The height a planet is seen from

A body in sunlight absorbs and radiates, and settles at whatever temperature makes those equal. The absorbed power is the sunlight arriving on its cross-section, less what it reflects; the radiated power is Stefan’s fourth-power law over its whole surface. Setting the two equal gives a temperature from two numbers — the sunlight and the albedo — and nothing else about the body at all.

That calculation takes one line and it is a genuine prediction. It is worth seeing how far it gets before asking what it misses.

The arithmetic, in full #

The whole calculation is short enough to write down, and writing it down shows what does and does not enter.

Sunlight arrives at a flux $S$ and the planet presents a disc of area $\pi R^2$ to it, so the absorbed power is $S(1-A)\pi R^2$ with $A$ the fraction reflected. The planet radiates from its whole surface, $4\pi R^2$, at $\sigma T^4$. Setting them equal cancels the radius entirely:

$$T_e = \left[\frac{S(1-A)}{4\sigma}\right]^{1/4}$$

For Earth, $S = 1361$ W/m² and $A = 0.29$, which gives 255 K. The factor of four is the ratio of the surface to the disc, and it is the reason a planet is colder than a flat plate facing the sun at the same distance would be.

Nothing about the planet’s size, mass, composition or rotation appears. That is what makes the prediction sharp and also what makes its failures informative: every departure from it must be caused by something that was left out, and the list of things left out is short.

The fourth root is worth noticing too. It means the temperature is insensitive: halving the sunlight lowers the temperature by only sixteen per cent, and an albedo wrong by a tenth moves the answer by under three per cent. A quantity that depends on the fourth root of everything is a quantity that is easy to predict and hard to change, which is the reason the balance temperature is the right thing to start from.

Two failures with opposite signs #

The figure has two kinds of departure in it and they should not be confused.

The airless bodies fall below the line, and the reason has nothing to do with atmospheres. Radiated power goes as the fourth power of temperature, so a surface running from noon heat to night cold radiates like its hottest parts while a thermometer averages like its coldest. The Moon’s mean surface is twenty kelvin below the temperature that matches its emitted flux; Mercury, which turns slowly enough to have a seven-hundred-kelvin day and a hundred-kelvin night, is ninety below. Neither is a failure of the balance argument — the flux does balance — but of the assumption that a planet has a temperature.

The bodies with atmospheres fall above it, always, and by amounts that are not small. That is a different question and it is the one this rung is about.

Mars sits almost exactly on the line, which is useful: its atmosphere is thin and its surface, unlike the Moon’s, is smoothed by what atmosphere there is. A body with neither an insulating atmosphere nor a wildly varying surface obeys the one-line argument, which is the argument’s credit.

The first correction, and why it overshoots #

The standard fix is to put a layer above the surface that lets sunlight through and stops the surface’s own radiation. The layer warms until it radiates away what it absorbs — half upwards, half back down — so the surface receives its sunlight plus that downward half and settles at $2^{1/4}$ times the balance temperature.

Getting the sign and the mechanism right from a model this crude is worth something, and the overshoot is worth more. The model’s error is the assumption of complete opacity, and correcting it is where the physics is.

The hole in the middle #

The atmosphere absorbs strongly at some infrared wavelengths and hardly at all at others.

That window is why a thermal camera in orbit can see the ground at all. An image taken between eight and thirteen micrometres shows the surface; the same scene at fifteen micrometres shows the upper atmosphere and nothing beneath it, because carbon dioxide is opaque there and the last thing to radiate towards space is high and cold.

The window’s width is itself a piece of physics rather than an accident of chemistry. It sits where it does because the molecules that make up most of the atmosphere — nitrogen and oxygen — are symmetric diatomics with no dipole moment, so they neither absorb nor emit in the infrared at all, and the absorption comes entirely from the small fraction of the air that is water, carbon dioxide, methane and ozone. An atmosphere of pure nitrogen would be perfectly transparent to its own planet’s radiation, and the balance temperature would be right.

It also settles which gases matter. A gas that absorbs where the atmosphere is already opaque adds almost nothing, because that radiation was not escaping anyway. A gas that absorbs inside the window intercepts radiation that was leaving, and counts for far more per molecule. That is why some industrial gases present at parts per trillion are worth worrying about and others present at parts per million are not.

