Revising core powered mass loss: A critical assessment of the "energy limited" argument

TL;DR

This study critically revisits the core powered mass loss framework for hydrogen-rich planetary atmospheres post-giant-impacts, demonstrating that the traditionally invoked energy-limited constraint is unnecessary. It shows that escape is governed by sonic-point conditions and the atmospheric hydrodynamics rather than interior cooling, with adiabatic cooling and density structure at the sonic point driving the outflow. Through broad synthetic-exoplanet simulations, the authors find that the energy-limited approach can underestimate mass loss by up to eight orders of magnitude, implying hydrodynamic escape can occur on dynamical timescales far shorter than 1 Myr. They introduce a revised, fully hydrodynamic model and an efficient analytic fit, enabling accurate, scalable estimates of atmospheric loss for population studies and impacting interpretations of exoplanet atmospheric evolution and radius distributions. $\dot{M}_{\rm atm}|_{GI} = 4 \pi R_{\rm s}^{2} \rho_{\rm s} c_{\rm s}$ and $\dot{M}_{\rm atm} \approx 1.26 \times 10^{15\pm1} (R_{\rm s}/R_{\oplus})^{2} \exp\left(-\tfrac{19}{27}\, \tfrac{R_{\rm s}}{R_{\oplus}}\right)$ kg s$^{-1}$ summarize the core results.

Abstract

The extreme conditions in the early stages of planetary evolution are thought to shape its subsequent development. High internal temperatures from giant impacts can provide sufficient energy to drive extreme volatile loss, with hydrogen being most readily lost. However, the conditions required for maintaining a primordial atmosphere over geological timescales remain enigmatic. This paper revisits the core powered mass loss model for hydrogen removal from planetary atmospheres. One popular approach is to combine mass continuity at the sonic point with an energy-based constraint. We demonstrate that the so-called ``energy limited'' component of this model is unnecessary because atmospheric loss following giant impacts is governed solely by conditions at the sonic point. By simulating a broad range of synthetic exoplanets, varying in planetary mass, atmospheric mass fraction, and temperature, we find that the ``energy limited'' model can underestimate the mass loss rates by up to eight orders of magnitude. Our findings suggest that, for sufficiently hot post-impact surface conditions, hydrogen rich atmospheres can be removed on dynamical timescales that are far shorter than one million years.

Figures (5)

• Figure 1: Schematic diagram showing the atmospheric structure adopted in this paper, which is based on the standard configuration used in the core powered mass loss model Ginzburg2016Ginzburg2018Gupta2018Biersteker2019Gupta2020. Temperature and distance are not to scale; $T_{\rm s}$, $T_{\rm eq}$, $T_{\rm ab}$, and $T_{\rm n}$ are the temperature of the sonic point, the radiative-convective boundary (approximated as the equilibrium temperature), the top of the atmospheric boundary layer, and the nucleus surface. The boundary layer thickness and temperature contrast are denoted by $\Delta R_{\rm ab}$ and $\Delta T_{\rm ab}$, respectively. The core powered mass loss framework does not incorporate a boundary layer analysis but it is included here because it constitutes an important part of our modeling. In addition, whereas the standard model assumes an isothermal upper atmosphere Ginzburg2018, our analysis accounts for cooling from advection. The temperature profile of the magma ocean is not shown because it is not directly required to model the atmospheric outflow; however, the thermal state of the magma ocean is ultimately what powers core powered mass loss and sets the timescale over which it operates.
• Figure 2: Schematic diagram showing the atmospheric temperature profile with respect to (a) radial distance and (b) pressure, before (orange; subscript 1) and immediately after (blue; subscript 2) experiencing mass loss. The orange dashed line (subscript 3) represents the profile after radiative heating dominates over advective cooling. The advection and diffusion timescales are $t_{\rm ad}$ and $t_{\rm df}$, respectively. Radii $R_{\rm s}$, $R_{\rm rcb}$, and $R_{\rm n}$ correspond to the sonic point, the radiative-convective boundary, and the surface of the planetary nucleus, respectively. The temperature of the sonic point is $T_{\rm s}$ and the temperature of the planetary nucleus is $T_{\rm n}$. The sonic point of profile 2 is located at a higher elevation because it is inversely proportional to temperature. The radius of the radiative-convective boundary is mostly independent of mass loss because, for an irradiated planet, it is thought to scale with the ratio of the incoming radiant flux to the internal heat flux. This ratio is unlikely to change when mass is lost. Profile 2 is unlikely to be realized because radiative heating dominates over advective cooling within the atmosphere (i.e., $t_{\rm df}{<}t_{\rm ad}$). The atmosphere will transition directly from profile 1 to 3 instead. Pressure profile 3 is smaller than 2 because the atmosphere adiabatically expands to restore the lost mass at the sonic point.
• Figure 3: (a) The "energy limited" (Equation \ref{['eq:GI_EL1']}) and the mass conservation models using Equation \ref{['eq:GI_EL2']} (blue) and the isothermal hydrodynamic approximation Biersteker2019 as a function of the Jeans parameter at the planetary nucleus (Equation \ref{['eq:Lambda']}). The "energy limited" model follows an exponentially decaying relation with the Jeans parameter ($R^{2}{=}0.53$ goodness fit). Mass loss rates below $5 {\times} 10^{5}~{\rm kg{\,}s^{-1}}$ are likely governed by Jeans escape rather than hydrodynamic outflow. (b) Comparison of mass loss rates calculated using the "energy limited" and our hydrodynamic mass conservation equation, plotted against the Jeans parameter at the planetary nucleus. We simulated $2500$ synthetic planets with masses uniformly distributed between $1{-}10$ Earth masses and atmospheric mass fractions uniformly sampled in the logarithmic space between $10^{-3}$ and $10^{-1}$. The atmospheric mass for each planet was calculated as the product of its planetary mass and atmospheric mass fraction. Surface temperatures were drawn uniformly from $3000{-}10{,}000~{\rm K}$, and equilibrium temperatures from $500{-}3000~{\rm K}$. The color shading shows relative probability density (note that only the samples with the mass loss ratio less than unity are considered here for clarity). (c) Marginalized cumulative distribution function (CDF) for the mass loss ratio of the "energy limited" and our hydrodynamic mass conservation equation.
• Figure 4: The mass loss rate vs the sonic point radius of 2500 synthetic planets. The red points are the mass loss rates from our hydrodynamic simulations, the black line is the analytic fit (Equation \ref{['eq:fit']}), and the blue shaded region is the one standard deviation scatter in the simulated mass loss rates relative to the fit. Mass loss rates below $5 {\times} 10^{5}~{\rm kg{\,}s^{-1}}$ are likely governed by Jeans escape rather than hydrodynamic outflow.
• Figure 5: (a) Velocity profiles for three temperature models: analytic solution (Equation \ref{['eq:T_solution']}; black), semi-isothermal approximation $T(r){=}T_{\rm s}$ (red), and linear approximation $T(r)/T_{\rm s}{=}(\gamma{+}1)/2 {-} (\gamma{-}1)(r/R_{\rm s})/2$ (blue) for $\gamma{=}5/3$. Orange diamonds indicate boundary condition at $r{=}R_{\rm s}$ and $v{=}v_{\rm s}$. The boundary conditions at $r{=}0$ ($v{=}0$) and $r{\rightarrow}\infty$ ($v{\rightarrow}\sqrt{(\gamma+1)/(\gamma-1)}v_{\rm s}$) are not shown. (b) Corresponding temperature profiles for the three models.