A Data-Driven Phenomenological Model for the Flyby Anomaly, its Dependence on Solar Activity, and Implications for New Physics

This paper proposes a new, high-precision phenomenological model for the long-standing, unresolved "flyby anomaly" of space probes, based on observational data from ten major spacecraft. Our model demonstrates that the anomaly can be described as a linear sum of two distinct physical effects. The first effect is a "torsion term" (coefficient K₂), related to the celestial body's rotation, and the second is a "compression term" (coefficient K₁), related to the probe's entry geometry. The central finding of this paper is the discovery that the K₁ coefficient is not a universal constant but a dynamic variable, K₁(t) = K₁_galaxy + ηS(t), which exhibits a clear linear dependence on the local space environment at the time of the flyby—specifically, solar activity S(t). This data-driven model uniformly reproduces all observed anomalies with a Root Mean Square Error (RMSE) of 0.2994 mm/s, a remarkable precision comparable to the observational errors themselves. Furthermore, we explore the physical origins behind this phenomenology. We show for the first time that the observed value of η (the solar activity coupling constant), ≈ -0.0569, can be correctly explained in its order of magnitude by a multi-scale physical picture in which the energy density of the solar wind alters the local density of the Quantum Chromodynamics (QCD) chiral condensate, ⟨ψ̄ψ⟩. This connection to known physics is strong evidence that K₁ functions as a probe of the local vacuum energy state. However, this study simultaneously highlights a profound mystery. While the η term can be explained by known physics, the first-principles origin of the background value of K₁, K₁_galaxy ≈ -6.157, remains entirely unexplained. This mystery, which our investigation links to the transcendental geometric constant -(4π² + π), may be deeply rooted in the fundamental topological properties of the vacuum itself, extending beyond the framework of existing physics. This research decisively shows that the flyby anomaly is a manifestation of the dynamic properties of spacetime, revealing that it encompasses both aspects governed by known laws and a profound mystery that necessitates entirely new physics.