The Ja Model: Methodology and Empirical Demonstration of the MUja Framework (Damped Oscillator Reconstruction and GRACE FO 1 Satellite Drag Analysis)

This paper presents the complete methodology and empirical evaluation of the MUja framework, a dimensionless comparative index for dissipative processes defined as: where F is force, d displacement, m mass, E energy, ΔS entropy change, m₀ a reference mass, and k_B Boltzmann’s constant. The framework replaces conventional time with a cycle count N, where N counts periods of a reference periodic process. The methodology is demonstrated on two physically distinct dissipative systems: 1. Damped harmonic oscillator – reconstructed from published fitted parameters (Sebastião 2014). The oscillator parameters are: m = 0.10219 kg, K = 10.392 N·m⁻¹, λ = 0.13878 s⁻¹. The amplitude decay is used to compute dissipated energy per cycle and . The reconstructed MUja evolution follows , with a dimensionless decay constant of 0.173 cycle⁻¹. Note: these numerical values are derived from the fitted model and are illustrative, not direct experimental measurements. 2. GRACE‑FO 1 satellite (NORAD 43476) under atmospheric drag – using 2680 orbital records from 30 November 2023 to 31 May 2026 (912.42 days). The semi‑major axis altitude decreased from 483.379 km to 451.615 km (total loss 31.764 km), corresponding to a total orbital energy loss ΔE_total = 5.836 × 10⁷ J. Over 14,034 orbital revolutions, the mean dissipated energy per orbit is 4.158 × 10³ J, yielding a mean MUja of 6.972 × 10²⁰. The full MUja distribution (n = 2608 valid adjacent intervals) gives: mean = 7.126 × 10²⁰, median = 6.421 × 10²⁰, standard deviation = 3.558 × 10²⁰, 95% bootstrap CI = [7.00 × 10²⁰, 7.29 × 10²⁰], skewness = 2.425, indicating a right‑tailed distribution consistent with episodic atmospheric disturbances (e.g., geomagnetic storms). The oscillator analysis is a proof‑of‑concept consistency test based on a fitted model. The satellite analysis uses real orbital observations and demonstrates operational feasibility. The framework does not claim new physical laws; it provides a dimensionless, cycle‑based language for comparing dissipative processes across vastly different scales. Limitations include the reliance on fitted parameters for the oscillator, the assumed constant thermospheric temperature (T = 1000 K) for the satellite, and the exclusion of conservative systems, massless particles, and relativistic effects.