This document is the single source of truth for all constants used in the Holistic Universe Model simulation. Other documents should reference this document rather than duplicating values.
Last synchronized with
tools/lib/constants.json 2026-03-29.
Constants in src/script.js are organized into 11 numbered sections with sub-headers:
- Foundational Model Constants — 2. Model Start & Physical Constants — 3. Sun & Moon Input Constants — 4. Planet Input Constants (4a. major planets, 4b. minor bodies, 4c. ascending nodes) — 5. Inclination System (5a–5d) — 6. Predictive Formula System — 7. Body Diameters — 8. Astronomical Reference Values / ASTRO_REFERENCE (8a–8g) — 9. Derived Constants (9a–9e) — 10. Mass Calculations (10a–10b) — 11. Orbital Formulas (11a–11c)
The shared tools module tools/lib/constants.js mirrors these with its own 14-section structure.
- Rule A — Formulas stay, computed numbers go. Write "H/13" not a specific year count.
- Rule B — Theory-intrinsic integers stay inline. Fibonacci numbers, cycle counts (13, 3, 16), and ratios are part of the theory.
- Rule C — Approximate values for readability. Use "H/13 (~25,794 years; see Constants Reference)" when a number aids understanding.
- Rule D — Tables reference this document. If a doc repeats planet parameters, add: "For current values see Constants Reference."
Per the Expanding Solar System Resonance Theory (Doc 99), the model has two distinct categories of "constants":
- Scale-invariant integers — Fibonacci divisors (3, 5, 8, 13, 16, 21, 34), L1 integer labels (n = 9, 12, ..., 65, 66, 68, ..., 185 — 32 components total), integer divisors of 8H. These are the same at every epoch — structural constants of the solar system.
- Epoch-dependent literal periods, lengths, and seconds-values — these scale with the current value of H(t). Most numeric values tabulated below (H = 335,317 yr; sidereal year = 365.25636 days; LOD = 86,400 s; Moon distance = 384,399 km; planet orbital periods in years; etc.) are J2000-epoch values — the model's primary calibration anchor. They apply to the modern era; for deep-time / future-projection work, use the epoch-dependent helpers in
src/script.js:
| J2000-anchored constant in this doc | Epoch-dependent helper (accepts t_Ma argument) |
|---|---|
holisticyearLength (H = 335,317 yr) |
meanHAtAge(t_Ma) |
meanLengthOfDay (~86,400.0 s) |
meanLodSecondsAtAge(t_Ma) |
meanSiderealYearSeconds (siderealYearJ2000 × 86400) |
meanSiderealYearSecondsAtAge(t_Ma) |
Tropical year (from meanSolarYearDays) |
meanTropicalYearSecondsAtAge(t_Ma) |
Moon distance (moonDistance = 384,399.07 km) |
meanMoonDistanceMetresAtAge(t_Ma) |
Planet orbital periods (Part 4 solarYearInput) |
meanPlanetOrbitalPeriodAtAge(t_Ma, T_p_J2000_s) |
| α(t) GIA correction → ΔT | meanDeltaTSecondsAtAge(t_Ma) |
H(t) evolves under two physically independent drivers: Driver 1 = Earth-Moon tidal evolution (LOD grows, Moon recedes); Driver 2 = solar mass loss (every planet's orbit slowly expands via Kepler's 3rd law). At Devonian (380 Ma) H ≈ 309,083 yr; at J2000 H = 335,317 yr; at +200 Myr H ≈ 350,665 yr. The integer divisors above are unchanged at every epoch — only the per-cycle period in years (or seconds) scales.
For the canonical 9-step derivation chain from t_Ma through LOD, H, AU, M_Sun, Kepler year, Moon distance, Moon period, anomalistic year, stellar/sidereal days, and planet orbital + synodic periods, see Doc 99 — ESSRT and the canonical reference at docs/hidden/old-documents/IP-deep-time-extension.md. For the Solar System Resonance Cycle (8H = 2,682,536 yr at J2000) period table covering all major planetary cycles as integer divisors of 8H, see Doc 55.
The six true degrees of freedom that define the model. Everything else is derived or taken from observations.
| # | Parameter | Variable | Value | DOF | Section |
|---|---|---|---|---|---|
| 1 | Earth Fundamental Cycle | holisticyearLength |
335,317 years | 1 | Part 1 — Earth Fundamental Cycle |
| 2 | Balanced year | balancedYear |
−302,635 (derived) | 0 | Part 2 — Time Constants |
| 3 | Fibonacci divisors | — | 3, 5, 8, 13, 21, 34 | 3 | Part 1 — Fibonacci Divisors |
| 4 | Mean obliquity | earthtiltMean |
23.41354° | 1 | Part 1 — Earth Parameters |
| 5 | Inclination amplitude | earthInvPlaneInclinationAmplitude |
0.63603° | 1 | Part 1 — Earth Parameters |
| 6 | Planet config | the default configuration | Unique mirror-symmetric solution | 0 | Part 1 — Fibonacci Divisors |
Total: 6 DOF (items 2 and 6 are derived/constrained, not independently free).
Reference values from astronomical observations (IAU, JPL, Meeus) used to anchor the model. All are in Part 3 — External Reference Values.
