Move the solar system on JPL's mean elements, so it stays right as the clock runs

Every body carried one set of osculating elements from Horizons at 2025-01-01, run forward by Kepler with a GM from a table of mass ratios. That set is exact at its instant and drifts from then on, and the clock now runs a month a second: the Moon, with Earth's mass ratio lacking its own and the osculating axis, went round in 27.70 days instead of 27.32, 66 degrees out after a year, and its locked face was spun at the same wrong rate. Planets and Pluto now take Standish's Table 2a/2b ("Keplerian Elements for Approximate Positions of the Major Planets"): elements against the J2000 ecliptic, their rates per century, and the b, c, s, f terms of Jupiter to Pluto, fit for 3000 BC to AD 3000. Table 1 is closer near the present (Saturn 0.23 degrees at worst 1950-2100, against 0.32 here) but is only fit for 1800-2050, and by AD 3000 has Saturn 4.3 degrees out where Table 2 holds every planet within 0.3. The moons take JPL SSD's satellite mean elements: sidereal mean motion n to ten figures, the periods of their node and periapsis, and each one's local Laplace plane by its pole. They propagate with n itself, never a GM: gmForParent and its mass table are gone. Horizons still gives size, spin and obliquity. Both tables are read from the Internet Archive's copy of JPL's pages, pinned to one capture: the live approx_pos page has dropped Pluto, and the live sats/elem page has dropped n and rounds the period to four or five figures (Phobos 0.3187 d, a revolution out within a decade). What the tables leave implicit, measured against Horizons before it was accepted: - The precession periods are magnitudes. A node regresses on a prograde orbit and advances on a retrograde one; a periapsis advances except where a resonance forces the eccentricity. Io's and Europa's follow their conjunction line backwards at 2 n(Europa) - n(Io) = 0.74 degrees a day, which is exactly the 1.625- and 1.394-year periods in the table. Read as advancing, Io was 0.9 degrees out and Europa 2.1. - On a retrograde orbit the node's turning is added back to the mean anomaly. Taken off, Triton drifted a degree a year, 105 degrees by 2100. - The Laplace frame's x axis is where the plane rises through the ICRF equator, RA of the pole plus 90. Read against the ecliptic, Io was 2.8 degrees out, Phobos 54 and Titan 127. Orbit lines are now drawn in their own plane and turned by a quaternion each tick, so a turning node carries the line with the body: fixed at one date, the Moon's line would be up to 69 000 km off it nine years on. The Earth row is the Earth-Moon barycentre, 4 700 km from Earth, 0.002 degrees from the Sun. A tidally locked moon's day is now 360 / n, its sidereal period (the Moon 27.321662 d), so it stays locked to the orbit it is drawn on. Angular error against Horizons VECTORS (ICRF, TDB; heliocentric for planets, planet-centred for moons), degrees, read from the live renderer's markers in the running app: body 1950-01-01 1975-01-01 1987-07-23 2000-01-01 2025-01-01 2037-03-06 2050-01-01 2075-01-01 2100-01-01 max mercury 0.004 0.002 0.003 0.002 0.002 0.001 0.000 0.002 0.000 0.004 venus 0.003 0.007 0.003 0.004 0.004 0.004 0.003 0.004 0.004 0.007 earth 0.003 0.008 0.002 0.005 0.004 0.009 0.003 0.002 0.003 0.009 mars 0.009 0.010 0.008 0.024 0.009 0.012 0.009 0.011 0.028 0.028 jupiter 0.063 0.030 0.171 0.135 0.013 0.020 0.056 0.041 0.075 0.171 saturn 0.080 0.064 0.018 0.320 0.066 0.114 0.044 0.164 0.177 0.320 uranus 0.018 0.169 0.068 0.050 0.101 0.015 0.141 0.017 0.114 0.169 neptune 0.070 0.028 0.004 0.021 0.036 0.037 0.013 0.029 0.072 0.072 pluto 0.045 0.054 0.041 0.033 0.019 0.020 0.023 0.027 0.026 0.054 moon 0.486 1.928 0.127 0.631 1.407 1.086 0.720 0.339 1.180 1.928 phobos 2.068 0.294 0.881 1.113 0.313 0.636 2.089 5.862 11.099 11.099 deimos 0.077 0.043 0.310 0.066 0.164 0.068 0.034 0.468 0.044 0.468 io 0.021 0.015 0.010 0.019 0.009 0.035 0.006 0.011 0.022 0.035 europa 0.036 0.039 0.053 0.064 0.078 0.032 0.006 0.034 0.044 0.078 ganymede 0.132 0.103 0.018 0.007 0.023 0.054 0.091 0.118 0.044 0.132 callisto 0.040 0.019 0.023 0.019 0.038 0.008 0.060 0.119 0.056 0.119 titan 0.003 0.019 0.023 0.023 0.027 0.028 0.048 0.008 0.014 0.048 triton 0.051 0.029 0.009 0.021 0.052 0.048 0.063 0.089 0.137 0.137 Three miss what was hoped for, and why: - Jupiter 0.17, Saturn 0.32, Uranus 0.17 against the 0.1 hoped for: short-period perturbations of the giants by one another, which no Keplerian fit carries. Standish states his own Table 2 errors as 600, 1 000 and 2 000 arcseconds (0.17, 0.28, 0.56 degrees). Out to AD 3000, measured at 1800, 2200, 2400, 2600 and 3000, every planet stays within 0.3. - The Moon, 1.9: evection (1.27) and variation (0.66), which a mean ellipse leaves out. - Phobos, 2.1 until 2050, then 5.9 in 2075 and 11.1 in 2100, growing as the square of the time: its tidal acceleration, which the table has no column for. Its elements are MAR080's, epoch 1950. The map's dates are also UTC where the elements are TDB, 69 s today, which is 0.9 degrees of Phobos and nothing for anything else. Held in place by: - build.ts: each body's mean elements against Horizons' own osculating elements on the ETL's 2025-01-01, at most 0.25 degrees for a planet and 2.5 for a moon (measured: Uranus 0.101, the Moon 1.407; a regressing Triton node reads 10.24 and fails), and every moon's day equal to its sidereal period (a 1% error fails). - Unit tests freezing nine Horizons vectors (Earth 2100, Jupiter 1950, Saturn 2075, Pluto 1975, the Moon 2050, Io and Europa 1950, Titan and Triton 2100) through SystemOrbitsRenderer, the Moon kept on its own turning line, the retrograde rule, the Standish terms, the Laplace frame, and both table parsers. Nine mutants each fail the test named for them, and the two validators each refuse a mutated build of the real catalogue. Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>
2026-09-24 20:18:57 +02:00
parent c38a42cbcb
commit 48319c3fe2
19 changed files with 1062 additions and 244 deletions
@@ -34,6 +34,9 @@ const earth: BodyRecord = {
  kind: 'planet',
  radiusKm: 6371,
  orbit: orbit(),
+  // Standish's mean longitude rate, 35 999.373 degrees a century.
+  rates: { meanMotionDegPerDay: 35999.37306329 / 36525, longitudeOfAscendingNodeDegPerDay: 0, argumentOfPeriapsisDegPerDay: 0 },
+  orbitSource: 'JPL approximate mean elements (Standish), fit for 3000 BC to AD 3000',
 };
 const luna: BodyRecord = {
  id: 'luna',
@@ -43,6 +46,9 @@ const luna: BodyRecord = {
  radiusKm: 1737,
  parentBodyId: 'earth',
  orbit: orbit({ semiMajorAxisAu: 0.00257 }),
+  // JPL SSD's sidereal mean motion for the Moon.
+  rates: { meanMotionDegPerDay: 13.176358, longitudeOfAscendingNodeDegPerDay: -0.05299, argumentOfPeriapsisDegPerDay: 0.16435 },
+  orbitSource: 'JPL SSD satellite mean elements, epoch 2000 Jan 1',
 };

