Turn each locked moon at its orbit's rate and Iapetus's pole round its orbit, so they face their planets at every date the clock reaches

The IAU gives a locked moon's W the mean motion of whichever orbit its authors had, and JPL's table has another. Near the present the difference is nothing; over the clock's AD 1 to 3000 it turned Proteus's far side to Neptune at AD 1 (146 degrees), Mimas 52 degrees from Saturn and Miranda 23. Iapetus was worse for another reason: its IAU pole is a straight line, 3.9 degrees a century in right ascension, through its orbit normal's 3 439-year circle round the Laplace pole, which by AD 1 has run past the celestial pole (Dec 97.9), 11 degrees off the orbit, with the face 87 degrees from Saturn. Mimas and Iapetus carry mission maps, so a wrong hemisphere was drawn facing Saturn. The ETL's lock check sampled only 1950-2100, so none of it failed. tools/etl/lib/locked-spin.ts, lockedToOrbit, called for every locked moon: - W's rate becomes the orbit's own mean motion, its constant moved so W is unchanged on 2025-01-01; the pole and every periodic term stay the IAU's. A W with a quadratic is left (Phobos's orbit already takes it; the Moon's is its tidal slowing, 0.75 degrees at AD 1). The kernel's rate must be within 1e-5 of the orbit's first (at most 3.4e-6, Iapetus). - Iapetus (poleFollowsOrbit): the pole follows its orbit normal, as a moon in a Cassini state does, in the IAU's own form: sines of the node's angle and four harmonics on right ascension, cosines on declination, fitted to the normal's circle and pinned to the IAU pole at the present; W takes sines of the same angles, fitted to hold the face where it is today. build.ts samples the lock over AD 1 to 3000 (8 114 dates, every 135 days) instead of 1950-2100, and subPlanetLongitudeDeg moved to the lib, shared by both. Measured on the real catalogue: at most 5.36 degrees (Titan) but the Moon 7.62 (its eccentricity, and W's quadratic at AD 1: a new named ceiling of 8), Mimas 8.94 (ceiling 11 -> 9.5) and Iapetus 15.95 (19 -> 16.5, 9.4 of it its row's lag); Proteus's own ceiling of 9 is gone, at 2.66. Iapetus's axis stays within 0.74 degrees of its orbit normal (11.06 before) and its pole is the IAU's at the present to 1e-4 degrees. Live on :4301, the longitude facing the planet at AD 1 / 1000 / 2025 / 2999: Proteus 2.6 / 2.6 / 2.6 / 2.6 (was -146.5 / -72.9 / 2.6 / 74.5), Iapetus -15.9 / -15.4 / -15.3 / -9.3 (-87.0 / -50.9 / -15.3 / 22.8), Mimas 4.1 / 6.8 / 5.7 / 8.4 (48.6 / 29.3 / 5.7 / -13.0), Miranda -0.1 / 2.2 / 0.0 / 1.4 (-23.4 / -9.6 / 0.0 / 12.6); the present is unchanged. The renderer spec now takes the rotational elements from bodies.json too, so no hand copy is left, and a new test turns Proteus, Miranda, Mimas and Iapetus to their planets at AD 1 and AD 3000. Guarded mutants, each run through the solar ETL and then the full suite on what it wrote: lockedToOrbit bypassed (validator: Mimas 52.30, ceiling 9.5; the new test fails), Iapetus on the IAU's straight pole (98.48), and its pole round the orbit without W's terms (73.13); each fails the new test and only it. Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>
2026-09-30 15:58:04 +02:00
parent 09eaf9532f
commit 3b4fd1af6c
6 changed files with 312 additions and 107 deletions
@@ -8,7 +8,7 @@ import { DEFAULT_EPOCH_JD, GM_SUN_AU3_PER_DAY2, ttMinusUtSeconds } from '../../s
 import { keplerRates } from '../../shared/astro/kepler';
 import { eclipticToEquatorial, laplacePlaneToEquatorial, OBLIQUITY_J2000_DEG } from '../../shared/astro/coordinates';
 import { orientationAt } from '../../shared/astro/rotational-elements';
-import { BodyRecord, RotationalElements } from '../../shared/models/body.model';
+import { BodyRecord } from '../../shared/models/body.model';
 import { ExoplanetRecord } from '../../shared/models/exoplanet.model';
 import { SystemOrbitsRenderer } from './system-orbits-renderer';
 import { bodyTexturePath, loadCachedTexture } from '../../shared/rendering/texture-catalog';
@@ -593,29 +593,18 @@ describe('exoplanet size without a measured radius', () => {
 });

 describe('solar-system bodies against Horizons', () => {
-  // The records the app ships, read from bodies.json, and Horizons' own positions for them (ICRF,
-  // AU; heliocentric for the planets, planet-centred for the moons) at dates across 1950-2100, so the
-  // whole path — the ETL's reading of the mean elements, their rates, the Laplace planes and the
-  // scene's frame — is checked against JPL's ephemeris rather than against itself. A hand copy of the
-  // records stood here, and an ETL that dropped Standish's a, e and i rates or Io's and Europa's
-  // backward periapses passed the whole suite on the data it wrote. Horizons' dates are TDB and the
-  // renderer's are the clock's UT, so each is handed over TT - UT earlier: 69.184 s today, 29 in 1950.
