A Master-Equation Reduction of the SGP4 Propagator from TLE to Topocentric Look Angle

Abstract

I present a fully expanded derivation of the Simplified General Perturbations 4 (SGP4) propagator as it appears in a working surveillance pipeline that ingests Two-Line Element (TLE) sets, propagates each catalog object, and reduces the result to topocentric azimuth and elevation at a fixed ground station. The classical SGP4 model is well documented in *Spacetrack Report No. 3* and in Vallado's revisitation, but the published descriptions are organized around a Fortran reference implementation, which obscures the algebraic structure of the propagator. I rewrite the entire chain as a sequence of seven composable operators, each with an explicit input and output signature, and I collapse the chain into a single boxed master equation per satellite. I also derive a compact implicit-triplet form, a Lyddane Kepler equation followed by a perturbed radius and a three-axis rotation, that is more tractable for inspection and debugging than the operator chain. The topocentric reduction is treated as a separate composition layer: a sidereal rotation into the Earth-fixed frame, a difference against the ground-station vector, and a projection onto the local south-east-zenith basis. I conclude with a complete variable-accounting table classifying every named scalar in the pipeline as either an external driver (TLE-derived element, evaluation-time clock, or ground-station configuration), a hard-coded physical constant, or a derived intermediate. I validate the implementation against live observations from my own earth station in Austin, Texas, comparing predicted azimuth and elevation against observed pass geometry for a representative set of low-Earth-orbit satellites. The paper is intended as a self-contained reference for engineers implementing SGP4 from scratch and for analysts who need to audit an existing propagator at the equation level.