S UBRAHMANYAN CHANDRASEKHAR was born into a free-thinking, Tamil-speaking Brahmin family in Lahore, India.
He was preceded into the world by two sisters and followed
by three brothers and four sisters. His mother Sitalakshmi
had only a few years of formal education, in keeping with
tradition, and a measure of her intellectual strength can be
appreciated from her successful translation of Ibsen and
Tolstoy into Tamil. His father C. S. Ayyar was a dynamic
individual who rose to the top of the Indian Civil Service. It
is not without interest that his paternal uncle Sir C. V. Raman
was awarded a Nobel Prize in 1930 for the discovery of the
Raman effect, providing direct demonstration of quantum
effects in the scattering of light from molecules.
Education began at home with Sitalakshmi giving instruction in Tamil and English, while C. S. