A shoreline measured in light already spent

The day sides measured so far say that the hottest rocky planets around red dwarfs are bare: their thermal emission at fifteen microns is as bright as rock with no air to carry heat away. That result was expected, and the question it leaves is where the bare planets stop. Somewhere between the scorched inner planets and the temperate ones there should be a boundary, and the rocky planets most worth studying — those in their stars’ habitable zones — lie on one side of it or the other.

A boundary of exactly this kind exists in the solar system and has been drawn. In 2017 Kevin Zahnle and David Catling plotted every planet and large moon by its escape velocity against the radiation it receives, and found that a single straight line on logarithmic axes, of slope four, separates the bodies with atmospheres from those without. They called it the cosmic shoreline.

Why the line has slope four #

The line is empirical, but its slope is not arbitrary, and the argument behind it says what the line is really about.

An atmosphere is lost when the energy deposited in its upper layers is enough to lift gas out of the planet’s gravity. The relevant energy is the star’s X-ray and extreme-ultraviolet light, absorbed high up where the gas is thin, because the visible light that sets the surface temperature passes through to the ground. In the simplest accounting — escape limited by the energy supply — the mass removed per unit time is the absorbed XUV power multiplied by an efficiency and divided by the work needed to lift a unit mass out of the potential well:

$$\dot M = \frac{\varepsilon\,\pi R^3 F_{\rm XUV}}{G M}.$$

Integrated over the planet’s history, the total mass removed is the same expression with the flux replaced by the fluence, the energy per unit area received over all time. Set that equal to the atmosphere’s mass, taken as a fraction $f$ of the planet’s, and the atmosphere survives only while

$$\Phi_{\rm XUV} < \frac{f\, v_{\rm esc}^4}{4\pi\,\varepsilon\, G R}.$$

At fixed density $R$ varies as $v_{\rm esc}$ and the condition is close to $\Phi \propto v_{\rm esc}^{4}$ apart from a factor of the radius; Zahnle and Catling’s line takes the fourth power as its slope. The empirical line is this inequality with the constants left to the data.

Two things about the derivation matter for what follows. The slope comes from the physics and the height does not: the height is set by $f/\varepsilon$, the atmosphere’s share of the planet divided by the efficiency of escape, and the solar system calibrates that ratio for the solar system. And the axis is the fluence, not the flux. What strips an atmosphere is the total energy delivered over the planet’s life, and a star’s XUV output is far from constant.

Eleven bodies and a ruler #

The solar-system line is drawn through a small sample, and it is worth seeing how small. Of the eleven bodies on the first plane, the ones that actually constrain the line’s position are those near it: Mars, with a thin atmosphere just above the threshold; Ganymede and Callisto, airless, just above it on the other side; Titan, with a thick atmosphere, a factor of three below; Triton and Pluto, with tenuous seasonal atmospheres that come and go with their distance from the Sun. Mercury, the Moon, Venus and the Earth are far from the line and would be separated by almost any line of positive slope.

So the line is defined by about five bodies, and with two free parameters — slope and height — five points do not determine a slope to better than about one. A line of slope three or five, adjusted in height, separates the solar system almost as cleanly as one of slope four. The physics prefers four, and the data permit it, but the data alone would not have produced it.

The solar system also contains bodies the line does not describe at all. Io is airless above the line, as it should be, but its tenuous sulphur dioxide atmosphere is continuously supplied by volcanoes driven by tidal heating, which is a source term the shoreline has no place for. Venus lies far below the line and has the thickest atmosphere of any rocky planet, but it has lost its water — the hydrogen escaped and the oxygen did not — so a line drawn for water would put Venus on the other side. The shoreline is a statement about whether a body has an atmosphere, not which one, and the composition of what survives is decided by which gases are heavy enough, and cold enough at the top, to stay.

What the fourth power rewards #

A fourth power is steep. Doubling a planet’s escape velocity raises the fluence it can withstand sixteen-fold, and that one fact organises the whole exoplanet search for rocky atmospheres.

Escape velocity grows with mass and density together. A rocky planet of five Earth masses with the Earth’s composition is about one and a half Earth radii across and has an escape velocity of about twenty kilometres a second — almost twice the Earth’s — which buys it a factor of eleven in the fluence it can survive. The super-Earths that red-dwarf surveys find in abundance are therefore much better candidates for keeping air than Earth-sized planets at the same orbits, and the line’s slope is what makes them so. LHS 1140 b, at six Earth masses, is the clearest case; TRAPPIST-1 d and h, at a third of an Earth mass each, are the opposite.

The fourth power also explains why the coolest planets of a system are not automatically the safest. The fluence falls with distance squared and the habitable zone’s outer planets receive a few times less than its inner ones — a factor easily cancelled by a modest difference in escape velocity. TRAPPIST-1 g, more massive than its neighbours and farther out, sits closer to the atmospheric side than h, which is farther still but much less massive.

