The 3D Cosmic Shoreline for Nurturing Planetary Atmospheres

Abstract

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.

Figures (10)

• Figure 1: To determine escape velocities for objects without measured masses, we derive an empirical mass-radius relationship from exoplanets (errorbars) and Solar System objects (squares). We use this relation, valid for rocky planets up to $1.8\rm{R}_\earth$, to estimate planet masses and uncertainties that incorporate the intrinsic scatter on the relation, the uncertainties on the model parameters, and uncertainties on the planet radii (https://github.com/zkbt/shoreline/blob/main/notebooks/1-fit-mass-radius-relation.ipynb).
• Figure 2: A cosmic shoreline dividing exoplanets (errorbars) and Solar System planets (squares) with any type of atmosphere or global surface volatiles (blue symbols, $A_{\sf{i}}=1$) from those without (brown symbols, $A_{\sf{i}}=0$). Magma ocean planets ($T_{\sf eq} > 1700$K) were excluded from the fit but are still shown for context, as are planets without definitive atmosphere constraints (gray symbols, $A_{\sf{i}}=?$). The shoreline defines a plane in the 3D space of ($f$, $v_{\sf esc}$, $L$); each row shows slices that consider a narrow range of stellar luminosity (top), planet escape velocity (middle), and planet flux (bottom). Background colors indicate the modeled probability of an atmosphere at each location (sandy brown for $p_{\sf i} = 0$, water blue for $p_{\sf i}$ = 1), marginalizing over the parameter uncertainties and the width of the slice; contours (dashed lines) highlight atmosphere probabilities of 5%, 50%, 95% (https://github.com/zkbt/shoreline/blob/main/notebooks/6-plot-shorelines-on-their-own.ipynb).
• Figure 3: The same cosmic shoreline as the top row of Figure \ref{['f:shoreline']}, but expressed as an animation that is available in the HTML version of the article, showing the 3D shape of the cosmic shoreline by stepping through slices of changing stellar luminosity (https://github.com/zkbt/shoreline/blob/main/notebooks/6-plot-shorelines-on-their-own.ipynb).
• Figure 4: Cosmic shoreline parameter posterior probabilities, including only planets cool enough to avoid global magma oceans. These parameters define the atmosphere probability shown in Figure \ref{['f:shoreline']}. Panels show marginalized 1D histograms (diagonal) and marginalized 2D distributions (off-diagonal) with contours that enclose 68.3% and 95.4% probability. Titles along the diagonal show central $68.3\%$ confidence intervals for the exoplanets + Solar System joint fit. The model parameters define a shoreline via $\log_{10} (f_{\rm shoreline}/f_\earth) = \log_{10} (f_{\sf 0}/f_\earth) + p \log_{10} (v_{\sf esc}/v_{\sf esc, \earth}) + q \log_{10} (L_\star/L_\sun)$, with $w$ representing the logistic width parameter setting the fuzziness of the shoreline (https://github.com/zkbt/shoreline/blob/main/notebooks/5-print-and-visualize-posteriors.ipynb).
• Figure 5: A cosmic shoreline exactly as in the top row of Figure \ref{['f:shoreline']}, but specifically targeting $\rm{CO}_2$-dominated atmospheres in the "no magma ocean, no CO$_2$ freezeout" sample with equilibrium temperatures warm enough for $\rm{CO_2}$ to exist as a gas (${\rm T_{CO_2}} > 194 \mathrm{K}$) and cool enough that a global magma ocean is less likely (${\rm T_{magma}} < 1700 \mathrm{K}$). This more narrowly defined cosmic shoreline matters because $\rm CO_2$ is both likely in warm secondary atmospheres and relatively easier to detect for exoplanets in thermal emission with JWST (https://github.com/zkbt/shoreline/blob/main/notebooks/6-plot-shorelines-on-their-own.ipynb).