## The Stefan-Boltzmann Law

The Stefan-Boltzmann Law was first discovered experimentally in 1879 by Stefan, then derived theoretically in 1884 by his student Boltzmann. It says:  Note the strong dependence on temperature: -- as the fourth power of the temperature. I.e., double the temperature, and the radiated power will increase by a factor 24 = 16. Triple the temperature, and the power will increase by a factor 34 = 81, etc. The Stefan-Boltzmann Law is valid only for perfect radiators (called "blackbodies"). Actual radiating surfaces are not perfect radiators, and will always radiate less than the luminosity given by the S-B Law -- typically some 10 - 80%. But the radiating surfaces (called "photospheres") of stars are fairly good approximations to black bodies, and typically radiate a luminosity of more than 90% of the value given by the S-B Law.

The value of the Stefan-Boltzmann constant, s = 5.67 x 10-5 ergs cm-2 s-1 K-4, is determined by experiments in Earth labs. We assume that the S-B Law, with the same value of s, is valid everywhere in the universe. That assumption is one example of the Principle of Universality of Physical Laws.

### Examples:

1. The Earth: The average temperature of the Earth's surface must be about 10 C. We must first convert this to the Kelvin scale: T(K) = T(C) + 273 = 283 K. The Earth's surface area is about A = 4p R2 = 4p (6.4 x 108 cm)2 = 5.15 x 1018 cm2. Now we can calculate the radiated luminosity of the Earth: L = 5.15 x 1018 x 5.67 x 10-5 x (283)4 = 1.87 x 1024 ergs s-1. To change ergs s-1 to Watts, multiply by 10-7. So, L = 1.87 x 1017 Watts. (Actually, the Earth radiates only about half this much luminosity into outer space because it is not a black body.)

2. The Sun: The temperature of the Sun's photosphere is 5800 K and its radius is 7 x 1010 cm. So the Stefan-Boltzmann Law gives a luminosity L = 4 x 1033 erg s-1 = 4 x 1026 Watts.

3. Your body: (This calculation will be a bit more subtle because you must correct for the temperature of the environment.)