An inner edge set by a ceiling on outgoing heat

The habitable zone is drawn as a band round a star between two distances, and one number sets it: the flux of starlight a planet receives, in units of what the Earth receives. Inside the band a planet with an Earth-like atmosphere and oceans could keep liquid water on its surface; outside it, the water either freezes or boils. The outer edge is set by how much carbon dioxide a planet can pile up before the gas starts reflecting more sunlight than it traps. The inner edge is set by something sharper, and it is the subject of this essay.

At the inner edge a planet does not merely become uncomfortably warm. It loses the ability to be in balance at all. A planet’s temperature is the one at which the heat it radiates to space equals the sunlight it absorbs, and for a dry rock the radiation rises steeply with temperature, so there is always a temperature that balances. For a planet with oceans the radiation rises with temperature only to a ceiling, and above the ceiling it stays flat. A planet absorbing more sunlight than the ceiling has no balancing temperature, and it heats until the oceans have boiled into the air — the runaway greenhouse.

The ceiling was found by the meteorologist George Simpson in the 1920s, while trying to understand the Earth’s own radiation balance, and rediscovered in the 1960s by Komabayashi and Ingersoll in the context of Venus. Its value can be estimated on the back of an envelope, and it depends on very little: the boiling curve of water, the strength of gravity, and how readily water vapour absorbs infrared light.

A planet that stops radiating more as it warms #

Heat leaves a planet as infrared radiation, and what an observer in space sees is not the surface but the level in the atmosphere from which radiation can escape directly — roughly the level above which the remaining atmosphere has an optical depth of about one in the infrared. On a dry planet that level is near the surface, and as the surface warms the escaping radiation warms with it.

Water vapour is a strong infrared absorber, and the amount the air can hold rises steeply with temperature. On a wet planet, as the surface warms, the vapour content rises, the atmosphere becomes more opaque, and the level from which radiation escapes moves upward to colder air. At modest temperatures the warmer surface still wins, and the emission rises, more slowly than for a dry planet. Beyond about 300 kelvin the moving level wins entirely: the emission flattens at a value that no further warming of the surface changes. That flat part is the ceiling.

The level space sees is pinned to the boiling curve #

Why flat, and why at that value? In a hot, wet atmosphere the upper layers are saturated with water vapour — the vapour pressure at each level is the saturation pressure at that level’s temperature. The level from which radiation escapes is where the column of vapour above weighs enough to be opaque: for an absorption coefficient $\kappa$, a column of mass $1/\kappa$ per unit area, which in a vapour-dominated atmosphere means a vapour pressure of $g/\kappa$. That pressure and the saturation curve together fix the temperature.

For the Earth’s gravity and an effective absorption coefficient of three hundredths of a square metre per kilogram — the value that makes this simple model agree with detailed calculations — the opaque level is at about 330 pascals of vapour, and the saturation curve puts it at 265 kelvin. A blackbody at 265 kelvin radiates 279 watts per square metre. Line-by-line radiative transfer through a real water-vapour spectrum, with all its windows and bands, gives a limit of about 282, and the simple picture explains why there is a limit at all: the temperature of the radiating level is the temperature at which water’s vapour pressure equals a number set by gravity and absorption, and the surface has no say.

The surprising connection is that the habitable zone’s inner edge is a property of the boiling curve of water — the same curve that sets the boiling point at the top of a mountain. Where the curve crosses the pressure $g/\kappa$ is to the runaway greenhouse what the crossing with atmospheric pressure is to a kettle.

Bigger planets have a higher ceiling, slowly #

The ceiling depends on the planet’s gravity through $g/\kappa$, and the dependence is weak.

The saturation pressure rises exponentially with temperature, so a tenfold change in the pressure at the opaque level moves its temperature by only about thirty kelvin. A planet with ten times the Earth’s surface gravity — a dense super-Earth several times its mass — has a ceiling about a third higher; a Mars-sized planet, a ceiling a sixth lower. The habitable zone’s inner edge moves inward for big planets and outward for small ones, but by less than their sizes would suggest.

