Probing Exoplanetary Rings with Asterodensity Profiling: A PhotoRing Analysis of Kepler-51
The Kepler-51 system hosts transiting super-puff planets whose extremely low bulk densities challenge structural models, possibly because unresolved rings inflate their inferred radii. We test this hypothesis by combining asterodensity profiling with TTV-aware transit fits of Kepler long-cadence photometry. For Kepler-51b we measure a density ratio $Ψ\equivρ_{\star,\mathrm{obs}}/ρ_{\star,\mathrm{true}} = 1.18\pm0.06$, a $2.9σ$ offset from $Ψ=1$; for Kepler-51d we find $Ψ=1.04\pm0.08$ ($0.6σ$), consistent with no anomaly. Unmodelled eccentricity is disfavoured for planet b by TTV-derived dynamical constraints ($p=0.007$) but remains viable for planet d ($p=0.33$). The offset of Kepler-51b is also compatible with the PhotoRing (PR) effect of unmodeled rings. Using a geometric ringed-transit forward model and Bayesian retrieval, we identify ring configurations that reproduce the transit observables and the measured density ratios. The preferred solutions are consistent with compact, moderately inclined rings that raise the true bulk densities by factors of $2-5$. Conversely, the absence of a large PR offset excludes extended, optically thick, high-obliquity rings at $99\%$ credibility for b and $70\%$ for d from archival photometry alone. These objects may thus be "puff ringed planets" although recent JWST observations of Kepler-51d show no direct signature of an opaque ring. While the required impact parameters and stellar densities show tension with independent constraints -attributable to Kepler's 30-minute cadence- this study presents the first complete application of the PR effect, offered as a methodological demonstration and exclusion experiment rather than as evidence that rings are present. All code is publicly available to support applications to other systems.