Stefan Boltzmann and the greenhouse effect

The Sun heats the surface. Greenhouse gases slow the rate at which that heat escapes to space, so the surface settles warmer. The equation used to deny this is the one that proves it.

Q = εσA (T24 − T14)
Q
net rate of heat loss (W)
ε
emissivity, 0 to 1
A
area (m²)
σ
Stefan–Boltzmann constant, 5.67×10−8 W m−2 K−4
T2
the surface (the warm body)
T1
the atmosphere the surface loses heat to

The constant σ. It sets how much power a perfect emitter radiates per square metre at a given temperature. In fundamental constants, σ = 2π⁵k⁴ / (15h³c²), linking temperature, quantum energy and light speed. Stefan measured the law in 1879; Boltzmann derived it in 1884.

Why the fourth power. A warming body radiates harder for two reasons at once: it emits more photons per second (their number rises roughly as T³), and each photon carries more energy as the spectrum shifts to shorter wavelengths (Wien’s law), so the average photon energy rises roughly as T. Multiply them: T³ × T = T⁴. Formally it is what you get by integrating Planck’s blackbody curve over all wavelengths. The steepness matters: a small change in temperature shifts the radiated power a great deal.

Same Sun in, three layers of reality

1  A model atmosphere with no greenhouse gases or clouds infrared escapes directly to space SPACE SURFACE  −18°C (255 K) 240 in 240 out straight to space 2  A model atmosphere with greenhouse gases but no clouds sunlight reaches the surface; infrared is absorbed and re-emitted SPACE GREENHOUSE GASES  (T₁ ≈ 277 K) SURFACE  +15°C (288 K) 240 in shortwave sunlight 390 ↑ 333 ↓ downward infrared net 57 ↑ 240 out 3  A fuller atmosphere with greenhouse gases, clouds and air motion clouds add albedo; evaporation and convection carry surface heat upward SPACE GREENHOUSE GASES  (T₁ ≈ 277 K) SURFACE  +15°C (288 K) 340 in 100 reflected cloud albedo 390 ↑ 333 ↓ ~100 ↑ evap + convection ~240 out

Reading the stages. The first two panels isolate the radiative greenhouse effect. The third adds the cloud albedo and non-radiative heat transport found in the real atmosphere. Values are W/m²; stage 3 figures are approximate global averages.

Why sunlight passes through more readily than heat

The reason CO₂ can warm while sunlight still reaches the ground is spectral. The Sun and Earth radiate in almost separate parts of the spectrum, and CO₂’s strongest absorption bands line up mainly with Earth’s outgoing infrared.

0.2 0.5 1 2 5 10 20 50 Wavelength (µm, log scale) emission (normalised) Sunlight shortwave, ~0.5 µm Earth’s heat longwave, ~10 µm CO₂ 15µm 4.3 2.0 · 2.7 Green bands: CO₂ absorption. The main overlap is the strong 15 µm band under Earth’s emission.

The spectral mismatch is the engine. The Sun radiates where CO₂ is transparent; Earth radiates where CO₂ absorbs. CO₂’s dominant band is at 15 µm, with a strong band at 4.3 µm and weak near-infrared bands near 2.0 and 2.7 µm. Curves are normalised to equal height so both are visible.

The numbers

Absorbed sunlight240 W/m²
Surface emission at 288 K, σT24390 W/m² ↑
Measured back-radiation, σT14≈ 333 W/m² ↓
Net surface radiative loss≈ 57 W/m²
Effective temperature without GHGs−18°C (255 K)
Surface temperature with GHGs+15°C (288 K)
Greenhouse effect33°C

Throughout, T2 > T1, so Q stays positive: net heat flows from the surface to the atmosphere and then to space. The second law is intact. CO₂ emitting infrared is how the insulation works; the σT14 term is the atmosphere emitting back towards the surface, reducing the surface’s net radiative loss.