**S. Chandrasekhar - The Scientist Who Worked Out How the Stars Find
Their Peace**

**A.N. Maheshwari**

**(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" [1]** **in the Indian Journal
of Physics when he was eighteen years old. His last published work,
"Newton's 'Principia' for the Common Reader" [2] 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 73^{rd} 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" [3]. The other is the book, "Chandrasekhar and His
Limit" [4], 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' [5]. It gives the
perspective of a pupil of his relationship with his teacher. In another article
entitled "Subrahmanyan Chandrasekhar: A Brilliant Star Without Limit"
[6] 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'être 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.

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.

**The 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" [7]
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" [8].

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.

**References **

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.* *