**Beyond White Dwarfs, Toward Black Holes**

*An Introduction to S. Chandrasekhar*

A.N. Maheshwari

S. Chandrasekhar was one of the most famous astrophysicists
of the twentieth century. He is recognised as the founder of the black hole
physics and the originator of the hypothesis of gravitational collapse. His
scientific career spanned over a period of sixty-six years. It is distinguished
by fundamental contributions to astrophysics, mathematical physics, plasma
physics, and general theory of relativity, science and aesthetics. The
consistent high quality of the professional output, specially the enormous
volume of the scientific work of Professor Chandrasekhar, had his students and
admirers in awe of the superhuman attributes endowed in him. On the contrary,
S. Chandrasekhar, who was affectionately called Chandra by the scientific
community, was down-to-earth in his approach and was a large-hearted teacher,
who would go out of the way to encourage and help his students. He would
invariably remind all those persons who had the good fortune to come near him
that by sheer dint of consistent hard work together with disciplined work
habits one can achieve highest peer recognition without being a Dirac or a
Heisenberg.

**Multi-faceted Personality**

It is not easy task to write an essay that can bring out
comprehensively the multifaceted personality of Chandrasekhar. Chandrasekhar
can be viewed from various perspectives. It is fortunate that by focussing on
narrow portions in the spectrum of his scientific contributions a glimpse of
the exceptional mind can be seen.

At the age of nineteen, Chandrasekhar did his monumental work
on the quantum mathematical description of white dwarf stars. He revealed the
possibility that massive stars towards the end of their life, because of the
inward gravitational pull, can undergo collapse and disappear into nothingness
and thus become black holes. But, what is a black hole? A black hole is a
region of space where the gravitational field is so strong that even light
cannot escape out of it because of the strong gravitational pull. This type of
a situation can be made plausible if we make use of the concept of escape velocity.

**Escape Velocity**

It is an everyday experience that when objects are thrown up
from the surface of the earth they fall back. We are not surprised by such
observations because Newton had explained that the gravitational force of the
earth pulls all objects. It is also a common experience that faster an object
is thrown up higher it goes before it falls back to the earth. The minimum
speed with which objects if thrown up so that they do not fall back at all and
may therefore appear to have escaped away from the earth to infinity is called
the escape velocity at earth.

To overcome gravitational potential energy the minimum
kinetic energy that the projectile must be given to enable it to escape to
infinity can be determined by imposing the condition that at infinity the total
energy, the sum of kinetic energy and potential energy, must be zero.

We next ask the question what should be the minimum radius R_{s
}of an object enclosing the mass *M *so that the gravitational pull
can be so strong that the escape velocity becomes equal to the speed of light
c? R_{s} is called the Schwarzschild radius of
an object of mass *M*. We can easily estimate the numerical value of R_{s}
for the sun and that of the earth by substituting for mass of the sun, M_{sun}_{ }= 1.988 x 10^{30} kg and
for mass of the earth, M_{earth} = 5.983 x 10^{24
}kg,and for the speed of
light c = 2.997 x 10^{8} m s^{-1}. The Schwarzschild radius of
the sun is 2.953 km and that of the earth is 0.89 cm. The equatorial radius of
the sun, Rsun
= 6.959 x 10^{8 }m. It is much bigger than its Schwarzschild radius,
2.953 km; and also the equatorial radius of the earth, R_{earth}
= 6.38 x 10^{6 }m is much bigger than its Schwarzschild radius, 0.89
cm. Light emitted at the surface of the sun reaches us and the earth is visible
from the moon. The natural curiosity would be to know whether there exist
objects in the universe whose outer surface is within their Schwarzschild radius? Such objects have been given the generic name black
holes. Recent astronomical observations have revealed that there are a large
number of strong candidates that fulfil the description of black holes.