Why the answer is per doubling #

The next question is how the effect grows as an absorber is added, and the answer is not proportionally.

The figure computes this rather than asserting it: doubling the amount from a hundred adds the same absorption as doubling it from a thousand, to within the tolerance the generator refuses to draw outside. That is the reason the effect of a greenhouse gas is quoted per doubling of concentration.

The mechanism is worth stating in words because it is a general one. A saturated absorber grows by widening rather than by deepening. The same argument appears wherever a quantity saturates locally and can only expand outwards — in a spectral line’s equivalent width, in the growth of a shadow, and in the exponential that decides everything run backwards, where a probability already near one has nothing left to gain.

The mechanism, stated as a height #

The layer models make it sound as though radiation is bouncing around and being trapped. The better statement uses a height and one non-radiative fact.

The temperature of a planet as seen from outside is fixed by the sunlight and cannot change: 255 K for Earth, whatever the atmosphere does. What the atmosphere decides is where that temperature is found. Add absorbing gas and the level at which radiation finally escapes moves upward. It is colder up there. So less is escaping, and the whole profile must warm until the escaping level is back at 255 K — with everything below it, including the ground, carried along.

The lapse rate is doing the work in that argument, and it is not a radiative quantity. Temperature falls with height because a parcel of air rising expands and cools, at a rate set by its heat capacity and gravity, and the layer a parcel cannot leave is where that comes from — as distinct from the fall in pressure with height, which is a static balance and is worked out in why the air thins with height. A purely radiative atmosphere would have a different and much steeper profile, and would give the wrong answer; convection is what flattens it to the observed rate.

So the effect requires both halves. Radiation decides the height at which the planet is seen. Convection decides how much colder that height is than the ground. Neither alone gives a number.

How the height is measured #

The emission height sounds like a modelling device and it is a measurement.

A spectrometer in orbit looking down records a flux at each wavelength, and each flux can be converted into the temperature a blackbody would need to produce it — a brightness temperature, one per wavelength. Where the atmosphere is transparent, that temperature is the ground’s. Where it is opaque, it is the temperature of whatever level the radiation last came from before escaping.

So a single spectrum is a set of temperatures at a set of heights, with the wavelength choosing the height. In the window the instrument reads about 288 K and sees the surface. In the middle of the carbon dioxide band at fifteen micrometres it reads about 220 K, which on the profile above is the tropopause — the radiation from lower down never got out. Between the two, the band’s wings read intermediate temperatures, and reading them in order is walking down through the atmosphere.

That spectrum is what makes the argument checkable rather than plausible. The dip in the middle of the band is measured, its depth is the difference between the surface temperature and the emission temperature at that wavelength, and it deepens when more absorber is present because the escaping level moves higher into colder air. The same instrument sees the dip fill in over a warm ocean and deepen over a cold desert, which is the profile changing rather than the gas.

What the arithmetic is worth #

Everything above is a one-dimensional, single-column, no-feedback calculation, and it is worth being explicit about which of its conclusions survive that and which do not.

The sign survives, and the order of magnitude. A greenhouse effect of tens of kelvin on Earth and hundreds on Venus follows from the emission height argument with no adjustable parameter beyond the lapse rate, which is measured.

The logarithmic law survives, because it comes from band saturation, which is a property of the absorption spectrum rather than of the atmosphere’s structure.

The number does not survive. The direct effect of moving the emission height is only part of the answer, because a warmer atmosphere holds more water vapour, which is itself an absorber, and clouds change both the albedo and the emission height in ways that partly cancel. Those feedbacks are larger than the effect that triggers them and they are where all the difficulty is.

That structure — a robust mechanism with a well-understood size, and a total that depends on responses to it — is common. The reliable part of a calculation and the interesting part are often different parts.

The same balance, elsewhere #

The argument is not about planets. It is about anything whose temperature is set by what it absorbs and what it radiates, and the same line does a lot of work in places with no atmosphere at all.

