Astronomy & Celestial Mechanics
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Exoplanets

A thermostat that only halves the error

The carbonate–silicate cycle is credited with keeping a planet's water liquid across the whole width of its habitable zone. Written down with its own measured exponents, it is a proportional controller that cancels about three-fifths of a change in sunlight, takes half a million years to do it — and reaches the published outer edge only on a planet with an order of magnitude more volcanism than the Earth.

Assumes Habitable zone, Atmospheric escape and Seasons.

16 min read 5 figures Small perturbations, large consequencesScales that defeat intuition

The outer edge of a habitable zone is not where a planet with the Earth’s atmosphere would freeze. With today’s 280 microbars of carbon dioxide, the Earth moved out to 1.15 AU would already be below freezing on average. The edge quoted in every survey — 1.67 AU for the Sun — is where a planet with as much carbon dioxide as can possibly help would freeze: several bar of it, enough that adding more scatters away more sunlight than it traps. The zone as a band of stellar flux takes that maximum as given.

Nothing gives a planet several bar of carbon dioxide on request. The mechanism invoked to supply it is the carbonate–silicate cycle: volcanoes release CO₂, rain dissolves it and weathers silicate rock, and the carbon ends up as carbonate on the sea floor. Weathering runs faster when the planet is warm, so a cooling planet weathers more slowly, CO₂ accumulates, and the planet warms again. It is a thermostat, and it is the reason the Earth’s oceans survived a Sun thirty per cent fainter in its first billion years.

The word “thermostat” suggests a set point: a temperature the planet returns to, whatever the sunlight does. A feedback loop does not have a set point unless its gain is very large. The carbonate–silicate cycle’s gain can be written down from three measured exponents, and it is not large.

A thermostat that passes 40 per cent of the change it is meant to cancel. Surface temperature against the flux a planet absorbs, in units of the Earth's, for a planet round a 1 M☉ star, with and without the carbonate–silicate cycle. With carbon dioxide held at 280 µbar the temperature follows the flux directly. With weathering allowed to adjust — rock dissolves faster when it is warm and when there is more CO₂, and in the steady state it must remove exactly what volcanoes supply at 1 times today's rate — a colder planet accumulates CO₂ until the balance is restored. The feedback is real and it is not a set point. Near S = 1 it passes 40 per cent of a flux change through to the surface: the loop gain is k s / β = 1.49, with weathering rising one e-fold for every 9.7 K, a greenhouse of 4.33 K per e-folding of CO₂ and a CO₂ exponent of 0.3. The required CO₂ would reach 8.8 bar — the point at which more of it scatters sunlight faster than it traps heat, and the controller has nothing left to add — at S = 0.249. The steady state reaches 273 K at S = 0.531, before the CO₂ has run out — the outer edge of this planet's habitable zone is where the thermostat saturates or freezes, whichever comes first. The climate law is logarithmic in CO₂, which is right near today's values and only a calibration at several bar; ice-albedo feedback, which makes a cooling planet able to jump to a frozen state, is left out.
Fig. 1 Surface temperature against the flux a planet absorbs, in units of the Earth’s, round a Sun-like star. The dashed curve holds carbon dioxide fixed at 280 µbar; the solid curve lets weathering balance volcanic outgassing at today’s rate. Near the Earth’s flux the feedback passes 40 per cent of any change through to the surface — the loop gain is 1.49 — and the planet freezes at an absorbed flux of 0.531, before the carbon dioxide has reached the 8.8 bar at which more of it stops warming. Weathering here rises one e-fold for every 9.7 K and as the 0.3 power of CO₂.

Three exponents and a loop

Everything in the figure follows from three relations, and each has a number that has been measured rather than chosen.

The greenhouse. Near present-day concentrations, each doubling of carbon dioxide warms a planet by about 3 K once the fast feedbacks — water vapour, lapse rate, clouds — have responded. That is a logarithmic law: TT rises by s=3/ln⁡2=4.33s = 3/\ln 2 = 4.33 K for each e-folding of CO₂.

Rock dissolution. Silicate minerals dissolve at a rate that follows an Arrhenius law with an activation energy of about 50 kJ/mol, which near 288 K is an increase of one e-folding for every 13.7 K. Rain also runs off faster on a warmer planet, by about three per cent per kelvin. Together the weathering rate rises as ek(T−T0)e^{k(T - T_0)} with k≈0.10k \approx 0.10 per kelvin.

