A Jacobsthal Window in Exoplanet Period Ratios:Derivation of the 71/35 Offset From Symplectic Depletion

Why It Matters

The finding provides a physics‑based explanation for the systematic resonance offset, reshaping theories of planetary migration and orbital stability across exoplanetary systems.

Key Takeaways

• Period ratios cluster at 71/35, not exact 2:1
• Jacobsthal map explains resonance offset analytically
• Model matches REBOUND simulations and TRAPPIST‑1 data
• No free parameters; predicts Δ = 1/21 window
• Edge depletion follows √x distribution across systems

Pulse Analysis

Mean‑motion resonances have long been a cornerstone of planetary dynamics, yet observations consistently show adjacent exoplanet pairs drifting slightly beyond the exact 2:1 commensurability. This subtle offset, first noted in Kepler data, challenges traditional migration models that predict tight resonant locking. By examining a large sample from the NASA Exoplanet Archive, researchers quantified the average period ratio at 2.0286, a value that cannot be attributed to random scatter. Understanding this systematic shift is crucial for refining formation scenarios and interpreting the architecture of distant worlds.

The breakthrough comes from recasting the near‑resonant dynamics in terms of a Jacobsthal map. Starting with the symplectic transfer matrix at the stability boundary, the authors introduce a linear dissipation term that transforms the Möbius map into f(r)=1+2/r. This Jacobsthal formulation yields a closed‑form expression for the offset, linking the dominant attractor to the fraction 43/21 and deriving the average deviation as 1/35. The model predicts a resonance window of width 1/21 and a probability density proportional to √x near the edge, all without adjustable parameters. Such an elegant analytical framework bridges abstract Hamiltonian mechanics with observable orbital spacings.

Empirical validation is compelling: REBOUND N‑body migration runs cluster at the predicted 43/21 ratio, TRAPPIST‑1’s non‑adjacent periods align with Jacobsthal values, and the Beta Pictoris system’s outer architecture mirrors the same pattern. Monte Carlo ensembles confirm the √ε edge exponent, and Kolmogorov–Smirnov tests show strong agreement across broader intervals. By providing a parameter‑free, testable prediction, this work reshapes how astronomers model planetary migration, resonance breaking, and long‑term stability, and it sets a clear target for future high‑precision timing surveys.