In the previous article of the series, we started our journey of core astrophysics by learning about the redshifts. Today I’m going to shed light on a very important law of physics that is 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 seventh article of the Basics of Astrophysics series, let us learn about the 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 per unit time by a black body at all the wavelengths is proportional to the fourth power of its absolute temperature.
L is the luminosity of the blackbody. Luminosity is actually the measure of total energy output of a the object. The law is also referred to as Stefan-Boltzmann Law. Our aim is to understand the basic concept behind it. So let us start from 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 is 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.
Related in this series:
Chapter 1: What is Astrophysics?
Chapter 2: The Importance of EM Spectrum in Astrophysics
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 the blackbody. So the 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 2,000 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 the 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 the surface of Sun is 2.57 times that of the temperature of the lamella. The exact answer comes out to be 5,700 K.
This was a remarkable result. It is off by just 1.3% of the current accepted value of 5,778 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. Before this, values ranging from as low as 1,800 °C to as high as 13,000,000 °C were claimed.
Stefan’s Law In Astrophysics
As we have already read, Stefan’s law was the first formula with which we estimated the temperature of the Sun. Not only the Sun, Stefan’s law can be used to calculate the surface temperature of the stars too. Once we know the luminosity and dimensions of the star, we can plug in the values and find the temperature. This formula for luminosity is also useful in calculating the stellar masses of galaxies, provided we know the luminosity of the Sun accurately (which we do!). Once we know the stellar mass of the galaxy, we can find its specific star formation rate too.
Stefan’s law is not very popular but it is a very important relation in astrophysics. It can be derived from thermodynamics and also from the Planck’s law. In the previous article, we saw how spectroscopy and atomic physics were at play in astrophysics. Today’s article gives the glimpse of the importance of thermodynamics in astrophysics.
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 be learning about stars. Stellar Astrophysics is one of the most important and widely studied topics in this field. To pursue this branch of Astrophysics in research, one requires a stronghold of physics. In this series, we will just have a taste of it. I hope you are enjoying the Basics of Astrophysics series.
Note: The images of the Sun used in this article were all captured by astrophotographer Tim Connolly. You can find more at his website www.astronorth.com