Direct dependence on CO₂. Weathering also rises with carbon dioxide itself, because the acid doing the dissolving is carbonic acid. The exponent, β\beta, is the least certain of the three: about 0.3 in the original 1981 formulation of the cycle, about 0.5 for purely abiotic weathering in laboratory measurements, and lower again on a planet where soil biology pumps CO₂ into the ground regardless of how much is in the air.

In a steady state, weathering must remove exactly what volcanoes supply: ek(T−T0) (p/p0)β=Ve^{k(T - T_0)}\,(p/p_0)^\beta = V. Combine that with the greenhouse law and the carbon dioxide concentration drops out, leaving a closed form. A change in flux that would move the temperature by ΔT\Delta T at fixed CO₂ moves it by

ΔTfeedback=βks+β  ΔT\Delta T_{\rm feedback} = \frac{\beta}{k s + \beta}\;\Delta T

The quantity G=ks/βG = ks/\beta is the loop gain — the factor by which the feedback opposes a disturbance — and the fraction passed through is 1/(1+G)1/(1+G). With k=0.103k = 0.103, s=4.33s = 4.33 and β=0.3\beta = 0.3, G=1.49G = 1.49 and the thermostat passes forty per cent of every change.

That is a very good controller by geophysical standards and a poor one by an engineer’s. A household thermostat has a gain of hundreds; it holds its set point to a degree whatever the weather. A loop gain of 1.5 halves the error and a little more.

A thermostat that passes 53 per cent of the change it is meant to cancel. Surface temperature against the flux a planet absorbs, in units of the Earth's, for a planet round a 1 M☉ star, with and without the carbonate–silicate cycle. With carbon dioxide held at 280 µbar the temperature follows the flux directly. With weathering allowed to adjust — rock dissolves faster when it is warm and when there is more CO₂, and in the steady state it must remove exactly what volcanoes supply at 1 times today's rate — a colder planet accumulates CO₂ until the balance is restored. The feedback is real and it is not a set point. Near S = 1 it passes 53 per cent of a flux change through to the surface: the loop gain is k s / β = 0.89, with weathering rising one e-fold for every 9.7 K, a greenhouse of 4.33 K per e-folding of CO₂ and a CO₂ exponent of 0.5. The required CO₂ would reach 8.8 bar — the point at which more of it scatters sunlight faster than it traps heat, and the controller has nothing left to add — at S = 0.155. The steady state reaches 273 K at S = 0.624, before the CO₂ has run out — the outer edge of this planet's habitable zone is where the thermostat saturates or freezes, whichever comes first. The climate law is logarithmic in CO₂, which is right near today's values and only a calibration at several bar; ice-albedo feedback, which makes a cooling planet able to jump to a frozen state, is left out.
Fig. 2 The same calculation with the CO₂ exponent at 0.5, the laboratory value for weathering with no biology. The gain falls to 0.89 and the thermostat passes 53 per cent of a change; the steady state now freezes at an absorbed flux of 0.624 instead of 0.531. The exponent sits in the denominator of the gain, so a planet whose weathering depends more strongly on CO₂ has a weaker thermostat — more carbon dioxide speeds up its own removal, and the concentration cannot build as far before the removal catches up.

The direction of that dependence surprises at first. A planet whose weathering is more sensitive to CO₂ is worse at holding its temperature, because the thing being used to warm the planet is also the thing that speeds up its own removal. What makes the thermostat strong is a weathering rate that is sensitive to temperature and indifferent to CO₂ — which is, roughly, what land plants do to the Earth’s cycle, and one reason the gain of the modern cycle may be larger than an abiotic planet’s.

How fast it answers

A gain is half of a controller’s description. The other half is its time constant, and for the carbonate–silicate cycle that is set by how much carbon has to move.

The atmosphere and the ocean together hold about 38,000 gigatonnes of carbon, almost all of it dissolved in seawater. Volcanoes supply roughly a tenth of a gigatonne a year. The reservoir is therefore about 380,000 years of outgassing, and a disturbance to the balance is corrected at a rate set by that reservoir divided by how hard the loop pushes back.

