An envelope spent so an ocean could stay

The smallest stars arrive on the main sequence slowly. A star of a tenth of the Sun’s mass contracts for several hundred million years before hydrogen burning holds it steady, and during that time it is several times brighter than it will be for the trillions of years afterwards. A planet that will one day sit in the middle of its habitable zone therefore begins life too close to a star that is too bright: for its first seventy-five million years it receives more light than the runaway-greenhouse limit, and any ocean on it is not an ocean but a thick atmosphere of steam. Ultraviolet light from the young star splits the water at the top of that atmosphere, the hydrogen escapes, and the oxygen is left behind. An Earth-mass planet there can lose more than ten Earth oceans of water before the star settles.

That calculation assumes the planet faces its star bare. A rocky planet that finishes growing while the gas of its birth nebula is still present does not; it captures a little hydrogen and helium from the nebula as a thin primordial atmosphere. That atmosphere is the first thing the star’s high-energy light meets, and while it lasts the energy goes into removing it. The water underneath — dissolved in a magma ocean, or condensed beneath the envelope, or simply below it in the atmosphere’s lower layers — is not where the escape is happening.

The hero figure follows four such envelopes, from a tenth of a per cent of the planet’s mass to three per cent, on an Earth-mass planet in the middle of a 0.1 solar-mass star’s eventual habitable zone. The shaded span is the runaway phase. The smallest envelope lasts 463 million years and the next 1.5 billion — both far longer than the seventy-five-million-year runaway. The two largest are still there after five billion years.

Why red dwarfs are where it matters #

The problem is peculiar to small stars, and the reason is arithmetic about where their habitable zones sit. The zone is set by stellar flux, so it lies at a distance proportional to the square root of the star’s luminosity. A star of a tenth of the Sun’s mass has a luminosity of about a thousandth of the Sun’s, and its habitable zone lies near three hundredths of an astronomical unit — closer to its star than Mercury is to the Sun by a factor of ten. Everything the star emits arrives concentrated by that closeness: its high-energy light, its flares, its wind.

The second ingredient is time. A Sun-like star reaches the main sequence in about forty million years, and its habitable zone moves outward only slowly afterwards. A red dwarf takes hundreds of millions of years to settle, so its eventual zone is too close for a long time, and it stays magnetically active — saturated in X-rays — for a billion years or more, far longer than the Sun did. The smallest stars are also the longest-lived and the commonest, which is why most of the temperate rocky planets known, and most that will ever be studied in detail, orbit them.

The combination is what makes the habitable zones of red dwarfs the places where water loss is a serious question rather than a footnote. And the same combination is what makes the primordial envelope matter more there than anywhere else: the envelope has to survive a long, intense, early exposure, and the exposure that wears it away is the one it shields the ocean from.

Why the envelope goes first #

Escape driven by a star’s X-rays and extreme ultraviolet is limited, in the regime that matters here, by the energy supplied. The light is absorbed high in the atmosphere, heats the gas there to thousands of kelvin, and the heated gas expands into a wind that lifts mass out of the planet’s gravity well at a rate set by the absorbed power: $\dot M \approx \varepsilon\,\pi R^3 F_{\rm XUV}/(GM)$, with $\varepsilon$ the fraction of the absorbed energy that goes into lifting gas rather than being radiated away. The rate depends on the planet’s size cubed — its cross-section to the light times its radius, which sets the depth of the well per unit mass — and on its mass.

The light does not choose what to lift. It heats whatever lies at the top of the atmosphere, and on a planet with a hydrogen envelope that is hydrogen, light and easily lifted. Water and its fragments sit lower and stay there until the hydrogen above them is thin enough for the ultraviolet to reach them. So the envelope acts as a sacrificial layer: every joule the star spends on the envelope is a joule not spent on the water.

The process has been watched happening on planets close to their stars, where a hydrogen exosphere makes a planet ten times larger in one ultraviolet line than in visible light, and where the metastable helium line measures the outflow from the ground. The rates measured there agree with the energy-limited estimate to within its own uncertainties.

