Admin and Founder of The Secrets of the Universe and former intern at Indian Institute of Astrophysics, Bangalore, I am a science student pursuing Master’s in Physics from India. I love to study and write about Stellar Astrophysics, Relativity& Quantum Mechanics.
Today I'm going to shed light on a very important law of physics that is very simple to understand and has many applications in astrophysics. This is not a pop science law and most of us fail to understand its importance. So, in the sixth article of Basics of Astrophysics Series, let us learn about Stefan's Law and its importance in Astrophysics.
Mathematical Form of Stefan's Law
The meaning of Stefan's law is simple: 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.
L is the luminosity of the star. Luminosity is actually the measure of total energy output of a star. The law is also referred to as Stefan-Boltzmann Law.
Our aim is to understand the basic concept behind this law. So let us start from the scratch.
What Is A Blackbody?
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. Such objects are hence 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 spectrum 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. So Sun is considered to be approximately a blackbody.
Temperature of Sun
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.
Importance of Stefan's Law In Astrophysics
By now, it must have been clear that Stefan's law has great significance in astrophysics. After all, we approximated the temperature of Sun for the first time, using this law. But Sun is not the limit of this equation. Using Stefan's law, we can calculate the temperature and size of other stars too.
We are now slowly diving deeper into Astrophysics. The aim of this article was to provide a taste of the concepts of luminosity and effective temperature of a star. In the coming articles of the series, we will learning about stars. Stellar Astrophysics is one of the most important and widely studied topic in the world. To pursue this branch of Astrophysics in research, one requires a strong hold of Physics. But here in this series, we will just have a taste of it. I hope you are enjoying Basics of Astrophysics series.