Hidden Oscillations in Planetary Motion: The Ef–FSI Framework

Gravitational waves are often regarded as rare signals from extreme cosmic events. This paper introduces the Ef–FSI framework, which demonstrates that their signature is continuously present in planetary orbits. The framework is built on two core quantities: the Foundational Energy Factor (Ef = vC), a constant energy-per-cycle per unit mass, and the Field Strength Index (FSI = v²C), which captures oscillatory Action–Reaction phases (perihelion = contraction, aphelion = expansion). Results confirm that Ef remains constant across perihelion, aphelion, and the mean orbital radius, with deviations typically <5%. FSI reveals systematic wave-like oscillations: Δ(+) > 1 at perihelion (Action) and Δ(–) < 1 at aphelion (Reaction). At the mean orbital radius, planetary FSIs converge to nearly identical values (~8.33 × 10²⁰ N·m/cycle/kg for Venus, Earth, and Mars), reflecting the symmetry of the solar field. Satellite systems (Jupiter and Saturn) also display internal convergence, confirming FSI as a system-level determinant. The framework reproduces Kepler’s laws directly, aligns with thermodynamics (Ef decreases systematically with radius), and extends naturally to the quantum scale, where Planck’s constant emerges as a special case of the cycle–energy law. Thus, planetary motion itself encodes continuous gravitational waves, making the Solar System a natural laboratory. Version notes (v4): This version updates and expands the framework with the following key additions: Quantum Bridge (revised): clarified derivation linking Ec=mvC with Planck’s h=mvλ, including closure to Newton’s Fd. Bohr–Ef–FSI Bridge (new section): demonstrates that the 2N² electron shell law arises directly from the field invariant (v²C), with tables comparing planetary and atomic systems. Complete numerical tables: added baseline constants (planetary and hydrogen), stepwise calculations (Ef, v, C, ne, ee), and normalization to show invariant patterns. Symbols and notes: appended additional symbol table and explanatory notes for new variables (N, ne, ee, ne/S). References updated: now include DOI links for all prior versions (v1, v2, v3), with v4 established as the canonical English release. Readers can immediately see how Version 4 strengthens the macro–micro bridge, adding full quantitative evidence for orbital stratification in both planetary and atomic domains. Note: In Version 3, the cover page mistakenly displayed ‘v2’. This has been corrected in Version 4.Project website (Ef–FSI Research Group):https://effsi.org