1The idea in one paragraph

The Earth is heated by sunlight and cools by radiating infrared into space. When the two powers are equal, its temperature stops changing: that is energy balance, which is conservation of energy applied to a planet. Some sunlight never counts, because it is reflected straight back; the fraction reflected is the albedo. Work the balance out for a bare rock at the Earth's distance and you get about 255 K, which is −18 °C. The real average surface is about 288 K. The difference is the greenhouse effect: certain gases absorb the infrared the surface emits and send part of it back down, so the surface must be warmer to get rid of the same energy. Add more of those gases, which is what burning fossil fuels does, and the surface has to warm further. That is the enhanced greenhouse effect.

2A planet in energy balance

Treat the Earth as a system: a region you draw a boundary around so you can count the energy crossing it. Two flows cross the boundary. Sunlight comes in; infrared radiation goes out. Nothing else of any size does.

Conservation of energy says the energy inside changes by whatever comes in minus whatever goes out. So:

• If the power absorbed is greater than the power emitted, the internal energy rises and the planet warms.
• If the power emitted is greater, the planet cools.
• If they are equal, the temperature is steady. The planet is in equilibrium.

The equilibrium is self-correcting, which is why it is stable. Suppose the planet is too cold. It emits less, because emitted power goes as T⁴, so it absorbs more than it loses and warms. Too hot, and it emits more than it absorbs and cools. It settles at the one temperature where the two match, and every calculation in this subtopic is a hunt for that temperature.

In equilibrium: power absorbed = power emitted. Every greenhouse calculation starts by writing this line.

3The solar constant, and why you divide it by four

The solar constant S is the intensity of the Sun's radiation arriving at the top of the Earth's atmosphere, measured on a surface held at right angles to the rays. Its value is about 1.36 × 10³ W m⁻². It is simply the Sun's apparent brightness b = L ÷ 4πd² at the Earth's distance.

Here is the step people find strange. The mean intensity arriving over the whole Earth is not S. It is S ÷ 4. Figure 1 shows why.

Seen from the Sun, the Earth is a flat disc. The rays it intercepts are the ones that would pass through a circle of the Earth's radius R, so the power it catches is the intensity times the area of that disc:

That power is then shared over the whole surface of the sphere, day side and night side, which has area 4πR². Averaged over the surface and over a day:

Two things hide in that factor of four: night, since half the sphere gets nothing at any instant, and slant, since away from the point under the Sun the same beam is spread over tilted ground. The disc is the planet's projected surface along the rays, and it handles both at once.

4Albedo: the energy that never gets in

Not all of the 340 W m⁻² is absorbed. Clouds, ice, snow, deserts and the sea all scatter some of it straight back into space. Albedo measures the fraction scattered:

albedo = total scattered power ÷ total incident power

It is a ratio of two powers, so it has no unit, and it lies between 0 (absorbs everything) and 1 (reflects everything). This page writes it as α. The Earth's average albedo is about 0.30, so roughly 30% of incoming sunlight is reflected and 70% is absorbed. Figure 2 gives rounded typical values for different surfaces.

The guide asks you to explain why the Earth's albedo is not fixed. It changes from day to day and from place to place.

• Cloud. Thick cloud reflects much more than the ground or sea beneath it. Cloud cover changes by the hour, so the Earth's albedo changes by the day.
• Latitude. Near the poles the ground is covered by ice and snow, which have high albedo. The Sun is also low in the sky there, and water reflects far more when light arrives at a glancing angle than when it arrives from overhead. Both push the albedo up at high latitudes.
• Season and surface. Snow cover comes and goes with the seasons; forest, desert and farmland reflect differently; sea ice grows in winter and shrinks in summer.

With albedo α, the mean power absorbed per square metre is:

Keep 238 W m⁻² in your head. It is the number the whole Earth has to get rid of again.

