The Stellar Death Clock: A Closed-Form Analytical Framework for Adiabatic Earth–Moon Co-migration Under a Transient Asteroid Flyby

N-body integration of the close encounter between asteroid 137 Meliboea and the Earth–Moon system — validated with REBOUND/IAS15 in Paper 5 of The Stellar Death Clock framework — revealed an unexpectedly precise co-migration between the two bodies: for a specific encounter velocity, the differential displacement between the Earth and the Moon after 892 days of secular drift falls to a fraction of a metre, against individual heliocentric displacements of order 6×10⁴ m — a relative cancellation of order 5 parts in 10⁶. This raises an immediate question: is this precision a physically meaningful property of the encounter geometry, or an artefact of the numerical setup? We answer this by deriving, from first principles, a closed-form analytical model of the encounter — combining Öpik's instantaneous-impulse approximation, Gauss's planetary equations, and exact two-body Keplerian propagation — that requires no fitted parameters and no numerical integration. The model reproduces the order of magnitude of both bodies' displacements to first order and, critically, reproduces the approximately linear sensitivity of the residual to the encounter velocity found numerically in Paper 5. We conclude that the co-migration effect reported in Paper 5 is not a numerical coincidence: it is a genuine consequence of Newtonian two-body dynamics acting differently on the Earth and the Moon, and we identify the two distinct physical mechanisms responsible for it. We further characterise the precision limits inherent to any first-order analytical treatment of this problem, clarifying the respective roles of analytical theory and N-body integration in this class of problem.