Plane Calibration

This page documents the refinement of the planets’ J2000 ascending nodes on the invariable plane — a geometric calibration that reproduces the observed ecliptic inclinations to better than 0.0001°.

Why the invariable plane?

The ecliptic is the obvious reference frame — it is Earth’s own orbital plane, and the classical elements are quoted against it. The difficulty is that it moves. The ecliptic precesses with a period of ~68,751 years, so an inclination measured against it is a reading taken from a turning platform.

Why ecliptic inclinations look disordered

An inclination measured to the ecliptic carries two signals at once: the planet’s own tilt to the invariable plane, and Earth’s. Both oscillate, on different periods, so the sum wanders even when neither orbit is doing anything of interest. Earth’s own tilt to the invariable plane swings between 0.845° and 2.117° — and that entire range is stamped onto every other planet’s ecliptic inclination as a wobble belonging to the observer, not the observed.

Measured against the invariable plane the contamination is gone, and most of the planets turn out to sit nearly still:

Jupiter, Uranus and Neptune each span a few hundredths of a degree across the whole cycle; Venus and Saturn roughly a tenth. Only Mercury and Mars move appreciably, and Mars is the one orbit that genuinely swings. The ecliptic column, by contrast, shows no such ordering — because each of its entries is a sum of two independent oscillations.

This is why the calibration below is done on the invariable plane and the ecliptic values are derived from it, rather than the other way round: the invariable-plane node is a property of the orbit, while the ecliptic node is a property of the orbit and of where Earth happens to be in its own cycle.

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” (~1.57869°) to the ecliptic, ascending node at 107°34’56” (~107.582°).
• 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. A systematic issue arises when combining S&S values with the model’s dynamic inclination framework.

The problem: epoch-specific vs mean inclinations

S&S ascending node values are calibrated to the J2000 epoch. The scene’s no-chain scaffolding uses mean inclinations (time-averaged over each planet’s inclination oscillation) as the baseline of its inclination law. Earth’s inclination illustrates the issue:

Using Earth’s mean inclination 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

Four parameters suffice: any planet’s ecliptic inclination is fully determined by its invariable-plane elements (i_inv, Ω_inv) and Earth’s invariable-plane elements (i_Earth, Ω_Earth). The angle 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 is the ecliptic inclination (the angle between the planet’s orbit and Earth’s), i_planet and i_earth are the planet’s and Earth’s inclinations to the invariable plane, and ΔΩ is the difference in ascending node longitudes (planet minus Earth).

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

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

The calibrated ascending nodes were computed using two independent methods — numerical optimisation (brute-force search: coarse ±10° in 0.01° steps, then fine ±0.1° in 0.0001° steps; script ) and analytical closed-form spherical trigonometry (arccos formula above; script ). The numerical method minimises the error between calculated and target ecliptic inclinations using dot products of orbital plane normal vectors; the analytical method solves the spherical triangle directly. Both produce identical results, confirming the geometric validity of the approach.

Results: calibrated ascending nodes

*Pluto is a dwarf planet (IAU 2006), included for completeness; it is one of the scene’s no-chain bodies.

Adjustments range from −6.00° (Pluto) to +5.97° (Jupiter). Mercury and Uranus need the smallest corrections (<1°); Jupiter the largest positive (+5.97°). The last column is the closure the two cited verification scripts report at J2000 — a figure from that calibration run, not a tolerance this site recomputes, and not a statement about any other epoch. Earth’s ascending node remains unchanged because Earth serves as the reference — the ecliptic is Earth’s own orbital plane.

Why this is not circular reasoning

A potential concern: are we simply fitting free parameters to match observations? No — 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 is resolved by choosing the solution closest to the S&S original — a physically motivated choice since the S&S values are already close.

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:

dΩ/dε = -sin(Ω) / tan(i)

where Ω is the ascending node longitude on the ecliptic, ε the obliquity, and i the orbital inclination to the ecliptic. As obliquity oscillates over the ~41,224-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.

The nodes of date

Earth’s node circulating on the invariable plane, with La2010 overlaid across the span that solution covers. The two agree through the overlap; the model continues either side of it, where no reference solution exists to check against.

The planets’ ascending nodes turn on the invariable plane over 50,000–2,000,000-year timescales — the nodal eigenfrequencies s₁…s₈, which the model’s own N-body engine reproduces from one cited J2000 state (N-body Exploration); the rendered planets’ nodes of date are read from that chain. With only ~4,000 years of recorded astronomy no complete cycle has been observed directly; the chain’s window rates are gated against JPL Horizons over 1800–2100.

Verification and reproducibility

All calibration work is fully reproducible through publicly available scripts:

Explore Formulas for the complete formula set.