| Parameter | Variable | Value |
|---|---|---|
| Perihelion-solstice alignment | perihelionalignmentYear |
1246.03125 AD |
| Long. perihelion (J2000) | perihelionLongitudeJ2000_deg |
102.947° |
| Obliquity (J2000) | obliquityJ2000_arcsec |
84381.406" (23.439279°) |
| Obliquity rate (J2000) | obliquityRate_arcsecPerCentury |
−46.836769"/cy |
| Obliquity range | — | ~22.1° to ~24.5° |
| Earth incl. (J2000) | earthInclinationJ2000_deg |
1.57869° |
| Eccentricity (J2000) | eccentricityJ2000 |
0.01671022 |
| Sidereal year (J2000) | siderealYearJ2000 |
365.25636301 days |
| Tropical year mean (J2000) | tropicalYearMeanJ2000 |
365.2421897 days |
| Tropical year VE (J2000) | tropicalYearVEJ2000 |
365.242374 days |
| Tropical year SS (J2000) | tropicalYearSSJ2000 |
365.241626 days |
| Tropical year AE (J2000) | tropicalYearAEJ2000 |
365.242018 days |
| Tropical year WS (J2000) | tropicalYearWSJ2000 |
365.242740 days |
| Anomalistic year (J2000) | anomalisticYearJ2000 |
365.259636 days |
| Tropical year rate | tropicalYearRateSecPerCentury |
−0.53 s/cy |
| Axial precession (J2000) | iauPrecessionJ2000 |
25,771.58 years |
| June Solstice 2000 JD | juneSolstice2000_JD |
2451716.575 |
| Solar day (J2000) | solarDayJ2000 |
86400.0 s |
| Sidereal day (J2000) | siderealDayJ2000 |
86164.090531 s |
| Stellar day (J2000) | stellarDayJ2000 |
86164.098904 s |
| Perihelion passage JD | perihelionPassageJ2000_JD |
2451547.042 |
| Moon mean anomaly (J2000) | moonMeanAnomalyJ2000_deg |
134.9634° |
| Moon mean anomaly rate | moonMeanAnomalyRate_degPerDay |
13.06499295°/day |
| Moon elongation (J2000) | moonMeanElongationJ2000_deg |
297.8502° |
| Moon elongation rate | moonMeanElongationRate_degPerDay |
12.19074912°/day |
| Sun mean anomaly (J2000) | sunMeanAnomalyJ2000_deg |
357.5291° |
| Sun mean anomaly rate | sunMeanAnomalyRate_degPerDay |
0.98560028°/day |
| Moon arg. latitude (J2000) | moonArgLatJ2000_deg |
93.2720993° |
These constants define the model. Changing any of them changes the theory.
| Constant | Variable | Value |
|---|---|---|
| Earth Fundamental Cycle | holisticyearLength (H) |
335,317 years |
| Perihelion alignment year | perihelionalignmentYear |
1246.03125 AD |
| Obliquity cycle position | temperatureGraphMostLikely |
14.5 (of 16) |
The Earth Fundamental Cycle is divided by Fibonacci-related integers to produce all Earth precession cycles (see Part 2 — Derived Constants).
| Constant | Variable | Value | Description |
|---|---|---|---|
| Base Eccentricity | eccentricityBase |
0.015386 | Base orbital eccentricity (derived from perihelion longitude) |
| Eccentricity Amplitude | eccentricityAmplitude |
0.001356 | Oscillation amplitude (derived from base + e(J2000) constraint) |
| Mean Obliquity | earthtiltMean |
23.41354 deg | Mean axial tilt (derived from obliquity at J2000) |
| RA Angle | earthRAAngle |
~1.254 | Derived: 2A − A²/ε where A = inclination amplitude, ε = earthtiltMean |
| Mean Inclination (inv. plane) | earthInvPlaneInclinationMean |
1.481134 deg | Mean orbital inclination to invariable plane (derived) |
| Inclination Amplitude | earthInvPlaneInclinationAmplitude |
0.63603 deg | Oscillation amplitude (derived from obliquity rate) |
| Inclination Cycle Anchor | earthInclinationCycleAnchor |
21.77 deg | ICRF perihelion longitude where Earth reaches MAX inclination (anchor for the oscillation) |
| Perihelion Ref JD | perihelionRefJD |
2451547.042 | JD of Earth perihelion 2000 (Jan 3.542) |
| Planet | Fibonacci Divisor (d) | Phase Group | Mirror Pair | EoC Type |
|---|---|---|---|---|
| Mercury | 21 | In-phase (234.52°) | Uranus | I |
| Venus | 34 | In-phase (218.64°) | Neptune | I |
| Earth | 3 | In-phase (21.77°) | Saturn | — |
| Mars | 5 | In-phase (236.07°) | Jupiter | II |
| Jupiter | 5 | In-phase (287.06°) | Mars | III |
| Saturn | 3 | Anti-phase (116.26°) | Earth | III |
| Uranus | 21 | In-phase (21.33°) | Mercury | III |
| Neptune | 34 | In-phase (174.04°) | Venus | III |
Phase angles re-anchored 2026-04-09 to balanced year n=7 (≈ -2,649,854 BC). See 10-fibonacci-laws.md § Phase Groups and 32-inclination-calculations.md § Per-Planet Phase Angles.
| Constant | Variable | Value | Description |
|---|---|---|---|
| Start Model JD | startmodelJD |
2451716.5 | June Solstice 2000 00:00 UTC |
| Start Model Year | startmodelYear |
2000.5 | Decimal year of model start |
| Start Angle | startAngleModel |
89.91949879 deg | Sun ecliptic longitude at model start |
| Correction Days | correctionDays |
-0.8288 | Correction for solstice alignment offset |
| Variable Speed | useVariableSpeed |
true | Enables equation of center (Kepler's 2nd law) |
| Constant | Variable | Value |
|---|---|---|
| Speed of Light | speedOfLight |
299,792.458 km/s |
| Astronomical Unit | currentAUDistance |
149,597,870.698828 km |
| Mean Sidereal Year | meanSiderealYearSeconds |
siderealYearJ2000 × 86400 (derived) |
| Gravitational Constant | G_CONSTANT |
6.6743 × 10⁻²⁰ km³/(kg·s²) |
| Earth/Moon Mass Ratio | MASS_RATIO_EARTH_MOON |
81.30056816 |
| Planet | massRatioDE440 (M_Sun / M_Planet) |
|---|---|
| Mercury | 6,023,657.94 |
| Venus | 408,523.72 |
| Mars | 3,098,703.59 |
| Jupiter | 1,047.348625 |
| Saturn | 3,497.9018 |
| Uranus | 22,902.944 |
| Neptune | 19,412.237 |
These are computed from foundational constants. The formula is the definition; the number is a convenience.
