The Cosmic Shoreline: The Evidence that Escape Determines which Planets Have Atmospheres, and what this May Mean for Proxima Centauri B

We consider isothermal atmospheres in which the major gas is in vapor pressure equilibrium with condensed volatiles at the surface. We call these Clausius-Clapeyron (CC) atmospheres because they are controlled by vapor pressure (Lehmer et al. 2017). Clausius-Clapeyron atmospheres are fairly common in the solar system, with Pluto, Triton, and Mars being notable examples. Our model resembles an earlier Jeans escape model used by Schaller & Brown (2007) and Brown (2012) to address thermal escape from Kuiper Belt objects. The chief difference is that we are addressing much faster escape rates, conditions where Jeans escape would be misapplied. Perez-Becker & Chiang (2013) investigated similar models for evaporation of very hot silicate planets. Limitations and uncertainties stemming from the isothermal approximation along with some alternative approximations to planetary winds that may seem more realistic are discussed in Section 9, and examined in more detail in the Appendices.

Isothermal hydrodynamic escape is described in terms of the isothermal sound speed ${c}_{\circ }^{2}={k}_{B}{T}_{c}/m$, where T_c is the temperature and m is the mean molecular mass of the atmosphere. Pressure p is that of a perfect gas $p=\rho \,{c}_{\circ }^{2}$ with density ρ. The outflow is described in spherical (radial) geometry in steady-state by continuity,

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial r}(\rho {{ur}}^{2})=0,\end{eqnarray} \tag{ 5 }$$

where u is the outflow velocity and r is the radial coordinate increasing outward, and by the force balance of inertia, pressure, and gravity,

$$\begin{eqnarray}&&u\displaystyle \frac{\partial u}{\partial r}+\displaystyle \frac{1}{\rho }\displaystyle \frac{\partial p}{\partial r}=-\displaystyle \frac{{GM}}{{r}^{2}},\end{eqnarray} \tag{ 6 }$$

in which M is the mass of the planet and G is the universal gravitational constant. These can be combined into a single isothermal planetary wind equation,

$$\begin{eqnarray}&&({u}^{2}-{c}_{\circ }^{2})\displaystyle \frac{1}{u}\displaystyle \frac{\partial u}{\partial r}=\displaystyle \frac{2{c}_{\circ }^{2}}{r}-\displaystyle \frac{{GM}}{{r}^{2}}.\end{eqnarray} \tag{ 7 }$$

In a strongly bound atmosphere, ${u}^{2}\ll {c}_{\circ }^{2}$ and $2{c}_{\circ }^{2}\ll {GM}/{r}^{2}$ near the surface, so that $\partial u/\partial r\gt 0$. At large radii the geometric term $2{c}_{\circ }^{2}/r$ term eventually surpasses the gravity term, and the right-hand side of Equation (7) changes sign. At the critical distance r_c where $2{c}_{\circ }^{2}/r={GM}/{r}^{2}$, zeroing the left-hand side of Equation (7) requires either that ${u}^{2}={c}_{\circ }^{2}$ or that $\partial u/\partial r=0$. The unique solution with $\partial u/\partial r\gt 0$, in which the velocity is equal to the sound speed ${u}_{c}={c}_{\circ }$ at the critical point ${r}_{c}={GM}/2{c}_{\circ }^{2}$, is the critical transonic solution for the wind and is the physical solution, provided that the ram pressure of the wind $\rho {u}_{c}^{2}$ at the critical point exceeds any background pressure exerted by interplanetary space.

Equation (7) is easily integrated for the upward velocity at the surface, u_s, in terms of the isothermal temperature T_c and the critical point conditions. Where escape is modest and ${u}_{s}^{2}\ll {c}_{\circ }^{2}$,

$$\begin{eqnarray}&&{u}_{s}\approx {c}_{\circ }{\left(\displaystyle \frac{{r}_{c}}{{r}_{s}}\right)}^{2}\exp \left(\displaystyle \frac{3}{2}-\displaystyle \frac{{GM}}{{c}_{\circ }^{2}{r}_{s}}\right)\end{eqnarray} \tag{ 8 }$$

gives a good approximation to u_s. The density and pressure at the surface of a CC atmosphere are determined by the saturation vapor pressure at the surface temperature T_s,

$$\begin{eqnarray}&&{\rho }_{s}=\displaystyle \frac{m\,{p}_{s}({T}_{s})}{{k}_{B}{T}_{s}}.\end{eqnarray} \tag{ 9 }$$

For display in Figure 1 we use empirical expressions for the vapor pressures of CH₄, N₂, and H₂O given by Fray & Schmidt (2009). For the isothermal CC atmosphere, the rate of mass loss is a function of T_s and u_s

$$\begin{eqnarray}&&\dot{M}=4\pi {\rho }_{s}{u}_{s}{r}_{s}^{2}.\end{eqnarray} \tag{ 10 }$$

The other determining equation is the balance between the power absorbed from sunlight and the sum of power radiated in the thermal infrared and power spent evaporating, heating, and lifting warm gas to space,

$$\begin{eqnarray}&&\displaystyle \frac{\pi {R}^{2}{L}_{\star }}{4\pi {a}^{2}}=\displaystyle \frac{4\pi {R}^{2}\sigma {T}_{s}^{4}}{1-\alpha }+\displaystyle \frac{{GM}\dot{M}}{R}+\dot{M}{L}_{v},\end{eqnarray} \tag{ 11 }$$

where α is the Bond albedo, which for icy worlds we take as $\alpha =0.4$. The rightmost term takes into account the latent heat of vaporization L_v, an important part of the energy budget for small icy bodies. Here we are interested in whether a planet can hold an atmosphere for billions of years. In these cases escape is slow enough that the terms in Equation (11) involving $\dot{M}$ are negligible, so that the surface temperature is just the effective temperature.

Empirically, upper atmospheres of the CH₄-rich planets in the solar system are roughly twice as hot at high altitudes as they are at the surface (Table 1). The upper atmospheres are warm because (i) CH₄ absorbs sunlight and (ii) CH₄ photolysis produces organic molecules and hazes that absorb sunlight, but at low temperatures neither CH₄ nor the hazes radiate as effectively as they absorb. The upper atmosphere temperatures are higher on Uranus and Neptune because the background gas is H₂, so that additional radiative coolants are limited to hydrocarbons like C₂H₂. On Titan, Triton, and Pluto, the background gas is N₂, which enables production of a wider variety of more efficient radiative coolants, HCN especially. For the CH₄–N₂ atmospheres of icy worlds we let reality be a guide and approximate the atmospheric temperature with ${T}_{s}=0.6\,{T}_{c}$.

Table 1.  Surface Temperatures and Mesosphere Temperatures of CH₄-rich Atmospheres

Planet	Surface Ts	Mesosphere Tc	${T}_{c}/{T}_{s}$	References
Titan	94	150–190	0.50–0.63	Koskinen et al. (2011)
Triton	38	50–60	0.63–0.76	Olkin et al. (1997)
Pluto	42	72	0.6	Gladstone et al. (2016)
Uranus	60a	110–150	0.4–0.55	Marten et al. (2005)
Neptune	60a	110–150	0.4–0.55	Marten et al. (2005)

Note.

^aThese are the effective temperatures.

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The black and green curves labeled "CH₄" and "N₂" are evaporation lines for the small icy planets. The density is set to $\rho =2$ g cm⁻³, similar to the densities of Triton, Titan, Ganymede, Callisto, and Pluto. The curves are computed for a star of age ${\tau }_{\star }=5$ billion years and an atmophile mass fraction ${\rm{\Delta }}M/M=0.01$, where ${\rm{\Delta }}M=\dot{M}{\tau }_{\star }$. It is interesting that the nominal curves "CH₄" and "N₂" resemble what is actually seen in the solar system. However, the results shown here are quite sensitive to ${T}_{c}/m$ (the sound speed, squared), and consequently they are insensitive to everything else. We can regard the CC model as a descriptive model, in the sense that it provides a plausible explanation of what is observed, but it may prove difficult to implement as a prescriptive model, because in general T_c and m are hard to predict theoretically.

The solid blue curve is the comparable ${\tau }_{\star }=5\,\mathrm{Gyr}$ evaporation line for H₂O from planets scaled from volatile-enriched versions of Earth ($\rho =5.5$, ${\rm{\Delta }}M/M=0.01$), Europa ($\rho =3$ g/cm³, ${\rm{\Delta }}M/M=0.1$), and Ganymede ($\rho =2$, ${\rm{\Delta }}M/M=0.5$)—by chance, these three cases are nearly indistinguishable in this plot, so we show them as one curve. For these models we set T_c = T_s as, unlike the case for N₂-CH₄ atmospheres, we know of no good reason nor have we seen much evidence to suggest that the upper atmospheres of watery worlds should be especially hot or cold.

The blue water line is to the asteroids as the methane line is to the KBOs, whether by accident or design, but unlike the case for the methane line, which approaches Pluto, Triton, and Titan, the water line comes nowhere close to explaining the terrestrial planets. The water line can be moved into the vicinity of the terrestrial planets by raising ${c}_{\circ }^{2}={k}_{B}T/m$ by an order of magnitude. This can be done either by converting the H₂O into H₂ or by invoking a hot upper atmosphere. The result of raising ${c}_{\circ }^{2}$ is illustrated by the dashed blue curve, computed from the same CC model as for water but with $m=2{m}_{{\rm{H}}}$. Such a model may also be relevant to Hayashi-like primary nebular atmospheres, as it is reasonable to anticipate chemical equilibration and exchange between H₂, H₂O, and silicates at the surface if the atmosphere is deep (Hayashi et al. 1979; Sekiya et al. 1980a, 1981; Ikoma & Genda 2006).

