The Local Star Formation Environment Predicts Young Star Cluster Radii: Evidence from PHANGS-MUSE H-alpha Maps

The effective radii of young star clusters correlate with cluster mass, but the large scatter about that relation is not well understood. Theory predicts that a cluster's initial radius depends on the gas surface density of its natal environment, so the star formation environment should leave a measurable imprint on cluster structure. In a companion analysis of 5,105 clusters across 31 LEGUS galaxies, we found that galaxy-wide specific star formation rate (sSFR) carries no predictive power for cluster radius. Here we test whether that null result reflects the spatial scale on which the environment was measured. We cross-match the LEGUS cluster catalog of Brown & Gnedin with H-alpha maps from the PHANGS-MUSE survey in the four galaxies common to both surveys, and measure the local star formation rate surface density, Sigma SFR, at the position of every cluster by aperture photometry. For the 325 clusters younger than 10 Myr with reliable radius and mass measurements, local Sigma SFR significantly predicts effective radius (beta = 0.079, p = 2 x 10-4) in a model containing mass and age, while the host galaxy's sSFR does not (beta = 0.042, p = 0.40) for those same clusters. The local model generalizes nearly three times better under cross-validation (CV R squared = 0.061 versus 0.022). The effect operates among clusters within a single galaxy: it survives galaxy fixed effects (beta = 0.068, p = 0.001) and appears independently in each of the two galaxies large enough to fit alone. Including local Sigma SFR also reduces the mass coefficient by 26%. The birth environment measurably shapes cluster structure, but only when it is measured on the scale on which clusters actually form.