The cosmological Barker equation: An extended analytical framework for local group dynamics and collision timing mechanisms

Abstract

This paper extends the classical Barker equation, a historical cornerstone of celestial mechanics, to encompass complex cosmological and galactic perturbation effects. The orbital dynamics of the Local Group, overwhelmingly dominated by the impending convergence of the Milky Way and Andromeda galaxies, are typically analyzed via computationally expensive N-body simulations or the highly idealized Timing Argument. In this study, we introduce a modified effective potential that simultaneously accounts for the universal expansion driven by the Hubble flow and the extended mass distributions of dark matter halos. The resulting Cosmological Barker Equation yields the ultimate collision timescale of the binary system in a closed-form analytic expression that depends exclusively on initial boundary conditions. The analytically derived expansion terms mathematically demonstrate precisely how the underlying cosmic flow delays gravitational collapse, manifesting as a higher-order perturbation. Concurrently, the inclusion of a dedicated dark matter parameter allows the theoretical architecture to be seamlessly calibrated against modern numerical simulations. Ultimately, this expanded analytical framework provides a transparent and intuitive mathematical alternative for investigating the orbital mechanics of macroscopic galactic systems, profoundly enhancing physical insight while entirely circumventing the traditional computational burden.