+  // The records the app ships, read from bodies.json with their IAU rotational elements, and
+  // Horizons' own positions for them (ICRF, AU; heliocentric for the planets, planet-centred for the
+  // moons) at dates across 1950-2100, so the whole path — the ETL's reading of the mean elements,
+  // their rates, the Laplace planes and the scene's frame — is checked against JPL's ephemeris rather
+  // than against itself. A hand copy of the records stood here, and an ETL that dropped Standish's a,
+  // e and i rates or Io's and Europa's backward periapses passed the whole suite on the data it
+  // wrote. Horizons' dates are TDB and the renderer's are the clock's UT, so each is handed over
+  // TT - UT earlier: 69.184 s today, 29 in 1950.
  const SHIPPED: BodyRecord[] = JSON.parse(readFileSync(`${process.cwd()}/src/assets/data/bodies.json`, 'utf8'));
  // Mimas and Phobos among them for the terms of their IAU W that are motion along the orbit: the
  // Mimas-Tethys libration and Phobos's tidal acceleration (see `orbitalTermsOfPrimeMeridian`).
  const IDS = ['earth', 'jupiter', 'saturn', 'neptune', 'pluto', 'moon', 'io', 'europa', 'titan', 'triton', 'uranus', 'titania', 'charon', 'venus', 'mars', 'mimas', 'phobos'];
-  // The IAU WGCCRE 2015 rotational elements bodies.json carries for them, from pck00011.tpc.
-  const ROTATION: Record<string, RotationalElements> = {
-    venus: {poleRaDeg: [272.76, 0, 0], poleDecDeg: [67.16, 0, 0], primeMeridianDeg: [160.2, -1.4813688, 0]},
-    earth: {poleRaDeg: [0, -0.641, 0], poleDecDeg: [90, -0.557, 0], primeMeridianDeg: [190.147, 360.9856235, 0]},
-    mars: {poleRaDeg: [317.269202, -0.10927547, 0], poleDecDeg: [54.432516, -0.05827105, 0], primeMeridianDeg: [176.049863, 350.891982443297, 0], terms: [{angleDeg: [79.398797, 0.5042615, 0], ra: 0.419057, dec: 0, pm: 0}, {angleDeg: [166.325722, 0.5042615, 0], ra: 0, dec: 1.591274, pm: 0}, {angleDeg: [95.391654, 0.5042615, 0], ra: 0, dec: 0, pm: 0.584542}]},
-    jupiter: {poleRaDeg: [268.056595, -0.006499, 0], poleDecDeg: [64.495303, 0.002413, 0], primeMeridianDeg: [284.95, 870.536, 0]},
-    io: {poleRaDeg: [268.05, -0.009, 0], poleDecDeg: [64.5, 0.003, 0], primeMeridianDeg: [200.39, 203.4889538, 0], terms: [{angleDeg: [283.9, 4850.7], ra: 0.094, dec: 0.04, pm: -0.085}, {angleDeg: [355.8, 1191.3], ra: 0.024, dec: 0.011, pm: -0.022}]},
-    saturn: {poleRaDeg: [40.589, -0.036, 0], poleDecDeg: [83.537, -0.004, 0], primeMeridianDeg: [38.9, 810.7939024, 0]},
-    uranus: {poleRaDeg: [257.311, 0, 0], poleDecDeg: [-15.175, 0, 0], primeMeridianDeg: [203.81, -501.1600928, 0]},
-    pluto: {poleRaDeg: [132.993, 0, 0], poleDecDeg: [-6.163, 0, 0], primeMeridianDeg: [302.695, 56.3625225, 0]},
-    moon: {poleRaDeg: [269.9949, 0.0031, 0], poleDecDeg: [66.5392, 0.013, 0], primeMeridianDeg: [38.3213, 13.17635815, -1.4e-12], terms: [{angleDeg: [125.045, -1935.5364525], ra: -3.8787, dec: 1.5419, pm: 3.561}, {angleDeg: [250.089, -3871.072905], ra: -0.1204, dec: 0.0239, pm: 0.1208}, {angleDeg: [260.008, 475263.3328725], ra: 0.07, dec: -0.0278, pm: -0.0642}, {angleDeg: [176.625, 487269.629985], ra: -0.0172, dec: 0.0068, pm: 0.0158}, {angleDeg: [357.529, 35999.0509575], ra: 0, dec: 0, pm: 0.0252}]},
-  };
  // Each ceiling sits just above what these elements measure on that date: Earth 0.003 degrees,
  // Jupiter 0.063, Saturn 0.164, Pluto 0.054, the Moon 0.72 (no mean ellipse has its evection or
  // variation), Io 0.021, Europa 0.036, Titan 0.014, Triton 0.137, Titania 0.62 (against Uranus's
