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>

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},
- * 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.
+ * W's rate is set to the orbit's here, its constant moved so W is unchanged at {@link PRESENT_JD}.
+ * Measured over AD 1-3000: Proteus 2.7 degrees, Mimas 8.9, Miranda 2.4, 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.
+ *
+ * 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;
 }