The greenhouse effect
🎯What you need to be able to do
- Use the solar constant and calculate the intensity received at a planet.
- Apply albedo and emissivity, and distinguish a black body from a grey body.
- Explain how greenhouse gases absorb infrared radiation, in terms of molecular resonance.
- Construct a simple energy balance model of a planet.
- Distinguish the natural greenhouse effect from the enhanced greenhouse effect.
📚The physics
Start with the incoming energy. The Sun radiates in all directions, so at a distance \(d\) the power is spread over a sphere of area \( 4\pi d^{2} \) and the intensity is
where \(L\) is the Sun’s luminosity. At Earth’s orbit this gives the solar constant, about 1360 W m\(^{-2}\).
Then correct it twice, and both corrections are easy to forget. First, the Earth intercepts sunlight as a disc of area \( \pi r^{2} \) but it radiates from its whole sphere of area \( 4\pi r^{2} \). The ratio is \( 1/4 \), so the average incoming intensity over the whole surface is only about 340 W m\(^{-2}\). Second, some of that is reflected straight back.
Albedo is the fraction of incident radiation reflected: for Earth as a whole roughly 0.30. So the absorbed intensity is \( 340 \times (1 - 0.30) \approx 238 \) W m\(^{-2}\). Fresh snow has an albedo near 0.85 and open ocean near 0.06, which is why melting ice accelerates warming — a bright reflector is replaced by a dark absorber, so more energy is absorbed, so more ice melts. That feedback loop is worth being able to state in one sentence.
Emissivity \(e\) is the ratio of the power a body radiates to the power a perfect black body at the same temperature would radiate, so Stefan’s law becomes \( P = e\sigma A T^{4} \). A black body has \( e = 1 \); anything real is a grey body with \( e < 1 \). Earth’s effective emissivity is about 0.61.
Why greenhouse gases behave differently from the rest of the air. The atmosphere is mostly nitrogen and oxygen, which are diatomic molecules of two identical atoms with no dipole moment — they barely interact with infrared at all. Greenhouse gases such as CO\(_2\), H\(_2\)O, CH\(_4\) and N\(_2\)O have vibrational modes whose natural frequencies lie in the infrared. Incoming radiation at those frequencies drives the molecules at resonance, so it is strongly absorbed and then re-radiated in all directions, including back downwards.
This is why the wavelength matters. The Sun, at about 5800 K, peaks in the visible by Wien’s law, and the atmosphere is largely transparent there, so sunlight reaches the ground. The Earth, at about 288 K, re-radiates in the infrared — exactly where greenhouse gases absorb. Energy comes in through an open window and tries to leave through a closed one. That asymmetry is the greenhouse effect.
Natural versus enhanced. The natural greenhouse effect is not a problem; without it Earth’s mean surface temperature would be about −18 °C rather than about +15 °C, and the oceans would be ice. The enhanced greenhouse effect is the additional warming from human-added greenhouse gases. Getting this distinction right is worth marks, and getting it wrong suggests the mechanism has not been understood.
✏️Worked example
Power absorbed = solar constant × disc area × (1 − albedo) \( = 1360 \times \pi r^{2} \times 0.70 \).
Power radiated = \( \sigma T^{4} \times \) sphere area \( = \sigma T^{4} \times 4\pi r^{2} \).
Set them equal. The \( \pi r^{2} \) cancels from both sides — the planet’s size is irrelevant, which is a satisfying result in itself. That leaves
So \( T^{4} = 238/(5.67 \times 10^{-8}) = 4.20 \times 10^{10} \), giving \( T = 255 \) K, or about −18 °C.
🔭See it happen
PhET — The Greenhouse Effect simulation. Add and remove greenhouse gases and watch the surface temperature follow.
📝Practise
Work through these, then reveal the answer. Each question targets a different objective from the list above.
1. The solar constant at Earth is 1360 W m\(^{-2}\). Mars orbits at 1.52 AU. Find the intensity of sunlight at Mars.
2. A planet receives an average 340 W m\(^{-2}\) over its whole surface and has an albedo of 0.45. Find the intensity it absorbs.
3. Earth absorbs about 238 W m\(^{-2}\) and has an effective emissivity of 0.61. Using \( P = e\sigma T^{4} \), find the predicted surface temperature.
4. Explain why nitrogen and oxygen, which make up most of the atmosphere, are not greenhouse gases.
5. State the ice–albedo feedback loop in one or two sentences.
6. Distinguish the natural greenhouse effect from the enhanced greenhouse effect.
🔗Go deeper — other people’s work
These are external resources, not mine. If one stops working, tell me and everything above it on this page still stands.
- PhET — The Greenhouse Effect simulation
- NASA — Global Climate Change, vital signs and evidence