@@ -644,8 +633,8 @@ describe('solar-system bodies against Horizons', () => {
  ];

  function record(id: string): BodyRecord {
-    const { kind, orbit, rates, laplacePole, parentBodyId, massRatio } = SHIPPED.find((body) => body.id === id)!;
-    return { id, systemStarId: 0, name: id, radiusKm: 1000, orbitSource: 'test', kind, orbit, rates, laplacePole, parentBodyId, massRatio, rotationalElements: ROTATION[id] };
+    const { kind, orbit, rates, laplacePole, parentBodyId, massRatio, rotationalElements } = SHIPPED.find((body) => body.id === id)!;
+    return { id, systemStarId: 0, name: id, radiusKm: 1000, orbitSource: 'test', kind, orbit, rates, laplacePole, parentBodyId, massRatio, rotationalElements };
  }

  const renderer = new SystemOrbitsRenderer(IDS.map(record), []);
@@ -796,7 +785,7 @@ describe('solar-system bodies against Horizons', () => {

  /** Where the IAU puts a body's prime meridian at a TDB date, in the scene. */
  function iauPrimeMeridian(id: string, jdTdb: number): THREE.Vector3 {
-    const { poleRaDeg, poleDecDeg, primeMeridianDeg } = orientationAt(ROTATION[id], jdTdb);
+    const { poleRaDeg, poleDecDeg, primeMeridianDeg } = orientationAt(SHIPPED.find((body) => body.id === id)!.rotationalElements!, jdTdb);
    const w = (primeMeridianDeg * Math.PI) / 180;
    const meridian = laplacePlaneToEquatorial({ x: Math.cos(w), y: Math.sin(w), z: 0 }, { raDeg: poleRaDeg, decDeg: poleDecDeg });
    return new THREE.Vector3(meridian.x, meridian.y, meridian.z);
@@ -907,3 +896,32 @@ describe('solar-system bodies against Horizons', () => {
    });
  }
 });
+
+describe('locked moons across the clock’s window', () => {
+  // As shipped, pole, W and all: the IAU gives each a W fitted near the present, and its rate is
+  // not quite its orbit's, nor Iapetus's pole a line for twenty centuries.
+  const shipped: BodyRecord[] = JSON.parse(readFileSync(`${process.cwd()}/src/assets/data/bodies.json`, 'utf8'));
+  const renderer = new SystemOrbitsRenderer(shipped.filter((body) => ['saturn', 'uranus', 'neptune', 'mimas', 'iapetus', 'miranda', 'proteus'].includes(body.id)), []);
+
+  /** East longitude, on its map, of the point on a moon's drawn sphere that faces its planet. */
+  function facingPlanet(id: string): number {
+    const moon = renderer.members.find((member) => member.id === id)!.marker;
+    // Its position is from the planet, which is its pivot; SphereGeometry wraps u = atan2(z, -x) / 2 pi.
+    const toPlanet = moon.position.clone().negate().applyQuaternion(moon.quaternion.clone().invert());
+    const u = Math.atan2(toPlanet.z, -toPlanet.x) / (2 * Math.PI);
+    return ((((u - 0.5) * 360) % 360) + 540) % 360 - 180;
+  }
+
+  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
+    // 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]) {
+      renderer.update(jd);
+      expect(Math.abs(facingPlanet('proteus'))).toBeLessThan(3);
+      expect(Math.abs(facingPlanet('miranda'))).toBeLessThan(3);
+      expect(Math.abs(facingPlanet('mimas'))).toBeLessThan(9.5);
+      expect(Math.abs(facingPlanet('iapetus'))).toBeLessThan(16.5);
+    }
+  });
+});
@@ -95,10 +95,12 @@ export interface BodyRecord {
  obliquityDeg?: number;
  /**
   * 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. 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.
+   * 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
+   * keep their faces to their planets 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.