| Cycle | Formula | Period (years) | Direction |
|---|---|---|---|
| Inclination Precession | H / 3 | ~111,772 | Counter-clockwise |
| Ecliptic Precession | H / 5 | ~67,063 | Counter-clockwise |
| Obliquity Cycle | H / 8 | ~41,915 | Clockwise (negative) |
| Axial Precession | H / 13 | ~25,794 | Clockwise (negative) |
| Perihelion Precession | H / 16 | ~20,957 | Both directions |
| Constant | Variable | Formula | Value |
|---|---|---|---|
| Mean Solar Year | meanSolarYearDays |
round(inputMeanSolarYear × H/8) / (H/8) | 365.242203646 days |
| Mean Sidereal Year | meanSiderealYearDays |
meanSolarYearDays × (H/13) / ((H/13) - 1) | 365.256364374 days |
| Mean Anomalistic Year | meanAnomalisticYearDays |
meanSolarYearDays / (H/16 - 1) + meanSolarYearDays | 365.259632390 days |
| Mean Length of Day | meanLengthOfDay |
meanSiderealYearSeconds / meanSiderealYearDays | ~86,400.0 s |
| Mean Sidereal Day | meanSiderealDay |
(meanSolarYearDays/(meanSolarYearDays+1)) × meanLengthOfDay | 86,164.0902 s |
| Mean Stellar Day | meanStellarDay |
(meanSiderealDay/(H/13)) / (meanSolarYearDays+1) + meanSiderealDay | 86,164.0993 s |
| Balanced Year | balancedYear |
perihelionalignmentYear - (14.5 × H/16) | -302,635 |
| Perihelion Alignment JD | perihelionalignmentJD |
startmodelJD - meanSolarYearDays × (startModelYearWithCorrection - perihelionalignmentYear) | ~2,176,153 |
| Perihelion Cycle Length | perihelionCycleLength |
H / 16 | ~20,957 years |
| Total Days in H | totalDaysInH |
H × meanSolarYearDays | ~122,471,920 days |
| J2000.0 epoch JD | j2000JD |
startmodelJD - (startmodelYear - 2000) × meanSolarYearDays | ~2451545.0 |
| Julian century | julianCenturyDays |
36525 (100 × 365.25) | 36,525 days (IAU Julian century) |
| Earth rotations/year | meanEarthRotationsPerYear |
meanSolarYearDays + 1 | 366.2422 |
| Start year corrected | startModelYearWithCorrection |
startmodelYear + correctionDays / meanSolarYearDays | ~2000.4977 |
| Years balanced→J2000 | yearsFromBalancedToJ2000 |
(startmodelJD - balancedJD) / meanSolarYearDays | ~302,635 |
Input constants used in the formulas above:
| Constant | Variable | Value | Description |
|---|---|---|---|
| Input Solar Year | inputMeanSolarYear |
365.2422 days | Input mean solar year |
| Sidereal Year (seconds) | meanSiderealYearSeconds |
siderealYearJ2000 × 86400 | Derived from IAU sidereal year reference |
| Constant | Variable | Formula | Value |
|---|---|---|---|
| Derived Mean Eccentricity | eccentricityDerivedMean |
sqrt(base² + amplitude²) | ~0.01545 |
| EoC Eccentricity | eocEccentricity |
derivedMean - base/2 | ~0.00776 |
| Perihelion Phase Offset | perihelionPhaseOffset |
(see constants.js derivation) | ~0.481 deg |
When orbital plane tilt is moved from RealPerihelionAtSun.containerObj (above annual rotation) to planet.containerObj (below), the ascending node direction changes reference frame. These corrections compensate for that shift. Derived, not tuned.
| Planet | ascNodeTiltCorrection |
Formula |
|---|---|---|
| Mercury | ~131.67 | 180 - ascendingNode |
| Venus | ~103.32 | 180 - ascendingNode |
| Mars | ~130.44 | 180 - ascendingNode |
| Jupiter | ~27.70 | 2 × startpos |
| Saturn | ~22.64 | 2 × startpos |
| Uranus | ~89.76 | 2 × startpos |
| Neptune | ~95.92 | 2 × startpos |
Type I/II (inner): 180 - ascendingNode (anti-node direction). Type III (outer): 2 × startpos (compensates orbital phase in tilt frame).
Non-Earth planets: massFraction[p] = 1 / massRatioDE440[p] (the GM chain cancels).
Earth mass is derived from Moon orbital mechanics:
GM_Earth_Moon = 4π²·d³ / T² (d = moonDistance, T = moonSiderealMonth × meanLengthOfDay)
GM_Earth = GM_Earth_Moon × (M_Earth / (M_Earth + M_Moon)) × (meanLengthOfDay / meanSiderealDay)
massFraction.earth = (GM_Earth / G) / M_Sun
| Planet | massFraction |
|---|---|
| Mercury | 1.660 × 10⁻⁷ |
| Venus | 2.448 × 10⁻⁶ |
| Earth | 3.004 × 10⁻⁶ |
| Mars | 3.227 × 10⁻⁷ |
| Jupiter | 9.548 × 10⁻⁴ |
| Saturn | 2.859 × 10⁻⁴ |
| Uranus | 4.366 × 10⁻⁵ |
| Neptune | 5.151 × 10⁻⁵ |
| Constant | Formula | Value |
|---|---|---|
| ψ (psi) | d_E × amp_E × √m_E (from Earth) | 3.307 × 10⁻³ |
| K | e_amp_E × √m_E / (sin(tilt_E) × √d_E) (from Earth) | 3.415 × 10⁻⁶ |
All 8 planets, combining inner planet J2000 values with outer planet pre-dual-balance values:
| Planet | eccJ2000 |
Source |
|---|---|---|
| Mercury | 0.20563593 | J2000 (same as model) |
| Venus | 0.00677672 | J2000 (same as model) |
| Earth | 0.01671022 | J2000 (from ASTRO_REFERENCE) |
| Mars | 0.09339410 | J2000 (same as model) |
| Jupiter | 0.04838624 | J2000 (base derived from phase) |
| Saturn | 0.05386179 | J2000 (base derived from phase) |
| Uranus | 0.04725744 | J2000 (base derived from phase) |
| Neptune | 0.00859048 | J2000 (base derived from phase) |
Amplitudes derived from Fibonacci Laws: amp = ψ / (d × √m). Means from J2000 constraint.
