Introduction

The greenhouse effect is a thermodynamic phenomenon where atmospheric gases absorb and re-emit outgoing infrared radiation, warming the Earth's surface. This article examines the fundamental physics underlying this process.

Blackbody Radiation

Planck's Law

The spectral radiance of a blackbody at temperature $T$ is given by Planck's law:

$B_\nu(T) = \frac{2h\nu^3}{c^2} \frac{1}{e^{h\nu/k_BT} - 1}$

where $h$ is Planck's constant, $\nu$ is frequency, $c$ is the speed of light, and $k_B$ is Boltzmann's constant.

Stefan-Boltzmann Law

Integrating Planck's law over all frequencies gives the total power radiated per unit area:

$j^\star = \sigma T^4$

where $\sigma = 5.67 \times 10^{-8} \, \text{W m}^{-2} \text{K}^{-4}$ is the Stefan-Boltzmann constant.

Radiative Balance

The Earth's effective temperature is determined by the balance between absorbed solar radiation and emitted thermal radiation:

$\frac{S(1-\alpha)}{4} = \sigma T_e^4$

where $S$ is the solar constant ($\approx 1361 \, \text{W/m}^2$), $\alpha$ is the albedo ($\approx 0.3$), and the factor of 4 accounts for the ratio of cross-sectional area to surface area.

This yields an effective temperature of approximately $T_e \approx 255 \, \text{K}$ ($-18°\text{C}$), significantly colder than the actual surface temperature of $\approx 288 \, \text{K}$ ($15°\text{C}$).

Greenhouse Gases

Key greenhouse gases and their absorption bands:

Climate Feedback Mechanisms

Ice-Albedo Feedback

As temperature rises, ice melts, reducing albedo and increasing absorbed solar radiation — a positive feedback loop.

Water Vapor Feedback

Warmer air holds more moisture, and water vapor is itself a greenhouse gas, amplifying the initial warming.

Source

Full LaTeX source and supporting materials available on GitHub.