The runaway greenhouse threshold is expected to be a weak function of planetary parameters. To illustrate, assume that the runaway greenhouse limit is set by a troposphere saturated with water vapor becoming optically thick (Nakajima et al. 1992). In the absence of pressure broadening, optical depth τ will scale as the column depth; with pressure broadening, this will be multiplied by the pressure to a power ξ on the order of unity (Pollack 1969; Robinson & Catling 2012; Goldblatt et al. 2013; Robinson & Catling 2014). For a pure water vapor atmosphere, these considerations imply that (Goldblatt 2015)

$$\begin{eqnarray}&&\tau =\kappa \displaystyle \frac{{p}_{\mathrm{vap}}^{1+\xi }}{g},\end{eqnarray} \tag{ 12 }$$

where κ is an opacity. If we approximate the vapor pressure of water by a simple exponential

$$\begin{eqnarray}&&{p}_{\mathrm{vap}}={p}_{w}\exp (-{T}_{w}/T),\end{eqnarray} \tag{ 13 }$$

with T_w = 6000 K and ${p}_{w}=2.1\times {10}^{7}$ bars, the runaway greenhouse limit should scale as

$$\begin{eqnarray}&&{I}_{r}\propto {T}^{4}\propto {\left(\displaystyle \frac{(1+\xi ){T}_{w}}{\mathrm{ln}(\kappa {p}_{w}^{1+\xi }/g)}\right)}^{4},\end{eqnarray} \tag{ 14 }$$

an expression that is well approximated over the range of interest by ${I}_{r}\propto {g}^{0.2/(1+\xi )}$. The surface gravity g is expressed in terms of ${v}_{\mathrm{esc}}$ and planet density ρ by ${g}^{2}\propto \rho {v}_{\mathrm{esc}}^{2}$. This leaves

$$\begin{eqnarray}&&{I}_{r}\propto {({v}_{\mathrm{esc}}^{2}\rho )}^{0.1/(1+\xi )}.\end{eqnarray} \tag{ 15 }$$

This relation is plotted with $\xi =1$ and $\rho =3$ in Figure 1 as the "H₂O runaway greenhouse." GCM-based estimates of the onset of the runaway range between 1.1 to 1.7 solar constants, with details of cloud structure and cloud physics and planetary rotation responsible for much of the variance (Abe et al. 2011; Leconte et al. 2013a, 2013b; Wolf & Toon 2015; Way et al. 2016). The range of uncertainty is encompassed by the thickness of the plotted line.

Figure 1 shows the Moon rising above the intersection of the water lines. In all likelihood, the apparent desiccation of the Moon is a memory of how the Moon was made rather than the ruin of a more promising world, but as plotted here, the Moon as a habitable world appears to be marginally unstable against both thermal escape and the runaway greenhouse. Today, most of the lunar surface is unstable to Jeans escape of water (e.g., Catling & Kasting 2017). When viewing the Moon from the viewpoint of terraforming it, both problems would be made more tractable by providing abundant heavy ballast gases to reduce the atmosphere's scale height.

Here we address thermal evaporation of EGPs. The topic has been addressed many times (e.g., Owen & Wu 2013) with much more sophisticated models than we employ here, but for our purposes, it seems best to hold to the isothermal approximation, which has the advantage of being analytic and easy to work with. Models of silicate-rich gas giants have tended to predict warm or even hot upper atmospheres, because small molecules made of rock-forming elements such as TiO are better absorbers of visible light than they are emitters of thermal infrared radiation. However, early reports of thermal inversions on hot Jupiters (e.g., on HD 209458b; Burrows et al. 2007; Knutson et al. 2008) have since been cast into doubt by analyses of more precise data of higher spectral resolution (Line et al. 2016). A difference from the icy worlds is that these planets are close enough to their stars that tidal truncation must be taken into account. This is conveniently done in terms of the Hill sphere distance, which is defined by

$$\begin{eqnarray}&&{r}_{h}=a{\left(\displaystyle \frac{M}{3{M}_{\star }}\right)}^{1/3},\end{eqnarray} \tag{ 16 }$$

where as above, a refers to the star-planet distance and M_⋆ to the mass of the star (Erkaev et al. 2007). Along the star-planet axis, Equation (6) becomes

$$\begin{eqnarray}&&u\displaystyle \frac{\partial u}{\partial r}+\displaystyle \frac{1}{\rho }\displaystyle \frac{\partial p}{\partial r}=-\displaystyle \frac{{GM}}{{r}^{2}}+\displaystyle \frac{{GMr}}{{r}_{h}^{3}},\end{eqnarray} \tag{ 17 }$$

and Equation (7) becomes

$$\begin{eqnarray}&&({u}^{2}-{c}_{\circ }^{2})\displaystyle \frac{1}{u}\displaystyle \frac{\partial u}{\partial r}=\displaystyle \frac{2{c}_{\circ }^{2}}{r}-\displaystyle \frac{{GM}}{{r}^{2}}+\displaystyle \frac{{GMr}}{{r}_{h}^{3}}.\end{eqnarray} \tag{ 18 }$$

Equation (18) assumes spherical symmetry for tidal forces, which is not a good assumption. Its crudeness is probably comparable to treating irradiation as globally uniform or the temperature as isothermal. The critical point is found by solving the cubic for r_c,

$$\begin{eqnarray}&&\displaystyle \frac{2{c}_{\circ }^{2}}{{r}_{c}}-\displaystyle \frac{{GM}}{{r}_{c}^{2}}+\displaystyle \frac{{{GMr}}_{c}}{{r}_{h}^{3}}=0.\end{eqnarray} \tag{ 19 }$$

Equation (18) is easily integrated analytically,

$$\begin{eqnarray}{{ur}}^{2} & = & {u}_{c}{r}_{c}^{2}\exp \left\{\displaystyle \frac{1}{2}\displaystyle \frac{{u}^{2}}{{c}_{\circ }^{2}}-\displaystyle \frac{1}{2}\displaystyle \frac{{u}_{c}^{2}}{{c}_{\circ }^{2}}+\displaystyle \frac{{GM}}{{c}_{\circ }^{2}{r}_{c}}\right.\\ & & -\left.\displaystyle \frac{{GM}}{{c}_{\circ }^{2}r}+\displaystyle \frac{1}{2}\displaystyle \frac{{{GMr}}_{c}^{2}}{{c}_{\circ }^{2}{r}_{h}^{3}}-\displaystyle \frac{1}{2}\displaystyle \frac{{{GMr}}^{2}}{{c}_{\circ }^{2}{r}_{h}^{3}}\right\}.\end{eqnarray} \tag{ 20 }$$

Note that u(r) is independent of density. Near the surface, where ${u}^{2}\ll {c}_{\circ }^{2}$, Equation (20) can be rewritten as an equation for the flow velocity at the surface u_s using the critical point conditions. The flux at the surface ${\rho }_{s}{u}_{s}$ is obtained by multiplying by the surface density ${\rho }_{s}$,

$$\begin{eqnarray}{\rho }_{s}{u}_{s} & = & {\rho }_{s}{c}_{\circ }\displaystyle \frac{{r}_{c}^{2}}{{r}_{s}^{2}}\exp \left\{-\displaystyle \frac{1}{2}+\left(1-\displaystyle \frac{{r}_{s}^{2}}{{r}_{c}^{2}}\right)\left(\displaystyle \frac{{GM}}{2{r}_{c}{c}_{\circ }^{2}}-1\right)\right.\\ & & +\left.\left(1-\displaystyle \frac{{r}_{s}}{{r}_{c}}\right)\displaystyle \frac{{GM}}{{r}_{c}{c}_{\circ }^{2}}\right\}.\end{eqnarray} \tag{ 21 }$$

To use Equation (21) requires choosing ${\rho }_{s}$ at the lower boundary using other information.

In EUV- and XUV-driven escape studies, the lower boundary is typically set at the homopause or at the base of the thermosphere, with the latter defined by a monochromatic optical depth in XUV radiation (e.g., Watson et al. 1981; Tian et al. 2005; Lammer et al. 2013). The homopause can be a reasonable a priori choice if escape is sluggish enough that a homopause exists. However, as we show below in Section 4.2, for a gas giant to evaporate in 5 Gyr, the flux of hydrogen to space is too great by orders of magnitude for a homopause to exist. In a full-featured hydrocode simulation of an XUV-heated wind, the lower boundary condition can be justified a postiori as self-consistent for a particular model by showing the model to be insensitive to changing it (e.g., Watson et al. 1981; Murray-Clay et al. 2009; Owen & Alvarez 2016).

In the simplest picture, a gas giant does not have a well-defined surface. Under these conditions, the whole planet takes part in the flow to space, with the source of the escaping gas being the shrinking or rarefaction of the interior. To first approximation, all EGPs have roughly the same radius; the typical radius of a hot Jupiter is 84,000 km (Fortney et al. 2008). Constant radius is a property of polytropes with a $p=K{\rho }^{2}$ equation of state. The radius is $R=\sqrt{2K/4\pi G}$, where G is the Newtonian gravitational constant (Hubbard 1973). Using observed radii of the known roster of transiting planets gives $K=3\pm 1\times {10}^{12}$ cm⁵ g⁻¹ s⁻². The interior is described by an analytic solution,

$$\begin{eqnarray}&&\rho (r)=\displaystyle \frac{\pi M}{4{R}^{3}}\displaystyle \frac{R}{\pi r}\sin \left(\displaystyle \frac{\pi r}{R}\right).\end{eqnarray} \tag{ 22 }$$

We set the lower boundary where the inner polytrope and the outer isothermal envelope meet; i.e., where the polytropic pressure equals the pressure of an ideal gas at the planet's effective temperature ${T}_{\mathrm{eff}}$. This gives

$$\begin{eqnarray}&&{\rho }_{\mathrm{lbc}}={c}_{\circ }^{2}/K.\end{eqnarray} \tag{ 23 }$$