 describe('heliocentricPeriodDays', () => {
@@ -6,6 +6,8 @@ import { afterEach, beforeEach, describe, expect, it, MockInstance, vi } from 'v
 import { DataLoaderService, StarField } from '../../core/data/data-loader.service';
 import { EngineService, EngineTickCallback } from '../../core/engine/engine.service';
 import { BodyRecord } from '../../shared/models/body.model';
+import { GM_SUN_AU3_PER_DAY2 } from '../../shared/astro/constants';
+import { keplerRates } from '../../shared/astro/kepler';
 import { DeepSkyRecord } from '../../shared/models/deepsky.model';
 import { ExoplanetRecord } from '../../shared/models/exoplanet.model';
 import { StarRecord } from '../../shared/models/star.model';
@@ -63,7 +65,8 @@ const EARTH: BodyRecord = {
    argumentOfPeriapsisDeg: 0,
    meanAnomalyAtEpochDeg: 0,
    epochJd: 2451545.0
-  }
+  },
+  rates: keplerRates(1, GM_SUN_AU3_PER_DAY2), orbitSource: 'test'
 };

 /** Minimal stand-in for `EngineService` that skips real WebGPU/WebGL initialization entirely,
@@ -1,7 +1,8 @@
 import * as THREE from 'three/webgpu';
 import { describe, expect, it } from 'vitest';

-import { DEFAULT_EPOCH_JD } from '../../shared/astro/constants';
+import { DEFAULT_EPOCH_JD, GM_SUN_AU3_PER_DAY2 } from '../../shared/astro/constants';
+import { keplerRates } from '../../shared/astro/kepler';
 import { eclipticToEquatorial, OBLIQUITY_J2000_DEG } from '../../shared/astro/coordinates';
 import { BodyRecord } from '../../shared/models/body.model';
 import { ExoplanetRecord } from '../../shared/models/exoplanet.model';
@@ -164,7 +165,8 @@ describe('SystemOrbitsRenderer exoplanet propagation', () => {
        argumentOfPeriapsisDeg: 0,
        meanAnomalyAtEpochDeg: 0,
        epochJd: DEFAULT_EPOCH_JD
-      }
+      },
+      rates: keplerRates(1, GM_SUN_AU3_PER_DAY2), orbitSource: 'test'
    };