See Fibonacci Laws, verified by Inclination Optimization and Inclination Verification.
| Planet | Mean (deg) | Amplitude (deg) | Range (deg) | Phase Angle | ICRF Period |
|---|---|---|---|---|---|
| Mercury | 6.703207 | 0.386478 | 6.32 - 7.09 | 234.52° | 8H/93 ≈ 28,844 yr |
| Venus | 2.151359 | 0.062165 | 2.09 - 2.21 | 218.64° | 8H/110 ≈ 24,387 yr |
| Earth | 1.481134 | 0.636032 | 0.85 - 2.12 | 21.77° | H/3 ≈ 111,772 yr |
| Mars | 1.833251 | 1.164218 | 0.67 - 3.00 | 236.07° | 8H/68 ≈ 39,449 yr |
| Jupiter | 0.321086 | 0.021404 | 0.30 - 0.34 | 287.06° | 8H/65 ≈ 41,270 yr |
| Saturn | 0.984965 | 0.065193 | 0.92 - 1.05 | 116.26° (anti-phase) | 8H/169 ≈ 15,873 yr |
| Uranus | 1.015182 | 0.023831 | 0.99 - 1.04 | 21.33° | H/16 ≈ 20,957 yr |
| Neptune | 0.743803 | 0.013551 | 0.73 - 0.76 | 174.04° | 2H/25 ≈ 26,825 yr |
| Pluto | 15.716200 | 0.717024 | 15.00 - 16.43 | 203.32° | H/14 ≈ 23,951 yr |
Formula: i(t) = mean + amplitude × cos(ω̃_ICRF(t) - cycleAnchor) (Saturn: sign flipped, anti-phase)
| Planet | Orbit Count in H | Distance (AU) | Period (years) |
|---|---|---|---|
| Mercury | 1,392,228 | 0.3871 | 0.2408 |
| Venus | 545,059 | 0.7233 | 0.6152 |
| Mars | 178,289 | 1.5237 | 1.8808 |
| Jupiter | 28,281 | 5.1996 | 11.8566 |
| Saturn | 11,396 | 9.5309 | 29.4243 |
| Uranus | 4,004 | 19.1414 | 83.7417 |
| Neptune | 2,048 | 29.9284 | 163.7270 |
| Planet | Formula | Period (years) |
|---|---|---|
| Mercury | H / (1 + 3/8) | ~243,867 |
| Venus | -8H / 6 | ~-447,089 (retrograde) |
| Earth | H / 16 | ~20,957 |
| Mars | H × 8/36 | ~74,515 |
| Jupiter | 8H / 39 | ~68,783 |
| Saturn | -8H / 65 | ~-41,270 (retrograde) |
| Uranus | H / 3 | ~111,772 |
| Neptune | H × 2 | ~670,634 |
All Moon cycles are derived from the 3 input months (sidereal, anomalistic, nodal) and H. Integer rounding to totalDaysInH = H × meanSolarYearDays ensures exact cycle closure over the Earth Fundamental Cycle.
| Cycle | Variable | Formula | Value (days) |
|---|---|---|---|
| Sidereal Month | moonSiderealMonth |
totalDaysInH / ceil(totalDaysInH / input) | ~27.3217 |
| Anomalistic Month | moonAnomalisticMonth |
totalDaysInH / ceil(totalDaysInH / input) | ~27.5545 |
| Nodal Month | moonNodalMonth |
totalDaysInH / ceil(totalDaysInH / input) | ~27.2122 |
| Synodic Month | moonSynodicMonth |
(from sidereal count - 1 + 13 - H) | ~29.5306 |
| Tropical Month | moonTropicalMonth |
(from sidereal count - 1 + 13) | ~27.3216 |
| Cycle | Variable | Value (days) | Value (years) |
|---|---|---|---|
| Apsidal Precession (Earth frame) | moonApsidalPrecessionDaysEarth |
~3,232 | ~8.851 |
| Apsidal Precession (ICRF) | moonApsidalPrecessionDaysICRF |
~3,233 | ~8.852 |
| Nodal Precession (Earth frame) | moonNodalPrecessionDaysEarth |
~6,793 | ~18.600 |
| Nodal Precession (ICRF) | moonNodalPrecessionDaysICRF |
~6,798 | ~18.613 |
| Apsidal-Nodal Beat | moonApsidalMeetsNodalDays |
~206 | — |
| Lunar Leveling Cycle | moonLunarLevelingCycleDays |
~61,250 | ~167.7 |
| Draconic Year (ICRF) | moonDraconicYearICRF |
~346.620 | — |
| Draconic Year (Earth frame) | moonDraconicYearEarth |
~346.597 | — |
| Full Moon Cycle (Earth frame) | moonFullMoonCycleEarth |
~411.78 | — |
| Full Moon Cycle (ICRF) | moonFullMoonCycleICRF |
~411.86 | — |
Eclipse cycles (from derived months): Saros = 223 synodic ≈ 6585.32 days, Exeligmos = 3 × Saros, Callippic = 940 synodic ≈ 76 solar years.
Year-length variations are modelled with Fourier harmonics. The means are derived from inputmeanlengthsolaryearindays via round(input × H/8) / (H/8) and the standard sidereal/anomalistic ratios. Only the harmonic coefficients are fitted (from data spanning full H at stepYears=23 steps).
Each array entry: [period_divisor, sin_coeff, cos_coeff] — period = H / divisor.
| Array | Terms | RMS | Dominant term |
|---|---|---|---|
TROPICAL_YEAR_HARMONICS |
12 | 0.002 s | H/8 (obliquity) |
SIDEREAL_YEAR_HARMONICS |
5 | 0.001 s | H/8 + H/3 |
ANOMALISTIC_YEAR_HARMONICS |
8 | 0.002 s | H/24 (beat) |
Note: The tropical year at runtime uses CARDINAL_POINT_HARMONICS (24 terms per type, derivative-based), not TROPICAL_YEAR_HARMONICS. The latter is used by pipeline tools only.
The PERI_HARMONICS array models Earth's perihelion longitude with 25 Fourier terms (RMSE 0.0006°, J2000 exact).
| Constant | Value | Description |
|---|---|---|
PERI_HARMONICS |
25-term array | [period, sin_coeff, cos_coeff] per term |
PERI_OFFSET |
~-0.260 deg | Global offset correction (J2000-anchored) |
| Constant | Variable | Value | Description |
|---|---|---|---|
| Initial Delta-T | deltaTStart |
63.63 s | Starting Delta-T value at model epoch |
| Constant | Variable | Value | Description |
|---|---|---|---|
| Mid-Eccentricity Amplitude | mideccentricitypointAmplitude |
2.4587 deg | Amplitude of mid-eccentricity-point variation |
| Helion Point Amplitude | helionpointAmplitude |
5.05 deg | Amplitude of perihelion-point variation |
Used by computeLongitudePerihelion() to estimate Earth's longitude of perihelion from the balanced year and precession cycle.