The other equation pertinent to escape is the global energy balance,

$$\begin{eqnarray}\displaystyle \frac{\pi {R}^{2}{L}_{\star }}{4\pi {a}^{2}} & = & \displaystyle \frac{4\pi {R}^{2}\sigma {T}_{s}^{4}}{1-\alpha }+\displaystyle \frac{{GM}\dot{M}}{R}\left(1-\displaystyle \frac{3}{2}\displaystyle \frac{R}{{r}_{h}}+\displaystyle \frac{1}{2}\displaystyle \frac{{R}^{3}}{{r}_{h}^{3}}\right)\\ & & +\dot{M}{c}_{p}({T}_{c}-{T}_{s}).\end{eqnarray} \tag{ 24 }$$

The terms involving $\dot{M}$ are the work done against gravity and any excess heat left in the gas as it escapes. As was the case in Section 2.1 above, if thermal escape is extended over 5 billion years, the $\dot{M}$ terms in Equation (24) are negligible, and Equation (24) reduces to the usual expression for effective temperature,

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{\star }}{4\pi {a}^{2}}\approx \displaystyle \frac{4\sigma {T}_{\mathrm{eff}}^{4}}{1-\alpha }.\end{eqnarray} \tag{ 25 }$$

For the EGPs we set the Bond albedo $\alpha =0.1$, $m=2.4{m}_{{\rm{H}}}$, ${R}_{s}=8.4\times {10}^{9}\,\mathrm{cm}$, and in keeping with the isothermal assumption, we set ${T}_{c}={T}_{s}={T}_{\mathrm{eff}}$. The magenta curve in the upper right-hand region of Figure 1 is computed for complete evaporation of the planet in ${\tau }_{\star }=5$ Gyr; i.e., we solve Equations (21), (23), and (24) for ${v}_{\mathrm{esc}}$ such that $\dot{M}{\tau }_{\star }=M$. The simple model bounds the population of EGPs (blue disks) rather nicely. We stress that this is not an XUV-driven escape model. The planet evaporates because the planet is thermally unstable, not because XUV heating is removing the outer atmosphere.

It is possible to quantify the predictions of the XUV hypothesis if the escape is energy limited. Energy-limited escape is expected if the XUV radiation is too great for the incident radiation that can be thermally conducted to the lower atmosphere (Watson et al. 1981). If tidal truncation is neglected for the moment, the energy-limited escape can be expressed as

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{el}}=\displaystyle \frac{\eta \pi {R}^{3}{L}_{\mathrm{xuv}}(t)}{4\pi {a}^{2}{GM}},\end{eqnarray} \tag{ 28 }$$

where $\eta$ is an efficiency factor that is usually taken to be $0.1\lt \eta \lt 0.6$ (e.g., Lammer et al. 2013; Owen & Wu 2013; Koskinen et al. 2014; Bolmont et al. 2017). The mass-loss efficiency η is lower than the heating efficiency (fraction of incident XUV energy converted into heat) because the escaping gas is hotter, more dissociated, and more ionized than it was before it was irradiated. The factor η is a function of T, being smaller in cooler gas (<3000 K) in which ${{\rm{H}}}_{3}^{+}$ is a major radiative coolant (Koskinen et al. 2014) and smaller in hot gas ($\sim {10}^{4}$ K) in which collisionally excited Lyα is a major coolant (Murray-Clay et al. 2009). In a steam atmosphere, the far-ultraviolet (FUV, $100\lt \lambda \lt 200$ nm) can also be important because it is absorbed by H₂O, O₂, and CO₂, provided irradiation is modest enough to leave the molecules intact (Sekiya et al. 1981). The contribution of the FUV is implicitly folded into η. The total mass loss

$$\begin{eqnarray}&&{\rm{\Delta }}{M}_{\mathrm{el}}={\int }_{0}^{{\tau }_{\star }}{\dot{M}}_{\mathrm{el}}\,{dt}\end{eqnarray} \tag{ 29 }$$

is obtained by integrating the star's XUV radiation history expressed in terms of the Sun's history using ${I}_{\mathrm{xuv}}\,=({L}_{\mathrm{xuv}}/{L}_{\mathrm{xuv}\odot }){({a}_{\oplus }/a)}^{2}$. For the XUV history of the Sun itself, we follow Ribas et al. (2005, 2016),

$$\begin{eqnarray}{L}_{\mathrm{xuv}\odot } & = & {L}_{x}{(t/{t}_{x})}^{-\beta }\qquad t\gt {t}_{x}\\ {L}_{\mathrm{xuv}\odot } & = & {L}_{x}\qquad t\lt {t}_{x},\end{eqnarray} \tag{ 30 }$$

where ${t}_{x}=0.1\,\mathrm{Gyr}$, $\beta =1.24$, and where

$$\begin{eqnarray}&&{L}_{x}=1.6\times {10}^{30}{({t}_{x}/{t}_{\odot })}^{-\beta }\quad \mathrm{ergs}\,{{\rm{s}}}^{-1}\end{eqnarray} \tag{ 31 }$$

is the saturated upper bound on the Sun's youthful excess. Cumulative escape is then

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}{M}_{\mathrm{el}}}{M}=\displaystyle \frac{\eta {L}_{x}{t}_{x}}{{a}_{\oplus }^{2}}\sqrt{\displaystyle \frac{3G}{8\pi }}\displaystyle \frac{{I}_{\mathrm{xuv}}}{{v}_{\mathrm{esc}}^{3}\sqrt{\rho }}\displaystyle \frac{1}{\beta -1}(\beta -{({t}_{x}/{\tau }_{\star })}^{\beta -1}),\end{eqnarray} \tag{ 32 }$$

which can be rearranged as a linear relation between ${I}_{\mathrm{xuv}}$ and x

$$\begin{eqnarray}&&{I}_{\mathrm{xuv}}=x\sqrt{\displaystyle \frac{8\pi }{3G}}\displaystyle \frac{{\rm{\Delta }}{M}_{\mathrm{el}}}{M}\displaystyle \frac{{a}_{\oplus }^{2}(\beta -1)}{\eta {L}_{x}{t}_{x}(\beta -{({t}_{x}/{\tau }_{\star })}^{\beta -1})},\end{eqnarray} \tag{ 33 }$$

with x defined by

$$\begin{eqnarray}&&x\equiv {v}_{\mathrm{esc}}^{3}\sqrt{\rho }.\end{eqnarray} \tag{ 34 }$$

Results are plotted in terms of the parameter x in Figure 3 for $\eta =0.2$ as green lines and labeled for a range of lost masses $2\times {10}^{-4}\leqslant {\rm{\Delta }}{M}_{\mathrm{el}}/M\leqslant 0.2$. As in Figure 2, we apply ${L}_{\mathrm{xuv}}\propto {L}_{\star }^{0.4}$ to the data in order to present a single relation that spans all the planets. The solar system is fit by ${\rm{\Delta }}{M}_{\mathrm{el}}/M\approx 0.001$, a trend that does not extend to the exoplanets, which are better matched by ${\rm{\Delta }}{M}_{\mathrm{el}}/M\approx 0.1$.

The diffusion-limited flux gives the upper bound on how quickly hydrogen can selectively escape by diffusing through a heavier gas that does not escape; more properly, it is the upper limit on the difference between hydrogen escape and heavy-constituent escape. It is often thought of in the context of vigorous hydrogen escape driven by XUV radiation (Sekiya et al. 1980b; Zahnle & Kasting 1986; Hunten et al. 1987), but it is quite general (Hunten & Donahue 1976). The diffusion-limited flux regulates hydrogen escape or H₂ abundance on Venus, Earth, Mars, and Titan today (see review by Catling & Kasting 2017). It is likely that, should the diffusion-limited flux be lower than the energy limit, the H₂ mixing ratio $f({{\rm{H}}}_{2})$ will increase until the two limits are equal, as seen on Titan and Mars. If $f({{\rm{H}}}_{2})\to 1$, and the energy limit still exceeds the diffusion limit, it is not obvious what happens. The atmosphere may either escape as a whole (although the heavier gases escape more slowly, so that the remnant atmosphere becomes mass fractionated), or hydrogen escape is throttled to the diffusion limit and the excess energy is radiated to space by the heavy gases. A possible example of escape at the diffusion limit among the EGPs is HD 209458b (Vidal-Madjar et al. 2003, 2004; Yelle 2004; Koskinen et al. 2013).

The upper bound on the escape flux of a gas species of molecular mass m_i from a static gas atmosphere of molecular mass m_j (${m}_{i}\lt {m}_{j}$) in the diffusion limit from an isothermal atmosphere is

$$\begin{eqnarray}&&{\phi }_{i,\mathrm{dl}}={f}_{i}\displaystyle \frac{{GM}({m}_{j}-{m}_{i})}{{R}^{2}}\displaystyle \frac{{b}_{{ij}}}{{kT}},\end{eqnarray} \tag{ 35 }$$

where f_i is the mixing ratio of the light gas and b_ij is the binary diffusion coefficient between the two species i and j. Typically, ${b}_{{ij}}\propto {T}^{0.75}$. The corresponding mass-loss rate is

$$\begin{eqnarray}&&\dot{M}=4\pi {R}^{2}{m}_{i}{\phi }_{i,\mathrm{dl}}.\end{eqnarray} \tag{ 36 }$$

The timescale for losing an atmosphere of mass ${\rm{\Delta }}{M}_{\mathrm{dl}}$ is ${\tau }_{\mathrm{dl}}={\rm{\Delta }}{M}_{\mathrm{dl}}/\dot{M}$. The upper bound that diffusion places on atmospheric escape can be written

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}{M}_{\mathrm{dl}}}{M}=\displaystyle \frac{4\pi {m}_{i}{f}_{i}({m}_{j}-{m}_{i}){b}_{{ij}}G{\tau }_{\mathrm{dl}}}{{kT}}.\end{eqnarray} \tag{ 37 }$$

It is notable that this ratio is very nearly independent of planetary parameters—i.e., this constraint is the same for all planets. For H₂ escaping through N₂, ${b}_{{ij}}\approx 1.5\times {10}^{19}{(T/300)}^{0.75}$, for which

