Novel Physics of Escaping Secondary Atmospheres May Shape the Cosmic Shoreline

Abstract

Recent James Webb Space Telescope observations of cool, rocky exoplanets reveal a probable lack of thick atmospheres, suggesting prevalent escape of the secondary atmospheres formed after losing primordial hydrogen. Yet, simulations indicate that hydrodynamic escape of secondary atmospheres, composed of nitrogen and carbon dioxide, requires intense fluxes of ionizing radiation (XUV) to overcome the effects of high molecular weight and efficient line cooling. This transonic outflow of hot, ionized metals (not hydrogen) presents a novel astrophysical regime ripe for exploration. We introduce an analytic framework to determine which planets retain or lose their atmospheres, positioning them on either side of the cosmic shoreline. We model the radial structure of escaping atmospheres as polytropic expansions - power-law relationships between density and temperature driven by local XUV heating. Our approach diagnoses line cooling with a three-level atom model and incorporates how ion-electron interactions reduce mean molecular weight. Crucially, hydrodynamic escape onsets for a threshold XUV flux dependent upon the atmosphere's gravitational binding. Ensuing escape rates either scale linearly with XUV flux when weakly ionized (energy-limited) or are controlled by a collisional-radiative thermostat when strongly ionized. Thus, airlessness is determined by whether the XUV flux surpasses the critical threshold during the star's active periods, accounting for expendable primordial hydrogen and revival by volcanism. We explore atmospheric escape from Young-Sun Mars and Earth, LHS 1140 b and c, and TRAPPIST-1 b. Our modeling characterizes the bottleneck of atmospheric loss on the occurrence of observable Earth-like habitats and offers analytic tools for future studies.

Figures (3)

• Figure 1: Illustration comparing hydrodynamic and hydrostatic escape of a secondary atmosphere (not to scale). In hydrodynamic escape, the net heating from XUV photoabsorption, reduced by recombination and atomic line cooling, drives a pressure gradient force to overcome gravity--- accelerating the flow radially outwards down the atmosphere’s density gradient. Jeans escape occurs from the hydrostatic exobase where the atmosphere becomes collisionless. If the thermal velocity of a particle directed outwards exceeds the escape velocity $v_{\mathrm{esc}}$ at the exobase radius $r_{\mathrm{exo}}$, then it can ballistically escape on a hyperbolic trajectory. The balance of XUV heating, conduction, and line cooling determines the escape rate via the exobase temperature.
• Figure 2: Contour plot of Mach number $\mathscr{M}$ against dimensionless radius $r/r_{sc}$ for a diatomic adiabatic expansion ($\gamma_a=1.4$) example solutions: transonic ($\delta =1$), subsonic/supersonic ($\delta=1.02$), and double-valued ($\delta=0.98$). The transonic escape solution is highlighted with red arrows. The transonic contours' upper and lower branches are highlighted with a dot-dashed line of cyan and green, respectively. $\delta$ relates to the entropy differences between different types of solutions (see Eqn \ref{['eqn: isenexp']}).
• Figure 3: Steady-state polytropic solutions space for vertical atmospheric equilibrium. For $\frac{\lambda_0}{\gamma} > \frac{1}{\gamma-1}$, the equilibrium is hydrostatic with density reaching zero at a finite height (pink). Implicit heating throughout the atmosphere reduces $\gamma$ from an adiabat $\gamma_a$. For $2<\frac{\lambda_0}{\gamma} <\frac{1}{\gamma-1}$, the thermal energy at the base in addition to the implicit heating is large enough to drive an accelerating transonic flow in principle (blue hydrodynamic region). The supersonic at the base constraint $\lambda_0=2$ is represented by the dashed vertical line and labelled 'unbound'. The blue region of consistent steady states does not extend all the way to this line, however, because the flow launch should have $\mathscr{M} \ll 1$, otherwise the escape is 'catastrophic' (illustrated in light blue). On failure of the hydrodynamic assumption at the sonic point, XUV-driven inversions ($\gamma < 1$) give way to hydrostatic atmospheres that would be infinite except for the formation of an exobase (brown region). The yellow line illustrates the Knudsen onset for collisional flow given by $\mathrm{Kn}_{sc}^{\mathscr{N}} \sim 1$ (see Section \ref{['sec: onset']}).