These come from external astronomical sources and do not change with the model.
| Constant | Variable | Value | Source |
|---|---|---|---|
| Obliquity | obliquityJ2000_arcsec |
84381.406" (23.439279 deg) | IAU 2006 (Capitaine et al. 2003) |
| Obliquity rate | obliquityRate_arcsecPerCentury |
-46.836769"/cy | IAU 2006 |
| Obliquity range | - | ~22.1 deg to ~24.5 deg | Laskar 1993 |
| Earth inclination | earthInclinationJ2000_deg |
1.57869 deg | Astronomical Almanac |
| Earth inclination rate | earthInclinationRate_arcsecPerCentury |
-18"/cy | Astronomical Almanac |
| Eccentricity | earthEccentricityJ2000 |
0.01671022 | JPL Horizons |
| Long. of perihelion | earthPerihelionLongitudeJ2000 |
102.947 deg | JPL Horizons |
| Perihelion passage J2000 | perihelionRefJD |
2451547.042 | USNO (2000 Jan 3 13:00 UTC) |
| IAU precession period | iauPrecessionJ2000 |
25,771.57634 years | IAU 2006 |
| June Solstice 2000 JD | juneSolstice2000_JD |
2451716.575 | USNO (June 21, 2000 01:48 UTC) |
| Constant | Variable | Value | Source |
|---|---|---|---|
| Tropical year (mean) | tropicalYearMeanJ2000 |
365.2421897 days | Meeus & Savoie 1992 |
| Tropical year (VE) | tropicalYearVEJ2000 |
365.242374 days | Meeus & Savoie 1992 |
| Tropical year (SS) | tropicalYearSSJ2000 |
365.241626 days | Meeus & Savoie 1992 |
| Tropical year (AE) | tropicalYearAEJ2000 |
365.242018 days | Meeus & Savoie 1992 |
| Tropical year (WS) | tropicalYearWSJ2000 |
365.242740 days | Meeus & Savoie 1992 |
| Tropical year rate | tropicalYearRateSecPerCentury |
-0.53 s/cy | Meeus & Savoie 1992 |
| Anomalistic year | anomalisticYearJ2000 |
365.259636 days | JPL Horizons |
| Sidereal year | siderealYearJ2000 |
365.25636301 days | JPL Horizons (adjusted for LOD=86400) |
| Solar day | solarDayJ2000 |
86400.0 s | SI definition |
| Sidereal day | siderealDayJ2000 |
86164.09053083288 s (~23h 56m 4.0905s) | IERS |
| Stellar day | stellarDayJ2000 |
86164.0989036905 s (~23h 56m 4.0989s) | IERS |
These offsets arise from the coin rotation paradox — precessing reference frames cause systematic measurement biases that cancel with multi-angle averaging.
| Constant | Variable | Formula | Value |
|---|---|---|---|
| Perihelion Coin Rotation | perihelionCoinRotationMs |
meanlengthofday / (H/16) / meansolaryearlengthinDays × 1000 | ~11.34 ms/day |
| Yearly accumulation | perihelionCoinRotationYearlySeconds |
offset × meansolaryearlengthinDays | ~4.14 s/year |
| Axial Coin Rotation | axialCoinRotationMs |
meanSiderealday / (H/13) / (meansolaryearlengthinDays + 1) × 1000 | ~9.12 ms/sidereal day |
| Yearly accumulation | axialCoinRotationYearlySeconds |
offset × (meansolaryearlengthinDays + 1) | ~3.34 s/year |
Perihelion Coin Rotation: Theoretical value derived from 1 extra solar day per H/16 cycle.
Axial Coin Rotation: Axial precession (H/13) causes the stellar day to exceed the sidereal day by ~9.1 ms. This accumulates to 1 extra sidereal day over one axial precession cycle. See Year and Day Length Formulas for details.
| Constant | Variable | Value | Description |
|---|---|---|---|
| Sidereal Month | moonSiderealMonthInput |
27.32166156 days | Return to same star |
| Anomalistic Month | moonAnomalisticMonthInput |
27.55454988 days | Perigee to perigee |
| Nodal Month | moonNodalMonthInput |
27.21222082 days | Node to node |
| Mean Distance | moonDistance |
384,399.07 km | Mean Earth-Moon distance |
| Orbital Eccentricity | moonOrbitalEccentricity |
0.054900489 | |
| Ecliptic Inclination | moonEclipticInclinationJ2000 |
5.1453964 deg | |
| Moon Tilt | moonTilt |
6.687 deg |
| Constant | Variable | Value |
|---|---|---|
| Moon Mean Anomaly J2000 | moonMeanAnomalyJ2000_deg |
134.9634 deg |
| Moon Mean Anomaly Rate | moonMeanAnomalyRate_degPerDay |
13.06499295 |
| Moon Mean Elongation J2000 | moonMeanElongationJ2000_deg |
297.8502 deg |
| Moon Mean Elongation Rate | moonMeanElongationRate_degPerDay |
12.19074912 |
| Moon Mean Elongation J2000 (full) | moonMeanElongationJ2000Full_deg |
297.8502042 deg |
| Moon Mean Elongation Rate | moonMeanElongationRate_degPerCentury |
445267.1115168 |
| Moon Arg. of Latitude J2000 | moonArgLatJ2000_deg |
93.2720993 deg |
| Moon Arg. of Latitude Rate | moonArgLatRate_degPerCentury |
483202.0175273 |
| Sun Mean Anomaly J2000 | sunMeanAnomalyJ2000_deg |
357.5291 deg |
| Sun Mean Anomaly Rate | sunMeanAnomalyRate_degPerDay |
0.98560028 |
The per-day rates are used for EoC phase computation; the per-century rates are used for the Meeus Ch. 47 ecliptic latitude correction (see Moon Meeus Corrections).
| Planet | Eccentricity (J2000) | Long. Perihelion (deg) | Source |
|---|---|---|---|
| Mercury | 0.20563593 | 77.4569131 | JPL J2000 |
| Venus | 0.00677672 | 131.5765919 | JPL J2000 |
| Mars | 0.09339410 | 336.0650681 | JPL J2000 |
| Jupiter | 0.04838624 | 14.70659401 | JPL J2000 |
| Saturn | 0.05386179 | 92.12794343 | JPL J2000 |
| Uranus | 0.04725744 | 170.7308251 | JPL J2000 |
| Neptune | 0.00859048 | 45.80124471 | JPL J2000 |
| Planet | Inclination (deg) | Ascending Node (deg) | Source |
|---|---|---|---|
| Mercury | 7.00497902 | 48.33033155 | JPL/SPICE |
| Venus | 3.39467605 | 76.67877109 | JPL/SPICE |
| Mars | 1.84969142 | 49.55737662 | JPL/SPICE |
| Jupiter | 1.30439695 | 100.4877868 | JPL/SPICE |
| Saturn | 2.48599187 | 113.6452856 | JPL/SPICE |
| Uranus | 0.77263783 | 74.00919023 | JPL/SPICE |
| Neptune | 1.77004347 | 131.7853754 | JPL/SPICE |
| Pluto | 17.14001 | 110.30393 | JPL Horizons |
| Moon | 5.1453964 | — | — |
JPL publishes these trends in the J2000-fixed ecliptic frame ("mean ecliptic and equinox of J2000"). The model error column below is the difference between the model's J2000-fixed-frame trend and JPL's catalog value, after the 2026-04-09 audit re-fit ascendingNodeCyclesIn8H and the n=7 phase anchor.