The gas matters too, in a way the line hides. An atmosphere of carbon dioxide radiates heat efficiently from its upper layers at fifteen microns, which keeps its thermosphere cool and slows escape; an atmosphere of nitrogen and oxygen does so less well, and one of hydrogen not at all. The same planet under the same star loses a hydrogen envelope readily, a nitrogen atmosphere more slowly, and a carbon-dioxide one most slowly of all. The empirical line averages over whatever atmospheres the solar system’s bodies happen to have; a red-dwarf planet with an atmosphere chiefly of carbon dioxide would sit, in effect, under a lower line.

A star that stays young #

Every Sun-like star begins its life spinning fast, magnetically active and bright in X-rays — a thousandth of its total light in the XUV, the saturated level, against a few millionths for the Sun today. Its wind removes its spin, the dynamo weakens, and the activity falls away. For the Sun the saturated phase lasted about a hundred million years, and most of its lifetime XUV was delivered in the first few hundred million.

Red dwarfs keep their activity much longer. Their winds are weaker relative to their spin, they brake more slowly, and the smallest of them stay saturated for billions of years. The duration is measured statistically — from the activity of red dwarfs in clusters of known age and of field stars with ages estimated from their kinematics — and it lengthens steeply toward low mass.

The factor seventeen is the essay’s central number. A planet around a 0.09-solar-mass star that receives the Earth’s sunlight has received, over its life, about seventeen times the Earth’s XUV energy. A planet that receives a tenth of the Earth’s sunlight has received nearly twice the Earth’s stripping light. The habitable zone is defined by the first quantity; the shoreline is about the second.

The temperate planets move #

The first plane, drawn in present light, was mildly embarrassing for the shoreline: two of the three bare planets sat on the side where air should survive. Redrawn in lifetime XUV, all three move across. That is not a proof the line works for red dwarfs, since three agreeing points cannot establish a boundary, but it removes the apparent contradiction, and it does so for a reason that has nothing to do with fitting — the axis was changed to the quantity the physics actually depends on.

The cost of the change is on the other side of the line. TRAPPIST-1 e, the planet in that system most often called potentially habitable, moves from comfortably on the side with air to just across the line. TRAPPIST-1 d moves further. Only f, g and h remain within a factor of a few of the line on the atmospheric side, and they are cold. The temperate planets of the most studied red-dwarf system are, by this reckoning, on the shoreline rather than safely behind it.

A line whose height nobody knows #

The shift by an order of magnitude would be decisive if the line’s height were known to better than an order of magnitude. It is not.

The atmosphere’s initial share of the planet’s mass, $f$, is the planet’s volatile inventory, and it is not a property the shoreline can supply. A planet that formed beyond its star’s snow line and migrated inward could carry a hundred times the Earth’s water; one that formed in place from dry material might carry a tenth. A planet with a magma ocean exchanges gas with its interior and can rebuild an atmosphere after losing one, which makes $f$ a rate rather than a stock — the most likely explanation for 55 Cancri e, which appears to have an atmosphere while sitting far on the airless side of every version of the line.

The efficiency $\varepsilon$ is no better known. It is the share of the absorbed XUV energy that ends up lifting gas rather than being radiated away, and it is not a constant: above a flux of about ten thousand erg per square centimetre per second the escaping gas is ionised, cools by line emission, and the efficiency falls as the flux rises. Close-in planets of active red dwarfs sit exactly in that regime during their stars’ saturated phase, which means their effective efficiency is lowest during the period that delivers most of their fluence.

Between them, $f/\varepsilon$ plausibly spans two orders of magnitude, and the solar system — where every body formed around one star, from one disc, and was stripped by one history — calibrates it only for itself.

Which worlds can test it #

The band changes what a measurement of any one planet can do. A planet far outside the band tests the line: if it is far on the airless side and has an atmosphere, the line is wrong, or at least the physics behind it is incomplete. A planet inside the band places the line: whatever its atmosphere turns out to be, the result moves the band’s edge rather than confirming or refuting anything.

That ordering is the logic behind the current observing programmes. LHS 3844 b’s bare surface, measured from its thermal phase curve, was the first confirmation of the airless side for a red-dwarf planet. LHS 1140 b, a dense planet with a high escape velocity on a cool orbit, is the nearest strong test of the other side: if it has no atmosphere, the line must be much lower for red dwarfs than for the Sun, and the temperate planets of every small star are likely bare. Transit spectra of LHS 1140 b so far allow a nitrogen-rich atmosphere and rule out a hydrogen-dominated one, without yet establishing that it has air at all.

A programme of several hundred hours on the James Webb Space Telescope has been set aside to test the line systematically, choosing rocky planets of red dwarfs that span the band. The selection is the shoreline argument turned into an observing strategy: planets above the band are expected bare and serve as controls, planets below are the hope, and the ones inside are the measurement of where the line is.

What follows if the line holds #

Suppose the shoreline drawn in lifetime XUV is right, with roughly the height the solar system gives it. The consequences reach well beyond one planetary system.

