Steam before the zone existed

Every argument about the habitable zone of a small star — its width, its long life, the time its planets stay inside it — begins on the main sequence. The luminosity used is the star’s hydrogen-burning luminosity, and for a Sun-like star that is a good description of the star’s history, because the Sun spent only a few tens of millions of years getting there. For a star a tenth of the Sun’s mass it is not. Such a star takes hundreds of millions of years to arrive, and for most of that time it is several to many times brighter than it will be for the next trillion years.

The habitable zone moves with the luminosity. While the star contracts, the zone lies further out than the zone catalogued for it today, and a planet that is in today’s zone was, for all that time, inside the inner edge. It was not merely warm. The inner edge of a habitable zone is where a runaway greenhouse begins, and a planet past it holds its ocean in its atmosphere as steam.

Why a small star takes so long #

A young star is not yet burning hydrogen. It shines by contracting: gravitational energy released as it shrinks, half of it radiated and half heating the interior — a system that gets hotter as it loses energy, which is the virial theorem at work and the reason a contracting star’s core eventually becomes hot enough to stop contracting. A low-mass star at this stage is convective throughout, and a fully convective star has a surface temperature pinned near 3,000 K by the opacity of its outer layers — its path on the Hertzsprung–Russell diagram, the Hayashi track, is nearly vertical. With the temperature fixed, luminosity goes as surface area.

The contraction then has a simple solution. For a fully convective star — a polytrope of index 3/2, whose structure is fixed by its equation of state alone — the binding energy is $\tfrac{6}{7}GM^2/R$, and radiating at fixed temperature,

$$\tfrac{3}{7}\,\frac{GM^2}{R^2}\left(-\frac{dR}{dt}\right) = 4\pi R^2 \sigma T^4 \quad\Rightarrow\quad R(t) = \left[\frac{GM^2}{28\pi\sigma T^4\,t}\right]^{1/3}$$

The radius falls as the cube root of the age and the luminosity as its two-thirds power. Contraction continues until the core is hot enough for hydrogen fusion to supply the luminosity, at which point the star has its main-sequence radius and stops.

The time to get there follows by setting $R$ equal to the main-sequence radius. The numerator is $M^2$; the denominator carries the main-sequence radius cubed and $T^4$, and for low-mass stars both fall steeply with mass. A 0.6 solar-mass star arrives in about twenty million years. A 0.08 solar-mass star — the smallest that burns hydrogen at all — needs more than three hundred million on this law, and detailed evolutionary models, which include the slow final approach as fusion gradually takes over, put it nearer a billion.

The law is crude in its details and robust in its shape. Detailed models give slightly different temperatures along the track and a deuterium-burning pause, but the luminosity of a 0.1 solar-mass star at ten million years is still about ten times its final value in every published set of tracks, and a factor of two or so at a hundred million. What matters here is that the excess lasts for a period comparable with the formation of the planets themselves, and that it is largest for exactly the stars that are otherwise the best targets.

From luminosity to steam #

A planet whose zone position is fixed by the star’s final luminosity receives, during the contraction, a flux larger by the ratio of the luminosities. It crosses back inside the runaway limit only once the luminosity has fallen to the runaway flux times the square of its orbit. The hero figure’s three curves are those crossing times, measured from an age of one million years.

The inner-edge planet round a 0.08 solar-mass star is too hot for three hundred million years. The planet in the middle of the zone — the one that would be described today as the most secure of the three — spends almost a hundred million. Even the outer-edge planet spends forty.

A runaway greenhouse is a specific state rather than a figure of speech. Water vapour is a greenhouse gas whose concentration rises with temperature, and past a critical absorbed flux the outgoing thermal radiation of a steam atmosphere stops rising with surface temperature — it saturates near 280 W/m² — so any extra absorbed flux cannot be radiated away and the surface heats until the ocean has evaporated. The planet then has a hot, deep atmosphere of steam, and its water is exposed at altitude to the star’s ultraviolet light instead of being protected as a liquid under a cold trap.

Two clocks that have to be compared #

The durations in the hero figure are measured from an age of one million years, which assumes the planet is there to be heated. Rocky planets are not built that quickly. The Earth took somewhere between thirty and a hundred million years to reach its final mass, most of it in giant collisions after the gas of the protoplanetary disc had gone. Whether a planet suffers the steam phase is a comparison between two clocks: how long the star stays too bright, and how long the planet takes to exist.