| Planet | Rate (deg/century) | Direction | Model Error |
|---|---|---|---|
| Mercury | -0.00595 | Decreasing | ~0.4"/cy |
| Venus | -0.00079 | Decreasing | ~1.7"/cy |
| Mars | -0.00813 | Decreasing | ~0.4"/cy |
| Jupiter | -0.00184 | Decreasing | ~0.0"/cy |
| Saturn | +0.00194 | Increasing | ~1.7"/cy |
| Uranus | -0.00243 | Decreasing | ~0.1"/cy |
| Neptune | +0.00035 | Increasing | ~0.0"/cy |
| Pluto | -0.00100 | Decreasing | (not fitted) |
Source: JPL Approximate Positions of the Planets
Note: All 7 fitted planets now match JPL trend direction in the J2000-fixed frame. Total trend error is ~4.3″/century across the 7 planets. See 32-inclination-calculations.md § Two Frames for the frame distinction. Model errors verified by Inclination Verification.
| Planet | Mean Anomaly (deg) | True Anomaly (deg) | Source |
|---|---|---|---|
| Mercury | 156.6364301 | 164.1669319 | JPL J2000 |
| Venus | 324.9668371 | 324.5198504 | JPL J2000 |
| Mars | 109.2630844 | 118.9501056 | JPL J2000 |
| Jupiter | 32.47179744 | 35.69428061 | JPL J2000 |
| Saturn | 325.663876 | 321.7910116 | JPL J2000 |
| Uranus | 145.7292678 | 148.5142459 | JPL J2000 |
| Neptune | 262.5003424 | 261.2242728 | JPL J2000 |
| Pluto | 15.55009 | 26.31965048 | JPL J2000 |
| Body | Value (deg) | Source |
|---|---|---|
| Sun | 7.155 | IAU |
| Mercury | 0.03 | IAU |
| Venus | 2.6392 | IAU (retrograde rotation) |
| Mars | 25.19 | IAU |
| Jupiter | 3.13 | IAU |
| Saturn | 26.73 | IAU |
| Uranus | 82.23 | IAU (near-sideways) |
| Neptune | 28.32 | IAU |
| Pluto | 57.47 | IAU |
| Planet | J2000 Value (deg) | Source |
|---|---|---|
| Mercury | 6.3472858 | S&S 2012 |
| Venus | 2.1545441 | S&S 2012 |
| Earth | 1.57866663 | S&S 2012 |
| Mars | 1.6311858 | S&S 2012 |
| Jupiter | 0.3219652 | S&S 2012 |
| Saturn | 0.9254704 | S&S 2012 |
| Uranus | 0.9946692 | S&S 2012 |
| Neptune | 0.7354155 | S&S 2012 |
| Pluto | 15.5639473 | S&S 2012 (adjusted) |
| Planet | Value (deg) | Source |
|---|---|---|
| Earth | 284.51 | S&S 2012 |
| Mercury | 32.22 | S&S 2012 |
| Venus | 52.31 | S&S 2012 |
| Mars | 352.95 | S&S 2012 |
| Jupiter | 306.92 | S&S 2012 |
| Saturn | 122.27 | S&S 2012 |
| Uranus | 308.44 | S&S 2012 |
| Neptune | 189.28 | S&S 2012 |
| Pluto | 107.06 | S&S 2012 |
| Ceres | 80.89 | S&S 2012 |
Observed linear trend rates from JPL SPICE/WebGeoCalc. These fluctuate over time and are not valid for long-term predictions.
| Planet | Rate (arcsec/cy) | Range | Source |
|---|---|---|---|
| Mercury | ~570 | single value | JPL SPICE |
| Venus | ~0 | single value | JPL SPICE |
| Earth | 1163 | single value | JPL SPICE |
| Mars | ~1550–1650 | min–max | JPL SPICE |
| Jupiter | ~800–1800 | min–max | JPL SPICE |
| Saturn | ~-3400 to -2000 | retrograde | JPL SPICE |
| Uranus | ~1100–1300 | min–max | JPL SPICE |
| Neptune | ~-200 to 200 | min–max | JPL SPICE |
Theoretical orbital inclination ranges from secular perturbation theory.
| Planet | Min (deg) | Max (deg) | Source |
|---|---|---|---|
| Mercury | 4.57 | 9.86 | Farside Table 10.4 |
| Venus | 0.00 | 3.38 | Farside Table 10.4 |
| Earth | 0.00 | 2.95 | Farside Table 10.4 |
| Mars | 0.00 | 5.84 | Farside Table 10.4 |
| Jupiter | 0.241 | 0.489 | Farside Table 10.4 |
| Saturn | 0.797 | 1.02 | Farside Table 10.4 |
| Uranus | 0.902 | 1.11 | Farside Table 10.4 |
| Neptune | 0.554 | 0.800 | Farside Table 10.4 |
Source: Farside physics textbook (Brouwer & van Woerkom)
Each planet has a per-planet cycle anchor (ICRF perihelion longitude where the planet reaches MAX inclination, evaluated at one of the eight balanced-year anchors). After the 2026-04-09 audit, the seven fitted planets share anchor n=7 (≈ -2,649,854 BC, the start of the current Solar System Resonance Cycle); Earth is locked to its IAU-derived n=0 reference.
| Planet | Cycle Anchor | Balanced Year | Notes |
|---|---|---|---|
| Mercury | 234.52° | n=7 | |
| Venus | 218.64° | n=7 | |
| Earth | 21.77° | n=0 | locked, IAU reference |
| Mars | 236.07° | n=7 | |
| Jupiter | 287.06° | n=7* | |
| Saturn | 116.26° | n=7* | anti-phase |
| Uranus | 21.33° | n=7* | |
| Neptune | 174.04° | n=7 | |
| Pluto | 203.32° | — | not fitted |
* Jupiter, Saturn, and Uranus have ICRF periods that divide H exactly; their phase at n=7 numerically coincides with their phase at n=0 (and any other anchor).
