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

What σ is. It sets how much power a perfect emitter radiates per square metre at a given temperature. It is not the Boltzmann constant k = 1.38×10⁻²³ J/K; rather σ is built from k with Planck’s constant h and the speed of light c: σ = 2π⁵k⁴ / (15h³c²). 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  No greenhouse gases air transparent to infrared SPACE SURFACE  −18°C (255 K) 240 in 240 out straight to space 2  Add greenhouse gases sunlight passes through, heat is intercepted SPACE GREENHOUSE GASES  (T₁ ≈ 277 K) SURFACE  +15°C (288 K) 240 in CO₂ transparent to shortwave 390 ↑ 333 ↓ back-radiation net 57 ↑ 240 out 3  The real atmosphere clouds reflect sunlight; the surface also loses heat by evaporation and convection SPACE GREENHOUSE GASES SURFACE  +15°C (288 K) 340 in 100 reflected by clouds (albedo) 390 ↑ 333 ↓ ~100 ↑ evap + convection ~240 out

Reading the stages. Reflection of incoming sunlight appears only at stage 3, and clouds do it, not CO₂. Greenhouse gases let sunlight through and act on the outgoing infrared alone. Values are W/m²; stage 3 figures are approximate global averages.

Why sunlight passes through but heat does not

The reason CO₂ can warm without blocking the Sun is spectral. The two bodies radiate in almost separate parts of the spectrum, and CO₂’s absorption bands fall under one of them, not the other.

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 strong 15 µm band sits under Earth’s emission; almost nothing sits under the Sun’s.

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 always flows surface to atmosphere to space. The second law is never broken. CO₂ being an emitter is not the opposite of insulation; the σT14 term is the atmosphere emitting, and that is what slows the loss.

Comparison is Earth with versus without greenhouse gases, same albedo and rotation, not the Moon, whose extreme cold owes more to having no atmosphere at all and a fortnight-long night. The two-body form treats the atmosphere as a single shell at T1; the real atmosphere radiates from many layers and a flat cartoon does not close the budget exactly. None of that changes the result.