    it('places an ecliptic orbit in the ecliptic plane of the equatorial scene', () => {
@@ -234,7 +236,9 @@ describe('SystemOrbitsRenderer exoplanet propagation', () => {
      name: 'Jupiter',
      kind: 'planet',
      radiusKm: 69911,
-      orbit: { semiMajorAxisAu: 5.2, eccentricity: 0.048, inclinationDeg: 1.3, longitudeOfAscendingNodeDeg: 100, argumentOfPeriapsisDeg: 275, meanAnomalyAtEpochDeg: 20, epochJd: DEFAULT_EPOCH_JD }
+      orbit: { semiMajorAxisAu: 5.2, eccentricity: 0.048, inclinationDeg: 1.3, longitudeOfAscendingNodeDeg: 100, argumentOfPeriapsisDeg: 275, meanAnomalyAtEpochDeg: 20, epochJd: DEFAULT_EPOCH_JD },
+      rates: keplerRates(5.2, GM_SUN_AU3_PER_DAY2),
+      orbitSource: 'test'
    };

    /** The grid and the tethers are the only line objects the renderer adds outside a pivot. */
@@ -383,6 +387,7 @@ describe('rotation', () => {
      kind: 'planet',
      radiusKm: 6371,
      orbit: { semiMajorAxisAu: 1, eccentricity: 0.0167, inclinationDeg: 0, longitudeOfAscendingNodeDeg: 0, argumentOfPeriapsisDeg: 0, meanAnomalyAtEpochDeg: 0, epochJd: DEFAULT_EPOCH_JD },
+      rates: keplerRates(1, GM_SUN_AU3_PER_DAY2), orbitSource: 'test',
      rotationPeriodHours: 23.934,
      obliquityDeg: 23.4392911,
      ...overrides
@@ -464,3 +469,63 @@ describe('exoplanet size without a measured radius', () => {
    expect(radiusOf({ radiusEarth: 1.88, massEarth: 2829 }) / EARTH_AU).toBeCloseTo(1.88, 2);
  });
 });
+
+describe('solar-system bodies against Horizons', () => {
+  // Real records 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 —
+  // mean elements, their rates, the Laplace planes and the scene's frame — is checked against
+  // JPL's ephemeris rather than against itself.
+  const RECORDS: Record<string, Pick<BodyRecord, 'kind' | 'orbit' | 'rates' | 'laplacePole' | 'parentBodyId'>> = {
+    earth: {kind: 'planet', orbit: {semiMajorAxisAu: 1.00000018, eccentricity: 0.01673163, inclinationDeg: -0.00054346, longitudeOfAscendingNodeDeg: -5.11260389, argumentOfPeriapsisDeg: 108.04266274, meanAnomalyAtEpochDeg: -2.4631431299999917, epochJd: 2451545}, rates: {meanMotionDegPerDay: 0.9856091187759068, longitudeOfAscendingNodeDegPerDay: -0.000006604751813826146, argumentOfPeriapsisDegPerDay: 0.000015309819575633124, semiMajorAxisAuPerDay: -8.213552361396303e-13, eccentricityPerDay: -1.002327173169062e-9, inclinationDegPerDay: -3.6609938398357287e-7}},
+    jupiter: {kind: 'planet', orbit: {semiMajorAxisAu: 5.20248019, eccentricity: 0.0485359, inclinationDeg: 1.29861416, longitudeOfAscendingNodeDeg: 100.29282654, argumentOfPeriapsisDeg: -86.0178741, meanAnomalyAtEpochDeg: 20.059839080000003, epochJd: 2451545}, rates: {meanMotionDegPerDay: 0.08309113532019165, longitudeOfAscendingNodeDegPerDay: 0.0000035659463381245725, argumentOfPeriapsisDegPerDay: 0.0000014167219712525667, semiMajorAxisAuPerDay: -7.841204654346339e-10, eccentricityPerDay: 4.935249828884326e-9, inclinationDegPerDay: -8.83501711156742e-8, meanAnomalyTerms: {b: -0.00012452, c: 0.0606406, s: -0.35635438, f: 38.35125}}},
+    saturn: {kind: 'planet', orbit: {semiMajorAxisAu: 9.54149883, eccentricity: 0.05550825, inclinationDeg: 2.49424102, longitudeOfAscendingNodeDeg: 113.63998702, argumentOfPeriapsisDeg: -20.778626390000014, meanAnomalyAtEpochDeg: -42.78564733999999, epochJd: 2451545}, rates: {meanMotionDegPerDay: 0.033459683702669406, longitudeOfAscendingNodeDegPerDay: -0.000006848734291581108, argumentOfPeriapsisDegPerDay: 0.000021682266940451745, semiMajorAxisAuPerDay: -8.391512662559891e-10, eccentricityPerDay: -8.773169062286106e-9, inclinationDegPerDay: 1.2374236824093085e-7, meanAnomalyTerms: {b: 0.00025899, c: -0.13434469, s: 0.87320147, f: 38.35125}}},
+    neptune: {kind: 'planet', orbit: {semiMajorAxisAu: 30.06952752, eccentricity: 0.00895439, inclinationDeg: 1.7700552, longitudeOfAscendingNodeDeg: 131.78635853, argumentOfPeriapsisDeg: -85.10477129, meanAnomalyAtEpochDeg: 257.54130563, epochJd: 2451545}, rates: {meanMotionDegPerDay: 0.005981249914852841, longitudeOfAscendingNodeDegPerDay: -1.6599644079397672e-7, argumentOfPeriapsisDegPerDay: 4.4250239561943875e-7, semiMajorAxisAuPerDay: 1.7650924024640657e-9, eccentricityPerDay: 2.2395619438740589e-10, inclinationDegPerDay: 6.132785763175907e-9, meanAnomalyTerms: {b: -0.00041348, c: 0.68346318, s: -0.10162547, f: 7.67025}}},
+    pluto: {kind: 'dwarf', orbit: {semiMajorAxisAu: 39.48686035, eccentricity: 0.24885238, inclinationDeg: 17.1410426, longitudeOfAscendingNodeDeg: 110.30167986, argumentOfPeriapsisDeg: 113.79534612000002, meanAnomalyAtEpochDeg: 14.86832412999999, epochJd: 2451545}, rates: {meanMotionDegPerDay: 0.003974823518959616, longitudeOfAscendingNodeDegPerDay: -2.2176071184120468e-7, argumentOfPeriapsisDegPerDay: -4.3489664613278575e-8, semiMajorAxisAuPerDay: 1.2313511293634495e-7, eccentricityPerDay: 1.6470910335386722e-9, inclinationDegPerDay: 1.3716632443531827e-10, meanAnomalyTerms: {b: -0.01262724, c: 0, s: 0, f: 0}}},