A −5 per cent step in sunlight, answered over 646 thousand years. Surface temperature after the flux a planet absorbs changes abruptly by −5 per cent at time zero, with the carbonate–silicate cycle left to respond. The climate itself adjusts within centuries, which on this axis is instantaneous: the temperature jumps from 288.0 K to 284.8 K. Then weathering, now running at a rate the volcanoes no longer match, lets CO₂ accumulate in the atmosphere and ocean, a reservoir that takes 380 thousand years of outgassing to fill. The temperature recovers with a time constant of 510 thousand years in the linear limit — the reservoir time divided by k s + β — and 646 thousand years read off the drawn curve, and settles at 286.7 K, having cancelled 60 per cent of the step and kept 40. Two conclusions follow and they pull in opposite directions. Against a star brightening over billions of years the thermostat is effectively instantaneous, so it always operates at its steady state; against anything faster than a few hundred thousand years — an impact, a large eruption, the onset of a glaciation — it is not there at all.
Fig. 3 Surface temperature after the absorbed flux falls abruptly by 5 per cent. The climate itself responds within centuries, which on this axis is a vertical drop from 288.0 K to 284.8 K. Weathering, now outpacing the volcanoes, slows until CO₂ accumulates; the temperature recovers with a time constant of 510 thousand years in the linear limit and 646 thousand years as drawn, and settles at 286.7 K — cancelling 60 per cent of the step and keeping 40. The drawn recovery is slower than the linear estimate because the reservoir grows as the CO₂ it holds, so the loop is working against a larger inventory by the time it finishes.

The time constant is the reservoir time divided by ks+βks + \beta, and it has exactly the form of the first-order lag that makes the hottest month later than the sunniest: a capacity divided by a restoring coefficient. There the capacity is heat in a mixed layer of ocean and the answer is weeks. Here it is carbon in the ocean-atmosphere system and the answer is half a million years.

Two consequences follow and they point in opposite directions. Against a star that brightens over billions of years, half a million years is instantaneous; the cycle is always at its steady state, and the gain figure is the whole story. Against anything faster — an asteroid impact, a flood basalt, a sudden release of carbon — the thermostat is not there at all on the timescale that matters to the climate.

The geological record contains a clean test. Fifty-six million years ago a large release of carbon warmed the Earth by five or more degrees within a few thousand years. The recovery took more than a hundred thousand years, and the shape of the recovery, recorded in the carbon isotopes of deep-sea sediments, is consistent with weathering drawing the excess down on a timescale of that order. The thermostat worked, and it worked on its own schedule.

The snowball episodes of seven hundred million years ago are the opposite test. Once ice reached the tropics, its reflectivity held the planet frozen at a temperature far below anything a small adjustment could reverse — an instability this simple figure leaves out, because it has no ice. Weathering on a frozen planet nearly stops, so volcanic CO₂ accumulated for millions of years until the greenhouse was strong enough to melt the ice from the tropics outward. The thermostat rescued the planet, but only after the controller’s inputs had been saturated for longer than most of the Earth’s species have existed.

What the faint young Sun asks of it

The case the cycle was proposed to explain is a useful calibration of the gain, because it can be checked in rocks.

The Sun arrived on the main sequence about thirty per cent fainter than it is now — a star’s core grows denser and hotter as hydrogen is converted to helium, and its luminosity rises with it. With carbon dioxide at today’s level, an absorbed flux of 0.7 would put the Earth’s mean surface temperature near 266 K: frozen, with nothing in the geological record to say it was. With the thermostat at a gain of 1.49, the same flux gives a drop of under nine kelvin rather than twenty-two, and a planet at about 279 K — cool, and liquid.

The price is carbon dioxide. The closed form says how much: the steady state needs about three e-foldings more than today, twenty times the present concentration, or roughly five millibars. Estimates of the Archean atmosphere from the chemistry of fossil soils and from the minerals that formed in ancient rivers come out in the range of tens to a few hundred times today’s level. The controller’s prediction lands at the low end of that range, which is where it should land if the young Earth’s hotter interior was also outgassing faster — raising VV and the curve with it — and if methane from early microbial life was adding a second greenhouse the figure does not include.

That is a reasonable success for a model with three exponents in it, and it says something specific about what the success does not require. It does not require a strong thermostat. A loop gain of one and a half, acting over hundreds of millions of years, with a modest boost from a younger interior, is enough to keep an ocean through a thirty per cent change in sunlight at the Earth’s distance. What it cannot do is extend that to the far side of the zone, where the change to be cancelled is a factor of three — and where the young star’s activity and the planet’s own history of escape have already decided how much atmosphere there is to control.

Where the thermostat runs out

The linear picture has a hard limit. Carbon dioxide cannot warm without bound: at pressures of a few bar it begins to condense in the upper atmosphere, and its Rayleigh scattering of incoming sunlight grows linearly with the column while its greenhouse effect grows only logarithmically. Past some pressure, more CO₂ cools. Radiative-convective climate models put that maximum greenhouse at a flux of 0.343 of the Earth’s for a Sun-like star, which is where the published outer edge comes from.

