Sun, Moon & Planets

The Sun, Moon, and planets are all modeled in the 3D simulation with high physical accuracy. All orbits include Kepler’s variable speed (equation of center) and parallax corrections for geocentric positions. The Moon adds full gravitational perturbation corrections (Meeus Ch. 47). All orbital periods fit into the 335,008-year Holistic-Year as whole-number multiples.

Sun: Equation of Center

The Sun’s orbit includes the equation of center — Kepler’s 2nd Law, which makes the Sun move faster near perihelion (January) and slower near aphelion (July). The correction uses the standard two-term series:

θ += 2e·sin(M) + 1.25e²·sin(2M)

where e is the orbital eccentricity and M is the mean anomaly from perihelion.

A key insight in the implementation: the model uses a circular orbit with an offset center to reproduce the Earth–Sun distance variation. This geometric offset already creates apparent speed variation (the Sun subtends a larger angle when closer). The equation of center uses a reduced eccentricity to add only the remaining speed variation not already provided by the geometry, avoiding double-counting.

Moon Movements

The Moon exhibits two primary precession cycles:

Lunar Standstill

The Moon’s nodal precession causes the phenomenon known as Lunar Standstill - when the Moon reaches its extreme northern or southern declination relative to Earth’s equator.

The Lunar Leveling Cycle (~16.88 years)

To correctly model all Moon movements in the 3D simulation, an additional cycle was needed: a leveling cycle with a duration of ~16.88 years. Without this third component, the Moon’s 3D position cannot be accurately reproduced.

The Moon’s nodal precession (~18.6 years, retrograde) and apsidal precession (~8.85 years, prograde) move in opposite directions. Because they move toward each other, they meet every ~6 years — this is the well-known apsidal-nodal synodic period. However, the two precessions have different angular velocities. The net angular displacement from their combined motion only completes a full 360° cycle after ~16.88 years:

ω_apsidal − ω_nodal = 2π/8.85 − 2π/18.6 = 0.3722 rad/yr
2π / 0.3722 = 16.88 years

A ~16.9-year period has been independently detected in Gulf of Alaska air temperatures by Royer (1993, J. Geophys. Res., 98, 4639–4644) and noted as a lunar tidal frequency by Pukite (2014) .

All Moon cycle durations come together in the Holistic-Year of 335,008 years.

Moon Position: Two Systems Working Together

The Moon’s position in the 3D simulation is determined by two systems:

1. 5-layer precession hierarchy (geometric): Five nested rotating containers in Three.js handle the Moon’s sidereal month, apsidal precession, nodal precession, leveling cycle, and orbital inclination (5.145°). This produces the circular orbit ring visible in the scene.

2. Meeus analytical corrections (perturbative): A table-driven implementation of Jean Meeus’s Astronomical Algorithms Ch. 47, using 60 longitude terms and 60 latitude terms. These capture the Sun’s gravitational perturbations on the Moon — evection, variation, annual equation, and dozens of smaller effects — shifting the Moon away from its geometric circle.

The visual result: the orbit ring shows the unperturbed circular path from the hierarchy, while the Moon itself shows the physically correct Meeus-corrected position. The difference between the ring and the Moon makes gravitational perturbation effects directly visible.

Moon Position Accuracy

The Moon’s position has been verified against 58 solar eclipses from the NASA GSFC catalog (2000–2025). At each known eclipse, the geocentric Moon–Sun angular separation was measured:

The 0.81° RMS is not an error — it is the theoretical geocentric limit. Solar eclipses are topocentric events: the Moon’s parallax (~0.95°) means a geocentric observer always sees a small offset. The correlation between this offset and NASA’s gamma parameter (r = 0.9945) confirms the model is correct. The true residual after accounting for parallax is just 0.04°.

Eclipse Cycles

The 3D simulation includes eclipse visualization using Three.js lighting and shadow functions. With the full Meeus Ch. 47 perturbation model, the Moon’s position is accurate to 0.04° (after parallax correction), making solar eclipses visible in the 3D scene at the correct dates.

Recent Eclipse Examples

Planetary Movements

All planets are configured in the 3D simulation with their perihelion precession fully modeled according to Kepler’s Third Law. Each planet includes an equation of center (variable orbital speed) and a parallax correction series for accurate geocentric positions.

Simulation Accuracy

All orbits in the simulation use circular geometry with variable-speed corrections (equation of center), approximating Keplerian elliptical motion without ellipse equations. The Sun includes the equation of center producing correct variable speed throughout the year. The Moon includes full Meeus Ch. 47 perturbations with eclipse-verified accuracy (see above). All planets include per-planet equation of center corrections and empirical parallax corrections fitted against JPL Horizons and historical transit/opposition reference data (~1800–2200 AD).

