933274
           ],
           "ra": -0.74,
           "dec": 0.16,
@@ -2064,8 +2061,8 @@
         },
         {
           "angleDeg": [
-            889.25,
+            889.1853112135495,
-            261.58
+            261.8387480616659
           ],
           "ra": -0.28,
           "dec": 0.05,
@@ -2073,8 +2070,8 @@
         },
         {
           "angleDeg": [
-            1067.1,
+            1067.0223734562594,
-            313.896
+            314.20649767399914
           ],
           "ra": -0.11,
           "dec": 0.02,
@@ -2082,8 +2079,8 @@
         },
         {
           "angleDeg": [
-            1244.95,
+            1244.8594356989695,
-            366.212
+            366.5742472863323
           ],
           "ra": -0.07,
           "dec": 0.01,
@@ -2091,8 +2088,8 @@
         },
         {
           "angleDeg": [
-            1422.8,
+            1422.6964979416794,
-            418.528
+            418.9419968986655
           ],
           "ra": -0.02,
           "dec": 0,
@@ -2100,8 +2097,8 @@
         },
         {
           "angleDeg": [
-            1600.65,
+            1600.5335601843892,
-            470.844
+            471.30974651099865
           ],
           "ra": -0.01,
           "dec": 0,
@@ -2193,8 +2190,8 @@
         },
         {
           "angleDeg": [
-            142.61,
+            141.60196187337888,
-            2824.6
+            2828.6320421151877
           ],
           "ra": -0.05,
           "dec": -0.04,

tools/etl/build.ts

@@ -1,6 +1,6 @@
 import { statSync } from 'node:fs';
 
-import { BodyRecord, OrbitalElements } from '../../src/app/shared/models/body.model';
+import { BodyRecord, OrbitalElements, RotationalElements } from '../../src/app/shared/models/body.model';
 import { eclipticToEquatorial, laplacePlaneToEquatorial, raDecToUnitVector } from '../../src/app/shared/astro/coordinates';
 import { meanElementsAt, positionAtEpoch } from '../../src/app/shared/astro/kepler';
 import { orientationAt } from '../../src/app/shared/astro/rotational-elements';
@@ -257,7 +257,7 @@ const MAX_OBLIQUITY_OFFSET_DEG = 0.1;
  *
  * Measured on this catalogue: at most 5.36 degrees (Titan) but for three. The Moon 7.62, at AD 1:
  * its longitude swings 6.3 either way with its eccentricity, Horizons' too, and W's quadratic, the
- * tidal slowing its orbit here does not carry, adds 0.75 by then. Mimas 8.94: about 6.3 off on
+ * tidal slowing its orbit here does not carry, adds 0.75 by then. Mimas 8.89: about 6.3 off on
  * average because the IAU's W and JPL's mean longitude disagree, and swung 2.3 either way (2e) by
  * its eccentricity. None of that is Mimas: its measured physical libration is 0.84 degrees
  * (Tajeddine et al. 2014, Science 346, 322), and W carries none; Horizons, on the same W against its
@@ -269,6 +269,22 @@ const MAX_OBLIQUITY_OFFSET_DEG = 0.1;
  */
 const MAX_SUB_PLANET_LONGITUDE_DEG = 7;
 const SUB_PLANET_CEILINGS_DEG: Record<string, number> = { moon: 8, mimas: 9.5, iapetus: 16.5 };
+/**
+ * How far a locked moon's spin axis may lean from the normal of the orbit it is drawn going round,
+ * over the same dates. A locked moon sits in a Cassini state, its axis on its orbit normal as the
+ * node carries both round the Laplace pole, and the IAU's pole goes round on a term of the node's
+ * angle; at the rate the IAU's source had for it and not the drawn orbit's, Miranda's axis was 7.89
+ * degrees off at AD 9 and Mimas's 2.63, and an Iapetus pole left on the Laplace pole is 8.30 off at
+ * every date (see `lockedToOrbit`).
+ *
+ * Measured on this catalogue: at most 0.97 degrees (Tethys, whose IAU pole sits 0.69 from its orbit
+ * normal today; Titan 0.94, whose pole the IAU holds still while its node turns in 705 years) but
+ * for four. The Moon 6.98, its real 6.7-degree tilt to its orbit. Phobos 1.81 and Deimos 1.74, and
+ * Proteus 1.09: their IAU poles nod with Mars's and Neptune's precessing poles, the Laplace poles
+ * their orbits are drawn round are fixed.
+ */
+const MAX_AXIS_FROM_ORBIT_DEG = 1;
+const AXIS_FROM_ORBIT_CEILINGS_DEG: Record<string, number> = { moon: 7.1, phobos: 2, deimos: 2, proteus: 1.2 };
 /** The clock's window, AD 1 to 3000 (`CLOCK_WINDOW` in `time.store.ts`), as Julian dates. */
 const CLOCK_START_JD = Date.parse('0001-01-01T00:00Z') / 86400000 + 2440587.5;
 const CLOCK_END_JD = Date.parse('3000-01-01T00:00Z') / 86400000 + 2440587.5;
@@ -279,6 +295,19 @@ function angleBetweenDeg(a: { x: number; y: number; z: number }, b: { x: number;
   return (Math.acos(Math.min(1, Math.max(-1, cosine))) * 180) / Math.PI;
 }
 