There is a feedback that makes the envelope more costly to keep than its mass suggests. A hydrogen envelope of even a fraction of a per cent of a planet’s mass inflates the planet considerably, because a hot, light gas has a large scale height. An Earth with one per cent of its mass in hydrogen has nearly twice the Earth’s radius, intercepts nearly four times the light, and loses mass about seven times as fast as the bare rock it surrounds. The envelope is a shield that wears itself away faster for being there. This is the same physics that strips sub-Neptunes down to rocky cores close to their stars, applied to an envelope a hundred times smaller.

How much water the shield saves #

The comparison that matters is between the envelope’s lifetime and the runaway’s.

A bare planet loses twelve oceans at this efficiency — more water than the Earth has ever had. An envelope of a hundredth of a per cent of the planet’s mass, a few Earth atmospheres of hydrogen, cuts that sharply, because it delays the moment the water is exposed until the star has faded somewhat. And at about four hundredths of a per cent the envelope outlasts the runaway entirely. By the time the hydrogen is gone, the star has contracted enough that the planet receives less light than the runaway limit, the steam can condense, and whatever ocean the planet started with is still there.

Four hundredths of a per cent of an Earth mass is about $2\times10^{21}$ kilograms of hydrogen — roughly ten times the mass of the hydrogen in the Earth’s oceans. It is a thin envelope by the standard of planet formation: the models of how much nebular gas an Earth-mass core captures before the nebula disperses give between a few hundredths of a per cent and about one per cent, depending on the core’s growth rate and on how quickly the nebula is removed. So the shielding envelope is not an exotic requirement. It is the expected outcome for a planet that finished growing early.

The envelope that never leaves #

There is a second boundary, and it is the one that makes the result interesting rather than merely reassuring. An envelope big enough to shield the water may also be big enough to survive the star’s entire active phase. A red dwarf’s X-ray output stays saturated — at about a thousandth of its total light — for a billion years or more, then declines; once it has declined, escape slows to a trickle, and whatever envelope remains is effectively permanent.

For the planet in the figure, an envelope of 0.69 per cent of its mass or more is never fully removed within five billion years. Such a planet is not a rocky world with an ocean. It is a small gas-rich planet, nearly twice the Earth’s radius, with a surface pressure of thousands of bars under a hydrogen atmosphere that traps heat far more effectively than carbon dioxide does — a sub-Neptune of the kind that lies above the gap in the radius distribution, not a habitable planet. Whatever water it has is buried under a lid that will not lift.

It has been argued that such a planet could still be habitable in a different way: a deep water layer under a modest hydrogen atmosphere, warm enough at its surface for liquid water, on planets sometimes called hycean worlds. The difficulty is the greenhouse effect of the hydrogen itself. Molecular hydrogen at high pressure absorbs infrared radiation through collisions between molecules, an opacity that grows as the square of the density, and a hydrogen atmosphere of even a few per cent of an Earth mass traps heat so effectively that the surface beneath it is hundreds or thousands of kelvin hot rather than temperate. Whether any balance of envelope mass, cloud cover and stellar flux leaves liquid water at the base is contested; for an envelope of the size that survives five billion years round a red dwarf at the middle of its habitable zone, most models put the surface well above the boiling point. The water is kept, but not as an ocean anyone would recognise.

So an envelope can fail in two ways. Too small, and it is gone before the runaway ends, and the water is exposed. Too large, and it outlasts the star’s active youth, and the planet never becomes rocky. Between the two, for this star, lie envelopes from 0.04 to 0.7 per cent of the planet’s mass — a factor of about eighteen — that are both used up and used up late enough.

A larger star, a shorter danger #

The runaway is shorter round a larger red dwarf, because its pre-main-sequence phase is shorter and its eventual habitable zone lies further out.

Round a star of 0.3 solar masses the bare planet loses three oceans rather than twelve, the shielding envelope needs to be only a hundredth of a per cent of the planet’s mass, and the envelope that never leaves is smaller too, because the star’s saturated X-ray phase is shorter and its planets are further out. The whole window has moved downward by a factor of about four and kept its width.

The timelines show why the window keeps its width. The runaway is over in twenty-three million years, a small fraction of the envelope’s life; what decides whether an envelope is spent is the long saturated phase that follows, and that phase is shorter round larger stars too. Both boundaries move together, set by two different parts of the same history — the first million years and the first billion.

The window across the red dwarfs #

Across the whole range of red dwarfs the window is about a factor of twenty wide, and it lies between a few thousandths and about one per cent of the planet’s mass. The nebular capture estimates span a comparable range. For the smallest stars, whose planets most need the shield, the window lies highest — from a few hundredths to most of a per cent — which is where the capture estimates for an early-forming Earth-mass planet also sit.

