Energy-limited escape

Energy-limited (EL) escape is one of the three limits on the hydrodynamic wind of the escape-regime dispatcher, valid when the atmosphere sustains a collisional, XUV-driven fluid wind. It is also available on its own through the entry point zephyrus.escape.EL_escape, which applies it to every state without checking the regime. This page defines the prescription itself, its radius scaling, and its tidal correction.

The physical idea is an energy budget. The stellar X-ray and extreme-ultraviolet (XUV) flux absorbed high in the atmosphere heats the gas. If a fixed fraction of that power goes into lifting gas out of the planet's gravitational well, the mass-loss rate follows from dividing the absorbed power by the escape energy per unit mass ¹². The rate is computed as

where \(\epsilon\) is the escape efficiency (the fraction of intercepted XUV power converted into work against gravity), \(R^3\) is the radius term selected below, \(F_\mathrm{XUV}\) is the XUV flux received at the planet's orbit (supplied by MORS in coupled runs), \(G\) is the gravitational constant, \(M_\mathrm{p}\) is the planetary mass, and \(K_\mathrm{tide}\) is the tidal correction of Eq. (2).

Canonical efficiencies for rocky planets lie between 0.1 and 0.3. The dispatcher's settings accept any \(\epsilon \in (0, 1]\) and reject values outside it, while EL_escape evaluates the formula for whatever efficiency it is given. Hydrodynamic simulations find that for strongly bound planets the effective efficiency collapses far below the canonical band, reaching of order \(10^{-2}\) above a threshold gravitational potential near \(\log_{10}(G M_\mathrm{p} / R_\mathrm{p}) \approx 12.9\) to \(13.2\) in cgs units ⁸. The threshold is on the untidal specific binding energy, which is the convention the screen in the diagnostics uses and the one Salz et al. (2016) ⁹ share, and the tidal factor enters the rate, not the potential the threshold is quoted against. The dispatcher offers that fitted efficiency as an option, while the entry point itself treats \(\epsilon\) as an input.

Two radii enter the problem. \(R_\mathrm{p}\) is the planetary (interior) radius. \(R_\mathrm{XUV}\) is the radius at which the atmosphere becomes optically thick to XUV photons. In PROTEUS it is recomputed at each time step from the atmospheric structure at the pressure set by escape.zephyrus.Pxuv (default \(5 \times 10^{-5}\) bar). The dispatcher takes it instead at its photospheric working level, 20 mbar by default, the photosphere-type level of Baumeister et al. (2023) ⁷. Either level is a bookkeeping radius for the intercepting area: the XUV heating is actually deposited, and the wind launched, at the far lower pressure (of order a nanobar) where the gas first reaches unit optical depth to ionising photons ⁶, and the escape-regime dispatcher locates that launching level on the profile when it needs it.

Radius scaling

The cubic radius term in Eq. (1) is selected at runtime by the scaling argument of EL_escape:

Both forms reduce to \(R_\mathrm{p}^3\) when \(R_\mathrm{XUV} = R_\mathrm{p}\), the conservative lower bound adopted by Luger & Barnes (2015) ¹⁰ and Moore et al. (2023) ¹¹. Allowing \(R_\mathrm{XUV} > R_\mathrm{p}\) increases the effective XUV-absorbing area and therefore the escape rate.

Tidal correction

When the tidal_contribution flag is True, the effective potential barrier is reduced by the host star's tidal field, following Erkaev et al. (2007), their Eq. (17) ³:

with the periapsis Hill radius

where \(a\) is the semi-major axis, \(e\) the orbital eccentricity, and \(M_\star\) the stellar mass. The radius \(R\) in \(\xi\) is the one that appears linearly in the \(R^3\) term of Eq. (1): \(R_\mathrm{p}\) for scaling=2 (the convention of Erkaev et al. 2007, whose own \(\xi\) is the Roche-lobe distance over the planetary radius) and \(R_\mathrm{XUV}\) for scaling=3, where the XUV radius is the only radius in the problem.

Factoring the numerator gives \(K_\mathrm{tide} = (\xi - 1)^2\,(2\xi + 1) / (2\xi^3)\), non-negative for every \(\xi > 0\) with a double root at \(\xi = 1\). In the physical regime \(\xi > 1\) it lies in \((0, 1)\), rising toward 1 for \(\xi \gg 1\) (the escape level deep inside the Hill sphere) and falling toward 0 as the atmosphere expands toward the Roche lobe at \(\xi = 1\). Because the rate divides by \(K_\mathrm{tide}\), the tidal correction enhances escape and diverges at the Roche lobe, so the tidally corrected rate is defined only for \(\xi > 1\): EL_escape raises a ValueError for \(\xi \le 1\), where the atmosphere reaches the Roche lobe and the energy-limited approximation no longer applies. When tidal_contribution is False, \(K_\mathrm{tide} = 1\). The dispatcher handles the \(\xi \le 1\) geometry itself, as the roche_overflow label.

When the prescription applies

The EL form is appropriate in the high-irradiation, collisional-wind regime that dominates the loss during roughly the first \(10^6\) to \(10^8\) years of a close-in planet's evolution ¹². Outside it, at lower XUV flux or for a less extended atmosphere, particle-by-particle (nonthermal and Jeans) escape becomes comparable or dominant and the bulk EL prescription no longer applies. The dispatcher classifies each state before choosing a rate. When using EL_escape alone, verify that the XUV-driven loss genuinely dominates (for scale, present-day nonthermal rates for an Earth-mass planet are of order \(10^4\) to \(10^5\) kg s\(^{-1}\), Kislyakova et al. 2014 ¹²).

Bulk removal is the second assumption. When the escaping particle flux drops below the critical value needed to drag a heavy species along, the outflow fractionates: hydrogen escapes preferentially and the residual atmosphere is enriched in heavy species ¹³¹⁴. For scale, the critical flux for water in a hydrogen background is about \(1.9 \times 10^{5}\) kg s\(^{-1}\) (Yoshida et al. 2022 ¹⁵). EL_escape removes everything in bulk, and the fractionation page describes the closure that resolves the partition when the regime framework confirms a wind.