Round a star of half a solar mass the exposure is a few million years, shorter than any plausible assembly time, and most planets in its zone finish forming after the star has already dimmed enough. Round a star of a tenth of a solar mass the exposure outlasts assembly by a wide margin, and a planet is exposed for most of its early life whenever it formed. The boundary between those regimes falls, on this law, somewhere between a quarter and a half of a solar mass — the same place the water-loss figure puts its crossing of one ocean, because both are the same comparison expressed in different units.

Planets round small stars also form faster, since orbital periods at their close-in distances are days rather than years and collisions happen correspondingly sooner. That shortens the second clock and makes the comparison worse, not better, for the smallest stars.

A planet can hide its water in its own rock #

A young rocky planet is not a solid body under an ocean. The energy of its final collisions leaves it with a surface of molten rock, and a magma ocean dissolves water readily — several times more water can be held in solution in a deep magma ocean than sits in the Earth’s oceans today. While the magma ocean lasts, most of the planet’s water is inside the planet rather than in its atmosphere, and ultraviolet light cannot reach it.

That reservoir changes the arithmetic in both directions. A magma ocean shields water from escape for as long as it stays molten. But a planet that is receiving more than the runaway flux cannot cool efficiently, because its steam atmosphere cannot radiate more than the saturation limit, and so its magma ocean stays molten for longer — tens of millions of years instead of one or two. Models of this coupling divide planets into two types: those beyond a critical orbital distance, which solidify quickly, condense their water into an ocean and keep it; and those inside it, which stay molten for as long as the runaway lasts and lose water throughout. The critical distance is where the absorbed flux equals the runaway limit, which for a contracting M dwarf is itself a moving line.

The planets in the eventual habitable zones of the smallest stars are, for tens of millions of years, on the wrong side of that line. Their water spends that time in a steam atmosphere over molten rock, released to space from above and replenished from below, and what survives to condense is whatever the two rates left.

What the steam costs #

Ultraviolet photons split water into hydrogen and oxygen high in a steam atmosphere, and the hydrogen, being light, can escape. The rate is limited by the energy available to lift it out of the planet’s gravity. In the energy-limited approximation, a fraction ε of the absorbed X-ray and extreme-ultraviolet flux drives the outflow:

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

Young low-mass stars are magnetically active and emit about a thousandth of their total light in X-rays and extreme ultraviolet, and they stay at that saturated level for longer than they take to arrive on the main sequence. Every kilogram of hydrogen removed is nine kilograms of water, if the oxygen stays behind.

The numbers are what the energy permits, and the real loss may be much smaller. At high fluxes an outflow radiates away part of its heating before it can use it, and the efficiency falls. Oxygen left behind accumulates, and hydrogen must diffuse through it to reach the escape region, which at some point limits the rate more than energy does. Detailed models of steam atmospheres on planets round late M dwarfs nonetheless find losses of several to tens of Earth oceans for the smallest stars, which is the order of the figure.

The robust conclusion is not a number of oceans. It is that a planet in the eventual habitable zone of a star below about a quarter of a solar mass had enough energy delivered to it during the star’s contraction to remove an Earth’s worth of water several times over, while smaller losses are the rule above half a solar mass.

The oxygen left behind #

Losing hydrogen and keeping oxygen has a second consequence that matters for the way such planets will be studied. If a steam atmosphere loses the hydrogen from several oceans and the oxygen stays, the planet is left with an enormous oxygen atmosphere — hundreds of bars from a few oceans, if nothing reacts with it.

In practice much of the oxygen would be absorbed by a molten surface or reacted into the crust, and some would escape with the hydrogen when the flow is vigorous enough to drag it along. How much remains is uncertain by orders of magnitude. But an M-dwarf planet with a thick, oxygen-rich atmosphere and no biology is a possible outcome of this history, and oxygen is the gas most often proposed as a sign of life. A spectrum flattened by cloud or by nothing is one ambiguity in reading a planet’s atmosphere; an oxygen line produced by a star is a worse one, because it is an unambiguous detection of the wrong thing.