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}{M}_{\mathrm{dl}}}{M}=0.001f({{\rm{H}}}_{2}){(1000/T)}^{0.25}{\tau }_{\mathrm{Gyr}},\end{eqnarray} \tag{ 38 }$$

with ${\tau }_{\mathrm{Gyr}}$ measured in Gyr. This means that it is difficult for a planet to selectively lose more than about 0.5% of its mass as H₂ in 5 Gyr, even if H₂ is a major constituent ($f({{\rm{H}}}_{2})\to 1$), as it often appears to be on exoplanets. In many XUV-limited escape scenarios, the time available for energy-limited escape is shorter than a few hundred million years, which reduces the maximum differential H₂ loss to less than 0.05% of the planet's mass. Whether this is an important constraint depends on several factors. If H₂ is overwhelmingly abundant and the heavy gases are inefficient radiative coolants, they can be carried along, and ${\phi }_{i,\mathrm{dl}}$ becomes the difference between H₂ escape and heavy gas escape (Sekiya et al. 1981; Zahnle & Kasting 1986). If the heavy gases condense, they can be separated from H₂ by precipitation and the gas diffusion limit does not apply. But for warm planets with considerable reservoirs of volatiles other than H₂, the constraint may set the boundary between planets that evolve to a vaguely Earth-like state versus those that never progress past a vaguely Neptune-like state. That XUV-driven escape should lead to such a bimodal distribution of planets is an observation that has also been made on the basis of the limited XUV energy available (Owen & Wu 2013).

The simple linear relation between ${I}_{\mathrm{xuv}}$ and x in Equation (33) does not apply for the close-in planets that are afflicted by tidal truncation. For these we need to include the Hill sphere terms, which break the ${I}_{\mathrm{xuv}}\propto x$ relation. For these planets we start with a tidally truncated XUV-heated energy-limited escape flux,

$$\begin{eqnarray}&&\displaystyle \frac{\pi {R}^{2}\eta {L}_{\mathrm{xuv}}}{4\pi {a}^{2}}=\displaystyle \frac{{GM}{\dot{M}}_{\mathrm{el}}}{R}\left(1-\displaystyle \frac{3}{2}\displaystyle \frac{R}{{r}_{h}}+\displaystyle \frac{1}{2}\displaystyle \frac{{R}^{3}}{{r}_{h}^{3}}\right).\end{eqnarray} \tag{ 39 }$$

As above, we treat EGPs as all having the same radius R. With R held constant, M can be replaced by ${v}_{\mathrm{esc}};$ the star-planet distance a can be replaced by ${I}_{\mathrm{xuv}};$ and M_⋆ is replaced M_⊙ by expressing ${L}_{x}/{L}_{x\odot }\propto {({L}_{\star }/{L}_{\odot })}^{0.4}\propto {({M}_{\star }/{M}_{\odot })}^{1.5}$, in which the stellar mass–luminosity relationship is conveniently written in the form ${L}_{\star }\propto {M}_{\star }^{3.75}$. With these relations, the ${L}_{x}/{L}_{x\odot }$ ratio cancels out of Equation (39). The resulting expression between ${I}_{\mathrm{xuv}}$ and ${v}_{\mathrm{esc}}$ should hold approximately for all main-sequence stars and their giant planets,

$$\begin{eqnarray}\displaystyle \frac{{a}_{3}{I}_{\mathrm{xuv}}}{{v}_{\mathrm{esc}}^{4}} & = & \displaystyle \frac{{\rm{\Delta }}{M}_{\mathrm{el}}}{M}\left(1-{a}_{1}\displaystyle \frac{{I}_{\mathrm{xuv}}^{1/2}}{{v}_{\mathrm{esc}}^{2/3}}+{a}_{2}\displaystyle \frac{{I}_{\mathrm{xuv}}^{3/2}}{{{Pv}}_{\mathrm{esc}}^{2}}\right)\\ {a}_{1} & = & \displaystyle \frac{3}{2}\displaystyle \frac{R}{{{Pa}}_{\oplus }}{\left(\displaystyle \frac{6{{GM}}_{\odot }}{R}\right)}^{1/3}\\ {a}_{2} & = & \displaystyle \frac{1}{2}\displaystyle \frac{{R}^{3}}{{a}_{\oplus }^{3}}\displaystyle \frac{6{{GM}}_{\odot }}{R}\\ {a}_{3} & = & \displaystyle \frac{{GR}}{{a}_{\oplus }^{2}}\displaystyle \frac{\eta {L}_{x}{t}_{x}}{(\beta -1)}(\beta -{({t}_{x}/{\tau }_{\star })}^{\beta -1}).\end{eqnarray} \tag{ 40 }$$

Equation (40) is readily solved for ${I}_{\mathrm{xuv}}$ as a function of ${v}_{\mathrm{esc}}$. Results are plotted for ${\rm{\Delta }}{M}_{\mathrm{el}}/M=0.2$ with $\eta =0.2$ as the dashed magenta lines in Figures 2 and 3. For the particular case with R held constant, $x\propto {v}_{\mathrm{esc}}^{4}$, so the curves are uniquely defined in both plots. As has been pointed out by others (e.g., Owen & Wu 2013), the quantitative predictions made by the simple XUV model are good enough to be intriguing.

Proxima b's escape velocity is uncertain because we do not know its radius; we do not know whether it is a globe of air, water, earth, or metal. The radial velocity gives $M\sin i=1.27\,{M}_{\oplus }$. We presume the median nominal mass of $M=1.27/\sin 60^\circ =1.47\,{M}_{\oplus }$. If rocky, using $M\propto {R}^{3.7}$ (Zeng et al. 2016), we estimate ${v}_{\mathrm{esc}}=12.7$ km s⁻¹. To set a rough upper uncertainty, we take a more face-on $\sin 30^\circ$ orbit, for which ${v}_{\mathrm{esc}}=15.3$ km s⁻¹. To set a rough lower uncertainty, we presume that Proxima b is ice-rich with a bulk density half that of a rocky world of the same mass, for which ${v}_{\mathrm{esc}}=10.2$ km s⁻¹.

For Proxima specifically, Ribas et al. (2016) estimate both the current and the cumulative relative XUV irradiations of Proxima b and Earth. For the present, they estimate that ${I}_{\mathrm{xuv}}/{I}_{\oplus }\approx 60$. For the cumulative total, they estimate that ${I}_{\mathrm{xuv}}/{I}_{\oplus }\approx 16$. We plot both estimates in Figures 2 and 3, each with a factor 3 uncertainty. In Figure 2, Proxima b appears relatively vulnerable to XUV-driven escape. Only a large and dense Proxima b orbiting an XUV-quiet Proxima plots with the terrestrial planets of our solar system. Otherwise, Proxima b plots with the least of the extrasolar Neptunes and a few other smaller planets. When the ensemble is replotted according to the expectations of energy-limited flux, however, Proxima b looks rather ordinary (Figure 3). Figure 3 also suggests that Proxima b at 0.05 au has intercepted enough XUV energy to drive off about 1% of its mass; a water-world Proxima b should be durable to XUV radiation if it could be made in the first place, but a more Earth-like hydrosphere could be vulnerable to being wholly lost.

After it formed, Proxima is presumed to have slowly faded to the main sequence like any small M dwarf. Figure 1 shows insolation levels from Ribas et al. (2016) at three times: when Proxima was just 10 Myr old, when Proxima was 100 Myr old, and today. With respect to total insolation, Proxima b is much like Earth or Venus in Figure 1. As Ribas et al. (2016) and Barnes et al. (2017) and many others have pointed out, when Proxima was young, insolation exceeded the runaway greenhouse threshold at Proxima b's current 0.05 au distance for the first ∼150 Myr or so of their mutual existence (Barnes et al. 2017). This means that Proxima b, if it had water when young, would have held it initially in the form of steam, which eliminates the cold trap as a bottleneck to hydrogen escape. (Other ways to eliminate the cold trap as a bottleneck to escape is to make water vapor a major constituent, as can happen if other gases are scarce (Wordsworth & Pierrehumbert 2013), or to invoke a so-called "moist greenhouse" stratosphere, the reality of which is in active debate (Kasting 1988; Kasting et al. 2015; Leconte et al. 2013a; Wolf & Toon 2015).) If hydrogen escape is restricted to the runaway greenhouse epoch, estimates of the total XUV-driven energy-limited escape range from less than an Earth ocean of water Ribas et al. (2016) to 3–10 oceans (Barnes et al. 2017), a difference that can be attributed to assumptions about Proxima as a young star.

It has been suggested that if the source of escaping hydrogen is water, O₂ might build up in the atmosphere at the diffusion-limited rate, and if the hydrogen from several oceans of water escaped, it might be possible for hundreds of bars of O₂ to accumulate in the atmosphere that is left behind (Luger & Barnes 2015; Schaefer et al. 2016; Barnes et al. 2017). However, these models do not account for atmospheric photochemical reactions between oxygen and hydrogen that can reduce the H₂ mixing ratio and thus throttle hydrogen escape. For example, the hydrogen escape rate from Mars is currently very slow because the strong negative feedback between oxygen and hydrogen ensures that both are lost from the atmosphere in the 1:2 ratio of the parent molecule (Hunten & Donahue 1976). The same 1:2 ratio holds for oxygen and hydrogen escape from Venus today (Fedorov et al. 2011). Moreover, iron in a vigorously convecting mantle has the capacity to consume thousands of bars of O₂. For example, Gillmann et al. (2009) and Hamano et al. (2013) dispose of the excess oxygen generated by hydrogen escape from Venus's accretional steam atmosphere by placing it into the mantle while still mostly molten under a steam atmosphere. In any event, O₂ on Venus has yet to be detected (Fegley 2014). Schaefer et al. (2016) include aspects of the kinetics of the mantle sink. If the XUV is very large, hydrogen escape can in principle be vigorous enough to drag the oxygen that is liberated by water photolysis into space (Zahnle & Kasting 1986; Luger & Barnes 2015; Schaefer et al. 2016). For example, Zahnle & Kasting (1986, Figure 8) showed that for conditions germane to a steam atmosphere on Venus, molecular diffusion ensures that oxygen escape must exceed the surface sink on oxygen, no matter how efficient the latter. Schaefer et al. (2016) reached a similar conclusion for GJ 1132b and other worlds. When hydrogen escape exceeds the diffusion limit, the diffusion limit becomes the rate at which oxygen is left behind to oxidize the planet or accumulate in the atmosphere. However, in addition to not including atmospheric chemistry, neither model took into account that because the mixed wind is heavier, it must be hotter than pure hydrogen and thus has more power to cool itself radiatively.