+/** Degrees between a body's spin axis — its IAU pole, turned over where W runs backwards — and the normal of the orbit it is drawn going round, at a TDB date. */
+function axisFromOrbitDeg(body: BodyRecord, rotation: RotationalElements, jd: number): number {
+  const pole = orientationAt(rotation, jd);
+  const pointing = raDecToUnitVector(pole.poleRaDeg / 15, pole.poleDecDeg);
+  const sense = Math.sign(rotation.primeMeridianDeg[1]);
+  const axis = { x: sense * pointing.x, y: sense * pointing.y, z: sense * pointing.z };
+  const { inclinationDeg, longitudeOfAscendingNodeDeg } = meanElementsAt(body.orbit, body.rates, jd);
+  const tilt = inclinationDeg * DEG_TO_RAD;
+  const node = longitudeOfAscendingNodeDeg * DEG_TO_RAD;
+  const normal = { x: Math.sin(tilt) * Math.sin(node), y: -Math.sin(tilt) * Math.cos(node), z: Math.cos(tilt) };
+  return angleBetweenDeg(axis, body.laplacePole ? laplacePlaneToEquatorial(normal, body.laplacePole) : eclipticToEquatorial(normal));
+}
+
 function validateBodies(bodies: BodyRecord[], horizonsOrbits: Map<string, OrbitalElements>, horizonsTracks: Map<string, TrackPoint[]>): void {
   assertCondition(bodies.length > 0, 'No solar-system bodies were produced.');
 