The absorption coefficient matters more, and it is less certain. It represents the whole infrared spectrum of water vapour as a single number, and that number depends on the pressure broadening of the lines and the continuum absorption between them, both of which depend on what else is in the atmosphere. An atmosphere with a large background of nitrogen broadens the lines and raises the absorption; one of nearly pure steam behaves differently again. Detailed models agree on a ceiling near 280 for an Earth-like planet; the spread among them comes mostly from here.

Where the sunlight crosses the ceiling #

The planet’s temperature is set where the absorbed sunlight meets the emission curve.

At the Earth’s present sunlight and albedo, the absorbed flux of about 240 watts per square metre meets the emission curve near 288 kelvin. At ten per cent more sunlight the crossing moves to a much warmer surface, because the curve is flattening; at seventeen per cent more, the absorbed flux reaches the ceiling, and beyond that the horizontal line passes above the whole curve. The planet heats, more water evaporates, the opaque level stays where it is, the emission stays at the ceiling, and the surplus keeps going into heating the surface and evaporating the ocean, until the ocean has gone into the atmosphere and the surface is hot enough — over a thousand kelvin — for the hot rock itself to radiate through the steam’s windows.

That is the runaway, and it is not gradual. A planet a little inside the critical flux has a warm climate; one a little outside it boils its oceans, and the difference is a few per cent in sunlight. Few boundaries in planetary science are this abrupt, and the abruptness comes entirely from the flat top of the emission curve. Before reaching it, a planet passes through a moist greenhouse phase in which water vapour reaches the upper atmosphere, is broken by ultraviolet light and loses its hydrogen to space — the route by which a planet can lose its water even before its star has settled.

Clouds set the inner edge more than the ceiling does #

The ceiling is a property of water. The flux at which a planet reaches it depends also on how much sunlight it reflects.

The absorbed flux is the incident flux times one minus the albedo, so the albedo moves the critical instellation directly, and over a far larger range than gravity does. A cloudless Earth with a dark ocean surface, albedo 0.1, would be past the runaway already at its present distance. The Earth’s clouds are part of what keeps it on the safe side. A planet tidally locked to a red dwarf, turning slowly with one face to its star, is expected from climate models to build a thick deck of cloud over its substellar point, raising its albedo enough to push the inner edge to twice the Earth’s flux. The ceiling stays where water puts it; how close a planet must be to its star to reach it is a question about weather.

That is also why a thermostat that only halves the error — the carbonate–silicate cycle that adjusts carbon dioxide to keep a planet temperate — cannot save a planet past the runaway. Removing carbon dioxide from the air lowers the greenhouse warming at moderate temperatures, but at the ceiling the emission is set by water alone, and no amount of carbon dioxide removal raises it.

The inner edge for stars of every brightness #

The critical flux becomes a distance through the star’s luminosity.

For the Sun, the critical flux of 1.17 times the Earth’s places the inner edge at 0.92 astronomical units with the Earth’s albedo — eight per cent inside the Earth’s orbit. Detailed models, with real clouds absent and a more careful treatment of the moist greenhouse, put it closer still, at about 0.95 to 0.99. The Earth is near the inner edge, and the Sun’s slow brightening, about one per cent every hundred million years, will bring it across in about a billion years.

Venus is the planet that crossed. It receives nearly twice the Earth’s sunlight, well beyond the critical flux for any albedo it could have had while it had oceans, and if it began with water it would have run away early, lost its hydrogen to space and kept the carbon dioxide that now gives it a surface at 740 kelvin. The deuterium in its atmosphere, enriched a hundredfold over the Earth’s — the heavier isotope escapes less easily — is the record of a great deal of water lost.

The band moves and the ceiling does not #

The critical flux is fixed; the flux a planet receives is not. Stars brighten as they age on the main sequence, the Sun by about forty per cent since it formed, and the habitable zone moves outward past planets that stay where they are. For the Earth the ceiling is the edge it is moving towards. Four billion years ago the young Sun was faint enough that the Earth was in no danger of running away and was in danger, instead, of freezing; a billion years from now it will be bright enough that the Earth’s absorbed sunlight, with its present albedo, crosses 280 watts a square metre.

