The 3D Cosmic Shoreline for Nurturing Planetary Atmospheres
Various ``cosmic shorelines" have been proposed to delineate which planets have atmospheres. The fates of individual planet atmospheres may be set by a complex sea of growth and loss processes, driven by unmeasurable environmental factors or unknown historical events. Yet, defining population-level boundaries helps illuminate which processes matter and identify high-priority targets for future atmospheric searches. Here, we provide a statistical framework for inferring the position, shape, and fuzziness of an instellation-based cosmic shoreline, defined in the three-dimensional space of planet escape velocity, planet bolometric flux received, and host star luminosity. We circumvent the need to estimate individual host stars' historical X-ray and extreme ultraviolet fluences by including luminosity in the definition of the shoreline, explicitly modeling how sharply such drivers of atmospheric escape intensify toward lower-luminosity M dwarf stars and marginalizing over the associated uncertainties. Using Solar System and exoplanet atmospheric constraints, under the assumption that one planar boundary applies across a wide parameter space, we find the critical flux threshold for atmospheres scales with escape velocity with a power-law index of $p=5.9_{-0.43}^{+0.61}$, steeper than the canonical literature slope of $p=4$, and scales with stellar luminosity with a power-law index of $q=1.17_{-0.20}^{+0.28}$, steep enough to disfavor atmospheres on Earth-sized planets out to the habitable zone for stars less luminous than $\log_{10} (L_\star/L_\odot) = -2.23_{-0.21}^{+0.18}$ (roughly spectral type M4V). This model provides quantitative predictions for the probability any planet may have an atmosphere, which can be rigorously tested by upcoming JWST Rocky Worlds observations.