+    moon: {kind: 'moon', orbit: {semiMajorAxisAu: 0.0025695552897999907, eccentricity: 0.0554, inclinationDeg: 5.16, longitudeOfAscendingNodeDeg: 125.08, argumentOfPeriapsisDeg: 318.15, meanAnomalyAtEpochDeg: 135.27, epochJd: 2451545}, rates: {meanMotionDegPerDay: 13.176358, longitudeOfAscendingNodeDegPerDay: -0.052990660396105185, argumentOfPeriapsisDegPerDay: 0.164353223839846}, parentBodyId: 'earth'},
+    io: {kind: 'moon', orbit: {semiMajorAxisAu: 0.0028195588481728304, eccentricity: 0.0041, inclinationDeg: 0.036, longitudeOfAscendingNodeDeg: 43.977, argumentOfPeriapsisDeg: 84.129, meanAnomalyAtEpochDeg: 342.021, epochJd: 2450464.5}, rates: {meanMotionDegPerDay: 203.4889583, longitudeOfAscendingNodeDegPerDay: -0.1328337309120696, argumentOfPeriapsisDegPerDay: -0.6065392513031117}, laplacePole: {raDeg: 268.057, decDeg: 64.495}, parentBodyId: 'jupiter'},
+    europa: {kind: 'moon', orbit: {semiMajorAxisAu: 0.004486026417754354, eccentricity: 0.0094, inclinationDeg: 0.466, longitudeOfAscendingNodeDeg: 219.106, argumentOfPeriapsisDeg: 88.97, meanAnomalyAtEpochDeg: 171.016, epochJd: 2450464.5}, rates: {meanMotionDegPerDay: 101.3747242, longitudeOfAscendingNodeDegPerDay: -0.03265393199600969, argumentOfPeriapsisDegPerDay: -0.7070489837643877}, laplacePole: {raDeg: 268.084, decDeg: 64.506}, parentBodyId: 'jupiter'},
+    titan: {kind: 'moon', orbit: {semiMajorAxisAu: 0.008167663044150534, eccentricity: 0.0288, inclinationDeg: 0.306, longitudeOfAscendingNodeDeg: 28.06, argumentOfPeriapsisDeg: 180.532, meanAnomalyAtEpochDeg: 163.31, epochJd: 2451545}, rates: {meanMotionDegPerDay: 22.5769756, longitudeOfAscendingNodeDegPerDay: -0.001398845136769169, argumentOfPeriapsisDegPerDay: 0.002799120423059061}, laplacePole: {raDeg: 36.214, decDeg: 83.949}, parentBodyId: 'saturn'},
+    triton: {kind: 'moon', orbit: {semiMajorAxisAu: 0.002371417442908832, eccentricity: 0, inclinationDeg: 156.865, longitudeOfAscendingNodeDeg: 177.608, argumentOfPeriapsisDeg: 66.142, meanAnomalyAtEpochDeg: 352.257, epochJd: 2451545}, rates: {meanMotionDegPerDay: 61.2572638, longitudeOfAscendingNodeDegPerDay: 0.001433750844964632, argumentOfPeriapsisDegPerDay: 0.0025509841146658433}, laplacePole: {raDeg: 299.456, decDeg: 43.414}, parentBodyId: 'neptune'},
+  };
+  // 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.
+  const HORIZONS: Array<[id: string, jd: number, x: number, y: number, z: number, maxDeg: number]> = [
+    ['earth', 2488069.5, -0.1574071329883954, 0.890666220858489, 0.3859132211165683, 0.02],
+    ['jupiter', 2433282.5, 3.406605247558555, -3.425997624196318, -1.551719750032203, 0.1],
+    ['saturn', 2478938.5, -3.51309768447752, -8.723317933082274, -3.452662390556131, 0.25],
+    ['pluto', 2442413.5, -29.2488165026956, -7.1421817246801, 6.58403957591589, 0.1],
+    ['moon', 2469807.5, 0.00240364781322315, 0.0006554283236619424, 0.0004472719300783614, 2],
+    ['io', 2433282.5, 0.0004488349204269952, 0.002519633434577752, 0.00120678715190893, 0.05],
+    ['europa', 2433282.5, 0.004084372287322533, -0.001665375585011311, -0.0007673072324795899, 0.1],
+    ['titan', 2488069.5, 0.007800850235156121, -0.001556932380983438, -0.0006078959246502567, 0.05],
+    ['triton', 2488069.5, -0.001421151845853369, -0.0001894510477241482, 0.001888790702926415, 0.2],
+  ];
+
+  function record(id: string): BodyRecord {
+    return { id, systemStarId: 0, name: id, radiusKm: 1000, orbitSource: 'test', ...RECORDS[id] };
+  }
+
+  const renderer = new SystemOrbitsRenderer(Object.keys(RECORDS).map(record), []);
+
+  for (const [id, jd, x, y, z, maxDeg] of HORIZONS) {
+    it(`puts ${id} within ${maxDeg} degrees of Horizons on JD ${jd}`, () => {
+      renderer.update(jd);
+      const drawn = renderer.members.find((member) => member.id === id)!.marker.position;
+      const angleDeg = (drawn.angleTo(new THREE.Vector3(x, y, z)) * 180) / Math.PI;
+      expect(angleDeg).toBeLessThan(maxDeg);
+    });
+  }
+
+  it('turns the Moon’s drawn orbit with its node, so the Moon stays on its own line', () => {
+    // Half the node's 18.6-year turn on, the ellipse drawn at the epoch has the Moon 10 degrees off
+    // its plane at the worst.
+    const moon = renderer.members.find((member) => member.id === 'moon')!.marker;
+    const line = moon.parent!.children.find((child) => child.name === 'orbit-line')!;
+    for (const days of [0, 1700, 3397, 3400]) {
+      renderer.update(DEFAULT_EPOCH_JD + days);
+      const normal = new THREE.Vector3(0, 0, 1).applyQuaternion(line.quaternion);
+      expect(Math.abs(moon.position.clone().normalize().dot(normal))).toBeLessThan(1e-9);
+    }
+  });
+});
@@ -1,13 +1,12 @@
 import * as THREE from 'three/webgpu';