The history and nature of impacts experienced by Proxima b are almost wholly conjectural (Coleman et al. 2017). Still, impacts happen. It may be helpful to divide impactors into three general classes: (i) material co-orbital with Proxima b; (ii) material in orbits about Proxima (analogous to the Sun's asteroid and Kuiper belts), and (iii) material in orbits about α Centauri A or B or both (Kuiper belts and Oort clouds would be solar system analogs). The first category is swept up very quickly and is better regarded as part of Proxima b's accretion; this is addressed separately below. In the second category we imagine bodies in prograde orbits roughly coplanar with Proxima b and perturbed from relatively distant orbits, with aphelia at 1 au, into highly elliptical Proxima b-crossing orbits. By analogy to impacts on Earth (Bottke et al. 1995), we estimate that typical encounter velocities would be on the order of $\langle {v}_{\mathrm{enc}}\rangle \approx 0.5\mbox{--}0.8\,{v}_{\mathrm{orb}}$, which corresponds to ${v}_{\mathrm{imp}}\approx 30\pm 15$ km s⁻¹. The third category is analogous to the comets and asteroids that strike the Galilean satellites (in mildly hyperbolic orbits with respect to Jupiter), with almost all of the velocity of the stray body attributable to the gravitational well of the central body. We have previously modeled this scenario for the Galilean satellites, taking into account the distribution of impact probabilities associated with the distribution of encounter orbits (Zahnle et al. 1998b). Generalizing from Zahnle et al. (1998b), we estimate that close encounters of the third kind would fall in the range ${v}_{\mathrm{enc}}\approx 1.4-2.4\,{v}_{\mathrm{orb}};$ i.e., we estimate that ${v}_{\mathrm{enc}}\approx 90\pm 25$ km s⁻¹. Cases (ii) and (iii) are plotted on Figure 4. It is apparent at a glance that Proxima b is more vulnerable to the negative consequences of impacts than are Earth and Venus, a not surprising observation that has been anticipated (cf. Lissauer 2007; Raymond et al. 2007). We conclude that almost all the collisions that matter to impact erosion and impact delivery must be from debris orbiting Proxima itself, at velocities that are marginally more erosive than what we see in the inner solar system.

If averaged over 100 Myr, the energy of accretion of a planet like Earth is comparable in magnitude to the insolation received over that same period. This was very important for Earth and Venus because they likely accreted on a 30–100 Myr timespan, and the added energy of accretion pushed both planets above their runaway greenhouse limits (Matsui & Abe 1986; Abe & Matsui 1988; Zahnle et al. 1988; Hamano et al. 2013). For Earth, the steam atmospheres were episodic transients after large impacts, but Venus's steam atmosphere was probably irreversible, and hence led directly to the profound desiccation of Venus's atmosphere and mantle (Hamano et al. 2013).

By contrast to Earth and Venus, Proxima b could have accreted very quickly. Lissauer (2007) showed that in basic Safronov accretion theory, habitable zone (HZ) planets of small M dwarfs are expected to accrete in less than 10⁵ years, orders of magnitude faster than Earth or Venus. Accretion in 10⁵ years implies a surface temperature of 2000 K if airless and perhaps 4000 K if the planet had an atmosphere, which at temperatures like these it most certainly would have had (Lupu et al. 2014). In its potential for rapid accretion, Proxima b is more like a Galilean satellite than a solar system planet. The comparable accretion time for Europa is 200 years, which scarcely seems credible for a small world that retains much water, to say nothing of icy Callisto, accreting in just 3000 years, yet never fully melting. Evidently, Europa's and Callisto's accretions were governed by the supply of new matter from the Sun's accretion disk to Jupiter's accretion disk, rather than by the properties of Jupiter's accretion disk. The ruling timescale then becomes that of forming the solar system as a whole, which appears to have been on the order of 3 million years (and slow enough to preserve a cold Callisto). But Proxima b is much bigger than Europa or Callisto. Even if material were supplied to the Proxima system from an unknown source on a more leisurely 10 million year timescale, Proxima b's accretion would not only be too rapid for water to condense, it would be too rapid for a magma surface to freeze solid unless there were no atmosphere (Lupu et al. 2014).

Here we ask if atmospheric erosion by Proxima's stellar wind has been important. In the solar system, the solar wind erodes through direct collisions (sputtering) and through its magnetic field (ion pickup). The latter in particular is important for Venus and Mars. Venus intercepts about 4000 grams of solar wind per second, estimated using ${\dot{M}}_{\odot }=2\times {10}^{-14}\,{M}_{\odot }$ yr⁻¹ (Wood et al. 2002). Average quiet-Sun observed rates of oxygen ion escape from Venus are much lower, about 150 g ${{\rm{s}}}^{-1}$ (Fedorov et al. 2011). Modeled O⁺ escape rates range from 150 to 800 g ${{\rm{s}}}^{-1}$ (Jarvinen et al. 2009). Escape driven by the solar wind at Mars is more efficient: Mars intercepts about 300 grams of solar wind each second, which is comparable to the observed O⁺ escape rate of 160 g ${{\rm{s}}}^{-1}$ (Brain et al. 2015) and to modeled O⁺ escape rates of 300–500 g ${{\rm{s}}}^{-1}$ (Lammer et al. 2003a). Apparently the solar wind impinging on Mars roughly erodes its own mass in Martian gas although Venus suggests that escape processes are, among other things, sensitive to ${v}_{\mathrm{esc}}$.

Suppose that the observed Martian regime approaches the asymptotic efficiency. This may be a reasonable bound on a process that in broad brush is a problem of turbulent mixing. If so, the planet's mass-loss rate would be equal to the mass of stellar wind intercepted,

$$\begin{eqnarray}&&\dot{M}=\displaystyle \frac{\pi {R}^{2}}{4\pi {a}^{2}}{\dot{M}}_{\mathrm{sw}},\end{eqnarray} \tag{ 42 }$$

where ${\dot{M}}_{\mathrm{sw}}$ is the star's mass-loss rate. The stellar wind is assumed to be strong enough that the planet's cross-section to the wind is comparable to its physical cross-section. Proxima b therefore intercepts $1\times {10}^{-6}$ of ${M}_{\mathrm{sw}}$. We estimate that ${\dot{M}}_{\mathrm{sw}}$ for Proxima today is roughly $3\times {10}^{-15}\,{M}_{\odot }$ yr⁻¹, in which we have scaled the solar wind by XUV (Wood et al. 2002). If we assume that ${\dot{M}}_{\mathrm{sw}}\propto {t}^{-1}$, we estimate that Proxima b has intercepted roughly $1\times {10}^{23}$ g of stellar wind since it first became potentially habitable at 150 Myr of age. This corresponds to $1\times {10}^{-5}$ of Proxima b's mass, or about 10 bars of atmosphere. Losses would be smaller than 1 bar if the extrapolation were based on Venus. These estimates are respectively 2 and 3 orders of magnitude smaller than what XUV can do in the first 150 Myr for an H₂-rich or H₂O-rich atmosphere (Barnes et al. 2017). On the hand, the stellar wind might pose the greatest existential threat to a CO₂ atmosphere at times after the first 150 Myr.

Proxima b may have no good analog in the solar system. Venus seems the best candidate. Venus retains very little water (in the atmosphere, $5\times {10}^{-6}$ of Earth, Fegley 2014) and probably very little in its interior. Gillmann et al. (2009) and Hamano et al. (2013) explain Venus as having been thoroughly desiccated by hydrogen escape from a steam atmosphere over a molten silicate surface that lasted for more than 100 Myr. In this picture, the mantle remained in equilibrium with the water vapor in the atmosphere, and thus the loss of all the water from the atmosphere also meant the loss of almost all the water from the mantle. Both early insolation and accretional energy are greater for Proxima b than for Venus, which makes Venus's story seem all too likely Proxima b's story as well.

Mercury and Io represent end members where the forces of escape (for Mercury, insolation and XUV; for Io, impact erosion and thermal radiation from Jupiter when young) are almost wholly victorious. Both retain sulfur. Notably, Mercury appears to retain about as much water as it can harbor in its shadowed craters, which is empirical evidence either that late high-speed impacts by comets or asteroids do not wholly preclude the accretion of small amounts of water, or that even a planet as blasted as Mercury can still degas a little water. Mars may be a guide to what Proxima b might look like if it were in equilibrium with a late bombardment (post-dating the runaway greenhouse phase) of volatile-rich bodies dislodged from cold distant orbits. Between Mercury and Mars there is a place for a habitable desert state for Proxima b (Abe et al. 2011; Turbet et al. 2016).

Europa and Ganymede are examples of planets born with too much water to lose. We argued with respect to Figure 3 that over 5 Gyr it is difficult for any planet, no matter how small or strongly irradiated, to selectively lose more than about 0.2% of its mass as hydrogen, so that an initial water inventory greater than 2% of the planet's mass is likely not to be lost save by impact erosion. Both Europa and Ganymede appear to have been heated well enough during accretion to be thoroughly differentiated, yet each retains large amounts of water. How a water-world Proxima b might accrete is a puzzle—perhaps a giant impact of a stray water-rich planet with a local planet could do it, or perhaps it migrated in from a colder place in what is to date a wholly conjectural planetary system—but given the current absence of a predictive model of planet formation, nothing should be ruled out (Lissauer 2007; Ribas et al. 2016; Barnes et al. 2017; Coleman et al. 2017).

