S. Chandrasekhar - The Scientist Who Worked Out How the Stars Find Their Peace
(S. Chandrasekhar 1910 - 1995)
Professor Chandrasekhar was one of those rare scientists who lived the life of a scientist from their youth till they breathed their last. He published his first scientific paper entitled "Thermodynamics of the Compton Effect with Reference to the Interior of the Stars"  in the Indian Journal of Physics when he was eighteen years old. His last published work, "Newton's 'Principia' for the Common Reader"  appeared in 1995, the year he died. He was eighty-four years old then. His professional output during his long scientific career, spanning nearly seventy years, was phenomenal. For his outstanding contributions to astrophysics and mathematical physics he received a large number of awards from professional societies of physics and astrophysics. Universities conferred upon him their honorary doctorate degree. The Presidents and the Heads of States felt honoured in honouring him. But he valued more the professional recognition from his peers and students. On his 73rd birthday in 1983 the Nobel Prize for Physics was announced for Chandrasekhar. Recently, the NASA has named its next satellite for scientific research "Chandra", the name by which his colleagues and admirers called him.
Chandrasekhar's life and work have been viewed with varied perspectives. Many books on Chandrasekhar have been written. There are two inspiring biographies of Chandrasekhar that the author of this article would like to recommend to the readers. One is the biography by Kameshwar C. Wali, entitled "Chandra" . The other is the book, "Chandrasekhar and His Limit" , written by G. Venkataraman. The author himself paid a tribute to Chandrasekhar in an article published in the University News in 1996. The title of that article is 'S. Chandrasekhar - As I Knew Him' . It gives the perspective of a pupil of his relationship with his teacher. In another article entitled "Subrahmanyan Chandrasekhar: A Brilliant Star Without Limit"  the author has worked out the calculation that Chandrasekhar at the age of 20 years might have carried out during his sea voyage to England. In that article an analysis of the concepts that Chandrasekhar made use of in working out the fundamental mass, known as the Chandrasekhar limit, has been given. Therefore, raison d'Ítre of a new article on Chandrasekhar can at best be the presentation of a perspective different from that of the articles already published by the author. As this article is being written for a special issue of the Indian Journal of Mathematics Education being brought out under the aegis of the Delhi Mathematics Teachers' Association the emphasis is on reasoning and not on the advanced mathematics used by Chandrasekhar in carrying out his calculations.
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Chandrasekhar is best known for his path breaking work on the physics of white dwarfs. He could explain how the stars would settle down after their burn out. He made brilliant use of the newly discovered special theory of relativity and the quantum statistical mechanics in the classical model of structure of stars and arrived at an expression of a critical stellar mass. In the following an attempt has been made to bring out the line of reasoning that led Chandrasekhar to reach his conclusion on the fate of stars.
There are innumerable numbers of stars in the universe. In our own galaxy the Milky Way it is estimated the number of stars is of the order of hundred thousand million. The Sun is a typical star of the Milky Way. It is located at a distance of about 30,000 light years from the centre of the galaxy. We at the Earth owe all the life that exits on our planet to the Sun. The Earth along with the other planets of the solar system was created out of the same debris of gas and dust that formed the Sun. It is, therefore, reasonable to hypothesise that the laws of physics that have been discovered by performing terrestrial experiments will also be applicable to the Sun, the other stars in the galaxy and even the universe itself.
In the twentieth century new types of astronomy such as radio, infra red, X-ray, gamma ray and the satellite-based telescopes became available as windows to view the universe. New astrophysical objects such as the pulsar, the black hole and the cosmic microwave radiation were discovered and interest increased in the study of stars, galaxies and the universe using the laws of physics. The special and the general theory of relativity, the quantum mechanics and the nuclear physics emerged as the new physics in the first half of the twentieth century. It was realised that though the new physics was discovered through the study of atomic and nuclear phenomena it is likely to be equally important in understanding the large scale structures such as the stars.
It has already been stated that the purpose of this article is to bring out the importance of the pioneering work of Chandrasekhar that he carried out during the period 1930 - 1935. He used the new physics, also known as the modern physics, in determining theoretically how the stars find their peace or in other words what happens when the stars die. As a by product of his work he could anticipate the existence of black holes forty years before these esoteric objects were discovered.
The astronomy has made it possible to see stars in different stages of their life cycle. Like the life we know on the Earth, each star is born, it evolves with time and ultimately dies either undergoing a catastrophic phase such as the supernova explosion or without it and settles down as a white dwarf or as a pulsar or as a black hole. The end of a star was expected to be decided by its mass, as all other properties such as its initial chemical composition would have been obliterated by the time the fusion process, the source of energy production, stops. As already mentioned Chandrasekhar found a critical mass called the Chandrasekhar limit, a benchmark that determines the ultimate fate of a star. The numerical magnitude of the Chandrasekhar limit is expressed in the unit of mass of the Sun. In this unit it is equal to 1.4 times the solar mass.
