The energy balance predicts −18 °C for an Earth with no greenhouse gases, yet the real average is about 15 °C. Here is the simplest model that explains the difference: one layer of air that absorbs infrared.
🔑 Key idea
An air layer that absorbs a fraction $\varepsilon$ of the ground's infrared glows both up and down. The ground must warm until the energy leaving to space again equals the sunlight absorbed: $$T_s = T_e\left(\frac{2}{2 - \varepsilon}\right)^{1/4}$$ With $\varepsilon = 0.78$ and $T_e = 255$ K, $T_s \approx 288$ K (15 °C).
💡 Why it works
Take a layer that absorbs all the ground's infrared ($\varepsilon = 1$). Seen from space, only the layer glows, so it must glow $\sigma T_e^4$ upward (about 240 W/m²). A layer glows equally from its top and bottom, so it also sends 240 W/m² down. Now the ground receives 240 W/m² of absorbed sunlight plus 240 W/m² from the layer, 480 W/m² in all, and must glow that much: $\sigma T_s^4 = 2\sigma T_e^4$, so $T_s = 2^{1/4}T_e \approx 1.19 \times 255 \approx 303$ K. A layer that absorbs only part of the infrared gives the formula in the key idea.
One air layer between the ground and space. Change how much infrared it absorbs (ε) and watch the ground's temperature: more absorbing gas, a warmer ground, with the same energy leaving to space.
✏️ Worked example: A perfectly absorbing layer
If the air absorbed all of the ground's infrared and $T_e = 255$ K, what would the surface temperature be?
With $\varepsilon = 1$: $T_s = 255 \times 2^{1/4}$.
$2^{1/4} \approx 1.189$, so $T_s \approx 303$ K, about 30 °C.
Too warm: the real air lets some infrared through the atmospheric window.
Answer: About 303 K
✏️ Worked example: A partly absorbing layer
With $T_e = 255$ K and $\varepsilon = 0.78$, find $T_s$.
$T_s \approx 255 \times 1.132 \approx 289$ K, close to the real 288 K.
Answer: About 289 K (about 15 °C)
⚠️ Common mistake
Thinking greenhouse gases stop heat from leaving at all. In the model, Earth still sends exactly as much energy to space as it absorbs from the Sun, about 240 W/m². What changes is where it leaves from: more of it comes from the air layer, high and cold, instead of the warm ground. In the real atmosphere the infrared that escapes comes, on average, from about 5 km up, where the air is near −18 °C. To push out 240 W/m² from up there, the ground below must be warmer. Check: energy out to space = energy absorbed, before and after.
Adding more carbon dioxide raises $\varepsilon$ a little and lifts the average height from which infrared escapes. For a while Earth sends out less than it absorbs, and it warms until the balance returns at a higher surface temperature. The model leaves out a lot (convection, clouds, the sunlight the air absorbs), but this core idea survives in every full climate model.
The greenhouse effect is not the ozone hole. Ozone loss lets in more ultraviolet light; it is a different problem with a different cause.
Practice
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