5Emissivity: the energy that gets out

A black body is a perfect emitter, and it radiates σT⁴ from each square metre. Real surfaces emit less than that at the same temperature. Emissivity, e, is the ratio:

emissivity = power radiated per unit area ÷ σT⁴

So a real surface radiates a power

Emissivity has no unit and lies between 0 and 1; a black body has e = 1. Water, soil and vegetation have infrared emissivities close to 1, so a question will often let you treat the surface as a black body. A good emitter at a wavelength is also a good absorber at that wavelength, so the emissivity of a layer of gas also tells you what fraction of incoming infrared it absorbs. Section 8 uses both ideas.

6The temperature of a planet with no atmosphere

Now put sections 3 to 5 together. For a planet with no atmosphere, every square metre on average absorbs (1 − α)S ÷ 4 and emits eσT⁴. In equilibrium the two are equal:

(1 − α) × S ÷ 4 = e σ T⁴

Worked example 1: the Earth without its atmosphere. Take α = 0.30, S = 1.36 × 10³ W m⁻², and treat the surface as a black body.

Figure 3 draws the same balance as arrows.

The measured mean surface temperature of the Earth is about 288 K, or 15 °C. So the bare-planet model is 33 K too cold. Everything that follows in this subtopic is the explanation for those 33 K.

Worked example 2: an invented planet, with emissivity. A rocky planet with no atmosphere receives S = 900 W m⁻². Its albedo is 0.25 and the emissivity of its surface is 0.95. Estimate its equilibrium temperature.

An emissivity below 1 makes the surface warmer than a black body would be, not cooler. It emits less effectively, so it has to be hotter to shed the same power.

Worked example 3: a planet around another star. The guide says questions may use "solar or other constants". A planet orbits a star of luminosity 2.0 × 10²⁶ W at a distance of 1.2 × 10¹¹ m. Its albedo is 0.35. Find S for this planet, then its black-body temperature.

7Why greenhouse gases absorb infrared

The Sun and the Earth radiate in different parts of the spectrum, and the whole greenhouse effect rests on that difference. Wien's law gives the peaks. The Sun's surface is about 5800 K, so λₘₐₓ = 2.9 × 10⁻³ ÷ 5800 = 5.0 × 10⁻⁷ m, which is 0.50 μm, in the visible. The Earth's surface is about 288 K, so λₘₐₓ = 2.9 × 10⁻³ ÷ 288 = 1.0 × 10⁻⁵ m, which is 10 μm, in the infrared. Figure 4 draws both spectra on one wavelength axis.

The atmosphere treats the two very differently. Most of the incoming visible light passes through it to the ground. Much of the outgoing infrared does not get through, because some gases in the air absorb it. Those are the greenhouse gases. The guide names four as the main ones, and each comes from both natural sources and human activity.

The two gases that make up 99% of dry air, nitrogen and oxygen, are not on the list. Their molecules vibrate in a way that does not shift the charge within them, so infrared radiation has nothing to push against and passes through.

The guide asks you to explain the absorption with two models. You need both. Figure 5 sets them side by side.

The molecular energy level model. A molecule can only hold certain amounts of vibrational energy: its energy levels are discrete. It can absorb a photon only if the photon's energy hf exactly matches the gap between two of its levels. For greenhouse gas molecules, some of those gaps correspond to infrared photons, so the molecule absorbs infrared and moves to an excited state (step ① in Figure 5). Shortly after, it drops back and emits a photon of the same energy, or passes the energy to neighbouring molecules in collisions. The emitted photon goes off in a random direction, not necessarily the direction the original was travelling (step ②). Some of that re-emitted radiation heads back down to the surface.

The resonance model. Picture the atoms in a molecule as masses and the bonds between them as springs. Such a system has natural frequencies at which it vibrates. For CO₂, one of those frequencies is about 2 × 10¹³ Hz, which is infrared radiation of wavelength close to 15 μm. When infrared of that frequency arrives, its oscillating electric field drives the vibration at the natural frequency, the amplitude builds up, and energy is transferred from the radiation to the molecule. That is resonance, the same effect as pushing a swing in time with its own rhythm, and it is why the absorption curve in Figure 5(b) has a sharp peak at f₀. The vibrating molecule then radiates, again in all directions.

