Ascending Node Calibration

The model uses two distinct ascending node calculation systems. This page documents the refinement of planetary ascending nodes to the invariable plane — a calibration that achieves sub-arcsecond accuracy for all planets.

Background: Souami & Souchay (2012)

Souami & Souchay (2012) published the definitive modern determination of the solar system’s invariable plane, including ascending node longitudes for all planets. Their work provided:

• The invariable plane orientation at J2000: inclination 1°34’43.3” (~1.578°) to the ecliptic, ascending node at 107°34’56” (~107.58°)
• Planetary inclinations to the invariable plane
• Ascending node positions on the invariable plane for each planet

These values form the foundation of the model’s invariable plane calculations. However, a systematic issue arises when combining S&S values with the model’s dynamic inclination framework.

The Problem: Epoch-Specific vs Mean Inclinations

The S&S ascending node values are calibrated to the J2000 epoch. The Holistic Universe Model uses mean inclinations (time-averaged over each planet’s full oscillation cycle) as the baseline for its dynamic inclination formula:

i(t) = i_mean + A × cos(Ω(t) - φ₀)

Earth’s inclination to the invariable plane illustrates the issue:

When using Earth’s mean inclination (1.482°) in the geometric relationship between orbital planes, the original S&S ascending nodes produce ecliptic inclinations that deviate from observed JPL values. The ascending nodes need recalibration to work correctly within the mean-inclination framework.

The Solution: J2000-Verified Ascending Nodes

Geometric Principle

The angle between any two orbital planes (the ecliptic inclination) is determined by three quantities:

1. Each plane’s inclination to the invariable plane
2. The difference in their ascending node positions

This relationship follows directly from spherical trigonometry:

cos(i_ecl) = cos(i_planet) · cos(i_earth) + sin(i_planet) · sin(i_earth) · cos(ΔΩ)

Where:

• i_ecl = ecliptic inclination (the angle between the planet’s orbit and Earth’s orbit)
• i_planet = planet’s inclination to the invariable plane (from S&S 2012)
• i_earth = Earth’s inclination to the invariable plane (1.57866° at J2000)
• ΔΩ = difference in ascending node longitudes (planet minus Earth)

Since i_ecl is known from JPL observations, and i_planet and i_earth are known from S&S, the ascending node difference ΔΩ is the only unknown. Solving:

cos(ΔΩ) = [cos(i_ecl) - cos(i_planet) · cos(i_earth)] / [sin(i_planet) · sin(i_earth)]

Two Independent Verification Methods

The calibrated ascending nodes were computed using two completely independent methods:

Both methods produce identical results, confirming the geometric validity of the approach. The numerical method minimizes the error between calculated and target ecliptic inclinations using dot products of orbital plane normal vectors. The analytical method solves the spherical triangle directly.

Results: Calibrated Ascending Nodes

Comparison Table

Key Observations

• Adjustments range from -6.00° (Pluto) to +5.97° (Jupiter)
• Mercury and Uranus require the smallest corrections (< 1°)
• Jupiter requires the largest positive adjustment (+5.97°)
• All verified values reproduce JPL ecliptic inclinations to better than 0.0001°

Why This Is Not Circular Reasoning

A potential concern: are we simply fitting free parameters to match observations?

The answer is no, because the three inputs come from independent sources:

The ascending node is not a free parameter — it is geometrically determined by the other two independently measured quantities. Given the planet’s tilt to the invariable plane and its tilt to the ecliptic, only one (or two, with a sign ambiguity) ascending node position is mathematically possible.

The sign ambiguity (two arccos solutions) is resolved by choosing the solution closest to the S&S original value — a physically motivated choice since the S&S values are already close to correct.

Dynamic Inclination: Coupling Node Precession to Tilt

The calibrated ascending nodes are not static — they precess around the invariable plane. This precession is directly coupled to each planet’s inclination oscillation through the formula:

i(t) = i_mean + A × cos(Ω(t) - φ₀)

Where:

• Ω(t) = Ω₀ + (360° / Period) × t — the precessing ascending node
• Ω₀ = J2000-verified ascending node value
• φ₀ = universal phase angle (~203.32° for prograde planets, ~23.32° for Saturn)

Universal Phase Angle

A notable finding: all planets share a single phase angle (γ₈ ≈ 203.32°), derived from the Laplace-Lagrange s₈ eigenmode. This is not fitted per planet — the same value works for all prograde planets. Saturn alone uses ~23.32° (offset by 180°) because its ascending node precesses in the opposite direction.

Complete Parameters

*Saturn’s period is retrograde (negative direction).

Period Ratios

All oscillation periods are simple fractions or multiples of the 333,888-year Holistic-Year (H):

Dual Coordinate Systems

The simulation tracks ascending nodes in two coordinate systems because height calculations require ecliptic-frame coordinates while long-term dynamics use inertial coordinates:

The relationship between the two rates follows from the precession meeting frequency:

1/P_ecliptic = 1/P_ICRF + 1/P_axial
1/20,868 ≈ 1/111,296 + 1/25,684

This is the same relationship that produces the perihelion precession period from the axial and inclination precession periods — another manifestation of the model’s unified framework.

Ecliptic Ascending Node System

Separate from the invariable plane system, the model also tracks how ascending nodes shift on the ecliptic as Earth’s axial tilt (obliquity) changes. This uses a different formula:

dΩ/dε = -sin(Ω) / tan(i)

Where:

• Ω = ascending node longitude on the ecliptic
• ε = obliquity (axial tilt)
• i = orbital inclination to the ecliptic

As the obliquity oscillates over the 41,736-year cycle, the ecliptic itself tilts slightly, shifting where planetary orbits cross it. This is an ecliptic-frame effect that operates independently of the invariable plane node precession.

Verification and Reproducibility

All calibration work is fully reproducible through publicly available scripts:

Summary

Return to Invariable Plane or explore Formulas for the complete formula set.