The logarithmic greenhouse law above, extrapolated, reaches that edge at 8.8 bar. That is not a derivation of the saturation — the law is only a calibration at bar-level pressures — but it lands where the detailed models find the maximum, which is a useful check that the extrapolation is not absurd.

The outer edge is set by the volcanoes until the CO₂ runs out. The carbon dioxide a planet round a 1 M☉ star holds in the steady state of the carbonate–silicate cycle, against the flux it absorbs, for volcanic outgassing at 0.25, 1, 4, 16 times the Earth's present rate. As the flux falls the planet cools, weathering slows, and CO₂ builds up until removal matches supply again — by about a factor of ten for every 0.26 fall in S near S = 1. The horizontal line is 8.8 bar, where more CO₂ stops warming. A planet with enough volcanism reaches that ceiling while still warm, and its edge is then the maximum-greenhouse flux S = 0.343, which depends on the star and not on the planet. A planet with less never reaches it: its equilibrium CO₂ is too low, its surface drops through 273 K first, and its edge sits nearer the star. At 0.25× the edge is at S = 0.755, 1.15 AU; at 1× the edge is at S = 0.531, 1.37 AU; at 4× the edge is at S = 0.361, 1.66 AU; at 16× the edge is at S = 0.343, 1.71 AU, at the ceiling. The published outer edge of the habitable zone is the ceiling, and it silently assumes a planet with enough volcanism to use the whole of the greenhouse the star permits.
Fig. 4 The steady-state carbon dioxide against absorbed flux for volcanic outgassing at 0.25, 1, 4 and 16 times the Earth’s rate. Near S = 1 the concentration rises tenfold for every 0.26 fall in flux. The horizontal line is 8.8 bar, where the greenhouse saturates. With a quarter of the Earth’s volcanism the planet freezes at S = 0.755, 1.15 AU from a Sun-like star; with the Earth’s, at 0.531 and 1.37 AU; at four times, 0.361 and 1.66 AU; and only at sixteen times does it reach the ceiling at 0.343 and 1.71 AU. The published outer edge is the last case.

The figure turns the outer edge from a property of the star into a property of the planet. A planet with vigorous volcanism holds enough carbon dioxide at every flux to reach the ceiling while still warm, and for that planet the edge really is the maximum-greenhouse flux, set by the star’s spectrum alone. A planet with less volcanism never gets there. Its steady-state CO₂ is lower at every flux, its surface drops through freezing first, and its edge sits closer in — for the Earth’s own outgassing rate, at 1.37 AU rather than 1.67.

That difference is the whole annulus between them, and in area it is about half of the conventional zone. Mars sits at 1.52 AU. With the Earth’s volcanism, on this accounting, it would be outside the reach of its own thermostat even if it had one; with sixteen times the Earth’s it would be inside. Mars has had very little volcanism for billions of years, and no plate tectonics to return carbonate to the mantle — which is a sufficient explanation for its present state that does not need the flux at all.

The part that needs plate tectonics

The cycle depends on the rock that weathers being renewed and the carbonate that forms being returned. On the Earth both are done by plate tectonics: mountain building exposes fresh silicate, and subduction carries carbonate down to be released again at volcanoes. The outgassing rate VV in all of the figures above is the output of the planet’s interior, and it is not constant.

A planet’s heat flow declines as its radioactive isotopes decay and its interior cools, so its volcanic outgassing declines too. A planet without plate tectonics — a “stagnant lid” planet, like Mars and Venus today — outgasses only as long as melt can reach the surface through a thick lithosphere, and models of such planets give outgassing that falls by orders of magnitude over a few billion years. On the second figure’s axes that planet moves steadily from the upper curves to the lower ones while its star brightens and moves it rightwards. Whether its temperature holds depends on which motion is faster.

That is the reason the habitable zone for a real planet is a band that depends on the planet’s mass, age and tectonic regime as well as on its star. A larger planet retains its heat longer and outgasses for longer. A planet with more water may have its volcanism suppressed by the weight of an ocean. An atmosphere a planet cannot keep is not available to a thermostat at all. None of those enter the flux-only zone, and all of them move the outer edge by more than the difference between competing climate models.

