A 1151-Year Quasi-Commensurability Among the Pre-Uranian Planets: Empirical Detection and Statistical Characterization

Abstract

We report the empirical detection of a multi-planet quasi-commensurability in the Solar System. A computational exhaustive search identifies T* = 420,403 days (~1,151 years) as the global minimum of a series-comparison similarity metric applied to daily heliocentric ecliptic longitudes of the six pre-Uranian planets (Mercury, Venus, Earth, Mars, Jupiter, and Saturn -- the planets known before the discovery of Uranus in 1781), computed from the DE441 ephemeris over a symmetric search range of +-1,300 years. At this interval, the mean simultaneous angular displacement of all six planets from their positions T* days earlier is 14.7 degrees, with a standard deviation of 0.75 degrees sustained over a century-long comparison window, and stable across any reference epoch spanning 1,200 years. The score of the optimal cycle lies 3.26 sigma below the mean of all 2,600 candidates, with every other candidate producing a worse score. The result is independent of the reference epoch and of the series length (T* is the global minimum for series as short as one year). No sub-multiple produces a comparable result, establishing T* as an irreducible primary quasi-period. The interval 1,151 years was known to Babylonian astronomers as the period after which Venus returns to the same position in the sky (de Jong 2019). The present work demonstrates that this same interval is simultaneously optimal for all six pre-Uranian planets, a property not previously documented in the astronomical literature. Geocentric consequences -- including the quasi-synchronisation of planetary retrograde motions -- are discussed. Source code and data are publicly available.

Figures (5)

• Figure 1: Score $S(T)$ for all candidate intervals in $[-1{,}300, +1{,}300]$ years. The vast majority of candidates cluster between $50^\circ$ and $140^\circ$. The two candidates at $\pm 1{,}151$ yr stand far below all others.
• Figure 2: Score distribution. The single candidate with $S < 16^\circ$ is $T^*$; no other falls below $16.9^\circ$.
• Figure 3: Daily angular offset $\delta_k(t) = \lambda_k(t) - \lambda_k(t - T^*)$ for each planet over the 100-year comparison series. Upper panel: Mercury, Venus, Earth, and Mars over 5 years (daily resolution). Dashed lines show the mean offset for each planet. Lower panel: Jupiter and Saturn over 100 years (weekly resolution). The annotated double arrow indicates the $\approx 12$-year Jupiter--Saturn conjunction period visible as a modulation in Jupiter's offset. Each planet oscillates around a stable mean, demonstrating that the quasi-commensurability is sustained dynamically over centuries --- as confirmed by the stability analysis across 1,200 years (Table \ref{['tab:stability']}) --- and is not merely a snapshot coincidence.
• Figure 4: Convergence of the result with series length. Left: scores of $T^*$ (red) and the second-best candidate (grey). Right: gap, stable at $\approx 1.45^\circ$.
• Figure 5: Heliocentric ecliptic positions at epochs 50, 200, 400, 600, and 800 CE (filled circles) and $T^* \approx 1{,}151$ yr earlier (open circles). Each planet is at its orbital radius (Mercury innermost, Saturn outermost). The near-coincidence persists across all five epochs.