Thermal Radiation: Blackbodies, Color and Temperature

Atoms and molecules are always jiggling, and the hotter something is, the harder they jiggle. Jiggling charges give off electromagnetic waves. So every object above absolute zero sends out thermal radiation, all the time.

Unlike conduction and convection, radiation needs no material to travel through. It crosses empty space at the speed of light, which is how the Sun's energy reaches Earth.

At room temperature almost all thermal radiation is infrared, invisible to us. Only above about 800 K (the Draper point, measured at 798 K, about 525 °C) do solid objects start to glow a faint, visible red. The glow you see from a toaster's wires or a campfire's coals is the visible tail of a much bigger infrared output.

Objects also absorb radiation from everything around them. Whether something warms up or cools down depends on the difference between what it emits and what it absorbs.

A blackbody is an ideal surface that absorbs every bit of radiation that lands on it, reflecting none. The name is a little misleading. A cold blackbody looks black, but a hot one glows: stars behave roughly like blackbodies.

A good real-life model is a closed box with a tiny hole. Light that gets into the hole bounces around inside until it's absorbed, so the hole is almost perfectly black. Heat the box, and the radiation that leaks out of the hole has a blackbody spectrum that depends only on the temperature, not on what the box is made of.

Figure 1 shows that spectrum at three temperatures. Each curve rises steeply, peaks and then falls away slowly toward long wavelengths. Two patterns jump out: hotter curves are taller at every wavelength, and their peaks sit farther to the left. Ideas 3 and 4 turn those two patterns into laws.

In 1893 Wilhelm Wien showed that the peak wavelength is inversely proportional to the temperature:

λ_max = b ÷ T where b ≈ 2.898 × 10⁻³ m·K

Always use kelvin for T. Double the temperature and the peak wavelength halves.

The Sun's surface is about 5772 K, so its glow peaks near 502 nm, in the green part of visible light. Your skin, at about 306 K (33 °C), peaks near 9.5 μm, far into the infrared, which is why thermal cameras that "see" people work at around 8 to 14 μm. The cosmic microwave background, the afterglow of the early universe at 2.725 K, peaks at about 1.06 mm, in microwaves.

Wien's law also explains the colors of glowing objects. As a steel bar heats up, its peak slides from the infrared toward visible light, so we first see the red end, then more and more of the spectrum, until it looks white. Very hot stars, with peaks in the ultraviolet, look blue-white.

The total power a blackbody radiates is the area under its spectrum curve. In 1879 Josef Stefan, working from John Tyndall's measurements, found that it grows with the fourth power of the absolute temperature. In 1884 Ludwig Boltzmann derived the same law from theory:

P = σAT⁴ where σ ≈ 5.670 × 10⁻⁸ W/(m²·K⁴)

P is the power in watts, A is the surface area in m² and σ (sigma) is the Stefan–Boltzmann constant. The fourth power makes temperature matter enormously: doubling the kelvin temperature multiplies the power by 2⁴ = 16, and tripling it multiplies the power by 81.

The Sun shows how big the numbers get. Each square meter of its 5772 K surface gives off about 63 million watts. Multiply by the Sun's whole surface, about 6.08 × 10¹⁸ m², and you get about 3.83 × 10²⁶ W, which matches the Sun's measured power output.

Real surfaces radiate less than a perfect blackbody at the same temperature. We fix the law with the emissivity e, a number between 0 and 1:

P = eσAT⁴

Human skin is nearly a blackbody in the infrared, with e ≈ 0.97, whatever its visible color. Carbon black, used in printer toner, reaches about 0.99. Polished metals are poor emitters, and a tungsten bulb filament is only about 0.5.

In 1860 Gustav Kirchhoff showed that, at each wavelength, a surface emits exactly as well as it absorbs. A good absorber is a good emitter, and a good reflector is a poor emitter. That's why a matte black can of hot water cools faster than a shiny one, and why emergency blankets and vacuum flasks are silvery.

Every object also absorbs radiation from its surroundings. If the surroundings are at temperature T_s, the net power the object loses is:

P_net = eσA(T⁴ − T_s⁴)

When the object is hotter than its surroundings, it loses energy and cools. When it's colder, the net flow is inward and it warms. At the same temperature, the flows balance.

The Stefan–Boltzmann law lets you estimate a planet's temperature from sunlight alone. At Earth's distance, sunlight delivers about 1361 W/m² to a surface facing the Sun. Earth reflects about 30% of it straight back to space; that fraction is its albedo.

Earth catches sunlight on a disk of area πR², but it radiates from its whole sphere, 4πR², four times as much area. So, averaged over the globe, it absorbs about 1361 × 0.70 ÷ 4 ≈ 238 W/m².

Earth warms until the infrared it radiates matches what it absorbs. Setting σT⁴ = 238 W/m² gives T ≈ 255 K, about −18 °C. This is Earth's effective temperature, the temperature of a blackbody that would radiate the same power. It's a self-correcting balance: a warmer Earth radiates more and cools back down, and a cooler Earth radiates less and warms up.

There's a puzzle: Earth's average surface temperature is about 15 °C (288 K), not −18 °C. The difference, about 33 °C, is the greenhouse effect.

Wien's law is the key. Sunlight comes from a 5772 K surface, so it's mostly visible and near-infrared, and most of it passes through the air. The ground, at around 288 K, radiates far-infrared near 10 μm. Water vapor, carbon dioxide and some other gases absorb those wavelengths strongly. They re-emit the energy in all directions, and part of it goes back down, so the surface must be warmer to push enough energy out to space.

Without it, Earth would be frozen. Adding more of these gases strengthens the effect, which is why rising carbon dioxide levels warm the climate.

Thermal radiation went from a curiosity with thermometers to the clue that launched quantum physics.