A thermostat that runs out of range

A thermostat that passes 40 per cent of the change it is meant to cancel. Surface temperature against the flux a planet absorbs, in units of the Earth's, for a planet round a 1 M☉ star, with and without the carbonate–silicate cycle. With carbon dioxide held at 280 µbar the temperature follows the flux directly. With weathering allowed to adjust — rock dissolves faster when it is warm and when there is more CO₂, and in the steady state it must remove exactly what volcanoes supply at 4 times today's rate — a colder planet accumulates CO₂ until the balance is restored. The feedback is real and it is not a set point. Near S = 1 it passes 40 per cent of a flux change through to the surface: the loop gain is k s / β = 1.49, with weathering rising one e-fold for every 9.7 K, a greenhouse of 4.33 K per e-folding of CO₂ and a CO₂ exponent of 0.3. The required CO₂ would reach 8.8 bar — the point at which more of it scatters sunlight faster than it traps heat, and the controller has nothing left to add — at S = 0.332. The steady state reaches 273 K at S = 0.361, before the CO₂ has run out — the outer edge of this planet's habitable zone is where the thermostat saturates or freezes, whichever comes first. The climate law is logarithmic in CO₂, which is right near today's values and only a calibration at several bar; ice-albedo feedback, which makes a cooling planet able to jump to a frozen state, is left out.
Fig. 5 A planet with four times the Earth’s volcanism, drawn to lower fluxes. The loop gain is unchanged at 1.49 — the outgassing rate shifts the steady state without changing its slope — and the curve now stays above freezing down to an absorbed flux of 0.361. The carbon dioxide would reach its 8.8-bar ceiling at 0.332, just beyond; a planet with slightly more volcanism would hit the ceiling first, and below it the solid curve would bend down to follow the fixed-CO₂ law, because a controller that has exhausted its range is no controller.

Every controller has a range, and the figure makes the two ways out of it visible. Outgassing sets the offset of the curve and the weathering exponents set its slope; the ceiling at 8.8 bar is where the actuator is fully open. A planet near the outer edge is either freezing because its loop gain is finite — the lower curves — or freezing because its actuator is at its stop — the upper ones. The published edge is the second failure. The first, which depends on the planet, comes earlier for most planets drawn.

At the inner edge the loop fails differently and more simply. Weathering can draw carbon dioxide down only to near zero, and once it has, the planet has no further way to cool. On a planet receiving slightly more than the Earth’s flux, water vapour then begins to dominate the greenhouse and the runaway that defines the inner edge takes over — a positive feedback no weathering can counter. The thermostat is a feature of the middle of the zone.

What the figures cannot show

The climate law is logarithmic in carbon dioxide and linear in nothing else. It has no ice, so it cannot produce the bistability in which a planet cooling past a threshold jumps to a frozen state and stays there; real planets near the outer edge are more likely to freeze suddenly than the smooth curves suggest. It has no clouds, whose response to warming is the largest uncertainty in the Earth’s own climate sensitivity and could move ss by a factor of two. And it treats the absorbed flux as the Earth’s albedo times the stellar flux, which is only right for a Sun-like spectrum: round a cooler star, redder light is absorbed more readily and scattered less by CO₂, and the same law puts the carbon dioxide ceiling at an unphysical pressure. The figures are therefore drawn for the Sun only.

The weathering law is a single global exponential. Real weathering depends on topography, on the area of exposed land, on how fast rock is uplifted and on whether soils are thick enough to limit how much fresh mineral the water can reach. A planet covered entirely by ocean has almost no continental weathering and must rely on reactions at the sea floor, whose temperature dependence is weaker. For such a planet the gain is lower still.

A feedback is a gain, not a set point

A negative feedback is not a guarantee. Its strength is a product of sensitivities, and whether it holds a system near a value depends on that product being large — which in a natural system is a question of measurement, not of principle. The carbonate–silicate cycle has a gain of one to two, which makes it one of the most important stabilising processes on any planet and nowhere near strong enough to hold a temperature constant across a factor of three in sunlight.

The same distinction runs through the tilt held steady by being too fast to resonate, where stability comes not from a restoring force but from a frequency separation, and through the line beyond which ice counts as rock, where a threshold that looks like a property of the disc is really a property of how fast the disc is changing. In each case the word that names the mechanism — thermostat, stabiliser, line — is a stronger claim than the arithmetic supports.

Still open: whether eccentricity is felt or averaged

The zone discussed so far is a band of mean flux. A planet on an eccentric orbit receives a mean that can be comfortable and extremes that are not, and whether its surface follows the mean or the extremes depends on a ratio of two times — how long its ocean takes to change temperature against how long its year lasts. That ratio is set by the star through the orbital period, and it runs the opposite way to most of the arguments about which stars are the best places to look.