 import { appearanceForBody, appearanceForExoplanet } from '../../shared/astro/body-appearance';
-import { gmForParent } from '../../shared/astro/constants';
 import { PlanetAppearance } from '../../shared/astro/planet-appearance';
 import { planetTexture } from '../../shared/rendering/procedural-planet-texture';
 import { bodyTexturePath, loadCachedTexture } from '../../shared/rendering/texture-catalog';
-import { isPropagatableOrbit, orbitEllipsePoints, propagateOrbit, resolveGravitationalParameter, resolveOrbitalElements } from '../../shared/astro/kepler';
-import { CartesianCoordinates, OBLIQUITY_J2000_DEG } from '../../shared/astro/coordinates';
-import { BodyRecord, OrbitalElements } from '../../shared/models/body.model';
+import { isPropagatableOrbit, keplerRates, meanElementsAt, orbitEllipsePoints, positionAtEpoch, resolveGravitationalParameter, resolveOrbitalElements } from '../../shared/astro/kepler';
+import { CartesianCoordinates, laplacePlaneToEquatorial, OBLIQUITY_J2000_DEG } from '../../shared/astro/coordinates';
+import { BodyRecord, MeanElementRates, OrbitalElements } from '../../shared/models/body.model';
 import { bodyMarkerRadiusAu, systemGridRingsAu } from './system-framing';
 import { PolarGridPlane, TetherField } from './grid-plane';
 import { ExoplanetRecord } from '../../shared/models/exoplanet.model';
@@ -45,11 +44,45 @@ const SYSTEM_TETHER_OPACITY = 0.3;

 /**
 * Rotation carrying the **ecliptic** frame into the scene's equatorial one — a turn of the
- * obliquity about the shared vernal-equinox axis. Solar-system elements come from Horizons
- * against the ecliptic, so this is their frame.
+ * obliquity about the shared vernal-equinox axis. The planets' and the Moon's mean elements are
+ * given against the J2000 ecliptic, so this is their frame.
 */
 const ECLIPTIC_FRAME = new THREE.Quaternion().setFromAxisAngle(new THREE.Vector3(1, 0, 0), OBLIQUITY_J2000_DEG * DEG_TO_RAD);

