Zero-dimensional Energy Balance Model

Global mean energy budget of a planet, treated as one uniform, well-mixed sphere:

$$C\frac{dT}{dt}=\frac{S_0}{4}(1-\alpha )-\epsilon \sigma T^4$$

where:

• $T$: global mean surface temperature (K),
• $C$: heat capacity per unit area (J m⁻² K⁻¹),
• $S_0$: solar constant (≈ 1361 W m⁻² for Earth), divided by 4 for the ratio of the planet’s cross-section to its surface area,
• $\alpha$: planetary albedo (≈ 0.3 for Earth),
• $\epsilon$: effective emissivity (≈ 0.61 gives Earth’s 288 K),
• $\sigma$: Stefan-Boltzmann constant.

Equilibrium ($dT/dt=0$):

$$T={\left[\frac{S_0(1-\alpha )}{4\epsilon \sigma }\right]}^{1/4}$$

With latitude: one-dimensional energy balance model.