This is one of the fundamental relationships of modern physics:
_Q__body = ε_A_σ_T_⁴.
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, Q_body, 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 ε_A_ 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, α_S_ 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_
= π_r_²) is one-fourth that of the sphere’s total surface area (_A_ =
4π_r_²), we know that the radiant energy from the sun that is absorbed
will be
_Q__sun = (_A_)/(4)α_S_.
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, _Q__body is equal to _Q__sun.
This means that we have the equality
ε_A_σ_T_⁴ = (_A_)/(4)α_S_.