Why the Sun’s planets were spared #

A Sun-like star contracts too, and for the first million years it is several times brighter than today. It leaves its convective track after a few million years, when its core becomes radiative, and reaches the main sequence in about forty million. A planet at the Earth’s distance would have been past the runaway limit only in the first few million years, when there was not yet a planet at the Earth’s distance: the Earth was still assembling from embryos, and the collision that made the Moon happened after the Sun had settled.

So the steam phase is a property of low-mass stars in a specific sense. It is not that small stars are brighter when young than large ones are — relative to their final luminosity they are, somewhat, but the real difference is time. The Sun’s excess was over before its rocky planets were finished; a late M dwarf’s excess lasts longer than its planets take to form, and far longer than the envelopes that young planets capture from their discs can be expected to protect them.

TRAPPIST-1 and Proxima #

The argument is not hypothetical, because the two best-studied systems in the habitable zones of nearby stars are round exactly the stars it applies to most strongly.

TRAPPIST-1 is a star of about 0.09 solar masses, with seven Earth-sized planets, three or four of them in the zone it has now. On the contraction law those planets spent roughly a hundred million years past the runaway limit, and those nearest the inner edge considerably longer. Proxima Centauri, at 0.12 solar masses, has an Earth-mass planet in its zone with the same history. Both stars are old — billions of years — so their planets are long past the runaway phase; the question is what it left.

Infrared observations of the two innermost TRAPPIST-1 planets, which are hotter than the zone, have found thermal emission consistent with bare rock and no thick atmosphere. Those planets were never in the zone, so the result does not test the habitable-zone planets directly, but it is consistent with a system whose inner planets lost what they had. The planets’ densities are all slightly lower than an Earth-like rocky composition, which could mean a few per cent of water by mass — far more than the Earth’s ocean — or could mean less iron. If it is water, the arithmetic here says that it was either there in great quantity to begin with, or delivered after the star had settled, or kept by a planet that formed beyond the line where ice counts as rock and migrated in after the runaway phase had passed.

What the figures leave out #

The contraction law holds the surface temperature at its main-sequence value, which is within a few hundred kelvin on a Hayashi track, and it omits both the deuterium-burning pause early on and the slow approach to the main sequence at the end. The second is the more important. Detailed tracks keep the smallest stars above their final luminosity for longer than this law does, so the durations for the lowest masses here are likely to be underestimates.

The runaway limit is the one for a planet with an Earth-like atmosphere round a main-sequence star of that temperature. A pre-main-sequence star is slightly hotter, and a young planet may have a thick hydrogen envelope left from its formation, which changes both the runaway threshold and what is lost first: escape strips an envelope before it reaches the water. A planet can be protected for a while by the envelope it would otherwise lose, and the order in which the two are removed is not something this figure can decide.

And the planet is assumed to be there from the start. Rocky planets take tens of millions of years to finish assembling, and a planet that grows by collisions after the most intense phase escapes part of it. In the other direction, the ultraviolet output of young M dwarfs includes flares — brief outbursts of hundreds of times the quiescent level — which a smooth saturation fraction averages away.

A zone drawn in the present tense, for a planet made by its history #

A habitable zone is a present-tense quantity: the band where water can be liquid given the star’s luminosity now. A planet is a product of its whole history, and the habitable zone at one time says nothing about the zone at another. For a Sun-like star the zone moves slowly outward over billions of years and the question is whether a planet stays inside; for a small star the zone moves rapidly inward in the first few hundred million years and the question is what the planet was before the zone reached it.

The same two-directional history appears in the time a surface needs to feel its own orbit: the stars whose planets are best buffered against eccentricity are the stars whose planets were most exposed to their star’s youth. The M dwarfs’ advantages — long lives, stable zones, short years, frequent transits — are real, and every one of them is a statement about the main sequence. The disadvantage is a statement about everything before it, and it is written in whatever water those planets have left.

Still open: whether a planet can hide from its star’s youth #

The escape estimates assume a planet exposed at its present orbit from the moment its star formed. A planet that formed beyond the snow line and migrated inward later would arrive with far more water and miss part of the runaway phase; a planet whose magnetic field or thick early envelope delayed the loss would keep more. Which of those actually happened is recorded in the densities of the planets now in these zones, and in whether their atmospheres, when they can be measured, contain water, oxygen, or nothing at all.