8.1 Radiation from Real Bodies

A perfect black body is an ideal reference object. Real materials emit less thermal radiation than a black body at the same temperature, so we introduce emissivity $\epsilon$, a number between 0 and 1. Emissivity tells us how effectively a surface emits thermal radiation relative to the ideal limit.

Albedo $\alpha$ is the reflected fraction of incident radiation. If transmission is negligible, emissivity and albedo are linked by energy partition at the surface: what is not reflected is absorbed. In many introductory models this leads to the approximation $\epsilon + \alpha \approx 1$ for the relevant wavelength band.

A warm surface in cooler surroundings both emits and absorbs radiation at the same time. Net radiative loss is the difference. This is crucial for Earth-system reasoning: outgoing longwave radiation can increase while incoming solar remains fixed, and the sign of the net determines whether cooling or warming follows.

The $T^4$ dependence means small changes in absolute temperature can produce large flux changes at high temperatures. That nonlinearity is why radiative equilibrium is a stabilizing mechanism: if a body warms, it radiates much more strongly, increasing outgoing energy and opposing further warming unless forcing also increases.

For Earth applications we care about global averages, so local variability in emissivity and cloud cover gets compressed into effective parameters. This simplification is useful for first-principles insight, but we must remember that effective averages hide strong regional contrasts.

Emissivity can be viewed in two complementary ways: as an overall scaling of total emitted intensity, and as a wavelength-dependent property that reshapes detailed spectra. In simplified IB-level models we often use one effective emissivity, which preserves the main energy-balance logic without resolving spectral lines explicitly.

A useful interpretation: changing emissivity usually changes how much radiation is emitted at all wavelengths combined, while peak wavelength location is still set primarily by temperature through Wien scaling. So two surfaces at the same temperature can have similar peak positions but different total intensity levels.

Suggested reading sequence in the simulation: first hold temperatures fixed and sweep emissivity to isolate scaling effects. Then hold emissivity fixed and vary temperature to see how strongly $T^4$ growth dominates. Finally raise surroundings temperature and watch net power shrink toward zero as equilibrium is approached.