Earth's Energy Balance

You now have the pieces: sunlight arrives, albedo decides how much is absorbed, and every surface glows by the Stefan–Boltzmann law. Put them together and you can predict the temperature of a planet from two numbers, using a fourth root.

Key idea

Earth warms until the infrared it sends to space equals the sunlight it absorbs: $$\sigma T_e^4 = \frac{S(1 - \alpha)}{4}$$ With $S = 1361$ W/m² (sunlight at Earth's distance) and albedo $\alpha = 0.30$, this gives $T_e \approx 255$ K, about −18 °C.

Why it works

If Earth absorbed more than it gave off, it would warm; warming raises its glow as $T^4$, until the two match. That matching temperature is the balance. The factor of 4 comes from geometry: Earth intercepts sunlight on a disk of area $\pi R^2$ (its shadow), but it glows from its whole sphere, area $4\pi R^2$. Spread the sunlight over the sphere and each square meter gets, on average, $1361 \div 4 \approx 340$ W/m².

Sunlight in, reflected sunlight and infrared out. Change the albedo or the sunlight and watch the balance temperature move.

Worked example: Earth with no greenhouse effect

Find Earth's balance temperature for $S = 1361$ W/m² and albedo 0.30.

Average sunlight per square meter: $1361 \div 4 \approx 340$ W/m².
Absorbed: $(1 - 0.30) \times 340 \approx 238$ W/m².
Set $\sigma T^4 = 238$: $T^4 = 238 \div (5.67 \times 10^{-8}) \approx 4.20 \times 10^{9}$.
$T = (4.20 \times 10^{9})^{1/4} \approx 255$ K, or about −18 °C.

Answer: About 255 K (−18 °C)

Worked example: A planet four times as far

A planet like Earth (same albedo) orbits four times as far from the Sun. What is its balance temperature?

Sunlight spreads as the square of distance, so $S$ is $4^2 = 16$ times smaller.
$T_e$ goes as $S^{1/4}$, so it changes by $(1/16)^{1/4} = 1/2$.
$T_e \approx 255 \div 2 \approx 127$ K. No calculator needed.

Answer: About 127 K

Worked example: A snowball Earth

If ice covered the planet and raised the albedo to 0.60, what would the balance temperature be?

Absorbed: $(1 - 0.60) \times 340 = 136$ W/m².
$T^4 = 136 \div (5.67 \times 10^{-8}) \approx 2.40 \times 10^{9}$, so $T \approx 221$ K (−52 °C).
A whiter planet absorbs less sunlight and settles much colder.

Answer: About 221 K

Common mistake

Forgetting to divide the sunlight by 4. Using the full 1361 W/m² would give about 360 K, near the boiling point of water. Earth catches sunlight on a disk but glows from a sphere four times the disk's area. Check: the average sunlight per square meter of Earth is about 340 W/m².

Earth's real average surface temperature is about 15 °C (288 K), some 33 °C warmer than this balance temperature. The difference is the greenhouse effect, the subject of the next lesson. Satellites confirm the outgoing side: Earth sends about 240 W/m² of infrared to space, just as this calculation predicts.

Earth's Energy Balance

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