   */
  rotationalElements?: RotationalElements;
 }
@@ -809,8 +809,8 @@
        0
      ],
      "primeMeridianDeg": [
-        79.39932954,
-        285.16188899,
+        79.49055322521608,
+        285.161879,
        0
      ],
      "terms": [
@@ -906,8 +906,8 @@
        0
      ],
      "primeMeridianDeg": [
-        200.39,
-        203.4889538,
+        200.34890824984421,
+        203.4889583,
        0
      ],
      "terms": [
@@ -971,8 +971,8 @@
        0
      ],
      "primeMeridianDeg": [
-        36.022,
-        101.3747235,
+        36.015607949990184,
+        101.3747242,
        0
      ],
      "terms": [
@@ -1045,8 +1045,8 @@
        0
      ],
      "primeMeridianDeg": [
-        44.064,
-        50.3176081,
+        44.07221835003116,
+        50.3176072,
        0
      ],
      "terms": [
@@ -1119,8 +1119,8 @@
        0
      ],
      "primeMeridianDeg": [
-        259.51,
-        21.5710715,
+        259.498129049991,
+        21.5710728,
        0
      ],
      "terms": [
@@ -1199,8 +1199,8 @@
        0
      ],
      "primeMeridianDeg": [
-        333.46,
-        381.994555,
+        334.00971630006546,
+        381.9944948,
        0
      ],
      "terms": [
@@ -1264,8 +1264,8 @@
        0
      ],
      "primeMeridianDeg": [
-        6.32,
-        262.7318996,
+        6.336436700062315,
+        262.7318978,
        0
      ]
    }
@@ -1315,8 +1315,8 @@
        0
      ],
      "primeMeridianDeg": [
-        8.95,
-        190.6979085,
+        8.928084400003424,
+        190.6979109,
        0
      ],
      "terms": [
@@ -1380,8 +1380,8 @@
        0
      ],
      "primeMeridianDeg": [
-        357.6,
-        131.5349316,
+        357.60821835003117,
+        131.5349307,
        0
      ]
    }
@@ -1425,8 +1425,8 @@
        0
      ],
      "primeMeridianDeg": [
-        235.16,
-        79.6900478,
+        235.1773498500081,
+        79.6900459,
        0
      ],
      "terms": [
@@ -1481,8 +1481,8 @@
        0
      ],
      "primeMeridianDeg": [
-        186.5855,
-        22.5769768,
+        186.5964577999983,
+        22.5769756,
        0
      ]
    }
@@ -1544,19 +1544,66 @@
    "rotationPeriodHours": 1903.9469348834284,
    "rotationalElements": {
      "poleRaDeg": [
-        318.16,
-        -3.949,
+        283.4205056238858,
+        0,
        0
      ],
      "poleDecDeg": [
-        75.03,
-        -1.143,
+        77.15116551386498,
+        0,
        0
      ],
      "primeMeridianDeg": [
-        355.2,
-        4.5379572,
+        388.27970766227594,
+        4.5379416,
        0
+      ],
+      "terms": [
+        {
+          "angleDeg": [
+            261.10499999998126,
+            -10.46898128087986
+          ],
+          "ra": -42.60490881707297,
+          "dec": 7.692796952777751,
+          "pm": 41.78612687698857
+        },
+        {
+          "angleDeg": [
+            522.2099999999625,
+            -20.93796256175972
+          ],
+          "ra": -15.535995393995726,
+          "dec": 1.2845933786338362,
+          "pm": 15.53892376069157
+        },
+        {
+          "angleDeg": [
+            783.3149999999438,
+            -31.406943842639578
+          ],
+          "ra": -7.628387941010053,
+          "dec": 0.45238994454766307,
+          "pm": 7.62837456853055
+        },
+        {
+          "angleDeg": [
+            1044.419999999925,
+            -41.87592512351944
+          ],
+          "ra": -4.2134447101107595,
+          "dec": 0.20197385694504744,
+          "pm": 4.213441809239813
+        },
+        {
+          "angleDeg": [
+            1305.5249999999064,
+            -52.344906404399296
+          ],
+          "ra": -2.482395388436937,
+          "dec": 0.10190190070186696,
+          "pm": 2.4609277085117798
+        }
      ]
    }
  },
@@ -1644,8 +1691,8 @@
        0
      ],
      "primeMeridianDeg": [
-        30.7,
-        -254.6906892,
+        30.41144460000184,
+        -254.6906576,
        0
      ],
      "terms": [
@@ -1727,8 +1774,8 @@
        0
      ],
      "primeMeridianDeg": [
-        156.22,
-        -142.8356681,
+        156.12685870007942,
+        -142.8356579,
        0
      ],
      "terms": [
@@ -1792,8 +1839,8 @@
        0
      ],
      "primeMeridianDeg": [
-        108.05,
-        -86.8688923,
+        108.00982140004953,
+        -86.8688879,
        0
      ],
      "terms": [
@@ -1857,8 +1904,8 @@
        0
      ],
      "primeMeridianDeg": [
-        77.74,
-        -41.3514316,
+        77.67607950003162,
+        -41.3514246,
        0
      ],
      "terms": [
@@ -1913,8 +1960,8 @@
        0
      ],
      "primeMeridianDeg": [
-        6.77,
-        -26.7394932,
+        6.729821400017093,
+        -26.7394888,
        0
      ],
      "terms": [
@@ -1969,8 +2016,8 @@
        0
      ],
      "primeMeridianDeg": [
-        296.53,
-        -61.2572637,
+        296.5309131499458,
+        -61.2572638,
        0
      ],
      "terms": [
@@ -2124,8 +2171,8 @@
        0
      ],
      "primeMeridianDeg": [
-        93.38,
-        320.7654228,
+        91.53817645008237,
+        320.7656245,
        0
      ],
      "terms": [
@@ -2190,8 +2237,8 @@
        0
      ],
      "primeMeridianDeg": [
-        122.695,
-        56.3625225,
+        122.70869724996541,
+        56.362521,
        0
      ]
    }
@@ -11,6 +11,7 @@ import { fetchDeepSky } from './fetchDeepSky';
 import { fetchExoplanets } from './fetchExoplanets';
 import { fetchSolarSystem, FREELY_SPINNING_MOONS, offsetFromTrackDeg } from './fetchSolarSystem';
 import { TrackPoint } from './lib/horizons';
+import { subPlanetLongitudeDeg } from './lib/locked-spin';
 import { BYTES_PER_STAR_META, BYTES_PER_STAR_POSITION, decodeStarCatalog, encodeStarCatalog } from '../../src/app/shared/models/star-catalog';
 import { fetchStars } from './fetchStars';
 import { describeSources } from './sources/registry';
@@ -223,37 +224,29 @@ const MAX_OBLIQUITY_OFFSET_DEG = 0.1;
 /**
 * How far from its planet a locked moon's drawn face may turn: the east longitude, on the IAU's
 * body-fixed frame, of the direction to the planet from where the mean elements put the moon,
- * sampled every 135 days from 1950 to 2100, where both the tables and the IAU's elements hold.
