How the Greenhouse Effect Works

Energy in, energy out

The Earth's temperature settles where the energy it absorbs from the Sun equals the energy it radiates to space. Sunlight arrives at S₀ = 1,361 W/m² at the top of the atmosphere. The Earth intercepts it on a disk but has four times that area as a sphere, so the average over the whole surface is S₀ ÷ 4 = 340 W/m². About 30% (the albedo, α) is reflected straight back by clouds, ice and bright ground, leaving 238 W/m² absorbed.

A body radiates σT⁴ per square meter (the Stefan–Boltzmann law, σ = 5.67 × 10⁻⁸ W/(m²·K⁴)). Setting σT⁴ = 238 gives

T_e = (S₀(1 − α) ÷ 4σ)^1/4 = 255 K = −18 °C

That is the temperature the Earth would have with no greenhouse effect. The actual average surface temperature is about 15 °C: 33 degrees warmer.

The greenhouse effect

Sunlight is mostly visible light, which the air lets through. The warmed ground radiates in the infrared, around 10 µm, and water vapor, carbon dioxide, methane and a few other gases absorb much of that infrared and radiate it again in every direction, including back down. The surface therefore receives energy from the Sun and from the atmosphere, and warms until it radiates enough to balance both. In a one-layer model where the atmosphere absorbs a fraction ε of the surface's infrared,

T_s = T_e × (2 ÷ (2 − ε))^1/4

and ε = 0.77 gives the observed 288 K. (A greenhouse keeps warm mainly by stopping warm air rising, not this way, but the name has stuck.)

Radiative forcing

Adding a greenhouse gas makes the atmosphere absorb more of the outgoing infrared, so the planet loses less energy than it gains until it warms up. The imbalance just after the change is the radiative forcing, ΔF. For CO₂ it grows with the logarithm of the concentration:

ΔF = 5.35 ln(C ÷ C₀) W/m², with C₀ = 280 ppm before industry

so each doubling adds 5.35 × ln 2 = 3.7 W/m², whatever the starting level. The logarithm appears because the strongest absorption bands are already nearly saturated and extra CO₂ works mostly at their edges. Methane adds about 0.036(√M − √M₀) W/m² with M in ppb, and a change in albedo adds −(S₀ ÷ 4) Δα.

Climate sensitivity

The temperature rises until the extra outgoing radiation cancels the forcing: ΔT = λ ΔF. With no feedbacks, from σT⁴ alone, λ ≈ 0.3 K per W/m², about 1.2 °C per doubling of CO₂. But warmer air holds more water vapor (7% more per degree), which is itself a greenhouse gas; shrinking snow and ice darken the surface; clouds change. Together these feedbacks raise λ to about 0.8, and the warming per doubling, the equilibrium climate sensitivity, is likely 2.5 to 4 °C with a best estimate of 3 °C.

A worked example

CO₂ rises from 280 to 420 ppm and methane from 722 to 1,920 ppb.

1. CO₂ forcing: 5.35 × ln(420 ÷ 280) = 5.35 × 0.405 = 2.17 W/m².
2. Methane forcing: 0.036 × (√1920 − √722) = 0.036 × (43.8 − 26.9) = 0.61 W/m².
3. Total: 2.78 W/m². With λ = 0.81 K per W/m², the warming once everything settles is 0.81 × 2.78 = 2.25 °C.

The observed warming so far is about 1.2 °C, less than this, because the oceans have not caught up yet and because aerosol pollution (left out here) reflects sunlight and offsets roughly a third of the greenhouse forcing.

How fast the temperature responds

The ocean's surface layer has a large heat capacity, about 8 W·year per m² per K for a 60 m mixed layer. The temperature then follows C dT ÷ dt = ΔF − (T − T₀) ÷ λ, approaching the new balance with a time constant Cλ ≈ 6.5 years. The deep ocean, which this model leaves out, takes centuries to warm and keeps the real climate catching up for a long time after the forcing stops rising.

Using the simulation

In the Greenhouse Effect Simulator, yellow sunlight photons come down and red infrared photons go up; a gray molecule that catches one flashes and sends it off in a random direction. The temperature starts at the pre-industrial 13.8 °C and moves toward the balance for the settings you choose. Things to try:

• Double CO₂ from 280 to 560 ppm with each set of feedbacks: the warming at balance is 1.2, 2, 3 or 4.5 °C.
• Halve it from 560 to 280: the forcing changes by the same 3.7 W/m², the logarithm at work.
• Raise the albedo from 0.30 to 0.31: the cooling of 3.4 W/m² outweighs today's CO₂ forcing of 2.2 W/m², which is why clouds and ice matter so much.
• Change the Sun by 1 W/m² (a typical solar cycle): the forcing is only 0.17 W/m², much smaller than CO₂'s.

What the model assumes

• A zero-dimensional planet: one global average temperature, with no day and night, seasons, latitudes, land and ocean.
• One mixed ocean layer with a fixed heat capacity, and no deep ocean.
• Simplified forcing formulas for CO₂ and methane, without the overlap between methane and nitrous oxide bands, and no other gases, aerosols, ozone or land use.
• A constant climate sensitivity λ, the same at every temperature, chosen from the menu.
• Photons that show the idea, not the numbers. The chance an infrared photon is caught rises with CO₂ for illustration; the temperature comes from the energy-balance equations, not from counting photons.

Edge cases

• CO₂ at 280 ppm, methane at 722 ppb, albedo 0.30: no forcing, and the temperature stays at 13.8 °C.
• Very low CO₂ gives negative forcing and cooling; below about 150 ppm plants struggle to photosynthesize, the slider's lower limit.
• No feedbacks means the Planck response alone: the warming is smallest and the approach to balance fastest, since the time constant is proportional to λ.
• Large albedo changes (ice ages, snowball Earth) swamp everything else: going from 0.30 to 0.60 removes about 100 W/m².

Where the model stops being right

• Feedbacks are not constant. Ice-albedo and cloud feedbacks change with the climate itself; at some thresholds (ice sheets, permafrost, ocean circulation) the response can be abrupt and slow to reverse.
• Regional climate. Warming is uneven: the Arctic warms three to four times faster than the average, land faster than ocean. A one-number model cannot say anything about rainfall, storms or local temperatures.
• Transient warming. Because the deep ocean is missing, the model reaches balance in decades; the real climate keeps warming for centuries after concentrations stabilize, and sea level rises for much longer.
• Carbon cycle. CO₂ is set by hand here; in reality oceans and forests absorb about half of what is emitted, and warming can weaken those sinks.
• Venus-like extremes. At very high CO₂ the logarithmic formula and the simple layer model fail; detailed radiative transfer is needed.

Related tools

The Blackbody Radiation Simulator shows why the Sun shines in visible light and the Earth in infrared, and the Seasons and Day Length Simulator shows how sunlight is spread over latitudes and the year. Convert temperatures with the Temperature Converter.