A spacecraft is designed by exactly this calculation. Its equilibrium temperature is set by the sunlight on its illuminated face, its absorptivity there, and the area and emissivity of its radiators; the whole discipline of spacecraft thermal control is the business of choosing surface coatings so that those two numbers land where they are wanted. A surface that absorbs sunlight well and radiates infrared badly runs hot, which is what a solar absorber is, and the reverse is what a radiator is. Both are the balance equation with the absorptivity and the emissivity allowed to differ, which the ideal blackbody argument forbids and the glow that says nothing about the surface explains: they must be equal at each wavelength, and the sun and the radiator work at different ones.

A star is the same equation with the energy arriving from inside rather than outside, and there the balance sets a limit rather than a temperature: past a certain luminosity, the outward push of the radiation exceeds gravity and the star cannot hold itself together, which is the brightness a mass cannot exceed. Even a greenhouse — the actual glass kind — is a case where the balance has to be redone, and where the popular explanation is wrong: a garden greenhouse works mostly by stopping the warm air from leaving, and only slightly by the mechanism this essay is about, which is why one with an open vent stops working.

Where the model stops #

One column, no geography. A planet has poles and a tropics, and the transport of heat between them is what sets most of the temperature pattern that anyone lives in. The balance argument constrains the global average and says nothing about the distribution.

No feedbacks at all. Water vapour, ice albedo and clouds are absent, and each is comparable in size with the effect they respond to.

The window is treated as fully open and the rest as fully closed. Real absorption is a spectrum of thousands of lines with wings that overlap, and the “window” leaks — significantly so at high humidity, which is why a humid night is warmer than a dry one at the same air temperature.

And the lapse rate is imposed rather than derived. It is close to the value moist convection produces and it is not a constant: it varies with humidity, with latitude and with height, and a model that solves for it rather than assuming it gets a different and better answer.

A number worth checking by hand #

The emission-height argument gives a prediction, and it is short enough to check.

A doubling of carbon dioxide raises the level from which the fifteen-micrometre radiation escapes by of order a couple of hundred metres. At six and a half kelvin per kilometre that is a bit over one kelvin at the surface, which is the figure usually quoted for the direct effect before any feedback. The same calculation done in energy rather than temperature gives a few watts per square metre of imbalance, and dividing that by the rate at which a warmer planet radiates more — the derivative of Stefan’s law, about 3.8 W m⁻² K⁻¹ at 255 K — returns the same kelvin.

Two routes, one number, and both are arithmetic anyone can do. That agreement is not a coincidence: they are the same statement expressed once as a height and once as a flux, and the fact that they can be checked against each other is the reason the direct effect is not in dispute while everything downstream of it is.

The thermal inertia of the ocean is the other quantity that belongs here and does not fit in any of these figures. It sets how long the surface takes to reach whatever temperature the balance demands, and the answer is decades — the summer that reaches the cellar in December is the same diffusion problem at a domestic scale. A balance argument gives a destination, never a date.

What the pictures cannot show #

The balance figure plots single numbers for whole planets. Venus’s five hundred kelvin is not five hundred kelvin of the same mechanism as Earth’s thirty-three; it is a qualitatively different regime, with an atmosphere ninety times as massive, opaque nearly everywhere, and a lapse rate acting over sixty kilometres. Putting the two on one plot shows that they are the same sign and invites the reading that they are the same size of thing.

The height figure draws a straight profile and a shifted copy of it, which is the argument’s skeleton and not a prediction. A real profile is not straight, the shift is not uniform, and the stratosphere above the level shown moves the other way — it cools when the surface warms, which is one of the observations that distinguishes this mechanism from a general warming of the whole column.

Where the ladder goes next #

The blackbody ladder began with the curve that would not come down, where classical physics predicted infinite radiation and quantisation was the price of fixing it, and went on to the glow that says nothing about the surface, where a cavity’s radiation forgets what it was made of, and to the photon gas nobody counted. This rung takes the same law outdoors and applies it to an object with an atmosphere, where the interesting quantity turns out not to be a temperature but a height.

The rung after it is the spectrum as an instrument: what an infrared spectrum taken from orbit says about the profile beneath it, since each wavelength sees down to a different depth and the brightness at that wavelength is the temperature there. The habit worth carrying is the one this rung is built on: when a balance argument fails, do not add a mechanism to it — ask which of its quantities is being evaluated in the wrong place.