Both models say absorption happens only at frequencies the molecule already "has". The resonance model is classical and easy to picture, but a classical spring can hold any energy, so it cannot explain why real absorption comes in discrete lines. The energy level model, a quantum picture, can. That is the limitation the guide's linking question is after.

What does absorption plus re-emission in all directions do to the surface? Radiation that would have gone straight to space is partly sent back down. The surface now receives sunlight and infrared from the atmosphere, so it warms until it emits enough to balance both. The next section puts numbers on that.

8The surface and the atmosphere: a two-part energy balance

The guide says energy-balance problems will include energy exchanged between the surface and the atmosphere. The simplest model that does this treats the atmosphere as a single layer. It is transparent to incoming sunlight, and it absorbs a fraction e of the infrared coming up from the surface, where e is the layer's emissivity. Being at a temperature of its own, it radiates eσT⁴ from both faces: up to space and down to the ground. Figure 6 shows the flows with e = 0.78, a value chosen because it reproduces the measured 288 K.

Check that each part balances, because that is exactly what an exam question asks you to do with one number missing. All values are in W m⁻².

The last row is worth seeing. From space, the planet still emits 238 W m⁻², exactly what the bare planet did. The atmosphere has not changed how much energy leaves; it has changed the surface temperature needed for that much to leave.

Worked example 4: find the surface temperature. In the model of Figure 6, the surface absorbs 238 W m⁻² of sunlight and 152 W m⁻² of radiation from the atmosphere. Treating the surface as a black body, determine its temperature.

Worked example 5: the emissivity of the Earth seen from space. A classic short question: the Earth's surface is at 288 K and the planet emits 238 W m⁻² to space. Treating the Earth as a single emitter at the surface temperature, determine its effective emissivity.

This effective 0.61 is not a property of any one surface. It is a way of saying that only 61% of what a black body at 288 K would radiate actually escapes.

How the model gives the surface temperature for any e. You will not be asked to derive this, but it shows why more absorption means a warmer surface. The atmosphere balances when eσTₛ⁴ = 2eσTₐ⁴, so it radiates half of what it absorbs in each direction. Putting that into the balance at the top of the atmosphere gives:

With e = 0 this is the bare planet, 255 K. With e = 0.78 it is 288 K. With e = 1 the surface reaches 303 K. Figure 7 plots the whole curve.

9The enhanced greenhouse effect

The natural greenhouse effect is good for life. Without it the mean surface temperature would be around −18 °C and most of the ocean would be frozen. The guide's term for the part added by people is the enhanced greenhouse effect: the augmentation of the greenhouse effect due to human activities.

The mechanism follows from Figure 7. Human activity adds greenhouse gases. The burning of fossil fuels is the primary cause, because it releases carbon that had been locked underground for millions of years as CO₂. Farming and landfill add methane; fertilisers add nitrous oxide. The concentration of CO₂ in the air has risen from about 280 parts per million before industrialisation to above 420 parts per million in the 2020s. More greenhouse gas means the atmosphere absorbs a larger fraction of the surface's infrared, so its effective emissivity e rises. More radiation is sent back down, and the surface warms until it again emits enough for the planet to balance. Figure 8 lays out the chain.

Two feedbacks make the warming larger than the gas alone would.

• Ice–albedo feedback. Warming melts ice and snow. The darker ground or sea underneath has a lower albedo, so more sunlight is absorbed and the surface warms further.
• Water vapour feedback. Warmer air can hold more water vapour, and water vapour is itself a greenhouse gas, so e rises again.

The way electricity is generated matters here: a coal or gas power station adds CO₂ with every joule it delivers, while wind, solar, hydroelectric and nuclear generation add almost none while running.

The one-layer model is only a model. The real atmosphere has many layers at different temperatures, clouds that reflect sunlight and absorb infrared, and heat carried by convection and evaporation. It is used because it captures the one idea that matters with arithmetic you can do in an exam.