@@ -364,14 +393,7 @@ function validateBodies(bodies: BodyRecord[], horizonsOrbits: Map<string, Orbita
         spins.push(`${body.id} day ${dayOffset.toExponential(1)}`);
       }
       if (body.obliquityDeg !== undefined) {
-        const pole = orientationAt(rotation, horizons!.epochJd);
+        const obliquity = axisFromOrbitDeg(body, rotation, horizons!.epochJd);
-        const pointing = raDecToUnitVector(pole.poleRaDeg / 15, pole.poleDecDeg);
-        const axis = { x: Math.sign(rate) * pointing.x, y: Math.sign(rate) * pointing.y, z: Math.sign(rate) * pointing.z };
-        const { inclinationDeg, longitudeOfAscendingNodeDeg } = meanElementsAt(body.orbit, body.rates, horizons!.epochJd);
-        const tilt = inclinationDeg * DEG_TO_RAD;
-        const node = longitudeOfAscendingNodeDeg * DEG_TO_RAD;
-        const normal = { x: Math.sin(tilt) * Math.sin(node), y: -Math.sin(tilt) * Math.cos(node), z: Math.cos(tilt) };
-        const obliquity = angleBetweenDeg(axis, body.laplacePole ? laplacePlaneToEquatorial(normal, body.laplacePole) : eclipticToEquatorial(normal));
         assertCondition(
           Math.abs(obliquity - body.obliquityDeg) <= MAX_OBLIQUITY_OFFSET_DEG,
           `${body.name}'s IAU spin axis is ${obliquity.toFixed(3)} degrees from its orbit's pole, where ${body.id === 'pluto' ? 'its IAU pole' : 'Horizons'} gives an obliquity of ${body.obliquityDeg} (at most ${MAX_OBLIQUITY_OFFSET_DEG} apart expected) — the pole or the sense of W was read wrongly.`
@@ -406,6 +428,13 @@ function validateBodies(bodies: BodyRecord[], horizonsOrbits: Map<string, Orbita
           `Moon ${body.id} turns its face up to ${worst.toFixed(2)} degrees from its planet between AD 1 and 3000 (at most ${ceiling} expected) — its orbit and its W disagree.`
         );
         spins.push(`${body.id} faces ${worst.toFixed(2)}`);
+        const axisCeiling = AXIS_FROM_ORBIT_CEILINGS_DEG[body.id] ?? MAX_AXIS_FROM_ORBIT_DEG;
+        const worstAxis = Math.max(...LOCK_DATES_JD.map((jd) => axisFromOrbitDeg(body, rotation!, jd)));
+        assertCondition(
+          worstAxis <= axisCeiling,
+          `Moon ${body.id}'s spin axis leans up to ${worstAxis.toFixed(2)} degrees from its orbit's normal between AD 1 and 3000 (at most ${axisCeiling} expected) — its pole does not go round with its node.`
+        );
+        spins.push(`${body.id} axis ${worstAxis.toFixed(2)}`);
       }
       if (body.massRatio !== undefined) {
         // The pair's barycentre, which the planet's elements place, must lie outside the planet —

tools/etl/lib/locked-spin.ts

@@ -21,6 +21,16 @@ const MAX_LOCKED_RATE_OFFSET = 1e-5;
 /** Harmonics of the node's angle that carry a pole round its orbit's; see {@link lockedToOrbit}. */
 const POLE_HARMONICS = 5;
 
