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Earth's energy budget: the maths

A one-layer greenhouse model: Earth absorbs sunlight and radiates as a black body, with one atmospheric layer absorbing a share ε of the outgoing infrared. Computed live.

Equations

Q = (1 − A) S₀ / 4

Sunlight absorbed per square metre, averaged over the whole globe (a disc's worth of sunlight spread over a sphere). A is the share reflected.

T_e = (Q / σ)^1/4

The temperature Earth radiates to space at.

T_s = T_e · (2 / (2 − ε))^1/4

The surface temperature under a layer absorbing a share ε of the infrared.

Constants

S₀	1361 W/m²	sunlight at Earth (Kopp & Lean 2011)
σ	5.670374 × 10⁻⁸ W m⁻² K⁻⁴	Stefan–Boltzmann, exact
A	0.30	Earth's albedo
ε	0.78	set so the surface comes out at today's 15 °C

Worked examples ✓ checked on every change

Earth's radiating temperature	255.0 K	the textbook 255 K
Surface with no greenhouse	-18.0 °C	the textbook −18 °C
Surface with today's greenhouse	15.0 °C	today's global average, about 15 °C

The app's test suite puts the lab in each setup, reads the lab's own result, and fails if it strays from these values.

What's simplified

• One layer, one temperature; no weather, oceans or feedbacks.
• ε is tuned to today's average, not calculated.

Where it breaks

For forecasts or feedbacks — ice, clouds, water vapour — which is what full climate models add. This is a teaching model, and says so.

Sources: Kopp & Lean, GRL 38 (2011); any climate textbook (e.g. Pierrehumbert, Principles of Planetary Climate, 2010).