**White Dwarf Star**

In 1929 When Chandrasekhar started his research studies on
stellar structures the most massive and tiny objects known to astronomers were
white dwarf stars. Arthur Eddington, the most famous
astronomer and astrophysicist of all times, had first observed the enormous
density of the tiny white dwarf star orbiting as a companion to Sirius - itself
the brightest star in the heavens. Eddington pointed
out that the white dwarf star is so dense that a ton of it would easily fit
into a matchbox. White dwarf stars like Sirius B and 40 Eridani
B have masses typically of the order of one solar mass, and radii of the order
of 1/10 to 1/100 of the sun's radius. The Schwarzschild radius of these dense
objects is well within their surface. They cannot be black holes. So, what is
the connection of the Chandrasekhar's work on stability of white dwarfs with
the hypothesis of gravitational collapse?

To fully understand the theoretical calculations made by
Chandrasekhar which led him to the discovery of a fundamental mass expressible
in terms of physical constants concepts both from classical and quantum physics
are required. These concepts are:

1. Equation of gravitational equilibrium of a stare,

2. Class of thermodynamical
configurations called the polytropic equilibrium,

3. Special theory of relativity and the Fermi-Dirac
statistics.

The 19-year-old boy not only knew these concepts he could
most effectively make use of them in working out the physics of white dwarfs
and made his fundamental discovery. A mathematical description of this work is
beyond the scope of this article and will not be attempted. What is being given
in the article further is a qualitative description of the stability of stars
based on Chandrasekhar's work.

According to Newton's universal law of gravitation an inward
gravitational pull will be experienced in the presence of a spherical
distribution of mass. But what prevents objects from being sucked inside by the
gravitational force?

Objects like moon prevent themselves from this type of
situation by revolving around so that the centrifugal force balances the inward
gravitational pull. On the surface of cold and solid objects like the earth it
is the elastic force that provides the normal force to balance the weight.
However, in gaseous objects like stars such as the sun the gravitational pull
is balanced by pressure due to kinetic motion of gaseous contents combined with
radiation pressure. In stars due to thermo-nuclear processes energy is
continuously generated. It provides pressure for the stability and is also
radiated away from the surface enclosing the star. What happens to stars like
sun when their fuel gets totally consumed by successive chain of thermo-nuclear
processes? Stars at that stage undergo gravitational collapse till new types of
physical processes provide the required pressure. It is now well understood
that on account of intense rise in temperature due to release of gravitational
potential energy when stars shrink the entire matter (hydrogen and other
elements) inside gets totally ionised and a gas of electrons comes into being.
Electron gas then contributes the dominant part of the pressure. It was well
understood by astrophysicists like A.S. Eddington and
R.H. Fowler that the main sequence stars towards the end of their life achieve
equilibrium due to electron pressure and settle down as white dwarfs. However,
their calculations for working out pressure used the approximation that
electron gas can be treated by non-relativistic quantum statistics.
Chandrasekhar estimated that electron gas in white dwarf stars could be both
degenerate and relativistic. He had learnt the physics of relativistic
Fermi-Dirac statistics from the lectures given by A. Sommerfeld
to the science students of the Presidency College, Madras in 1928.
Chandrasekhar made the calculation of pressure due to relativistic and
degenerate electron gas and studied the question of stability against
gravitational force in objects like sun when they reach the white dwarf stage.

The expression for the Chandrasekhar limit is completely
given in terms of fundamental physical constants, Planck constant, speed of
light , Newton's constant of gravitation, Mass of a nucleon, and the average
number of nucleons per electron. The numerical value of M_{cs}
can be calculated. It is

M_{cs}_{ }= 1.4 M_{sun}_{.}

This mass is known by astronomers and astrophysicists
throughout the world as the Chandrasekhar limit.

**Chandrasekhar Limit**

The significance of the Chandrasekhar limit is that if a star
has mass less than this limit it can settle down for eternity as a white dwarf,
but if its mass is greater than this limit the
electron pressure will not be sufficient to prevent further gravitational
collapse. This discovery of Chandrasekhar was so revolutionary and startling
that Eddington challenged the correctness of this
work and asserted that there must be a more complicated sequence of thresholds
determined by the complexity and quantum nature of events that can get around
the Chandrasekhar's inevitability that stars with mass more than M_{cs} can eventually collapse to nothingness. It
is now recognised that Eddington's strong personality
and scientific stature retarded for about a quarter century further research in
this field. In the history of science the confrontation between Chandrasekhar and
Eddington has a special place.