+ * sampled every 135 days over the clock's AD 1 to 3000. Every locked moon's W turns at its orbit's
+ * own rate (see `lockedToOrbit`); at the IAU's own rates, and sampled only from 1950 to 2100, this
+ * let Proteus turn its far side to Neptune at AD 1 (146 degrees), Iapetus 87 degrees, Mimas 52 and
+ * Miranda 23, on dates the clock offers.
 *
- * Measured on this catalogue: at most 6.70 degrees (the Moon, whose longitude swings 6.3 either
- * way with its eccentricity; Horizons has the same). Three need their own. Mimas 10.15: its drawn
- * face runs from 2.5 to 10.15 degrees, about 6.3 off on average because the IAU's W and JPL's mean
- * longitude disagree, drifting 3.3 over the span because W turns 6.0e-5 degrees a day faster than
- * the row's n, 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 integrated orbit, runs from -2.7 to 12.7
- * degrees over 1950-2100 with the 71-year S5 term the orbit here cancels. Iapetus
- * 18.33, whose row sits 9.4 degrees behind Horizons; and Proteus 8.18, whose W turns 6.3e-7 of
- * its rate slower than its orbit, a drift of 74 degrees by AD 3000. What this catches is an orbit
- * and a W that go round at different rates: the tidal acceleration W carried and the orbit did not
- * turned Phobos 13.8 degrees from Mars by 2100, and the Mimas-Tethys libration Mimas 54.5.
+ * 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
+ * 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
+ * integrated orbit, runs from -2.7 to 12.7 degrees over 1950-2100 with the 71-year S5 term the
+ * orbit here cancels. Iapetus 15.95, whose row sits 9.4 degrees behind Horizons. What this catches
+ * is an orbit and a W that go round at different rates: the tidal acceleration W carried and the
+ * orbit did not turned Phobos 13.8 degrees from Mars by 2100, and the Mimas-Tethys libration Mimas
+ * 54.5.
 */
 const MAX_SUB_PLANET_LONGITUDE_DEG = 7;
-const SUB_PLANET_CEILINGS_DEG: Record<string, number> = { mimas: 11, iapetus: 19, proteus: 9 };
-const LOCK_DATES_JD = Array.from({ length: 407 }, (_, index) => 2433282.5 + index * 135);
-
-/** The planet's east longitude on a moon's IAU body-fixed frame, from the moon's mean place, at a TDB date. */
-function subPlanetLongitudeDeg(body: BodyRecord, jd: number): number {
-  const own = positionAtEpoch(meanElementsAt(body.orbit, body.rates, jd));
-  const place = body.laplacePole ? laplacePlaneToEquatorial(own, body.laplacePole) : eclipticToEquatorial(own);
-  const { poleRaDeg, poleDecDeg, primeMeridianDeg } = orientationAt(body.rotationalElements!, jd);
-  const pole = { raDeg: poleRaDeg, decDeg: poleDecDeg };
-  const w = primeMeridianDeg * DEG_TO_RAD;
-  const meridian = laplacePlaneToEquatorial({ x: Math.cos(w), y: Math.sin(w), z: 0 }, pole);
-  const east = laplacePlaneToEquatorial({ x: -Math.sin(w), y: Math.cos(w), z: 0 }, pole);
-  const along = (axis: { x: number; y: number; z: number }) => -(place.x * axis.x + place.y * axis.y + place.z * axis.z);
-  return Math.atan2(along(east), along(meridian)) / DEG_TO_RAD;
-}
+const SUB_PLANET_CEILINGS_DEG: Record<string, number> = { moon: 8, mimas: 9.5, iapetus: 16.5 };
+/** 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;
+const LOCK_DATES_JD = Array.from({ length: Math.floor((CLOCK_END_JD - CLOCK_START_JD) / 135) + 1 }, (_, index) => CLOCK_START_JD + index * 135);

 function angleBetweenDeg(a: { x: number; y: number; z: number }, b: { x: number; y: number; z: number }): number {
  const cosine = (a.x * b.x + a.y * b.y + a.z * b.z) / (Math.hypot(a.x, a.y, a.z) * Math.hypot(b.x, b.y, b.z));
@@ -368,10 +361,10 @@ function validateBodies(bodies: BodyRecord[], horizonsOrbits: Map<string, Orbita
        // have to agree, or its face turns away from its planet.
