How We Measure the Radiation of Heat
This is one of the fundamental relationships of modern physics:
Qbody = εAσT⁴.
It was discovered experimentally by Josef Stefan in 1879, and verified theoretically by Ludwig Boltzmann; it is known as the Stefan-Boltzmann Law. This formula tells us the amount of radiant energy, Qbody, that will be emitted by a body having the surface area A when it is at the temperature T. Temperature, here, is measured in degrees Rankine (°R), or Fahrenheit temperature above absolute zero (to calculate degrees Rankine, add 460 to the temperature in degrees Fahrenheit). The expression εA is used to show that only a certain fraction of the energy that would leave a perfect black body of area A will actually leave a real body of the same size; the size of this fraction is determined by the body’s emissivity. The symbol σ is a quantity we call the Stefan-Boltzmann constant.
We can also calculate the heat from the sun that will be absorbed by a body. If we let S be the total amount of solar energy that would be absorbed by a perfect black body, αS will be the amount that is actually absorbed by a body with an absorptivity of α for solar radiation. If our body is a spherical satellite, the sun’s rays will only strike it from a single direction. Thus only an area equivalent to the sphere’s cross-section (largest inscribed circle) will receive energy at any one time. Since, as shown in the [sketch], this area (a = πr²) is one-fourth that of the sphere’s total surface area (A = 4πr²), we know that the radiant energy from the sun that is absorbed will be
Qsun =
| A |
| 4 |
A man-made satellite’s position relative to the earth is very like that of the earth in relation to the sun; the earth, after all, is itself a satellite of the sun. And during most of its useful life a satellite will be in thermal equilibrium—it will be losing just as much heat energy by its own radiation into space as it will be gaining from other sources, primarily the sun. Since the total amount of energy it absorbs is equal to the amount of energy it emits, Qbody is equal to Qsun. This means that we have the equality
εAσT⁴ =
| A |
| 4 |