Earth would be the best of all possible analogs, but the real Earth hydrosphere is too thin to have survived Proxima's youthful luminosity and the onslaught of XUV radiation and flares that continues today (Davenport et al. 2016). Both Ribas et al. (2016) and Barnes et al. (2017) explicitly construct habitable states by starting with just enough water that the loss of the hydrogen from some 1–10 oceans of water leaves an ocean or so behind after the early steam atmosphere condensed. A decade ago, Lissauer (2007) had pointed out that an Earth-like outcome in this scenario, or in any of several other scenarios he considered, requires "precisely the right amount of initial water or just the right dynamics." This means that Earths are unlikely outcomes for Proxima b compared to a vastly greater phase space of less happy outcomes. When conceiving Kepler, Borucki et al. (1996) set out to determine the number of Earth-like planets in the cosmos by direct observation. With all apologies to theory, this remains the path forward.

With T constant, Equation (43) reduces to

$$\begin{eqnarray}&&\displaystyle \frac{\phi {r}_{s}^{2}{c}_{\circ }^{2}}{{r}^{2}}\left(\displaystyle \frac{2}{r}+\displaystyle \frac{1}{u}\displaystyle \frac{\partial u}{\partial r}\right)={{\rm{\Gamma }}}_{{\rm{h}}}-{{\rm{\Gamma }}}_{{\rm{c}}}.\end{eqnarray} \tag{ 44 }$$

We have written this in terms of the flow velocity u and the constant mass flux $\phi =\rho {{ur}}^{2}$. The velocity gradient $(1/u)(\partial u/\partial r)$ for the isothermal wind is given by Equation (7) of the main text. Equation (44) allows us to compare the energy spent on escape to radiative heating at all heights.

The most important heating elements are stellar UV and XUV radiation and thermal radiation from the planet's surface or lower atmosphere. The importance of UV heating of H₂-rich planetary atmospheres has been widely discussed in the literature since Urey (1952).

Planetary thermal radiation can also be important if the atmosphere contains IR-active molecules such as H₂O. In the plane-parallel gray approximation, thermal radiation tends to drive the upper atmosphere to an asymptotic isothermal state with ${T}_{\infty }={T}_{\mathrm{eff}}/{2}^{0.25}$. In reality, if planetary thermal radiation is the main driver, the temperature declines monotonically with increasing altitude, in part because a declining temperature is a general property of a non-gray atmosphere (Marley & Robinson 2015), and in part because we are considering very extended atmospheres.

We do not attempt here to compute the radiative transfer or solve the temperature structure of the atmosphere self-consistently; our present purpose is to determine how greatly escape affects the energy budget. Thus we wish to estimate the magnitudes of the chief sources of diabatic heating (${{\rm{\Gamma }}}_{{\rm{h}}}$) and compare them to the energy required to maintain isothermal escape.

B.1.1. Alternative Europa

B.1.2. Alternative Pluto

A different way to look at energy is through the global energy budget obtained by integrating Equation (43) over the whole atmosphere, from a lower boundary at r_s to the critical distance r_c. For a one-component fluid with constant m and constant mass flux ϕ, the integral is

$$\begin{eqnarray}&&\phi \left(\displaystyle \frac{{u}_{c}^{2}}{2}-\displaystyle \frac{{u}_{s}^{2}}{2}+\displaystyle \frac{\gamma }{\gamma -1}\left(\displaystyle \frac{{p}_{c}}{{\rho }_{c}}-\displaystyle \frac{{p}_{s}}{{\rho }_{s}}\right)-\displaystyle \frac{{GM}}{{r}_{c}}+\displaystyle \frac{{GM}}{{r}_{s}}\right.\\ &&-\left.{k}_{c}{\left(\displaystyle \frac{\partial T}{\partial r}\right)}_{c}+{k}_{s}{\left(\displaystyle \frac{\partial T}{\partial r}\right)}_{s}\right)={\displaystyle \int }_{{r}_{s}}^{{r}_{c}}{r}^{2}{dr}({{\rm{\Gamma }}}_{{\rm{h}}}-{{\rm{\Gamma }}}_{{\rm{c}}}).\end{eqnarray} \tag{ 45 }$$

Note that the energy required to evaporate a volatile from the condensed phase is not included in the energy budget of the atmosphere.

Equation (45) is a statement of conservation of energy for the escaping atmosphere as a whole. Setting the upper bound at r_c is appropriate to the transonic wind because nothing that happens above r_c can be communicated to r_s. This includes any XUV heating. This restriction does not apply to subsonic winds, for which the energy balance needs to explicitly account for the upper boundary condition that prevents the wind from freely expanding, as well as any radiative heating or cooling that takes place before the wind reaches the upper boundary. On the left-hand side, Equation (45) accounts for the net gains of kinetic, thermal, and potential energy, and the flow of heat into or out of the volume by thermal conduction. The right-hand side includes radiative heating and cooling, and it implicitly includes other heating terms (e.g., breaking waves and friction) that may be pertinent. If there are chemical changes induced by photolysis, the γ would differ at the top and bottom, and the net endothermicity of the chemical changes would need to be taken into account. This is usually included as a component of an empirical "heating efficiency."

Figure 6 depicts a continuum of evaporating Europas as a function of surface temperature. For illustration, we estimate the order of magnitude of the thermal energy term by setting $\gamma =1.4$ (the room temperature value for a diatomic molecule) and assuming that ${p}_{s}/{\rho }_{s}-{p}_{c}/{\rho }_{c}\approx {p}_{s}/{\rho }_{s};$ we are agnostic with respect to the sign. The approximate magnitude of thermal conduction is estimated by setting the temperature gradient equal to ${T}_{c}/({r}_{c}-{r}_{p})$. The thermal conductivity of water vapor is approximated by extrapolating a fit to values measured between 300 and 600 K, $k({{\rm{H}}}_{2}{\rm{O}})=1.1\,{T}^{1.3}$ ergs cm⁻¹ K⁻¹ s⁻¹. It is generally true that thermal conduction can only be important over distances that are very small compared to the extended atmospheres considered here, and thus thermal conduction is not a major player when escape is taking place.

The Europas of Figure 6 naturally divide into cold long-lived oceans with energy budgets dominated by solar FUV and XUV heating, and warm short-lived oceans with energy budgets dominated by planetary thermal radiation and escape. The latter can be conventionally habitable with liquid water at the surface, but they do not last long. For the colder planets, the relatively high levels of FUV and XUV heating suggest that the isothermal approximation probably underestimates the temperature and the escape rate, while for the warmer planets the energy demands set by rapid escape suggest that the isothermal approximation overestimates the temperature and the escape rate.

The polytropic equation of state can be defined by

$$\begin{eqnarray}&&p\propto {\rho }^{n+1},\end{eqnarray} \tag{ 46 }$$

where n is the polytropic index. For an ideal gas with constant mean molecular mass m, the temperature goes as

$$\begin{eqnarray}&&T\propto {\rho }^{n},\end{eqnarray} \tag{ 47 }$$

although strictly it is the ratio T/m that goes as ${\rho }^{n}$. The polytrope generalizes the adiabatic relation $p\propto {\rho }^{\gamma }$ to values of n that can be positive or negative. The formal relation between γ and the polytropic index n is $\gamma =n+1$.

Equations (47), (5) and (6) of the main text can be combined into a planetary wind equation that for the polytrope looks superficially like the isothermal wind of Equation (7), differing only by a factor $n+1$ in the sound speed,

$$\begin{eqnarray}&&\left({u}^{2}-\displaystyle \frac{(n+1)p}{\rho }\right)\displaystyle \frac{1}{u}\displaystyle \frac{\partial u}{\partial r}=\displaystyle \frac{2(n+1)p}{\rho r}-\displaystyle \frac{{GM}}{{r}^{2}}.\end{eqnarray} \tag{ 48 }$$

The critical point conditions for the transonic solution also look like those of the isothermal wind,

$$\begin{eqnarray}{u}_{c}^{2} & = & \displaystyle \frac{(n+1){p}_{c}}{{\rho }_{c}}\\ \displaystyle \frac{2(n+1){p}_{c}}{{\rho }_{c}} & = & \displaystyle \frac{{GM}}{{r}_{c}}.\end{eqnarray} \tag{ 49 }$$

However, unlike in the isothermal wind, the sound speed is not a constant. Equation (48) can be integrated analytically (Parker 1963; Zahnle & Kasting 1986) in spherical geometry,

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}\displaystyle \frac{{u}^{2}}{{c}_{s}^{2}}-\displaystyle \frac{{GM}}{{c}_{s}^{2}r}+\displaystyle \frac{n+1}{n}\displaystyle \frac{T}{{T}_{s}}=\displaystyle \frac{1}{2}\displaystyle \frac{{u}_{s}^{2}}{{c}_{s}^{2}}-\displaystyle \frac{{GM}}{{c}_{s}^{2}{r}_{s}}+\displaystyle \frac{n+1}{n}.\end{eqnarray} \tag{ 50 }$$

In Equation 50, the constant of integration has been evaluated at the surface. It is also evaluated at the critical point, and the equations combined to obtain a transcendental expression for the velocity u_s at the surface as a function of n and the other properties at the surface,

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}\displaystyle \frac{{u}_{s}^{2}}{{c}_{s}^{2}}+\displaystyle \frac{n+1}{n}-\displaystyle \frac{{GM}}{{c}_{s}^{2}{r}_{s}}=\displaystyle \frac{1}{\beta }{\left(\displaystyle \frac{4{c}_{s}^{2}{r}_{s}}{{GM}}\right)}^{4\beta }\\ &&\quad \times {\left(\displaystyle \frac{n+1}{2}\right)}^{(2\beta /n)}{\left(\displaystyle \frac{1}{2}\displaystyle \frac{{u}_{s}^{2}}{{c}_{s}^{2}}\right)}^{\beta },\end{eqnarray} \tag{ 51 }$$

with

$$\begin{eqnarray}&&\beta =\displaystyle \frac{n}{2-3n}.\end{eqnarray} \tag{ 52 }$$

As in the isothermal wind, the solution for u(r) is independent of density or flux. If the surface density ${\rho }_{s}$ is known independently, the escape flux is $\phi (r)={\rho }_{s}{u}_{s}{r}_{s}^{2}/{r}^{2}$.