The eigenmode-cluster claim from earlier model versions (cycle anchors ≈ Laplace-Lagrange γ₁–γ₈) no longer holds in detail under the n=7 anchor — the alignment was specific to the n=0 anchor set. See 10-fibonacci-laws.md § Phase Groups.
The EIGENMODE_PHASES array in script.js provides Laplace-Lagrange reference values:
| Value | Label | Source |
|---|---|---|
| 202.8° | γ₈ | Farside Table 10.1 |
| 20.23° | γ₁ | Farside Table 10.1 |
| 255.6° | γ₃ | Farside Table 10.1 |
| 296.9° | γ₄ | Farside Table 10.1 |
| 127.3° | γ₆ | Farside Table 10.1 |
| 315.6° | γ₇ | Farside Table 10.1 |
| 318.3° | γ₂ | Farside Table 10.1 |
f₅ = 0 (invariable plane, no evolution) is excluded — 7 active Laplace-Lagrange modes remain.
| Parameter | Variable | Value | Source |
|---|---|---|---|
| Orbital Period | solarYearInput |
90,465 days | JPL Horizons |
| Eccentricity | orbitalEccentricityBase |
0.2488273 | JPL Horizons |
| Long. Perihelion | longitudePerihelion |
224.06891 deg | JPL Horizons |
| Sun/Pluto Mass Ratio | MASS_RATIO_SUN_PLUTO |
136,045,556 | DE440 |
| GM | GM_PLUTO_ALONE |
~870 km³/s² | Pluto alone (matches Wikipedia / DE440 BODY999). System value ~975.5 km³/s² also stored as GM_PLUTO_SYSTEM. |
| Parameter | Variable | Value | Source |
|---|---|---|---|
| Ecliptic Inclination | eclipticInclinationJ2000 |
162.26269 deg | JPL (retrograde orbit) |
| Ascending Node (Ecliptic) | ascendingNode |
58.42008 deg | JPL Horizons |
| Orbital Period | solarYearInput |
27,503 days | JPL |
| Eccentricity | orbitalEccentricityBase |
0.96714291 | JPL |
| Long. Perihelion | longitudePerihelion |
111.33249 deg | JPL Horizons |
| Mass | M_HALLEYS |
~2.2 × 10¹⁴ kg | Estimated (~11×8×8 km, ~0.6 g/cm³) |
| Parameter | Variable | Value | Source |
|---|---|---|---|
| Ecliptic Inclination | eclipticInclinationJ2000 |
10.82760 deg | JPL Horizons |
| Ascending Node (Ecliptic) | ascendingNode |
304.30993 deg | JPL Horizons |
| Orbital Period | solarYearInput |
642.93 days | JPL |
| Eccentricity | orbitalEccentricityBase |
0.2229512 | JPL |
| Long. Perihelion | longitudePerihelion |
178.81322 deg | JPL Horizons |
| Mass | M_EROS |
6.687 × 10¹⁵ kg | NEAR Shoemaker (2000–2001) |
| Parameter | Variable | Value | Source |
|---|---|---|---|
| Ecliptic Inclination | eclipticInclinationJ2000 |
10.59407 deg | JPL Horizons |
| Inv. Plane Inclination | invPlaneInclinationJ2000 |
0.4331698 deg | S&S 2012 |
| Ascending Node (Ecliptic) | ascendingNode |
80.30533 deg | JPL Horizons |
| Orbital Period | solarYearInput |
1,680.5 days | JPL |
| Eccentricity | orbitalEccentricityBase |
0.0755347 | JPL |
| Long. Perihelion | longitudePerihelion |
73.59769 deg | JPL Horizons |
| Orbit Distance | orbitDistanceOverride |
2.76596 AU | JPL Horizons |
| GM | GM_CERES |
62.6274 km³/s² | Dawn spacecraft (2015–2018) |
These values result from the optimization campaign (2025-2026) and may change in future campaigns. For the optimization process and history, see Optimization Tool Overview and Optimization Execution Plan.
| Constant | Variable | Value | Description |
|---|---|---|---|
| Correction Sun | correctionSun |
0.49552 deg | Degree correction for Sun position (dynamic EoC); optimizer step 1 |
| Planet | Variable | Value (days) |
|---|---|---|
| Mercury | solarYearInput |
87.9683 |
| Venus | solarYearInput |
224.695 |
| Mars | solarYearInput |
686.93 |
| Jupiter | solarYearInput |
4330.53 |
| Saturn | solarYearInput |
10747 |
| Uranus | solarYearInput |
30586 |
| Neptune | solarYearInput |
59800 |
Values from astro-reference.json. These are the official JPL solar year periods.
Base eccentricities represent the long-term oscillation midpoint. They are derived at runtime from the balanced-year phase (same principle as Earth). The eccentricity balance (Law 5) emerges naturally at ~99.9%.
Note: these values are computed at runtime by constants.js — not stored in JSON.
Eccentricity oscillation amplitudes from the tilt formula: e_amp = K × sin(tilt) × √d / (√m × a^(3/2)). See doc 35 §4-5.
| Constant | Formula | Value | Description |
|---|---|---|---|
| K | e_amp × √m × a^1.5 / (sin(tiltMean) × √d) | 3.4149 × 10⁻⁶ | Universal eccentricity amplitude constant (from Earth) |
All 8 planet amplitudes are derived at runtime from K using model mean obliquity:
e_amp = K × sin(meanObliquity) × √d / (√m × a^1.5). See The Closed Loop.
Phase angles for the eccentricity oscillation are now derived at runtime from the balanced-year phase: phase = (2000 - balancedYear) / wobblePeriod × 360°. Earth's phase is independently determined by the Sun optimizer. The phases are no longer stored in JSON — they are computed by constants.js and script.js (section E2d).