        assertCondition(rotation !== undefined, `Moon ${body.id} is locked but has no W to keep its face to its planet by.`);
        const ceiling = SUB_PLANET_CEILINGS_DEG[body.id] ?? MAX_SUB_PLANET_LONGITUDE_DEG;
-        const worst = Math.max(...LOCK_DATES_JD.map((jd) => Math.abs(subPlanetLongitudeDeg(body, jd))));
+        const worst = Math.max(...LOCK_DATES_JD.map((jd) => Math.abs(subPlanetLongitudeDeg(body, rotation!, jd))));
        assertCondition(
          worst <= ceiling,
-          `Moon ${body.id} turns its face up to ${worst.toFixed(2)} degrees from its planet between 1950 and 2100 (at most ${ceiling} expected) — its orbit and its W disagree.`
+          `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)}`);
      }
@@ -9,6 +9,7 @@ import { MeanOrbit, parsePlanetMeanElements, parseSatelliteMeanElements, parseSm
 import { fetchPlanetMeanElementsText, fetchSatelliteMeanElementsHtml, fetchSmallBodyAnswer } from './lib/mean-elements';
 import { MIN_PERIODIC_TERM_DEG, orbitalTermsOfPrimeMeridian, parsePckRotationalElements, SUN_ROTATIONAL_ELEMENTS } from '../../src/app/shared/astro/rotational-elements';
 import { fetchPckText } from './lib/pck';
+import { lockedToOrbit } from './lib/locked-spin';
 import { dataPath, ensureDataDir } from './lib/paths';

 const HOURS_PER_DAY = 24;
@@ -54,6 +55,8 @@ interface BodySpec {
   * given. See `orbitalTermsOfPrimeMeridian`.
   */
  orbitFromW?: { angleRateDegPerCentury?: number };
+  /** A locked moon whose pole is carried round with its orbit's, as Iapetus's; see `lockedToOrbit`. */
+  poleFollowsOrbit?: boolean;
  /**
   * Days between the Horizons positions the orbit is checked against from 1950 to 2100; 2 unless
   * the error changes faster than that. Nereid, at an eccentricity of 0.75, sweeps through its
@@ -131,7 +134,7 @@ const BODY_SPECS: BodySpec[] = [
  // Hyperion tumbles ("Rotational period = Chaotic") and Phoebe, captured, turns in 9.27 hours.
  // Hyperion's eccentricity is 0.105 in JPL's current table (ssd.jpl.nasa.gov/sats/elem, SAT441).
  { id: 'hyperion', name: 'Hyperion', kind: 'moon', horizonsCommand: '607', center: '500@699', parentBodyId: 'saturn', spinsFreely: true, measuredEccentricity: 0.105, trackStepDays: 1 },
-  { id: 'iapetus', name: 'Iapetus', kind: 'moon', horizonsCommand: '608', center: '500@699', parentBodyId: 'saturn' },
+  { id: 'iapetus', name: 'Iapetus', kind: 'moon', horizonsCommand: '608', center: '500@699', parentBodyId: 'saturn', poleFollowsOrbit: true },
  // Phoebe's row gives a mean motion of 0.6569114 degrees a day, a 548.02-day year, where its
  // Horizons page and JPL's current table (SAT441) give 550.30: the table's own note warns that
  // its source misstated the mean motions of retrograde moons. On the row's figure Phoebe was
@@ -245,10 +248,11 @@ export async function fetchSolarSystem(): Promise<{ bodies: BodyRecord[]; horizo
    // A moon listed here is tidally locked unless its spec says otherwise, so its day is its
    // orbit: the sidereal period from the same mean motion that carries it round. Not every page
    // says so — the Moon's gives a rate, Titan's and Proteus's nothing. Every locked moon here is
-    // turned by its IAU W rather than by this day, and `build.ts` checks that W and the orbit keep
-    // its face to its planet from 1950 to 2100; this day is what the renderer would turn a moon
-    // without W by.
-    const rotationPeriodHours = result.tidallyLocked || (spec.kind === 'moon' && !spec.spinsFreely)
+    // turned by its IAU W, at this same rate (see `lockedToOrbit`), and `build.ts` checks that W and
+    // the orbit keep its face to its planet from AD 1 to 3000; this day is what the renderer would
+    // turn a moon without W by.
