The Benchmark Equation for Celestial Orbital Quantization: Verification and Applications

Abstract Are the orbital scales of planetary systems continuously variable, or discretized into a quantum-like ladder? Following the semiclassical procedure of Bohr's quantum theory, in which quantization is postulated before the dynamics is derived, we introduce the quantum number n as a postulate and propose a benchmark equation that fixes the reference orbital radius of a celestial body directly from its radius and density, then use it as an empirical ruler for a decisive large-sample test. Applied to a large exoplanet sample from the NASA archive (2026-02-26 snapshot, cross-checked against a 2026-10-25 re-retrieval), the quantum-number indicators decompose into integer and fractional parts: the integer parts form a clear ladder for Solar-System and Kuiper-belt bodies (four Kuiper-belt dwarfs within 0.03% of the nearest integer; the remaining nine bodies deviate by 0.25–5.2%, with Venus at 18.6% as the benchmark-inner boundary case), while no significant integer clustering is detected in the exoplanet sample at current precision (the fractional part is statistically indistinguishable from uniform, χ² = 4.29, df = 9, p ≈ 0.89); within the Solar System, the same ladder extends from Mercury (n ≈ 1.04) and Earth (n ≈ 1.02) through Ceres (n ≈ 96.6) to the Kuiper-belt dwarfs Pluto (n ≈ 640), Eris (n ≈ 826), Haumea (n ≈ 1.02×10³) and Makemake (n ≈ 1.29×10³), spanning three orders of magnitude. When an orbit occupies the n ≠ 1 rung, the inversion from the orbital radius yields the equivalent value of that rung rather than the true value, which is the normal state of ladder occupation. The derived relations allow the radius, density and mass of a celestial body to be retrieved from its orbital radius. We further present a method for determining the orbital quantum number of a planet in combination with celestial mechanics: the orbital radius is determined by gravitational perturbation inversion (a Le Verrier-type method), the quantum number is then calibrated by the benchmark equation, and the admissible range is finally bounded by Newtonian gravitational stability analysis. Taking Neptune as an example, the stability criterion confines its quantum number to an outer admissible band with n ≥ 21 (n = 17–20 forming a dynamical forbidden band), while the perturbation inversion gives n ≈ 26.6, restoring its actual orbital position, whereas the corresponding prediction of the Titius–Bode rule deviates by +29%. The method is not thereby exclusive: had Le Verrier's historical estimate of 36 AU been adopted, the resulting n ≈ 31.9 would still lie within the admissible band n ≥ 21, while the Titius–Bode value implies n ≈ 34.4; what distinguishes the present procedure is that it yields a quantum number lying inside the dynamically admissible range rather than a bare distance. The equation may be regarded as an empirical hypothesis of discrete orbits, whose universality and dynamical foundation remain to be tested; this determination method has also been extended to the TRAPPIST-1 multi-planet system and to the inner Solar System.