Ordered Radial Trajectories in Acceleration Space Reveal the Internal Structure of the Radial Acceleration Relation

Abstract

Barry Kieran Radcliffe

The radial acceleration relation (RAR) establishes a remarkably tight empirical correlation between the observed gravitational acceleration in galaxies and that predicted from their baryonic mass distributions. Previous studies have primarily examined the RAR as a population-level relation constructed from measurements spanning many galaxies. However, galaxy rotation curves contain an intrinsic radial ordering that permits investigation of how the relation is realized within individual galaxies. Using rotation curve data from the SPARC database, we reconstruct trajectories in acceleration space by connecting successive measurements of baryonic acceleration and observed acceleration in order of increasing galactocentric radius. This approach transforms the conventional point-based representation of the RAR into a set of ordered radial paths that trace the internal dynamical evolution of individual galaxies. We find that galaxies follow coherent and continuous trajectories in the (log gbar, log gobs) plane, evolving systematically from high- acceleration inner regions toward low-acceleration outer regions. Despite substantial diversity in morphology, mass, and surface brightness, these trajectories closely follow the mean radial acceleration relation. The observed population- level RAR therefore emerges not merely as a statistical correlation across galaxies, but as the collective expression of ordered radial evolution within individual systems. These results provide a complementary perspective on the origin and interpretation of the radial acceleration relation. Any successful explanation of galaxy dynamics—whether based on dark matter, modified gravity, or alternative frameworks—must account not only for the existence of the population-level relation but also for its realization as a structured radial trajectory within individual galaxies.

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