The Cosmic Shoreline Revisited: A Metric for Atmospheric Retention Informed by Hydrodynamic Escape

TL;DR

This work revisits the cosmic shoreline concept by integrating hydrodynamic escape modeling across Earth-like, Venus-like, and steam atmospheres and by incorporating CH4 in energy-limited escape scenarios to derive time-integrated atmospheric retention thresholds. It combines stellar XUV evolution models, multiple atmospheric compositions, and Monte Carlo statistics to estimate a critical bolometric instellation $S^*_{bol}$ for atmospheric retention, unveiling a transition-zone rather than a sharp boundary that depends on stellar mass, planetary mass, and initial volatile inventory. A new target-priority metric is introduced, ranking rocky exoplanets by their proximity to the 90% retention shoreline, and population-level tests using ratio'd densities offer a potential observational diagnostic for shoreline validity. The study also analyzes the trend of ratio'd density against instellation and mass, highlighting how JWST observations can constrain or falsify the shoreline framework, while acknowledging substantial uncertainties in atomic-line cooling, XUV spectral details, and mantle outgassing dynamics. Overall, the paper provides a foundation for probabilistic atmosphere-retention assessments and practical guidance for forthcoming atmospheric detections or non-detections on rocky exoplanets.

Abstract

The "cosmic shoreline," a semi-empirical relation that separates airless worlds from worlds with atmospheres as proposed by K. J. Zahnle & D. C. Catling, is now guiding large-scale JWST surveys aimed at detecting rocky exoplanet atmospheres. We expand upon this framework by revisiting the shoreline using existing hydrodynamic escape models applied to Earth-like, Venus-like, and steam atmospheres for rocky exoplanets, and we estimate energy-limited escape rates for CH4 atmospheres. We determine the critical instellation required for atmospheric retention by calculating time-integrated atmospheric mass loss. Our analysis introduces a new metric for target selection in the Rocky Worlds Director's Discretionary Time and refines expectations for rocky planet atmosphere searches. Exploring initial volatile inventory ranging from 0.01% to 1% of planetary mass, we find that its variation prevents the definition of a unique clear-cut shoreline, though nonlinear escape physics can reduce this sensitivity to initial conditions. Additionally, uncertain distributions of high-energy stellar evolution and planet age further blur the critical instellations for atmospheric retention, yielding broad shorelines. Hydrodynamic escape models find atmospheric retention is markedly more favorable for higher-mass planets orbiting higher-mass stars, with carbon-rich atmospheres remaining plausible for 55 Cancri e despite its extreme instellation. We caution that our estimates are sensitive to processes with poorly understood dynamics, such as atomic line cooling. Finally, we illustrate how density measurements can be used to statistically test the existence of the cosmic shorelines, emphasizing the need for more precise mass and radius measurements.

Figures (24)