+    const locked = spec.kind === 'moon' && !spec.spinsFreely;
+    const rotationPeriodHours = result.tidallyLocked || locked
      ? (360 / mean.rates.meanMotionDegPerDay) * HOURS_PER_DAY
      : (spec.rotationPeriodHours ?? (smallBody ? smallBody.rotationPeriodHours : result.rotationPeriodHours));
    const parentGm = spec.barycentric && spec.parentBodyId ? gmById.get(spec.parentBodyId) : undefined;
@@ -274,7 +278,7 @@ export async function fetchSolarSystem(): Promise<{ bodies: BodyRecord[]; horizo
      ...(parentGm !== undefined ? { massRatio: result.gmKm3PerS2! / parentGm } : {}),
      ...(rotationPeriodHours !== undefined ? { rotationPeriodHours } : {}),
      ...((result.obliquityDeg ?? spec.obliquityDeg) !== undefined ? { obliquityDeg: result.obliquityDeg ?? spec.obliquityDeg } : {}),
-      ...(rotation ? { rotationalElements: rotation.elements } : {})
+      ...(rotation ? { rotationalElements: locked ? lockedToOrbit(rotation.elements, mean, spec.name, spec.poleFollowsOrbit) : rotation.elements } : {})
    });
  }

@@ -0,0 +1,141 @@
+import { eclipticToEquatorial, laplacePlaneToEquatorial } from '../../../src/app/shared/astro/coordinates';
+import { meanElementsAt, positionAtEpoch } from '../../../src/app/shared/astro/kepler';
+import { MeanOrbit } from '../../../src/app/shared/astro/mean-elements';
+import { orientationAt } from '../../../src/app/shared/astro/rotational-elements';
+import { BodyRecord, RotationalElements } from '../../../src/app/shared/models/body.model';
+
+const J2000_JD = 2451545;
+const DAYS_PER_JULIAN_CENTURY = 36525;
+const DEG_TO_RAD = Math.PI / 180;
+
+/** 2025-01-01, the date the ETL asks Horizons about: where a locked moon's W is left as the IAU has it. */
+export const PRESENT_JD = 2460676.5;
+
+/**
+ * How far the IAU's W rate for a locked moon may be from the mean motion its orbit is drawn at, as
+ * a fraction of it, before it is taken for the orbit's. Measured: at most 3.4e-6 (Iapetus; Proteus
+ * 6.3e-7). What this catches is a rate read for the wrong body: Oberon's for Titania's is 55 per cent out.
+ */
+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;
+
+/** 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));
+  const place = body.laplacePole ? laplacePlaneToEquatorial(own, body.laplacePole) : eclipticToEquatorial(own);
+  const { poleRaDeg, poleDecDeg, primeMeridianDeg } = orientationAt(elements, jd);
+  const pole = { raDeg: poleRaDeg, decDeg: poleDecDeg };
+  const w = primeMeridianDeg * DEG_TO_RAD;
+  const meridian = laplacePlaneToEquatorial({ x: Math.cos(w), y: Math.sin(w), z: 0 }, pole);
+  const east = laplacePlaneToEquatorial({ x: -Math.sin(w), y: Math.cos(w), z: 0 }, pole);
+  const along = (axis: { x: number; y: number; z: number }) => -(place.x * axis.x + place.y * axis.y + place.z * axis.z);
+  return Math.atan2(along(east), along(meridian)) / DEG_TO_RAD;
+}
+
+/**
+ * A locked moon's IAU elements, turned at the rate its orbit is drawn at, so it keeps its face to
+ * its planet over the clock's AD 1 to 3000 and not only near the present its W was fitted to.
+ *
+ * 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},
+ * and the pole and every periodic term are the IAU's. Measured over AD 1-3000: Proteus 2.7 degrees,
+ * Mimas 8.9, Miranda 2.8, Ariel 1.0. A W with a quadratic is left: Phobos's orbit already takes the
+ * quadratic from W (see `orbitalTermsOfPrimeMeridian`), and the Moon's, its tidal slowing, is 0.75
+ * degrees at AD 1.
+ *
+ * `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
+ * 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
+ * {@link POLE_HARMONICS} harmonics, the IAU's own form for a precessing pole, with its constants
+ * set so the pole is the IAU's at the present. W counts from where the equator crosses the ICRF
+ * equator, which swings as the pole goes round, so W takes sines of the same angles, fitted to hold
+ * the face where it is today. Measured over AD 1-3000: the axis within 0.74 degrees of the orbit
+ * normal (the IAU's line, 11.06), and the face within 16 of Saturn (87), which is what the row's own
+ * 9.4-degree lag and its eccentricity make it from 1950 to 2100 as well (15.9).