The ceiling also explains why the faint young Sun matters asymmetrically for the Earth and for Venus. Venus, receiving nearly twice the Earth’s sunlight now, received about 1.3 times the Earth’s present flux when the Sun was young — already past the critical value for a planet with the Earth’s albedo, and close to it even for a much cloudier one. Whether Venus ever had oceans on its surface, or ran away before its steam atmosphere could rain out, depends on whether its first atmosphere was cloudy enough on the side facing the Sun, and that in turn depends on how fast it was spinning, because a slow rotation favours a thick dayside cloud deck. Venus’s present day longer than its year is the end state of a long evolution, and its early spin is unknown; the possibility of an early habitable Venus rests on that unknown.

The young Sun’s activity enters too. A young star is bright in ultraviolet and X-rays, and a planet in its moist-greenhouse phase loses hydrogen at a rate set by that high-energy light. How fast the young Sun spun set how active it was, and through that how much water an Earth-like planet near the inner edge could have lost before the Sun calmed down. The ceiling decides whether the water reaches the upper atmosphere; the star’s activity decides how quickly it then leaves.

An eccentric orbit and the ceiling #

A planet on an eccentric orbit receives more sunlight at closest approach than its average, and could in principle cross the ceiling for part of each orbit while staying below it on average. Whether that is dangerous depends on how fast the planet’s climate responds. Oceans have large heat capacity, and a year too short for the planet to feel its own eccentricity averages the flux over the orbit: the runaway is decided by the mean flux, which for an orbit of eccentricity $e$ is larger than at the semi-major axis’s distance by a factor of $1/\sqrt{1-e^2}$. An orbit of eccentricity 0.3 raises the mean by five per cent — enough to move a planet across the edge if it was close to it.

A planet with a long year and shallow seas would respond to each passage of periastron, and might briefly reach the moist-greenhouse regime every orbit, losing a little water each time. That makes the inner edge somewhat fuzzier for eccentric planets than for circular ones. The ceiling itself does not move; the planet’s path across it does.

What the model leaves out #

The atmosphere is grey: one absorption coefficient for the whole infrared spectrum, where water vapour’s spectrum has windows through which the surface can radiate even through a thick atmosphere, and through which a very hot surface eventually does. The emission curve below the ceiling is schematic, chosen to pass through the present Earth’s balance; real curves depend on the profile of relative humidity, the lapse rate and clouds. Clouds are reduced to an albedo, although they also trap infrared radiation and shift the emission.

The ceiling also assumes an atmosphere dominated by water vapour at the radiating level. A planet with a very thick background of other gases — tens of bars of nitrogen or carbon dioxide — radiates from higher, colder levels for other reasons, and its limit is different. And the model assumes a planet that has oceans to evaporate; a desert planet with little water, its surface water confined to the poles, has no runaway of this kind and can stay habitable much closer to its star.

Still open: whether rocky planets at the inner edge have water #

The ceiling makes a sharp prediction that observations can test. Planets just inside the critical flux should be temperate or warm; planets just outside should be steam worlds or, after losing their hydrogen, dry with thick carbon dioxide or oxygen atmospheres. A population of rocky planets spanning the inner edge would show a transition — in their atmospheres, in their measured temperatures, perhaps even in their radii, since a planet with its ocean in its atmosphere is several per cent larger than the same planet with its ocean on the surface.

The rocky planets round nearby red dwarfs, whose atmospheres the James Webb Space Telescope is beginning to probe, straddle that edge, and so do planets that may have spent an envelope to keep their oceans.

There is also a test closer to home. The ceiling predicts how the Earth’s own emission should respond to warming: as the surface warms, the outgoing radiation should rise less and less steeply, flattening towards 280 watts a square metre. Satellites measuring the Earth’s outgoing infrared radiation across seasons and regions see exactly that flattening in the warmest, wettest parts of the tropics, where the clear-sky emission barely rises with surface temperature above about 300 kelvin — the local runaway regime, held in check only because the atmosphere carries heat away to cooler latitudes. The tropics are a small, permanent experiment in the physics that sets the inner edge of every habitable zone.

Whether the exoplanets beyond the critical flux turn out to have lost their water, and those inside it to have kept it — a difference that should be visible as the presence or absence of water vapour and of the oxygen that a planet losing its hydrogen leaves behind — is the test of the habitable zone’s inner edge against other worlds rather than against models — and of the claim that the boiling curve of water draws a line round every star.