• Figure 1: Atmospheric loss rates for planet of different masses orbiting a Sun-like star at $t = 100$ Myr for a range of escape models/compositions. For CO2-dominated, and CH4-dominated, the escape rates for varying $M_p$ are color-coded, as indicated in the legend. For N2-dominated atmospheres, we use results from two different studies. CP2024 refers to the analytical framework developed by chatterjee_novel_2024 which provides rates for varying $M_p$, while N2022 is an 1D numerical model from nakayama_survival_2022, which only applies to Earth-mass planets. The shaded regions represent the range of atmospheric loss rates, including uncertainties in XUV flux. Additional uncertainties are included for specific atmospheres, such as CO2 interpolation methods, the $F_X/F_{EUV}$ ratio (H2O atmosphere), and efficiency ($\epsilon$) (CH4 atmosphere) (Table \ref{['tab:monte_carlo']}). For CH4 atmospheres, the loss rate weakly depends on $M_p$, whereas CO2 loss is strongly mass-dependent. For N2-dominated atmospheres, the only two studies that consider atomic line cooling, show the escape rates stay constant at higher XUV flux, but the values differ significantly: nakayama_survival_2022 suggests that N2-dominated atmospheres escape very slowly, whereas chatterjee_novel_2024 predicts high escape rates at high XUV flux.
• Figure 2: Critical instellations for atmospheric retention vs stellar mass for Earth-mass planets across different escape models/compositions and probing a range of volatile abundances. The critical bolometric instellation ($S^*_{bol}$) is normalized to Earth's insolation and stellar mass relative to the Sun. The right y-axis also shows the corresponding equilibrium temperature ($T_{eq}$), assuming no albedo and perfect redistribution. Blue, yellow, and red lines correspond to initial volatile fractions of $10^{-4}$, $10^{-3}$, and $10^{-2}$ in planetary mass, respectively. The shaded regions represent a 50–99% probability range of atmospheric retention output by our model, and the solid curves represent 90% probability of atmospheric retention. For H2O atmospheres, the blue line lies below $1\, S_{\oplus}$. For an N2-dominated atmosphere, predictions from nakayama_survival_2022 (N2022) suggest that all three cosmic shorelines exceed $10^4\, S_{\oplus}$, indicating extreme resilience to atmospheric escape. In contrast, chatterjee_novel_2024 (CP2024) predicts a less permissive cosmic shorelines. The horizontal dashed line marks the runaway greenhouse limit from kopparapu_habitable_2013.
• Figure 3: Cosmic Shoreline Revisited: critical instellations for atmospheric retention vs. planetary escape velocity across different escape models/compositions binned by host-star mass and probing a range of volatile abundances. The right y-axis also shows the corresponding equilibrium temperature ($T_{eq}$), assuming zero albedo and full heat redistribution. The top row shows results for CO2-dominated atmospheres, the middle row shows CH4-dominated atmospheres, and the bottom row shows N2-dominated atmospheres. Each column corresponds to a specific range of host stellar masses. Blue, yellow and red lines indicate a 90% probability of atmosphere retention with initial volatile fractions of $10^{-4}$, $10^{-3}$ and $10^{-2}$ of the planetary mass, respectively. The shaded regions in blue, yellow and red represent a 50–99% probability of atmosphere retention. The hatched region above the 50% red line represents conditions where planets are unlikely to retain an atmospheres even if volatile-rich. For comparison, the thick gray line reproduces the XUV-driven cosmic shoreline from Fig. 2 of zahnle_cosmic_2017, with cumulative XUV flux converted to bolometric flux to match our y-axis, using the relation $S = F_{\text{XUV}} (L_*/L_\odot)^{0.6}$ (their Eq. 27). The horizontal dashed gray lines mark the runaway greenhouse limit from kopparapu_habitable_2013. Venus ($\venus$), Earth($\oplus$) are plotted for reference. The symbols mark exoplanet targets from four different samples (Sec. \ref{['sec:data']}). For planets with both radius and mass measurements, the density—scaled to that of a planet with Earth-like composition ($\rho_{\oplus,s}(R_p)$)—is color-coded. Lower densities may suggest thicker atmospheres or higher volatile content. Planets with confirmed (55 Cnc e) and tentative (TOI-431 b, LHS 1478 b) atmosphere detections are labeled. Crosses denote planets with thick atmosphere ruled out: TRAPPIST-1 c, b, GJ 1132 b, GJ 1252 b, LTT 1445 A b, GJ 486 b, TOI-1468 b, GJ 367 b and TOI-1685 b (left to right). See Table \ref{['tab:targets']} and Sec. \ref{['sec:metric']} for planet-by-planet details.
• Figure 4: Upper panel: Escape rate at high-XUV plateau of N2-dominated atmospheres as a function of planetary mass, based on chatterjee_novel_2024. The right y-axis represents the total atmospheric loss over 3 Gyr, which also defines the critical initial volatile content required for atmospheric retention over this timescale—planets with an initial volatile mass above this threshold can retain their atmosphere regardless of past XUV flux intensity. Lower panel: the critical initial volatile fraction (relative to planetary mass) ($f^*_{initial}$) sufficient to sustain an atmosphere over 3 Gyr, independent of historical XUV flux intensity, shown as a function of planetary mass. The shaded region represents uncertainty in the critical flux at which the escape rate plateaus (see Table \ref{['tab:CP24']}).
• Figure 7: Planetary thermal structure and two end-member estimates for transit radius. The thin black line denotes an isothermal atmosphere in the radiative layer, while the thick black line represents an adiabatic profile in the convective layer. The white line corresponds to the rheological transition ($\sim$40% melt fraction) P-T relation for rock. If $T_{\text{surf}} > T_{\text{rheo}}(P = P_{\text{surf}})$, the surface is a low-viscosity magma ocean. The intersection of the thick black line with the white line marks the depth at which magma crystallizes. Volatiles are assumed to be partitioned between the magma and atmosphere, with surface pressure governed by the solubility law. The pressure level probed during a transit ($P_{\text{transit}}$) is set to a constant value. (1) Upper-limit estimate of atmospheric height: Assumes the entire atmosphere below the RCB is fully convective, following a dry adiabatic profile. $P_{\text{transit}} = 10^{-6} \text{ bar}$. (2) Lower-limit estimate of atmospheric height: Assumes a radiative layer extends from the surface up to $P = 100$ bar. $P_{\text{transit}} = 0.02$ bar.