+ */
+export function lockedToOrbit(elements: RotationalElements, mean: Pick<MeanOrbit, 'orbit' | 'rates' | 'laplacePole'>, name: string, poleFollowsOrbit = false): RotationalElements {
+  const [w0, w1, w2 = 0] = elements.primeMeridianDeg;
+  if (w2 !== 0) {
+    return elements;
+  }
+  const n = Math.sign(w1) * mean.rates.meanMotionDegPerDay;
+  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] };
+  return poleFollowsOrbit ? poleRoundOrbit(locked, mean) : locked;
+}
+
+function poleRoundOrbit(elements: RotationalElements, mean: Pick<MeanOrbit, 'orbit' | 'rates' | 'laplacePole'>): RotationalElements {
+  if (!mean.laplacePole) {
+    throw new Error('A pole that follows its orbit is carried round the orbit\'s Laplace pole, and this orbit has none.');
+  }
+  const laplacePole = mean.laplacePole;
+  const normalAt = (jd: number) => {
+    const { inclinationDeg, longitudeOfAscendingNodeDeg } = meanElementsAt(mean.orbit, mean.rates, jd);
+    const tilt = inclinationDeg * DEG_TO_RAD;
+    const node = longitudeOfAscendingNodeDeg * DEG_TO_RAD;
+    const normal = laplacePlaneToEquatorial({ x: Math.sin(tilt) * Math.sin(node), y: -Math.sin(tilt) * Math.cos(node), z: Math.cos(tilt) }, laplacePole);
+    return { raDeg: Math.atan2(normal.y, normal.x) / DEG_TO_RAD, decDeg: Math.asin(normal.z) / DEG_TO_RAD };
+  };
+  // The node's angle, T in centuries, turned so that 0 is where the normal is furthest north: the
+  // circle is then even in declination and odd in right ascension about it, as the form requires.
+  const nodeRate = mean.rates.longitudeOfAscendingNodeDegPerDay * DAYS_PER_JULIAN_CENTURY;
+  const nodeAtJ2000 = meanElementsAt(mean.orbit, mean.rates, J2000_JD).longitudeOfAscendingNodeDeg;
+  const jdAtAngle = (angleDeg: number, phaseDeg: number) => J2000_JD + ((angleDeg - phaseDeg - nodeAtJ2000) / nodeRate) * DAYS_PER_JULIAN_CENTURY;
+  let phase = 0;
+  let northmost = -Infinity;
+  for (let candidate = 0; candidate < 360; candidate += 0.01) {
+    const dec = normalAt(jdAtAngle(0, candidate)).decDeg;
+    if (dec > northmost) {
+      northmost = dec;
+      phase = candidate;
+    }
+  }
+  const centre = laplacePole;
+  const samples = 3600;
+  const ra = new Array<number>(POLE_HARMONICS + 1).fill(0);
+  const dec = new Array<number>(POLE_HARMONICS + 1).fill(0);
+  for (let sample = 0; sample < samples; sample++) {
+    const angle = (sample / samples) * 360;
+    const normal = normalAt(jdAtAngle(angle, phase));
+    const raOffset = ((((normal.raDeg - centre.raDeg) % 360) + 540) % 360) - 180;
+    for (let k = 0; k <= POLE_HARMONICS; k++) {
+      ra[k] += (2 / samples) * raOffset * Math.sin(k * angle * DEG_TO_RAD);
+      dec[k] += ((k === 0 ? 1 : 2) / samples) * (normal.decDeg - centre.decDeg) * Math.cos(k * angle * DEG_TO_RAD);
+    }
+  }
+  const terms = Array.from({ length: POLE_HARMONICS }, (_, index) => {
+    const k = index + 1;
+    return { angleDeg: [k * (nodeAtJ2000 + phase), k * nodeRate], ra: ra[k], dec: dec[k], pm: 0 };
+  });
+  const round: RotationalElements = { ...elements, poleRaDeg: [centre.raDeg, 0, 0], poleDecDeg: [centre.decDeg + dec[0], 0, 0], terms: [...(elements.terms ?? []), ...terms] };
+  // The IAU's pole at the present, exactly: the fitted circle's constants moved onto it.
+  const iau = orientationAt(elements, PRESENT_JD);
+  const fitted = orientationAt(round, PRESENT_JD);
+  round.poleRaDeg = [round.poleRaDeg[0] + iau.poleRaDeg - fitted.poleRaDeg, 0, 0];
+  round.poleDecDeg = [round.poleDecDeg[0] + iau.poleDecDeg - fitted.poleDecDeg, 0, 0];
+  // W's sines on the same angles, fitted over a turn of the node to what the face drifts by.
+  const pm = new Array<number>(POLE_HARMONICS + 1).fill(0);
+  const wSamples = 36000;
+  for (let sample = 0; sample < wSamples; sample++) {
+    const angle = (sample / wSamples) * 360;
+    const drift = subPlanetLongitudeDeg(mean, round, jdAtAngle(angle, phase));
+    for (let k = 1; k <= POLE_HARMONICS; k++) {
+      pm[k] += (2 / wSamples) * drift * Math.sin(k * angle * DEG_TO_RAD);
+    }
+  }
+  const ownTerms = elements.terms?.length ?? 0;
+  const turned: RotationalElements = { ...round, terms: round.terms!.map((term, index) => (index < ownTerms ? term : { ...term, pm: pm[index - ownTerms + 1] })) };
+  // And W the IAU's at the present.
+  const shift = orientationAt(turned, PRESENT_JD).primeMeridianDeg - orientationAt(elements, PRESENT_JD).primeMeridianDeg;
+  turned.primeMeridianDeg = [turned.primeMeridianDeg[0] - shift, turned.primeMeridianDeg[1], 0];
+  return turned;
+}