+/**
+ * How far a periodic term's angle may turn from a multiple of the node's rate, as a fraction of it,
+ * and still be taken for the node's angle as the IAU's source had it. Measured: at most 3.4e-2
+ * (Ganymede's J5), then Rhea's R4 1.2e-2 and Miranda's U11 3.2e-3; the nearest that is not a node is
+ * a term of Miranda's W alone, 6.0e-2 from three times it. Multiples go up to the ninth, the most the
+ * report takes (Triton's N7); past that, Umbriel's W has a term 1.0e-2 from ten times its node's.
+ */
+const MAX_NODE_RATE_OFFSET = 0.05;
+const MAX_NODE_HARMONIC = 9;
+
 /** The planet's east longitude on a moon's IAU body-fixed frame, from the moon's mean place, at a TDB date. */
 export function subPlanetLongitudeDeg(body: Pick<BodyRecord, 'orbit' | 'rates' | 'laplacePole'>, elements: RotationalElements, jd: number): number {
   const own = positionAtEpoch(meanElementsAt(body.orbit, body.rates, jd));
@@ -41,15 +51,25 @@ export function subPlanetLongitudeDeg(body: Pick<BodyRecord, 'orbit' | 'rates' |
  * The report gives a locked moon's W the mean motion of whichever orbit its authors had, and JPL's
  * table has another: Proteus's W turns 6.3e-7 of its rate slower than its row, which turned its far
  * side to Neptune at AD 1 (146 degrees), Mimas's 1.6e-7 faster (52 at AD 1) and Miranda's (23).
- * W's rate is set to the orbit's here, its constant moved so W is unchanged at {@link PRESENT_JD},
+ * W's rate is set to the orbit's here, its constant moved so W is unchanged at {@link PRESENT_JD}.
- * and the pole and every periodic term are the IAU's. Measured over AD 1-3000: Proteus 2.7 degrees,
+ * Measured over AD 1-3000: Proteus 2.7 degrees, Mimas 8.9, Miranda 2.4, Ariel 1.0. A W with a
- * Mimas 8.9, Miranda 2.8, Ariel 1.0. A W with a quadratic is left: Phobos's orbit already takes the
+ * quadratic is left: Phobos's orbit already takes the quadratic from W (see
- * quadratic from W (see `orbitalTermsOfPrimeMeridian`), and the Moon's, its tidal slowing, is 0.75
+ * `orbitalTermsOfPrimeMeridian`), and the Moon's, its tidal slowing, is 0.75 degrees at AD 1.
- * degrees at AD 1.
+ *
+ * The node's angle goes the same way. A moon in a Cassini state keeps its axis on its orbit normal,
+ * which goes round the Laplace pole with the node, and the IAU's pole goes round with it on a term
+ * of the node's angle, at the node's rate as its source had it: Miranda's U11 at -2024.22 degrees a
+ * century, where the table the orbit is drawn from has -2030.80. On a 4.3-degree circle that parted
+ * the axis from the drawn orbit by 7.9 degrees at AD 1, and Mimas's by 2.6. Every term whose angle
+ * turns within {@link MAX_NODE_RATE_OFFSET} of a multiple of the node's rate is set to that multiple,
+ * its constant moved so the angle is unchanged at the present, and the pole with it: over AD 1-3000
+ * Miranda's axis stays within 0.42 degrees of its orbit normal and Mimas's within 0.47, and the
+ * Io's, Europa's, Ganymede's, Rhea's and Triton's within 0.17. The rest of the pole and its terms
+ * are the IAU's: Callisto's J6 turns 40 per cent slower than its node, and is left.
  *
  * `poleFollowsOrbit` is for Iapetus, whose IAU pole moves 3.9 degrees a century in right ascension
  * and 1.1 in declination: a straight line through its orbit normal's 3 439-year circle round the
- * Laplace pole, 8.3 degrees across, which by AD 1 has run past the celestial pole (Dec 97.9) and 11
+ * Laplace pole, 8.3 degrees in radius (16.6 across), which by AD 1 has run past the celestial pole (Dec 97.9) and 11
  * degrees off the orbit, and turned its face 87 degrees from Saturn. Its axis sits on its orbit normal
  * (0.04 degrees apart today), as a moon in a Cassini state keeps it, so its pole is given the circle: the
  * normal's right ascension as sines and declination as cosines of the node's angle and its first
@@ -69,7 +89,17 @@ export function lockedToOrbit(elements: RotationalElements, mean: Pick<MeanOrbit
   if (Math.abs(w1 / n - 1) > MAX_LOCKED_RATE_OFFSET) {
     throw new Error(`${name}'s IAU W turns at ${w1} degrees a day, ${Math.abs(w1 / n - 1).toExponential(2)} of its orbit's ${n}: not the rate of the orbit it keeps its face to.`);
   }
-  const locked: RotationalElements = { ...elements, primeMeridianDeg: [w0 + (w1 - n) * (PRESENT_JD - J2000_JD), n, 0] };
+  const nodeRate = mean.rates.longitudeOfAscendingNodeDegPerDay * DAYS_PER_JULIAN_CENTURY;
+  const present = (PRESENT_JD - J2000_JD) / DAYS_PER_JULIAN_CENTURY;
+  const terms = elements.terms?.map((term) => {
+    const [constant, rate, quadratic = 0] = term.angleDeg;
+    const k = Math.round(rate / nodeRate);
+    if (quadratic !== 0 || k === 0 || Math.abs(k) > MAX_NODE_HARMONIC || Math.abs(rate / (k * nodeRate) - 1) > MAX_NODE_RATE_OFFSET) {
+      return term;
+    }
+    return { ...term, angleDeg: [constant + (rate - k * nodeRate) * present, k * nodeRate] };
+  });
+  const locked: RotationalElements = { ...elements, primeMeridianDeg: [w0 + (w1 - n) * (PRESENT_JD - J2000_JD), n, 0], ...(terms ? { terms } : {}) };
   return poleFollowsOrbit ? poleRoundOrbit(locked, mean) : locked;
 }