The efficiency of escape is the least certain number in the calculation, and the window moves with it.

Doubling the efficiency roughly doubles the shielding requirement and raises the never-lost boundary by more, because a larger envelope that would have survived now does not. The window widens rather than shifting. The result is robust in the way that matters: across the plausible efficiencies, a planet that captured an ordinary amount of nebular gas has a reasonable chance of keeping its ocean through its star’s youth and losing its hydrogen afterwards.

Two records the window predicts #

If envelopes did shield the oceans of habitable-zone planets round red dwarfs, two things should follow, and both can be looked for.

The first is in the planets’ densities. A planet that kept most of its water is less dense than a bare rock of the same mass — a few per cent of the mass in water lowers the bulk density noticeably — while a planet that lost its water through a runaway is dense and dry. The seven planets of the TRAPPIST-1 system, whose masses and radii are measured to a few per cent, are all slightly less dense than an Earth-like composition would make them, which is consistent with a few per cent water, or with less iron, or with both; the measurements do not separate the cases yet.

The second is in the atmospheres. A planet that lost its water leaves oxygen behind, potentially hundreds of bars of it, which would make an oxygen atmosphere a sign of past water loss rather than of life. A planet whose envelope shielded its water should have no such oxygen excess. A planet whose envelope never left should have a hydrogen-rich atmosphere with a large scale height, which is the easiest kind to detect in transmission. And a planet that lost both its envelope and its water, and the oxygen afterwards, would be bare rock — on the airless side of the shoreline drawn in a star’s lifetime ultraviolet, where the innermost TRAPPIST-1 planets appear to sit. The first spectra of temperate rocky planets round red dwarfs, now being taken, have so far found no thick hydrogen envelopes — which rules out the never-lost case for those planets, and leaves the other two.

Where the calculation stops #

The envelope is a clean lid. In reality hydrogen escaping fast enough can drag heavier species with it, including water vapour and oxygen, so a vigorous hydrogen wind does not leave the water untouched. The loss of water during the envelope’s removal is not zero; it is reduced by however much the hydrogen dominates the upper atmosphere, which is likely to be most of the time but not all of it.

The water is under the envelope and stays there. An ocean in contact with a hot hydrogen envelope may react with it — water and hydrogen exchange with a molten surface, and some of the hydrogen can end up as water and some water as hydrogen — and the surface under a thick envelope is hot enough to keep rock molten. Whether the ocean survives as an ocean, or as water dissolved in rock that degasses later, is a further question the calculation does not address.

The star’s X-ray history is a simple law. Saturation for 300 million years times half a solar mass divided by the star’s mass, then a decline as the age to the power $-1.5$, follows the measured average; individual stars differ by factors of a few, and young red dwarfs flare frequently in ways the average smooths over.

The envelope’s radius is a fitted law from models of planets with hydrogen envelopes, and the planet’s mass, its core and its position in the zone are held fixed. A planet further out in the zone, or more massive, needs less envelope to shield it and keeps more.

The runaway threshold assumes no hydrogen. A planet with a hydrogen envelope has a different runaway limit and a different surface temperature; the timing used here is that of a bare steam atmosphere, which is the timing that matters once the envelope is gone.

Still open: whether the shield and the planet are the same planets #

The window says that the envelopes planets are expected to capture are the envelopes that would save their oceans. It does not say that any particular planet captured one, and the same nebular gas that a slow-growing planet misses entirely, a fast-growing one captures too much of. An Earth-mass planet round a red dwarf that formed in a few million years would sit near the top of the window or above it; one that formed in a few tens of millions, after the nebula had gone, would have no envelope and no shield. The planets in the habitable zones of red dwarfs could therefore be a mixture — some dry, some wet, some still wrapped in hydrogen — sorted by how quickly they formed.

The same arithmetic applied to the Earth turns on how fast the young Sun spun, because that set its early X-ray output. Round red dwarfs it turns on how fast the planets grew, because that set how much gas they caught. Both are questions about the first few million years that a planet’s present density and atmosphere may still record, and the planets round the nearest red dwarfs are the ones whose densities and atmospheres can now be measured.