A planet has no way to lose heat except by glowing. Sunlight arrives, some bounces straight off, the rest warms the surface — and the surface radiates until it is losing exactly as much as it gains. That balance sets the temperature, and you can compute it in one line. Then the atmosphere gets in the way of the outgoing half, and everything else on this page follows from that.
Start with the simplest possible planet: a sphere with no atmosphere. It catches sunlight across its disc, reflects a fraction straight back, and radiates from its whole surface. Set those two equal and rearrange for temperature — that is the entire calculation, and it is the one line of physics the rest of this page is built on.
σ is the Stefan–Boltzmann constant, 5.67 × 10⁻⁸. The ¼ is there because a sphere intercepts sunlight over πr² but radiates over 4πr².
That bare-rock answer is about −18 °C. The actual surface averages about +15 °C, so something is adding roughly 33 degrees. The atmosphere is transparent to incoming sunlight but not to outgoing infrared: it absorbs the heat leaving and radiates half of it back down. Add one such layer and the arithmetic gives 2¼ × 255 K — which overshoots, and the reason it does is the honest start of every real climate model.
CO₂ does not warm in proportion to how much there is. Each band it absorbs in saturates, so the extra warming from more gas goes as the logarithm of the concentration — which is why the standard measure is not "per ppm" but "per doubling". Every doubling adds about the same 3.7 watts per square metre, whether you go from 280 to 560 or from 560 to 1120.
| CO₂ | vs 278 ppm | forcing | warming, Planck only | with feedbacks |
|---|
A watt of forcing does not become a fixed amount of warming. Warming changes the planet, and the changes feed back. Warmer air holds more water vapour, which is itself a greenhouse gas — that roughly doubles the effect. Melting ice exposes dark ocean, which absorbs more. Clouds do both and are the reason the range below is as wide as it is. Turn the feedbacks off to see the bare physics, and on to see the number that actually matters.
Everything here is real physics computed live: Stefan–Boltzmann for the radiating, the standard logarithmic expression for CO₂ forcing (Myhre et al. 1998), and feedback parameters in the range the IPCC reports. Check it against the numbers you can look up — a bare rock at 255 K, a 33 K greenhouse effect, 3.7 W/m² per doubling, and about 3 °C of eventual warming for that doubling. They all come out.
But this is a planet with no geography, no seasons, no day, no ocean currents and no clouds — a single number for the whole Earth. It cannot tell you what happens to the Sahel, when an ice sheet passes a threshold, or how fast any of it arrives, because it has no time in it: every answer here is the temperature the planet would eventually settle at. Real models divide the atmosphere into millions of boxes and step them forward, and they need supercomputers precisely because none of that simplification is available to them. What a zero-dimensional model is good for is showing that the big answer does not depend on the complexity — you get 3 °C either way.