Photon-Ring Subring Widths: The Exponential Ladder Set by the Lyapunov Exponent

Light that orbits a black hole once before escaping forms a ring nested just outside the critical curve; light that orbits twice forms another, thinner one, and the sequence converges exponentially. For a non-rotating hole the convergence factor per half-orbit is exp(-pi) = 0.0432, set by the Lyapunov exponent of the photon orbit. Under the stated model, in which the first subring sits ten per cent outside the critical diameter and carries eight per cent of the diameter in width, the M87* subrings have mean diameters of 43.7, 39.86 and 39.697 microarcseconds for the first three orders, with widths falling from 3.18 to 0.137 microarcseconds between the first and second. The Sagittarius A* widths fall from 4.26 to 0.184 microarcseconds on the same ladder. The second subring is therefore thinner than any current or planned Earth-based resolution element by more than an order of magnitude, and its detection must be interferometric, through visibility-domain signatures, rather than by imaging. Computed under a stated model; simulated, not measured.