All 9 targets within 0.22°, seven under 0.06°. The accuracy holds over ~200–400 years for most planets.

Planetary Perihelion Data

All perihelion calculations are grounded in data from NASA and WebGeocalc.

Perihelion Precession: Ecliptic and ICRF

The ecliptic frame is the natural frame for solar system dynamics — secular perturbation theory, the Laplace-Lagrange eigensystem, and angular momentum conservation all operate in this plane. No physical mechanism couples the solar system’s internal precession to the external ICRF; the ICRF rates are a kinematic consequence of the ecliptic dynamics (see Fibonacci Laws — ICRF Perspective).

The conversion to ICRF subtracts H/13 (the general precession). Only Earth’s ecliptic rate exceeds this threshold — making Earth the sole prograde planet in ICRF. All others precess retrograde:

Earth, Jupiter, and Saturn have pure Fibonacci ICRF denominators (3, 8, 21). The frame transformation produces the Fibonacci subtraction identities 16−13=3, 5−13=−8, and −8−13=−21 — generating the complete chain 3 → 5 → 8 → 13 → 21. Earth is the sole prograde planet in ICRF; Saturn is the sole retrograde planet in the ecliptic — mirror exceptions created by the same Fibonacci number (13). See Why Earth Is Special for what this mirror symmetry means for Earth’s inclination dynamics.

Mercury

Mercury’s model is fully aligned with NASA transit data.

Perihelion precession: ~575 arcseconds/century observed

Perihelion and ascending node movement relative to the invariable plane (the ascending node determines the inclination tilt):

For details on Mercury’s “missing” perihelion precession, see Mercury Precession.

Venus

Venus is fully aligned with NASA transit data.

Perihelion precession: ~400 arcseconds/century

Perihelion and ascending node movement relative to the invariable plane (the ascending node determines the inclination tilt):

Mars

Mars is aligned with opposition data.

Perihelion precession: ~1600 arcseconds/century

Perihelion and ascending node movement relative to the invariable plane (the ascending node determines the inclination tilt):

Jupiter

Perihelion precession: ~1800 arcseconds/century (varies over longer periods)

Perihelion and ascending node movement relative to the invariable plane (the ascending node determines the inclination tilt):

Saturn

Perihelion precession: ~-3400 arcseconds/century (retrograde, varies over time)

Perihelion and ascending node movement relative to the invariable plane (the ascending node determines the inclination tilt):

Uranus

Perihelion precession: ~1100 arcseconds/century

Perihelion and ascending node movement relative to the invariable plane (the ascending node determines the inclination tilt):

Neptune

Perihelion precession: ~200 arcseconds/century

Perihelion and ascending node movement relative to the invariable plane (the ascending node determines the inclination tilt):

How Planetary Calculations Work

All calculations in the 3D simulation follow three principles:

1. Grounded in scientific data: Ascending/descending nodes, eccentricity values, etc. from official sources
2. Transparent perihelion locations: Positions are calculated directly, not layered approximations
3. Kepler’s Third Law: Orbital elements follow the period-distance relationship

Example: Jupiter Calculation Structure

barycenterEarthAndSun.pivotObj.add(jupiterPerihelionDurationEcliptic1.containerObj);
jupiterPerihelionDurationEcliptic1.pivotObj.add(jupiterPerihelionFromEarth.containerObj);
jupiterPerihelionFromEarth.pivotObj.add(jupiterPerihelionDurationEcliptic2.containerObj);
jupiterPerihelionDurationEcliptic2.pivotObj.add(jupiterRealPerihelionAtSun.containerObj);
jupiterRealPerihelionAtSun.pivotObj.add(jupiter.containerObj);

The calculation chain:

1. Start at PERIHELION-OF-EARTH (barycenterEarthAndSun)
2. Add planet’s perihelion precession speed
3. Set perihelion location
4. Add counter-movement correction
5. Move to Sun-centered reference
6. Apply orbital elements and nodes

Summary

1. Moon’s nodal precession: ~18.6 years (causes Lunar Standstill)
2. Moon’s apsidal precession: ~8.85 years
3. All Moon cycles align with the 335,008-year Holistic-Year
4. Moon position accuracy: 0.04° residual (at geocentric limit), verified against 58 NASA eclipses
5. Meeus Ch. 47 perturbations: 60 longitude + 60 latitude terms for solar gravitational effects
6. Eclipse visualization is included in the 3D simulation with visible solar eclipses
7. Planetary perihelions form a spiral pattern when viewed from Earth
8. All planet orbits follow Kepler’s Third Law with data from NASA/WebGeocalc
9. The model matches transit, opposition, and conjunction observations

Continue to Mercury Precession for a detailed analysis of the “missing” perihelion precession of Mercury.