Where this goes next

The habitable zone of a 1 M☉ star, sweeping outwards. The inner and outer edges of the liquid-water zone against time, for a 1 solar-mass star whose main sequence lasts 10.0 Gyr. The star brightens as it burns hydrogen — a heavier core needs a hotter centre to hold the star up — so both edges move outward by a factor of 1.63 across the whole main sequence, and the band drawn here sweeps past any fixed orbit rather than containing it. Two quite different zones can be read off. The instantaneous zone at the age of the present-day Sun is 0.99 to 1.71 AU, which is the band a survey means by "in the habitable zone". The continuously habitable zone over the 10.0 Gyr drawn is the overlap of every instant in it — outside the inner edge at the end and inside the outer edge at the beginning — which is 1.37 to 1.44 AU, 10 per cent of the instantaneous width. The horizontal line is an orbit at 1 AU. It leaves the zone at 4.72 Gyr, when the inner edge overtakes it. None of these edges is a measurement: both come from one-dimensional climate models, and the inner one in particular is where a runaway greenhouse begins in a model whose clouds are prescribed. The band moves and the orbit does not A star brightens as it burns, so the distance at which water can be liquid sweeps outwards by a factor of one and a half across a main sequence. The band a survey quotes is an instant; the band a planet needs is the overlap of every instant, and for the Sun it is a tenth as wide and does not contain the Earth. The habitable zone, and the radius inside which a day never ends. The conservative habitable zone — the runaway-greenhouse and maximum-greenhouse limits of Kopparapu's parameterisation — as a band in stellar mass against orbital distance, both logarithmic. For the Sun it runs from 0.99 to 1.71 AU, which is the published result and the check on the arithmetic here. The band moves inward far faster than the mass falls, because luminosity goes as roughly the fourth power of mass: a 0.2 M☉ star's zone is at 0.082–0.158 AU. The dashed line is the distance inside which a planet is tidally locked within 4.5 billion years, computed from τ = ω₀αmQa⁶/(3GM⋆²k₂R³) with an initial ten-hour spin, Q = 100 and k₂ = 0.3. It crosses the inner edge of the zone at 0.67 M☉ — so around every star below that, which is the great majority of stars, a planet in the habitable zone has one hemisphere in permanent daylight. The Earth is outside its own locking radius of 0.53 AU, and TRAPPIST-1e at 0.029 AU is inside its star's by a factor of 8. One number sets the zone The habitable zone is a band of stellar flux, so it scales as the square root of luminosity and moves inward far faster than mass falls. For most stars it lies inside the radius at which a planet is tidally locked. The hottest month is not the sunniest. The annual cycle of insolation at latitude 45°, and the temperature three surfaces of different heat capacity answer it with — each curve scaled to its own peak, because what is being compared is timing and not degrees. A surface losing 2 W m⁻² per kelvin above equilibrium obeys C dT/dt = Q − λT, and for a sinusoidal forcing that is one line of algebra: the response lags by arctan(ωC/λ) and is reduced by (1 + (ωC/λ)²)^(−½). a continental interior, 2.4 m of water, lags by 45.6 days and swings by 71 per cent of what a massless surface would; a shallow shelf sea, 10 m of water, lags by 77.6 days and swings by 23 per cent of what a massless surface would; a deep ocean mixed layer, 50 m of water, lags by 88.5 days and swings by 4.8 per cent of what a massless surface would. The lag can never reach a quarter of a cycle — three months — because an arctangent cannot reach 90°, which is why no inhabited place has its warmest month in December in the north, and why the sea comes closest. And the two numbers are one number: the same ωC/λ that carries the phase towards the quarter cycle divides the amplitude, so a long lag is bought with a small swing and cannot be had any other way. The solstice falls on day 170 and the peak insolation with it; the warmest day at this latitude follows between 46 and 89 days later depending on what is underneath. What no curve here can show is the atmosphere's own transport, which carries heat sideways between the surfaces drawn and makes each one's effective capacity partly its neighbour's. The hottest month is not the sunniest A surface with a heat capacity answers a sinusoid late and small, and the two are the same number. The lag can never reach a quarter of a cycle — three months for a year, six hours for a day — because an arctangent cannot reach ninety degrees.

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Carbonate silicate cycleClimate sensitivityEquilibrium temperatureFaint young sunHabitable zoneLoop gainMaximum greenhouseNegative feedbackRunaway greenhouseSilicate weatheringVolcanic outgassing