Greenhouse effect

Physics · The particulate nature of matter. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

The Sun delivers a steady intensity at the top of the atmosphere, called the solar constant S. The Earth catches this on a disc of area πR² but its whole surface is 4πR², so the mean intensity over the surface is S/4. Part of it is scattered back to space. This fraction is the albedo.

The warm surface emits infrared radiation. Greenhouse gas molecules have natural frequencies of vibration in the infrared range, so they absorb this radiation by resonance and re-emit it in all directions, including back down. The surface is warmer than it would be without them.

The enhanced greenhouse effect is the extra warming caused by greenhouse gases added by human activity.

Rules to remember

Solar constant S is about 1360 W m⁻²; mean incoming intensity over the whole Earth = S/4
albedo = total scattered power ÷ total incident power (no unit; Earth's mean is about 0.30; ice and cloud high, ocean and forest low)
mean absorbed intensity = (1 − albedo) × S/4
emissivity e = power radiated per unit area ÷ σT⁴, so power = eσAT⁴; a black body has e = 1
Radiative equilibrium: power absorbed = power radiated, so the mean temperature stays constant
Main greenhouse gases: water vapour, carbon dioxide, methane, nitrous oxide. Nitrogen and oxygen are not greenhouse gases
Human sources: burning fossil fuels and deforestation (CO₂); cattle, rice fields and landfill (methane); fertilisers (nitrous oxide)

Common mistake

Students say greenhouse gases trap sunlight or reflect heat. The gases let most sunlight through; they absorb the infrared radiation emitted by the Earth's surface and re-emit it in all directions.

Worked example

The Earth absorbs a mean solar intensity of 240 W m⁻². Treating the Earth as a black body in radiative equilibrium with no atmosphere, what would its mean surface temperature be? (σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴)

In equilibrium, power radiated per m² = power absorbed per m²: σT⁴ = 240.
T⁴ = 240 ÷ (5.67 × 10⁻⁸) = 4.23 × 10⁹ K⁴.
Take the fourth root: T = 255 K.

Answer: About 255 K (−18 °C), which is colder than the real Earth because this model has no greenhouse effect.

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B.2 Greenhouse Effect

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