Watch sunlight stream in, heat the ground, leave again as infrared — and watch greenhouse-gas molecules grab some of that IR on the way out and re-radiate it back. Crank the CO₂. See the planet warm. Feel the math behind every news headline about climate.
Imagine the atmosphere as a one-way window. Sunlight (mostly visible & UV) passes in easily — the window is transparent to it. The ground absorbs that energy and re-emits it as invisible infrared (IR) heat. Greenhouse gases — CO₂, methane, water vapour — are like tiny sticky patches on the window: they grab outgoing IR and re-radiate half of it back down. The more sticky patches, the harder it is for heat to escape, so the surface has to get hotter until outgoing IR finally matches incoming sunlight again. Without these gases, Earth would be −18 °C. With them, +15 °C. Add more, it gets hotter still. That's the whole story.
| Slider | Real-world meaning |
|---|---|
| CO₂ | The headline greenhouse gas. Goes from 280 ppm (pre-industrial) → 420 today → potentially 1000+ if we keep burning. Logarithmic effect on temperature. |
| CH₄ | Methane. Per molecule, ~80× stronger than CO₂ over 20 years. Cattle, rice paddies, leaking gas pipes, melting permafrost. |
| N₂O | Nitrous oxide from agricultural fertiliser. Long-lived, ~300× CO₂ per molecule. |
| Water vapour | Strongest greenhouse gas by mass — but it's a feedback, not a forcing. Hot air holds more vapour, which traps more heat, which heats the air… |
| Albedo α | Fraction of sunlight reflected straight back to space. Ice ≈ 0.85, ocean ≈ 0.06, Earth average ≈ 0.30. Snowball Earth happens at α ≈ 0.7+. |
| Cloud cover | Clouds are double-edged: white tops reflect sun (cooling) but trap IR (warming). Net effect ≈ cooling, but it depends on cloud type. |
| Climate sensitivity λ | Best estimate ≈ 0.8 K per W/m². IPCC gives 2.5–4 K per CO₂ doubling. Uncertainty is from clouds and ice feedbacks. |
Leave a car in a sunny car park with the windows up. Come back an hour later and the inside is an oven. Sunlight poured in through the glass, the seats turned it into heat, and that heat had a hard time getting back out. A planet wrapped in greenhouse gases pulls the same trick — except the "glass" is just a few invisible gases mixed into the air.
To put numbers on it, picture the energy the Sun delivers. The full beam at Earth's distance is the solar constant, about $1361$ W/m², but the planet is a spinning ball, so that power is shared over a surface four times the size of the disk the Sun sees. The average sunlight per square metre is therefore $S_0/4 \approx 340$ W/m². Some bounces straight back off ice and clouds — that reflected fraction is the albedo $\alpha$. The rest, $(1-\alpha)\,S_0/4 \approx 239$ W/m², soaks into the ground and oceans. To hold a steady temperature, the surface must hand exactly that much back to space as infrared. A bare rock with no atmosphere would balance at about $-18\,^\circ$C. Greenhouse gases slow the infrared escaping, so the surface has to warm until enough leaks out — landing near $+15\,^\circ$C. That 33-degree gap is the blanket the gases provide.
The precise statement is just energy in equals energy out: $(1-\alpha)\tfrac{S_0}{4} = \varepsilon\sigma T_s^4$. Adding CO₂ does not add a fixed number of degrees; it adds a forcing that grows with the logarithm of concentration, $F = 5.35\,\ln(C/C_0)$ W/m², and the temperature change follows $\Delta T = \lambda F$. The control panel maps straight onto these symbols: the CO₂, CH₄ and N₂O sliders set $C$, the albedo slider sets $\alpha$, the solar-constant slider sets $S_0$, and the sensitivity slider sets $\lambda$.
Try this in the sim above. First set CO₂ to 280 ppm, then to 560 ppm (one doubling) and watch the surface temperature climb roughly 3 °C. Next push CO₂ to 1120 ppm: the jump is about the same size again, not double — that is the logarithm at work. Finally drag the albedo slider toward 0.7 (or pick the "Snowball Earth" preset) and watch the planet freeze even with greenhouse gases present, because now too much sunlight is reflected before it can ever warm the ground.
Any object above 0 K emits thermal radiation. For a blackbody at temperature T (kelvin):
The Sun's photons peak in the visible (T ≈ 5778 K). The Earth's emission peaks at ~10 µm (T ≈ 288 K) — in the thermal infrared. That mismatch is what makes the greenhouse effect possible: glass and gases that are transparent in visible can be opaque in IR.
At equilibrium, energy in equals energy out:
where the factor of 4 comes from spreading sunlight over the whole sphere. With S₀ = 1361 W/m² and α = 0.30, the absorbed flux is ~ 239 W/m². If Earth were a bare rock (ε = 1, no atmosphere), Tₛ = 255 K = −18 °C. The atmosphere's IR opacity reduces effective ε to ~ 0.61, raising Tₛ to ~ 288 K = +15 °C. That 33 °C is the natural greenhouse effect.
Adding more CO₂ traps additional outgoing IR. The forcing relative to a reference concentration C₀ is:
So doubling CO₂ from 280 to 560 ppm gives F ≈ 3.7 W/m². Climate sensitivity converts forcing to temperature: ΔT = λ · F. With λ ≈ 0.8 K/(W/m²), doubled CO₂ ⇒ ~ 3 K warming.
(These are the IPCC AR4 simplified expressions; the sim uses them for fast updates.)
F = 5.35 · ln(560/280) = 5.35 · 0.693 ≈ 3.71 W/m². ΔT = λ · F = 0.80 · 3.71 ≈ +3.0 °C. Add water-vapour feedback (~+50 %) and you land at ~ +4.5 °C — squarely inside the IPCC sensitivity range. Try it: select "Doubled CO₂ (560 ppm)" preset, press Run, and watch ΔT climb.