+/**
+ * Rotation carrying a moon's element frame into the scene: its local Laplace plane where JPL
+ * gives one, the ecliptic otherwise. Built from the three axes {@link laplacePlaneToEquatorial}
+ * sends, so the scene and the ETL's check against Horizons share the one conversion.
+ */
+function moonFrame(body: BodyRecord): THREE.Quaternion {
+  const pole = body.laplacePole;
+  if (!pole) {
+    return ECLIPTIC_FRAME.clone();
+  }
+  const axis = (x: number, y: number, z: number): THREE.Vector3 => {
+    const turned = laplacePlaneToEquatorial({ x, y, z }, pole);
+    return new THREE.Vector3(turned.x, turned.y, turned.z);
+  };
+  return new THREE.Quaternion().setFromRotationMatrix(new THREE.Matrix4().makeBasis(axis(1, 0, 0), axis(0, 1, 0), axis(0, 0, 1)));
+}
+
+const X_AXIS = new THREE.Vector3(1, 0, 0);
+const Z_AXIS = new THREE.Vector3(0, 0, 1);
+const scratchTurn = new THREE.Quaternion();
+
+/**
+ * Sets `target` to the rotation carrying an orbit's own plane, periapsis along +X, into the
+ * scene: the argument of periapsis, then the inclination, then the node, as
+ * `positionAtTrueAnomaly` turns a point, and then the frame the elements are measured in.
+ */
+function orientOrbit(target: THREE.Quaternion, elements: OrbitalElements, frame: THREE.Quaternion): THREE.Quaternion {
+  return target
+    .copy(frame)
+    .multiply(scratchTurn.setFromAxisAngle(Z_AXIS, elements.longitudeOfAscendingNodeDeg * DEG_TO_RAD))
+    .multiply(scratchTurn.setFromAxisAngle(X_AXIS, elements.inclinationDeg * DEG_TO_RAD))
+    .multiply(scratchTurn.setFromAxisAngle(Z_AXIS, elements.argumentOfPeriapsisDeg * DEG_TO_RAD));
+}
+
 /**
 * Rotation carrying the frame an **exoplanet's** elements are measured in into the scene.
 *
@@ -94,17 +127,23 @@ function colorForKind(kind: SystemMemberKind): THREE.Color {
 /** Marks orbit lines so the whole layer can be toggled without touching the bodies. */
 const ORBIT_LINE_NAME = 'orbit-line';

+/**
+ * The orbit's ellipse, drawn in its own plane and turned into place by the line's quaternion (see
+ * {@link orientOrbit}), which `update` sets again each tick: a node and a periapsis that turn cost
+ * a quaternion rather than a new geometry. The Moon's node goes right round in 18.6 years, so an
+ * ellipse fixed at one date has the Moon up to 2 sin 5.16° of its distance, 69 000 km, off its own
+ * line nine years on.
+ *
+ * The shape is the epoch's. The planets' axes and eccentricities do drift, but Pluto's axis, the
+ * fastest, moves 0.0045 AU a century against 39.5, which no drawn line shows.
+ */
 function buildOrbitLine(elements: OrbitalElements, kind: SystemMemberKind, frame: THREE.Quaternion): THREE.Line {
-  const points = orbitEllipsePoints(elements);
+  const points = orbitEllipsePoints({ ...elements, inclinationDeg: 0, longitudeOfAscendingNodeDeg: 0, argumentOfPeriapsisDeg: 0 });
  const positions = new Float32Array(points.length * 3);
-  const scratch = new THREE.Vector3();
  points.forEach((point, index) => {
-    // Elements are measured against their source's own reference plane; `frame` rotates that
-    // plane into the scene's equatorial one.
-    const { x, y, z } = scratch.set(point.x, point.y, point.z).applyQuaternion(frame);
-    positions[index * 3] = x;
-    positions[index * 3 + 1] = y;
-    positions[index * 3 + 2] = z;
+    positions[index * 3] = point.x;
+    positions[index * 3 + 1] = point.y;
+    positions[index * 3 + 2] = point.z;
  });

  const geometry = new THREE.BufferGeometry();
@@ -118,6 +157,7 @@ function buildOrbitLine(elements: OrbitalElements, kind: SystemMemberKind, frame

  const line = new THREE.Line(geometry, material);
  line.name = ORBIT_LINE_NAME;
+  orientOrbit(line.quaternion, elements, frame);
  return line;
 }

