The Morphology of Saros Series: An Anomalistic Phase-Space Model

This paper presents a new mathematical framework for understanding the long-term morphology of solar eclipse saros series. The study demonstrates that saros evolution can be interpreted as the projection of a continuous trajectory through anomalistic phase space together with a geometric Earth-surface gate that determines whether the evolving shadow geometry intersects the Earth. A principal contribution of the work is the development of the underlying geometry required to compute the continuous intersection of the lunar shadow axis with the Earth's surface. The paper derives the mathematical formulation for the Earth-surface intersection on a WGS-84 ellipsoid, and uses this continuous treatment to distinguish the intrinsic evolution of the eclipse shadow from the subset of events observable from Earth. This geometric formulation provides a rigorous basis for continuous eclipse analysis and forms the foundation for the Earth-surface gate introduced in the paper. The analysis demonstrates that many of the observed characteristics of saros series—including the progression among total, annular, hybrid, and partial eclipses, asymmetries in eclipse duration, and the appearance of shoulders and turning structures—arise naturally from the interaction of two anomalistic phase variables and the finite size of the Earth. A simple two-phase surrogate reproduces the principal morphological features observed in complete saros series while preserving the underlying physical interpretation. Saros 137 serves as the primary case study throughout the paper. Its complete evolutionary history is analyzed to show how an apparently irregular sequence of eclipse classifications and durations emerges from a smooth trajectory in anomalistic phase space. The resulting model provides a unified explanation for the large-scale morphology of the series and clarifies the respective roles of the lunar and terrestrial anomalistic cycles. Particular attention is devoted to Eclipse 32 of Saros 137, whose geometry lies near the transition between eclipse classifications. Continuous analysis of this event demonstrates the operation of the Earth-surface gate and illustrates how modest changes in the adopted solar-radius convention can alter the observable eclipse classification while leaving the underlying continuous shadow geometry essentially unchanged. Eclipse 32 therefore provides an important validation of the continuous geometric formulation developed in the paper. All eclipse calculations are performed using NASA NAIF SPICE with the DE441 planetary ephemeris, replacing traditional interpolation techniques with direct high-precision numerical ephemerides. The continuous geometric model is validated against published eclipse predictions and demonstrates agreement with established eclipse calculations while providing additional insight into the underlying mathematical structure governing Saros evolution. The paper introduces the concepts of anomalistic phase space and the Earth-surface gate as complementary components of a unified mathematical description of saros-series morphology. Together, these provide a new framework for interpreting eclipse evolution, understanding the origin of hybrid eclipses and eclipse-type transitions, and analyzing the long-term dynamics of solar eclipse series. UPDATE 1 Sep 2026: Version 2.0.0 update This version incorporates corrections identified during post-publication verification: Corrected the mathematical statement of recurrence and density for the discrete anomalistic translation on the two-dimensional torus. Recurrence depends on the individual phase advances modulo 2π, and density requires rational independence of 1, the normalized lunar phase advance, and the normalized terrestrial phase advance. The winding ratio alone is not sufficient to establish either recurrence or density. Revised the associated continued-fraction discussion so that the convergents are used to characterize the absence of low-order resonances, without asserting irrationality of the physical recurrence. Removed corresponding unsupported statements that the anomalistic phase advances are irrationally related or incommensurate. Corrected the projection-angle geometry used in the observer-dependent Earth-intersection formulation, including the sign of the cos⁡χ expression and the associated Moon–Sun vector convention. Clarified selection of the observer-facing root of the WGS-84 ellipsoid-intersection quadratic and the distinction between signed and unsigned Earth-surface coordinates. Clarified that the WGS-84 Earth-surface intersection requires the Besselian coordinates x, y, and shadow-axis declination d; γ alone is insufficient. Stated explicitly the collinear approximation used in the umbral-apex surrogate. Corrected the indexed anomalistic sample count from N to N+1. Corrected Appendix cross-references and the ordering of operations in the Earth-surface intersection algorithm. Corrected the UT1-based longitude subsection structure and associated computational description. Corrected a duplicated bibliography key and its affected citation, removed a duplicated subsection heading, and corrected figure/caption numbering and related presentation issues. These corrections do not alter the numerical results, figures, principal conclusions, or the two-phase anomalistic model presented in the paper.