The Equation of Center fraction determines how much of a planet's Keplerian variable-speed behavior is captured by the EoC formula vs the geometric offset. See Equation of Center.
| Planet | eocFraction |
Type | Description |
|---|---|---|---|
| Mercury | -0.527 | I | Negative (inferior planet geometry) |
| Venus | 0.436 | I | Below geometric 0.50 |
| Mars | -0.066 | II | Near-zero (Earth-crossing) |
| Jupiter | 0.484 | III | Near geometric prediction of 0.50 |
| Saturn | 0.543 | III | Above 0.50 |
| Uranus | 0.50 | III | Exactly geometric |
| Neptune | 0.50 | III | Exactly geometric |
| Planet | angleCorrection (deg) |
startpos (deg) |
|---|---|---|
| Mercury | 0.971049 | 83.53 |
| Venus | -2.784782 | 249.312 |
| Mars | -2.107087 | 121.47 |
| Jupiter | 0.92974 | 13.85 |
| Saturn | -0.17477 | 11.32 |
| Uranus | -0.733732 | 44.88 |
| Neptune | 2.33091 | 47.96 |
Calibrated to reproduce exact J2000 ecliptic inclinations (optimized by Ascending Node Optimization):
| Planet | ascendingNodeInvPlane |
Delta from S&S |
|---|---|---|
| Earth | 284.51 | 0.00 deg (S&S 2012) |
| Mercury | 32.83 | +0.61 deg |
| Venus | 54.70 | +2.39 deg |
| Mars | 354.87 | +1.92 deg |
| Jupiter | 312.89 | +5.97 deg |
| Saturn | 118.81 | -3.46 deg |
| Uranus | 307.80 | -0.64 deg |
| Neptune | 192.04 | +2.76 deg |
| Pluto | 101.06 | -6.00 deg |
See 34-j2000-calibration.md for the methodology. Verification: Ascending Node Verification verifies correct J2000 ecliptic inclinations. Ascending Node Souami-Souchay compares S&S vs Verified values.
| Planet | perihelionRef_JD |
Notes |
|---|---|---|
| Mercury | 2460335.9 | Phase-optimized |
| Venus | 2455464.42 | Re-optimized with 42p correction |
| Mars | 2456505.6 | Re-optimized with Dec correction |
| Jupiter | 2464224.5 | Phase-optimized (-6 deg from 2023-Jan-21) |
| Saturn | 2452875.9 | Phase-optimized (+1 deg from 2003-Jul-26) |
| Uranus | 2439699.8 | Phase-optimized (+5 deg from 1966-May-20) |
| Neptune | 2409432.4 | Phase-optimized (+17 deg from 1876-Aug-27) |
| Constant | Variable | Value | Description |
|---|---|---|---|
| Apsidal Start | moonStartposApsidal |
347.622 deg | Apsidal precession start |
| Nodal Start | moonStartposNodal |
-83.630 deg | Nodal precession start |
| Moon Start | moonStartposMoon |
131.930 deg | Orbital position start |
Per-planet empirical parallax correction for geocentric RA and Dec. Coefficients stored in ASTRO_REFERENCE.raCorrection and ASTRO_REFERENCE.decCorrection in tools/lib/constants.js.
| Planet | RA Parameters | Dec Parameters | Total | Selection |
|---|---|---|---|---|
| Mercury | 42 | 42 | 84 | Full 42p tier |
| Venus | 42 | 42 | 84 | Full 42p tier |
| Mars | 30 | 30 | 60 | 30p tier |
| Jupiter | 42 | 42 | 84 | Full 42p tier |
| Saturn | 36 | 36 | 72 | 36p tier |
| Uranus | 24 | 24 | 48 | 24p tier |
| Neptune | 24 | 24 | 48 | 24p tier |
Tier selection by per-planet LOOCV/k-fold cross-validation. See Planet Parallax Corrections.
Per-planet configuration for the predictive perihelion precession formula:
| Planet | Period Formula | Theta0 (deg) | Baseline (arcsec/cy) |
|---|---|---|---|
| Mercury | H × 8/11 | 77.4569131 | 1296000/period×100 |
| Venus | -8H / 6 | 131.5765919 | -1296000/period×100 (retrograde) |
| Mars | H × 8/35 | 336.0650681 | 1296000/period×100 |
| Jupiter | H / 5 | 14.70659401 | 1296000/period×100 |
| Saturn | H / 8 | 92.12794343 | -1296000/period×100 (retrograde) |
| Uranus | H / 3 | 170.7308251 | 1296000/period×100 |
| Neptune | H × 2 | 45.80124471 | 1296000/period×100 |
| Constant | Variable | Value | Description |
|---|---|---|---|
| Obliquity mean | PREDICT_OBLIQ_MEAN |
23.414 deg | Normalization center for obliquity features |
| Eccentricity base | PREDICT_ECC_BASE |
0.015321 | Training normalization (frozen at training-time value) |
| Eccentricity amplitude | PREDICT_ECC_AMP |
0.0014226 | Training normalization (frozen at training-time value) |
| Eccentricity mean | PREDICT_ECC_MEAN |
sqrt(base² + amp²) ≈ 0.01539 | Normalization center |
7 arrays of 429 trained coefficients each, one per planet. These are the regression weights from the Python training pipeline (tools/lib/python/coefficients/*_coeffs_unified.py). The dot product of the 429-term feature vector with the coefficient array gives the geocentric precession fluctuation above/below the heliocentric baseline.
- Souami, D. & Souchay, J. (2012): "The solar system's invariable plane"
- Publication: Astronomy & Astrophysics, 543, A133
- URL: https://www.aanda.org/articles/aa/full_html/2012/07/aa19011-12/aa19011-12.html
- Used for: Invariable plane inclinations and ascending nodes for all planets
-
JPL Horizons System
- URL: https://ssd.jpl.nasa.gov/horizons/
- Used for: J2000 ecliptic orbital elements (inclination, eccentricity, ascending node, mean anomaly)
-
JPL Approximate Positions of the Planets
- URL: https://ssd.jpl.nasa.gov/planets/approx_pos.html
- Used for: Longitude of perihelion reference values
-
JPL SPICE Toolkit
- URL: https://naif.jpl.nasa.gov/naif/toolkit.html
- Used for: High-precision ascending node values (preferred over Horizons where available)
-
IAU (International Astronomical Union)
- Used for: Astronomical Unit definition, epoch definitions, obliquity (IAU 2006 precession model)
- J2000 Epoch: January 1, 2000, 12:00 TT (JD 2451545.0)
-
IERS (International Earth Rotation Service)
- Used for: Length of day, Earth rotation parameters, sidereal/stellar day definitions
-
Meeus & Savoie (1992)
- Publication: "The history of the tropical year", J. British Astronomical Association, 102(1), 40-42
- Used for: Cardinal tropical year lengths, tropical year rate
-
USNO (United States Naval Observatory)
- Used for: Solstice timing reference (June Solstice 2000 JD)
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