@@ -201,8 +241,9 @@ const HOURS_PER_DAY = 24;
 * The obliquity fixes how far the pole leans from the orbit normal, and nothing more: which way
 * it leans needs the pole's right ascension, which the Horizons pages this reads do not carry. The
 * lean is taken about the orbit's ascending node because that is the one line the elements name,
- * not because the data says so — so the tilt is real and its azimuth is not. Likewise the phase:
- * each body starts at its elements' epoch (2025-01-01 here) in an arbitrary orientation, the
+ * not because the data says so — so the tilt is real and its azimuth is not. It is the node on the
+ * date drawn, so the lean follows the orbit as the node turns. Likewise the phase: each body
+ * starts at its elements' epoch (J2000 for the planets) in an arbitrary orientation, the
 * shortest rotation of +Y onto its axis, and turns from there. The rate and the sense are real;
 * the face towards the camera is not.
 *
@@ -212,7 +253,7 @@ const HOURS_PER_DAY = 24;
 * obliquity is given it carries the sense, and the period is taken as a magnitude; the sign of the
 * period is only read for a body with no obliquity at all.
 */
-function spinFor(elements: OrbitalElements, frame: THREE.Quaternion, rotationPeriodHours: number, obliquityDeg: number | undefined, epochJd: number): THREE.Quaternion {
+function spinFor(elements: OrbitalElements, frame: THREE.Quaternion, rotationPeriodHours: number, obliquityDeg: number | undefined, daysSinceEpoch: number): THREE.Quaternion {
  const node = elements.longitudeOfAscendingNodeDeg * DEG_TO_RAD;
  const inclination = elements.inclinationDeg * DEG_TO_RAD;
  const nodeDirection = new THREE.Vector3(Math.cos(node), Math.sin(node), 0);
@@ -220,7 +261,7 @@ function spinFor(elements: OrbitalElements, frame: THREE.Quaternion, rotationPer
    .applyAxisAngle(nodeDirection, (obliquityDeg ?? 0) * DEG_TO_RAD)
    .applyQuaternion(frame);
  const period = obliquityDeg === undefined ? rotationPeriodHours : Math.abs(rotationPeriodHours);
-  const turns = ((epochJd - elements.epochJd) * HOURS_PER_DAY) / period;
+  const turns = (daysSinceEpoch * HOURS_PER_DAY) / period;
  return new THREE.Quaternion()
    .setFromUnitVectors(SPIN_AXIS, axis)
    .multiply(new THREE.Quaternion().setFromAxisAngle(SPIN_AXIS, turns * 2 * Math.PI));
@@ -230,8 +271,9 @@ interface TrackedTopLevelBody {
  id: string;
  kind: SystemMemberKind;
  elements: OrbitalElements;
-  gmAu3PerDay2: number;
+  rates: MeanElementRates;
  marker: THREE.Mesh;
+  orbitLine: THREE.Line;
  /** Rotation from this body's own element frame into the scene's equatorial one. */
  frame: THREE.Quaternion;
  /** AU position last computed for this body; moons read their parent's here. */
@@ -244,8 +286,9 @@ interface TrackedTopLevelBody {
 interface TrackedMoon {
  id: string;
  elements: OrbitalElements;
-  gmAu3PerDay2: number;
+  rates: MeanElementRates;
  marker: THREE.Mesh;
+  orbitLine: THREE.Line;
  frame: THREE.Quaternion;
  pivot: THREE.Group;
  parentId: string;
@@ -322,7 +365,7 @@ export class SystemOrbitsRenderer {
      }
      // A body reaches here only when it has no parentBodyId, so `kind` is 'planet' or 'dwarf'.
      const kind: SystemMemberKind = body.kind;
-      const tracked = this.addTopLevelBody(body.id, kind, body.orbit, gmForParent(undefined), body.radiusKm, ECLIPTIC_FRAME, appearanceForBody(body, bodies, hostLuminositySolar), { periodHours: body.rotationPeriodHours, obliquityDeg: body.obliquityDeg });
+      const tracked = this.addTopLevelBody(body.id, kind, body.orbit, body.rates, body.radiusKm, ECLIPTIC_FRAME, appearanceForBody(body, bodies, hostLuminositySolar), { periodHours: body.rotationPeriodHours, obliquityDeg: body.obliquityDeg });
      members.push({ id: body.id, kind, marker: tracked.marker });
    }

@@ -335,7 +378,7 @@ export class SystemOrbitsRenderer {
      if (!parentTracked) {
        continue; // orphaned moon reference; skip rather than crash.
      }
-      const moon = this.addMoon(body.id, body.orbit, gmForParent(body.parentBodyId), body.radiusKm, parentTracked, ECLIPTIC_FRAME, appearanceForBody(body, bodies, hostLuminositySolar), { periodHours: body.rotationPeriodHours, obliquityDeg: body.obliquityDeg });
+      const moon = this.addMoon(body.id, body.orbit, body.rates, body.radiusKm, parentTracked, moonFrame(body), appearanceForBody(body, bodies, hostLuminositySolar), { periodHours: body.rotationPeriodHours, obliquityDeg: body.obliquityDeg });
      members.push({ id: body.id, kind: 'moon', marker: moon.marker, parentId: parent.id });
    }

@@ -353,21 +396,21 @@ export class SystemOrbitsRenderer {
      const elements = resolveOrbitalElements(exoplanet.orbit);
      const radiusEarth = exoplanet.radiusEarth ?? radiusFromMassEarth(exoplanet.massEarth);
      const radiusKm = radiusEarth ? radiusEarth * EARTH_RADIUS_KM : undefined;
-      // Not `gmForParent(undefined)`: that assumes a solar-mass host for every system, and
-      // most exoplanet hosts are red dwarfs a fraction of the Sun's mass.
+      // Not the Sun's: that assumes a solar-mass host for every system, and most exoplanet hosts
+      // are red dwarfs a fraction of the Sun's mass.
      const gm = resolveGravitationalParameter({
        semiMajorAxisAu: exoplanet.orbit.semiMajorAxisAu,
        periodDays: exoplanet.periodDays,
        hostStarMassSolar: exoplanet.hostStarMassSolar
      });
-      const tracked = this.addTopLevelBody(exoplanet.id, 'exoplanet', elements, gm, radiusKm, exoplanetFrame, appearanceForExoplanet(exoplanet, hostLuminositySolar));
+      const tracked = this.addTopLevelBody(exoplanet.id, 'exoplanet', elements, keplerRates(elements.semiMajorAxisAu, gm), radiusKm, exoplanetFrame, appearanceForExoplanet(exoplanet, hostLuminositySolar));
      members.push({ id: exoplanet.id, kind: 'exoplanet', marker: tracked.marker });
    }

    this.members = members;

