'Month of Equations': What Does Stefan's Law Really Mean?

                                          October 15: Stefan-Boltzmann's Law

15. Stefan's Law

Meaning of Equation

The total energy radiated per unit surface area of a black body at all wavelengths is proportional to the fourth power of its absolute temperature.

I'm now going to shed light on a very important law of physics that is very simple to understand and has many applications in astrophysics. This equation is not a pop science equation and most of us fail to understand its importance. So I decided to include it in my series. I will spend less time on describing the meaning of equation as it is very easy to understand. I'll rather emphasise how this equation led to important breakthroughs in the field of solar and stellar astrophysics. As usual, lets start from the scratch.

A blackbody is a perfect absorber and emitter of light. It absorbs any light that falls on it. A perfect blackbody is also a perfect radiator.  In general, the better an object is at absorbing light, the better it it at emitting it, so a perfect absorber should be the most efficient radiator possible; but at the same time,  if an object is a perfect absorber it will not reflect any radiation, and so it will look black. That's why such objects are known as Black Bodies.

Then how come Sun is a blackbody? Actually, Sun has no solid surface. So any radiation that strikes the Sun is scattered and absorbed until it is completely lost. This makes it a perfect absorber. But, Sun is not a perfect emitter. It is evident from the graph below:


The orange line curve shows the spectrum of a perfect blackbody and the red curve shows the spectrum of Sun. The latter has many overshoots and dips from the ideal curve of blackbody and hence Sun is considered to be approximately a blackbody.

I have completed my main task. I just wanted to tell how this law is applicable to the Sun and other stars and for that I had to tell how Sun and other stars can be considered a black body. Now comes the interesting part of this article. It's a small story that you'll relish.

Also See: What Does The Schrodinger Wave Equation Really Mean?

This law was first deduced experimentally by Josef Stefan in 1879. Before him, another scientist named J. Soret conducted a beautiful experiment in which he took a lamella (thin plate) and heated it to about 2000 K. He then kept the lamella at such a distance that it subtended an angle same as that subtended by the Sun. From his experiment, he inferred that the energy flux density (energy radiated) of Sun is 29 times that of the lamella. Stefan used this data and went further. He added another factor. He predicted that about 1/3 of the energy of Sun is absorbed by the Earth's atmosphere. So the actual energy flux is not 29 times, but 29 x 3/2 times that of lamella. The number comes out to be 43.5.

He then plugged this value in his formula (given above). The energy radiated (L) is 43.5 times that of lamella. This means that its temperature must be fourth power root of 43.5 times that of lamella (its easy mathematics. Just plug in the values in the equation above). Now (43.5)^1/4 = 2.57 and hence the final result obtained was that the temperature of surface of Sun is 2.57 times that of temperature of lamella. The exact answer comes out to be 5700 K. This was a remarkable result. It is just off by 1.3% of the current accepted value of 5778 K. Remember, Stefan assumed the quantity of energy absorbed by atmosphere to be 1/3 of the emitted energy. It was later found that his assumption was also correct. This was the first sensible approximation of the temperature of Sun in the history of mankind.

Now it should be clear why I was overemphasizing the importance of this simple equation. Take any star and you can measure its temperature, luminosity and surface area using this law. This law can be derived from thermodynamics as well as from Planck's law of blackbody radiation.

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