It is often easier to obtain the desired transonic solution numerically. The best approach is to obtain the slope ${({du}/{dr})}_{c}$ at the critical point by writing it as a ratio,

$$\begin{eqnarray}&&\displaystyle \frac{1}{u}\displaystyle \frac{\partial u}{\partial r}=\displaystyle \frac{N(r,\rho ,p)}{D(u,\rho ,p)},\end{eqnarray} \tag{ 53 }$$

in which the numerator and denominator simultaneously pass through zero at the critical point,

$$\begin{eqnarray}N(r,\rho ,p) & = & \displaystyle \frac{2(n+1)p}{\rho r}-\displaystyle \frac{{GM}}{{r}^{2}}\\ D(u,\rho ,p) & = & {u}^{2}-\displaystyle \frac{(n+1)p}{\rho }.\end{eqnarray} \tag{ 54 }$$

The slope at the critical point is obtained by applying L'Hôpital's rule,

$$\begin{eqnarray}&&\displaystyle \frac{{N}_{c}}{{D}_{c}}=\displaystyle \frac{{({dN}/{dr})}_{c}}{({dD}/{{dr}}_{c}}.\end{eqnarray} \tag{ 55 }$$

After some manipulation using the critical point conditions (Equation (49)) and continuity, we obtain a quadratic expression for $x\equiv {u}_{c}^{-1}{(\partial u/\partial r)}_{c}$,

$$\begin{eqnarray}&&(n+2){x}^{2}+\displaystyle \frac{4n}{{r}_{c}}x+\displaystyle \frac{4n-2}{{r}_{c}^{2}}=0,\end{eqnarray} \tag{ 56 }$$

that is easily solved to give the slope ${(\partial u/\partial r)}_{c}$ at the critical point. (The corresponding result for the isothermal wind is ${(\partial u/\partial r)}_{c}=\pm {u}_{c}/{r}_{c}$.) The positive root is appropriate for an expanding flow. The negative root corresponds to accretion. There are no accelerating planetary wind solutions with $n\gt 1/2$.

Equation (53) is integrated inward and outward from the critical point. The inward integration terminates at the surface at r_s. For a Clausius-Clapeyron atmosphere, the pressure p_s and density ${\rho }_{s}$ at the surface r_s are unique functions of the surface temperature T_s. The velocity u_c is iterated until the desired lower boundary conditions are met.

Next we consider the escape of an atmosphere that is in saturation vapor pressure equilibrium at all altitudes. We have described such atmospheres elsewhere (Lehmer et al. 2017). The hypothesis is that such an atmosphere is the coldest and therefore least able to escape of any atmosphere composed of a single substance. The saturated wind takes much of the energy it needs from the latent heat of condensation. The caveat is that condensation must take place at all heights, a condition that may not be met by a low-density vapor as it accelerates in the vicinity of the critical point.

We again assume an ideal gas, and we also assume that the usual continuity and force equations apply. The polytropic relation is replaced by an equation of vapor pressure equilibrium; i.e., saturation provides an additional relation between p and T that is functionally equivalent to the polytropic relation or the isothermal assumption. Saturation vapor pressure is approximated by the familiar two-parameter CC relation

$$\begin{eqnarray}&&p={p}_{w}{e}^{-{T}_{w}/T}.\end{eqnarray} \tag{ 57 }$$

A very good approximation for $130\lt T\lt 270$ K over ice takes T_w = 6120 K and ${p}_{w}=3.2\times {10}^{7}$ bars. The two-parameter fit agrees with the seven-parameter approximation (Fray & Schmidt 2009) that we used for the surface pressure of the isothermal atmosphere to better than 10% for $T\gt 120$ K. The simple two-parameter expression for saturation vapor pressure is desirable here because we wish to work with analytic expressions for dT/dp.

Applying Equation (5) and (6) to a condensing wind implicitly assumes that the condensate has negligible mass and is also stationary, so that there there is no radial transport and hence no net sink on the vapor. Saturation presumes that the increasingly unfavorable kinetics in a rapidly accelerating wind will not preclude the presence of condensates at all heights. These assumptions describe a thin fog extending from the surface to the critical point.

It is convenient to write the perfect gas law in the same form as we used for the isothermal atmosphere,

$$\begin{eqnarray}&&p=\rho {c}^{2},\end{eqnarray} \tag{ 58 }$$

where c is defined as

$$\begin{eqnarray}&&{c}^{2}\equiv \displaystyle \frac{{kT}}{m}.\end{eqnarray} \tag{ 59 }$$

As with the polytrope, c as defined by Equation (59) is not a constant.

The goal is to combine Equations (5), (6), (57), and (58) into a single planetary wind equation analogous to Equation (48) that expresses the slope $\partial u/\partial r$ as a function of r, u, and T. One can eliminate p from Equation (6) using Equations (58) and (59):

$$\begin{eqnarray}&&u\displaystyle \frac{\partial u}{\partial r}+\displaystyle \frac{{c}^{2}}{\rho }\displaystyle \frac{\partial \rho }{\partial r}+\displaystyle \frac{{c}^{2}}{T}\displaystyle \frac{\partial T}{\partial r}=-\displaystyle \frac{{GM}}{{r}^{2}}.\end{eqnarray} \tag{ 60 }$$

Saturation can be used to express $\partial T/\partial r$ in terms of $\partial \rho /\partial r$,

$$\begin{eqnarray}&&\displaystyle \frac{{c}^{2}}{\rho }\displaystyle \frac{\partial \rho }{\partial r}=\left(\displaystyle \frac{{T}_{w}-T}{T}\right)\displaystyle \frac{{c}^{2}}{T}\displaystyle \frac{\partial T}{\partial r},\end{eqnarray} \tag{ 61 }$$

and continuity eliminates $\partial \rho /\partial r$ in terms of $\partial u/\partial r$:

$$\begin{eqnarray}&&u\displaystyle \frac{\partial u}{\partial r}-{c}^{2}\left(\displaystyle \frac{{T}_{w}}{{T}_{w}-T}\right)\left(\displaystyle \frac{1}{u}\displaystyle \frac{\partial u}{\partial r}+\displaystyle \frac{2}{r}\right)=-\displaystyle \frac{{GM}}{{r}^{2}}.\end{eqnarray} \tag{ 62 }$$

The result is the desired planetary wind equation in the familiar Parker form of Equations (7) and (48),

$$\begin{eqnarray}&&\left(u-\displaystyle \frac{{c}^{2}}{u}\left(\displaystyle \frac{{T}_{w}}{{T}_{w}-T}\right)\right)\displaystyle \frac{\partial u}{\partial r}=\displaystyle \frac{2{c}^{2}}{r}\left(\displaystyle \frac{{T}_{w}}{{T}_{w}-T}\right)-\displaystyle \frac{{GM}}{{r}^{2}}.\end{eqnarray} \tag{ 63 }$$

Written explicitly as an expression for $\partial u/\partial r$,

$$\begin{eqnarray}&&\displaystyle \frac{1}{u}\displaystyle \frac{\partial u}{\partial r}=\displaystyle \frac{N(r,T)}{D(r,T,u)},\end{eqnarray} \tag{ 64 }$$

where here the numerator $N(r,T)$ is

$$\begin{eqnarray}&&N(r,T)\equiv \displaystyle \frac{2{c}^{2}}{r}\left(\displaystyle \frac{{T}_{w}}{{T}_{w}-T}\right)-\displaystyle \frac{{GM}}{{r}^{2}}\end{eqnarray} \tag{ 65 }$$

and the denominator $D(r,T,u)$ is

$$\begin{eqnarray}&&D(r,T,u)\equiv {u}^{2}-{c}^{2}\left(\displaystyle \frac{{T}_{w}}{{T}_{w}-T}\right).\end{eqnarray} \tag{ 66 }$$

The critical point conditions for the transonic wind, N_c = 0 and D_c = 0, link r_c, T_c, and u_c:

$$\begin{eqnarray}{r}_{c} & = & \displaystyle \frac{{GMm}({T}_{w}-{T}_{c})}{2{{kT}}_{c}{T}_{w}}\\ {u}_{c}^{2} & = & {c}_{c}^{2}\left(\displaystyle \frac{{T}_{w}}{{T}_{w}-{T}_{c}}\right)=\displaystyle \frac{{GM}}{2{r}_{c}^{3}}.\end{eqnarray} \tag{ 67 }$$

The transonic solution is obtained by integating Equation (64) numerically inward and outward from the critical point. For the first step we need to know the slope ${(\partial u/\partial r)}_{c}$ at the critical point. This is obtained from Equation (64) by using L'Hôpital's rule,

$$\begin{eqnarray}&&\displaystyle \frac{1}{{u}_{c}}{\left(\displaystyle \frac{\partial u}{\partial r}\right)}_{c}=\displaystyle \frac{{N}_{c}}{{D}_{c}}=\displaystyle \frac{{({dN}/{dr})}_{c}}{{({dD}/{dr})}_{c}}.\end{eqnarray} \tag{ 68 }$$