    // Which plane the system is read against follows from where its elements came from. Only the
-    // Sun has Horizons bodies and no system has both, so this is a choice between the two rather
+    // Sun has JPL bodies and no system has both, so this is a choice between the two rather
    // than a compromise: the ecliptic if there are solar-system bodies, the sky plane otherwise.
    this.referenceFrame = bodies.some((body) => !body.parentBodyId) ? ECLIPTIC_FRAME.clone() : exoplanetFrame;

@@ -403,11 +446,13 @@ export class SystemOrbitsRenderer {
  /** Recomputes every marker's position for the given Julian date. Call once per tick. */
  update(epochJd: number): void {
    for (const body of this.topLevelBodies) {
-      const orbital = propagateOrbit(body.elements, body.gmAu3PerDay2, epochJd);
+      const current = meanElementsAt(body.elements, body.rates, epochJd);
+      const orbital = positionAtEpoch(current);
      body.position.set(orbital.x, orbital.y, orbital.z).applyQuaternion(body.frame);
      body.marker.position.copy(body.position);
+      orientOrbit(body.orbitLine.quaternion, current, body.frame);
      if (body.rotationPeriodHours) {
-        body.marker.quaternion.copy(spinFor(body.elements, body.frame, body.rotationPeriodHours, body.obliquityDeg, epochJd));
+        body.marker.quaternion.copy(spinFor(current, body.frame, body.rotationPeriodHours, body.obliquityDeg, epochJd - body.elements.epochJd));
      }
    }

@@ -417,10 +462,12 @@ export class SystemOrbitsRenderer {
        continue;
      }
      moon.pivot.position.copy(parent.position);
-      const orbital = propagateOrbit(moon.elements, moon.gmAu3PerDay2, epochJd);
+      const current = meanElementsAt(moon.elements, moon.rates, epochJd);
+      const orbital = positionAtEpoch(current);
      moon.marker.position.set(orbital.x, orbital.y, orbital.z).applyQuaternion(moon.frame);
+      orientOrbit(moon.orbitLine.quaternion, current, moon.frame);
      if (moon.rotationPeriodHours) {
-        moon.marker.quaternion.copy(spinFor(moon.elements, moon.frame, moon.rotationPeriodHours, moon.obliquityDeg, epochJd));
+        moon.marker.quaternion.copy(spinFor(current, moon.frame, moon.rotationPeriodHours, moon.obliquityDeg, epochJd - moon.elements.epochJd));
      }
    }

@@ -473,7 +520,7 @@ export class SystemOrbitsRenderer {
    id: string,
    kind: SystemMemberKind,
    elements: OrbitalElements,
-    gmAu3PerDay2: number,
+    rates: MeanElementRates,
    radiusKm: number | undefined,
    frame: THREE.Quaternion,
    appearance?: PlanetAppearance,
@@ -485,7 +532,7 @@ export class SystemOrbitsRenderer {
    this.trackDisposable(orbitLine.geometry, orbitLine.material as THREE.Material);
    this.trackDisposable(marker.geometry, marker.material as THREE.Material);

-    const tracked: TrackedTopLevelBody = { id, kind, elements, gmAu3PerDay2, marker, frame, position: new THREE.Vector3(), rotationPeriodHours: rotation?.periodHours, obliquityDeg: rotation?.obliquityDeg };
+    const tracked: TrackedTopLevelBody = { id, kind, elements, rates, marker, orbitLine, frame, position: new THREE.Vector3(), rotationPeriodHours: rotation?.periodHours, obliquityDeg: rotation?.obliquityDeg };
    this.topLevelBodies.push(tracked);
    return tracked;
  }
@@ -493,7 +540,7 @@ export class SystemOrbitsRenderer {
  private addMoon(
    id: string,
    elements: OrbitalElements,
-    gmAu3PerDay2: number,
+    rates: MeanElementRates,
    radiusKm: number | undefined,
    parent: TrackedTopLevelBody,
    frame: THREE.Quaternion,
@@ -508,7 +555,7 @@ export class SystemOrbitsRenderer {
    this.trackDisposable(orbitLine.geometry, orbitLine.material as THREE.Material);
    this.trackDisposable(marker.geometry, marker.material as THREE.Material);

-    const moon: TrackedMoon = { id, elements, gmAu3PerDay2, marker, frame, pivot, parentId: parent.id, rotationPeriodHours: rotation?.periodHours, obliquityDeg: rotation?.obliquityDeg };
+    const moon: TrackedMoon = { id, elements, rates, marker, orbitLine, frame, pivot, parentId: parent.id, rotationPeriodHours: rotation?.periodHours, obliquityDeg: rotation?.obliquityDeg };
    this.moons.push(moon);
    return moon;
  }
@@ -34,7 +34,9 @@ const IO: BodyRecord = {
    argumentOfPeriapsisDeg: 0,
    meanAnomalyAtEpochDeg: 0,
    epochJd: 2451545.0
-  }
+  },
+  rates: { meanMotionDegPerDay: 203.4889583, longitudeOfAscendingNodeDegPerDay: 0, argumentOfPeriapsisDegPerDay: 0 },
+  orbitSource: 'test'
 };

 const PROXIMA_B: ExoplanetRecord = {