The Energy Balance Model: Earth's Temperature in One Equation

A planet as a single point with one temperature

A zero-dimensional energy balance model throws away geography, seasons and weather and keeps exactly one number: the planet's average temperature. It works because, over the long run, a planet in equilibrium must radiate away exactly as much energy as it absorbs — if it did not, its temperature would keep changing until it did. That single constraint, absorbed power equals emitted power, is enough to solve for a temperature with nothing more than a calculator.

Absorbed sunlight: the solar constant and albedo

The Sun delivers a flux S ≈ 1361 W/m² at Earth's distance (the solar constant), spread over the disc the planet presents to it, πR². A fraction α, the albedo, bounces straight back to space off clouds, ice and bright surfaces without being absorbed — Earth's albedo is about 0.30, mostly from clouds. The absorbed power is:

P_absorbed = (1 − α) · S · πR²          // intercepted disc, not the sphere

Emitted radiation: the Stefan-Boltzmann law

A body at temperature T radiates power proportional to the fourth power of its temperature, over its full sphere's surface area 4πR² — the Stefan-Boltzmann law, P = σT⁴εA, where σ ≈ 5.67×10⁻⁸ W/m²K⁴ and ε is the emissivity (how close to a perfect blackbody the surface radiates; treated as 1 in the simplest version). Setting absorbed power equal to emitted power and cancelling πR² gives the planet's effective temperature:

(1 − α) · S / 4 = σ · T_eff⁴

T_eff = [ (1 − α) · S / (4σ) ] ^ (1/4)

for Earth: α ≈ 0.30, S ≈ 1361 W/m²  →  T_eff ≈ 255 K ≈ −18 °C

The factor of 4 appears because the disc that intercepts sunlight has area πR² while the whole sphere that radiates it away has area 4πR² — a sphere's surface is exactly four times its shadow's area. −18°C is well below Earth's actual global average of about 15°C, and that 33°C gap is not an error in the model — it is the size of the greenhouse effect.

Adding the greenhouse term and feedback

Greenhouse gases absorb outgoing infrared and re-emit part of it back down, which the model captures with a forcing term ΔF added to the balance (ΔF ≈ 5.35·ln(CO₂/CO₂₀) W/m² is the standard logarithmic approximation for CO₂ doubling). A more complete version also lets albedo depend on temperature — more ice at lower T means higher α, which reflects more sunlight and cools further, a positive feedback. This ice-albedo feedback is why simple energy balance models, despite their crudeness, can show two stable equilibria (a warm "ice-free" state and a cold "snowball" state) separated by an unstable point, a toy version of the same bistability that shows up in full climate models of the "Snowball Earth" episodes roughly 700 million years ago.

Why the model still matters next to full GCMs

A modern general circulation model (GCM) solves fluid dynamics, radiative transfer and chemistry on a 3D grid with millions of cells. The energy balance model cannot reproduce regional detail or day-to-day weather, but it isolates the mechanism — radiative forcing versus emitted power — that ultimately sets the global mean temperature in every one of those larger models too. It is also fast enough to explore parameter space (different albedos, different solar outputs, different forcings) in real time, which is exactly what the interactive version on this page is for.

Frequently asked questions

Try it live

Everything above runs in your browser — open Simple Climate Model and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.