The numerator becomes

$$\begin{eqnarray}{\left(\displaystyle \frac{{dN}}{{dr}}\right)}_{c} & = & \displaystyle \frac{2{GM}}{{r}_{c}^{3}}-\displaystyle \frac{2{c}_{c}^{2}}{{r}_{c}^{2}}\displaystyle \frac{{T}_{w}}{{T}_{w}-{T}_{c}}+\displaystyle \frac{2{c}_{c}^{2}}{{r}_{c}}\displaystyle \frac{{T}_{w}}{{({T}_{w}-{T}_{c})}^{2}}\\ & & \times {\left(\displaystyle \frac{\partial T}{\partial r}\right)}_{c}+\displaystyle \frac{2{c}_{c}^{2}}{{r}_{c}{T}_{c}}\displaystyle \frac{{T}_{w}}{{T}_{w}-{T}_{c}}{\left(\displaystyle \frac{\partial T}{\partial r}\right)}_{c},\end{eqnarray} \tag{ 69 }$$

from which ${(\partial T/\partial r)}_{c}$ is eliminated in favor of ${(\partial u/\partial r)}_{c}$ by using Equation (61),

$$\begin{eqnarray}{\left(\displaystyle \frac{{dN}}{{dr}}\right)}_{c} & = & \displaystyle \frac{{GM}}{{r}_{c}^{3}}-\displaystyle \frac{4{c}_{c}^{2}}{{r}_{c}^{2}}\displaystyle \frac{{T}_{w}^{2}{T}_{c}}{{({T}_{w}-{T}_{c})}^{3}}\\ & & -\displaystyle \frac{2{c}_{c}^{2}}{{r}_{c}}\displaystyle \frac{{T}_{w}^{2}{T}_{c}}{{({T}_{w}-{T}_{c})}^{3}}\displaystyle \frac{1}{{u}_{c}}{\left(\displaystyle \frac{\partial u}{\partial r}\right)}_{c}.\end{eqnarray} \tag{ 70 }$$

The denominator becomes

$$\begin{eqnarray}{\left(\displaystyle \frac{{dD}}{{dr}}\right)}_{c} & = & 2{u}_{c}{\left(\displaystyle \frac{\partial u}{\partial r}\right)}_{c}-\displaystyle \frac{{c}_{c}^{2}}{{u}_{c}{T}_{c}}\,\displaystyle \frac{{T}_{w}}{{T}_{w}-{T}_{c}}{\left(\displaystyle \frac{\partial T}{\partial r}\right)}_{c}\\ & & -\displaystyle \frac{{c}_{c}^{2}}{{u}_{c}}\,\displaystyle \frac{{T}_{w}}{{({T}_{w}-{T}_{c})}^{2}}{\left(\displaystyle \frac{\partial T}{\partial r}\right)}_{c},\end{eqnarray} \tag{ 71 }$$

from which ${(\partial T/\partial r)}_{c}$ is eliminated in favor of ${(\partial u/\partial r)}_{c}$ by using Equation (61) and u_c from Equation (67)

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{dD}}{{dr}}\right)}_{c}=\left(2+\displaystyle \frac{{T}_{w}{T}_{c}}{{({T}_{w}-{T}_{c})}^{2}}\right){u}_{c}{\left(\displaystyle \frac{\partial u}{\partial r}\right)}_{c}+\displaystyle \frac{2{u}_{c}^{2}}{{r}_{c}}\displaystyle \frac{{T}_{w}{T}_{c}}{{({T}_{w}-{T}_{c})}^{2}}.\end{eqnarray} \tag{ 72 }$$

Equation (68) can then be written as a quadratic equation for $x\equiv {u}_{c}^{-1}{(\partial u/\partial r)}_{c}$,

$$\begin{eqnarray}&&\left(2+\displaystyle \frac{{T}_{w}{T}_{c}}{{({T}_{w}-{T}_{c})}^{2}}\right){x}^{2}+\displaystyle \frac{4}{{r}_{c}}\displaystyle \frac{{T}_{w}{T}_{c}}{{({T}_{w}-{T}_{c})}^{2}}\,x\\ &&\quad +\displaystyle \frac{4}{{r}_{c}^{2}}\displaystyle \frac{{T}_{w}{T}_{c}}{{({T}_{w}-{T}_{c})}^{2}}-\displaystyle \frac{2}{{r}_{c}^{2}}=0.\end{eqnarray} \tag{ 73 }$$

Again, the positive root corresponds to an accelerating flow and is the appropriate solution here. If there is no real root to Equation (73), there is no planetary wind solution. In these cases, the atmosphere must either be static or ill described by a fluid because escape occurs directly from the surface.

Comparison of Equations (56) and (73) reveals that the saturated wind and the polytrope correspond exactly at the critical point for an effective value of n,

$$\begin{eqnarray}&&n=\displaystyle \frac{{T}_{w}{T}_{c}}{{({T}_{w}-{T}_{c})}^{2}}.\end{eqnarray} \tag{ 74 }$$

Equation (74) is more an illustration than a useful relation, because T_c must be solved for, and T_c varies from case to case. But for the purposes of the illustration, with T_c = 183, the effective value of n is 0.03. In other words, the fully vapor-saturated wind—which we suggest is as cold a wind as a pure substance can support—is not extremely different from the isothermal wind for which n = 0.

Europa, the smallest of the Medicean planets, is currently of great interest to NASA as a local example of an ocean world. Europa in its current state at its current location does not qualify as a conventionally habitable planet. Its surface is too cold ($T\approx 100$ K), it has no significant atmosphere, and its liquid water is hidden under some 10 km of ice. But in the future, when the Sun has evolved to burn helium as a so-called "horizontal branch" star, our Sun will be roughly 25 times brighter than it is today and Europa will be in the middle of the habitable zone (Nisbet et al. 2006). It is also possible that Europa was, for a brief moment when both it and Jupiter were young, a warm wet world heated by thermal radiation from Jupiter itself.

Europa is very nearly the smallest world that can retain a liquid water surface for any significant period, which makes Europa an excellent candidate for a study of the smallest habitable world. In Figure 7 we compare four alternative atmospheres for a planet with the size and weight of Europa. For Figure 7, Europa is placed at 1 au from the Sun. The Bond albedo is set to 50%. This makes the surface temperature 234 K. Figure 7 compares vertical structures of two polytropes and the saturated atmosphere to the isothermal atmosphere. All the atmospheres are very large compared to the planet itself. The upper atmosphere of the $n=-0.04$ polytrope is either about twice as hot as the surface, or its mean molecular weight is about half what it is at the surface; the polytrope cares only about $p/\rho$. The saturated atmosphere resembles an n = 0.03 polytrope, as Equation (74) suggested it might. In this case, the upper atmosphere is about half the temperature of the surface.

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Figure 8 illustrates that alternative Europa's long-term survival as a conventionally habitable world is sensitive to details. The left-hand panel of Figure 8 compares evaporation lifetimes at 1 au predicted by the several models as a function of escape velocity. These Europas have been scaled from the true Europa (${v}_{\mathrm{esc}}=2.0$ km s⁻¹) at constant density, so that ${v}_{\mathrm{esc}}$ is directly proportional to the diameter of the planet. The escape velocities from the billion-year survivors shown here range from 1.9–2.7 km s⁻¹. The uncertainty is asymmetric. The isothermal approximation predicts that a Europa with ${v}_{\mathrm{esc}}=2.2$ km s⁻¹ would retain its ocean for a billion years. The coldest case has ${v}_{\mathrm{esc}}=1.9$ km s⁻¹, while the hottest case shown ($n=-0.04$) has ${v}_{\mathrm{esc}}=2.7$ km s⁻¹. Still "hotter" cases are possible if photochemistry significantly reduces the molecular weight m. The swath of uncertainty in ${v}_{\mathrm{esc}}$ that envelopes the isothermal approximation appears to range between −15% to at least +30%.

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The right-hand panel of Figure 8 moves the true Europa with respect to the Sun. Here the billion-year survivors are ice-covered Europas, at distances ranging between $0.85\mbox{--}1.4\,\mathrm{au}$. This corresponds to an uncertainty in insolation I that is on the order of ±60%. If photochemistry greatly reduces m, the evil I could be smaller still. We expect in reality that the swath of uncertainty in I would be tighter than in Figure 8, based on energy considerations. Figure 6 above suggests that Europas with the warmest surfaces will be more strongly influenced by condensation and will therefore have cooler temperature structures approaching the saturation curve, and thus will live longer lives as marginally habitable planets than predicted by the isothermal approximation, while the colder Europas with thinner atmospheres will be more strongly influenced by solar EUV and FUV heating, and thus will be hotter and have shorter lifetimes than predicted by the isothermal approximation.

It seems obvious that adding an abundant heavy constituent will reduce the scale height and reduce the escape rate. For a tiny ocean world like Europa, the likeliest ballast gas is probably O₂, which can be made directly from H₂O by selective hydrogen escape. But the build-up of O₂ is also determined atmospheric chemistry. We will address this matter elsewhere. For the present, we simply consider a 1-bar N₂ atmosphere as a placeholder for examining the efficacy of a heavy gas as ballast. With $p{{\rm{N}}}_{2}\gg p$ H₂O, the mean molecular weight to first approximation is constant and equal to the mean molecular weight of nitrogen of 28. Water vapor pH₂O is assumed to be saturated at the surface.

For Figure 9 we set the albedo to 20% and compare the pure water vapor atmospheres m = 18 to the N₂-dominated atmospheres m = 28. Both models use n = 0.02 polytropes. Adding the heavy gas has an important stabilizing effect, corresponding to an escape velocity that is 15% lower than if N₂ were not present. The swath of uncertainty resulting from the presence or absence of other gases is comparable to the uncertainty resulting from the use of the isothermal approximation.

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The uncertainty in evaporation lifetimes of small icy worlds is asymmetric. It is unlikely that our coldest model is too cold. That is, it is unlikely that the true experience could be much more optimistic for continuous habitability than our most long-lived models. But it is entirely possible that our hottest model falls well short of the true escape rate, especially in the presence of a more active Sun. This is only partly because of temperature. The greater uncertainty is photochemistry, which can reduce the mean molecular mass m by a factor of three or more, to the detriment of an alternative Europa as an abode of life.