**Tribute to Chandrasekhar**

Bernard Lovell in the obituary to S. Chandrasekhar in the *Guardian*
(newspaper published from London, U.K.) of August 24, 1995 has beautifully
described the events of that time. We reproduce Lovell's words:

There are only a few epic confrontations in the history of
astronomy and Chandrasekhar was involved in one of these 60 years ago. He had
arrived in Cambridge in the autumn of 1930 as a 20-year-old post-graduate
student. During the long voyage from India he had reached a remarkable
conclusion about the final collapsed state of a massive star.

It was not a good time to arrive in Cambridge with
iconoclastic views about stars. Eddington dominated
astronomy and his influence was so great that no one seriously questioned his
views on the internal constitution of stars. The young Chandrasekhar did so and
the conflict with Eddington erupted in public at a
memorable meeting of the Royal Astronomical Society in January 1935. At that
time the conventional opinion was that when a star had exhausted its
energy-reproducing process it would collapse into a white dwarf with its entire
mass concentrated in a volume less that of the earth. Eddington
maintained this was the only possible final state in which the nuclei of atoms
were crushed almost in contact by the enormous force of the stellar collapse.

On that voyage from India Chandrasekhar concluded that this
explanation could be given a more generalised form by applying the concepts of Einstein's
special relativity theory. He reached the startling conclusion that there was a
limit to the mass of a star which could collapse into a white dwarf. He
believed that if a star was more massive than 1.4 times the mass of the sun it
would collapse catastrophically beyond the white dwarf stage to states of
incredibly high density. That was the origin of the modern concepts of neutron
stars and black holes.

In that famous January 1935 meeting of the Royal Astronomical
Society, Eddington ridiculed Chandrasekhar's ideas.
He thought Chandrasekhar had made a fundamental error of principle and said,
"I think there should be a law of nature to prevent a star from behaving
in this absurd way." The young Chandrasekhar was humiliated and depressed,
but he remained certain that his ideas were correct and sought the opinion of
the great physicists whose concepts he had used. They agreed with him. But,
such was Eddington's influence, not one of them was
willing to enter the public dispute. It was a time when specialisation led to
the belief that physics and astrophysics had little in common. Chandrasekhar's
youthful work and all his subsequent career emphasised
the falsity of this supposition.

Thirty-three years later the discovery of pulsars and the
recognition that there were neutron stars produced the observational
confirmation that Chandrasekhar's revolutionary ideas were correct.

We continue with the description of developments that
followed Chandrasekhar's classic work. It was soon recognised that stars on collapse
beyond the white dwarf size can reach another threshold where equilibrium is
possible. Through radioactive process, commonly called beta-decay and inverse
beta-decay, electrons and protons can combine to form neutrons. In such
situations matter largely consists of neutrons and has properties of a giant
nucleus. It may be possible for the pressure due to neutrons to balance the
gravitational pull and for the star to reach equilibrium. The existence of such
stars was predicted by Oppenheimer and his collaborators in 1930 and these were
named neutron stars. However, these exotic objects remained a textbook
curiosity until the 1960s, when radio and optical astronomers discovered in
1968 a pulsar at the centre of crab nebula in the Constellation of Taurus.
Pulsars have since been identified as rotating neutron stars. The neutron stars
also reach a maximum limit in mass for equilibrium, which is of the order of
solar mass, and a radius of the order of 10 km.

The mental block for a star like the sun to shrink to a few
kilometres was overcome with the discovery of neutron stars. The possibility of
stars more massive than the Chandrasekhar limit collapsing into black holes
when they are neither able to reach equilibrium as a white dwarf or as a
neutron star became a likely possibility. The field of black holes has since
been taken up for serious scientific studies by astronomers, physicists and
astrophysicists alike. In 1960s Chandrasekhar himself revived his interest in
the general theory of relativity and its application to astrophysics in general
and black hole physics in particular. Chandrasekhar wrapped up his work on the
stability of massive objects like stars in his seminal work called *The
Mathematical Theory of Black Holes** (1987).* Though belated, in
1983 Chandrasekhar shared the Nobel Prize for his quantum mathematical
prediction that large stars must undergo gravitational collapse as they burn
out.

To conclude this article it would only be apt to recognise S.
Chandrasekhar as a brilliant star without limit.