Stars have masses varying from a fraction of the solar mass to hundreds of solar mass. When we see the night sky we see stars of different masses and of different ages. Measurable quantities that describe the state of a star are its absolute luminosity and its surface temperature. Hertzsprung and Russell made a scatter plot of absolute luminosity and temperature of stars. They saw that most of the stars in their graph, the H-R diagram, lie in a narrow band. This band is called the main sequence. It was noticed that some of the stars lie in a corner of the H-R diagram away from the main sequence. These stars are the dwarfs, as the estimate of their radius show that in size they are a fraction of the size of the Sun. Typically the size of white dwarfs is of the order of that of the Earth. Their mass is about that of the Sun. Therefore the white dwarf are highly dense objects that were stars before they reached this stage. Prior to the work of Chandrasekhar it was generally believed that all stars when they exhaust their nuclear fuel would become white dwarfs. The astrophysicists speculated that after a star has burnt out its nuclear fuel some new physics might help it in settling down as a white dwarf. In this case the new physics was the degeneracy pressure exerted by the electron gas. Unlike the thermodynamic pressure due to gas and radiation that provide equilibrium to a star against the inward gravitational pull as long as a star produces energy by burning its nuclear fuel, the degeneracy pressure does not require energy production. It is a quantum mechanical effect.
By the first quarter of the twentieth century astrophysicists, most well known of whom was Eddington, had worked out the theory of stellar structure using the Newtonian theory of gravitation, the equation of state of a polytrope and the equations of hydrostatic equilibrium. Using his model Eddington could explain the main sequence band in the H-R diagram. But the Eddington's model could not explain the white dwarfs as their positions in the H-R diagram were outside this band. This situation became a pointer to the need of invoking some new physics for going beyond the limitations of the classical model.
The genius of Chandrasekhar was that using the special theory of relativity and the quantum statistical mechanics as keys he explored the physics of the white dwarf and discovered new features. He had learnt the quantum statistical mechanics required for working out the equation of state of an electron gas from the lectures given by Arnold Sommerfeld in Madras in 1928. The electrons are Fermions; fundamental particles with spin 1/2. They cannot occupy the same state. When electrons are compressed they resist being squeezed and provide a pressure called the degeneracy pressure. The electron gas will therefore obey the Fermi - Dirac statistics. His estimate of the velocity of electrons inside a white dwarf indicated that the electron gas could be relativistic. Therefore the pressure of the electron gas may have to be worked out using the Fermi - Dirac statistics in the relativistic limit. He used the theoretical model for white dwarf stars developed by Fowler, the person under whom he later did his Ph.D. research. But he used in his calculations the equation of state for a relativistic degenerate electron gas instead of the non-relativistic case considered by Fowler. The details of this calculation can be seen from the author's article and also from Chandrasekhar's book "Introduction to the Theory of Stellar Structure".
Chandrasekhar found that if a star has a mass less than the critical mass it would find peace as a white dwarf. But if the star has a mass greater than the critical mass it will become unstable against the inward gravitational pull and it will collapse. The possibility that an object that initially had a mass and size more than that of the Sun will shrink in size and disappear as a point was revolutionary and mind boggling. His startling findings were ridiculed by no less a person than Eddington. The fall out of this controversy was that for nearly two decades scientific interest in the physics of massive astrophysical objects remained dormant.
Chandrasekhar pursued the physics of black hole in the last quarter of his scientific career. He published his magnum opus "Mathematical Theory of Black Holes"  in 1983. An elementary introduction to black holes has been given by the author in his article "Beyond White Dwarfs, Toward Black Holes - An Introduction to S. Chandrasekhar" .
As this article has been written with the limited purpose of giving a non-mathematical account of the fundamental work carried out by Chandrasekhar in the initial years of his scientific career, it is difficult to go further without bringing in the mathematics. It is hoped that the references will help the readers in finding answers to questions that will arise in their mind after reading this elementary introduction to the life and work of Chandrasekhar.
1.Chandrasekhar, S. (1928), Thermodynamics of the Compton effect with reference to the interior of the stars, Indian Journal of Physics.
2.Chandrasekhar, S. (1995), Newton's 'Principia' for the Common Reader, Oxford University Press.
3.Wali, K.C. (1990), Chandra, Penguin Books India.
4.Venkatraman, G. (1992), Chandrasekhar and his limit, Universities Press.
5.Maheshwari, A. N. (1996), S. Chandrasekhar - As I knew him, University News Vol. 34, No.10.
6.Maheshwari, A. N. (1996), Subrahmanyan Chandrasekhar: A brilliant star without limit, The Mathematics Student Vol. 65, No. 1-4.
7.Chandrasekhar, S. (1983), Mathematical Theory of Black Holes, Oxford University Press.
8.Maheshwari, A. N. (1996), Beyond white dwarfs, Toward black holes: An introduction to S. Chandrasekhar, School Science Vol. 34, No. 1.