Three-quarters of the stars in the Galaxy are red dwarfs, and the surveys that count planets find that small stars carry rocky planets at least as often as Sun-like ones do. Most of the rocky planets in temperate orbits in the Galaxy therefore orbit red dwarfs. If the temperate planets of the smallest red dwarfs sit at or above the line, as TRAPPIST-1 e does, then most of the Galaxy’s temperate rocky planets lost their primary atmospheres, and whether they have air now depends on whether they rebuilt it — from volcanic outgassing, as the Earth’s own atmosphere was largely rebuilt after its formation, or from impacts of volatile-rich bodies arriving late.

That changes what a habitable-planet survey is looking for. A temperate red-dwarf planet with air would most likely have a secondary atmosphere, made of what its interior released, rich in carbon dioxide, nitrogen and water and poor in hydrogen. Its existence would say that the planet’s interior is active enough to replace what the star removes, which is itself a statement about its geology. The shoreline then becomes a boundary not between planets with and without atmospheres but between planets whose atmospheres are primordial and planets whose atmospheres are maintained.

It also changes which stars are worth the time. The early-M dwarfs, half the Sun’s mass, leave saturation within a few hundred million years and deliver a lifetime fluence only a few times the Sun’s per unit of light. Their temperate planets sit well below the line in the second plane even with modest escape velocities. They are rarer than the smallest red dwarfs, their planets’ transits are shallower, and their habitable zones are farther out — every observational property is worse — and the shoreline says they are where the odds of air are best.

The habitable zone and the shoreline answer different questions #

The habitable zone and the shoreline look like they should agree, since both ask whether a planet can have a surface environment, and they do not agree, because they are measured in different light.

The habitable zone asks whether a planet’s surface could hold liquid water today, and the answer depends on the bolometric light the planet receives now, filtered through the atmosphere it is assumed to have. The shoreline asks whether the planet still has an atmosphere at all, and the answer depends on the XUV the planet has received over its life, which the zone’s definition never consults.

For the Sun’s planets the two quantities are nearly proportional, because every planet orbits the same star with the same history. For a red dwarf’s planets they differ by an order of magnitude, and the difference is the essay’s surprising connection: a temperate orbit around a small star is temperate in visible light and scorched in the light that strips. The planet at the right distance for liquid water is, by the same arithmetic, a planet that has absorbed ten times the Earth’s stripping energy — before counting the further hundreds of millions of years when the young star was several times brighter than it is now, and before counting flares.

So “in the habitable zone of a red dwarf” is two statements, not one, and the second — that the planet kept the atmosphere the zone assumes — is the one the shoreline addresses and the one no observation has yet confirmed for a single temperate planet.

What the figures assume #

The saturation times are a scaling. The duration of each star’s saturated phase is drawn from a power of its mass chosen to give about a hundred million years for the Sun and a couple of billion for the smallest red dwarfs; measured saturation times scatter by factors of several among stars of one mass, because stars born spinning slowly leave saturation sooner. A planet’s fluence inherits that scatter, and the young Sun’s own spin is unknown to within a factor of a few.

The saturated level is a single number. A thousandth of the bolometric luminosity is typical; it depends on the bands included, and the extreme ultraviolet, which does most of the stripping, cannot be observed directly for any star but the Sun, because the interstellar hydrogen absorbs it. It is inferred from X-rays and from ultraviolet lines formed at similar temperatures.

The pre-main-sequence phase is left out. Small red dwarfs take hundreds of millions of years to contract onto the main sequence, shining several times brighter than their final luminosity in all light, so the XUV factors drawn here are underestimates for their planets, by perhaps a factor of two.

Flares and coronal mass ejections are left out. Red dwarfs flare frequently, and the energy in their flares adds to the XUV fluence; the particles in their ejections can strip an atmosphere directly, by a mechanism that the energy-limited formula does not describe.

The line is drawn as a single fourth-power law. Escape that is limited by the supply of hydrogen diffusing upward, or by the ionisation of the escaping gas, has different scalings, and a planet’s atmosphere may change regime over its history. The slope is a good description of the solar system; whether it holds over five orders of magnitude of fluence is part of what the observations will test.

Still open: the first temperate red-dwarf planet with air #

The single most informative measurement would be an atmosphere — any atmosphere — on a temperate rocky planet of a small red dwarf. One such detection would show that the band reaches high enough to include those planets, and the planet’s escape velocity and fluence would fix a point on the line’s lower edge. A string of bare temperate planets would push the line down, and would mean that the commonest rocky planets in the Galaxy, orbiting the commonest stars, are without air.

The tests are thermal emission at fifteen microns, which senses whether heat is carried to the night side, and transmission spectra, which sense molecules directly but are complicated by the spots and faculae of the star itself. TRAPPIST-1 b and c have been answered as bare; d, e and f are the planets inside the band at the temperatures that matter, and their eclipses are a few tens of parts per million deep. When they are measured, the result will be where the shoreline lies for stars unlike the Sun — and with it whether the habitable zones of most of the Galaxy’s stars contain anywhere with air.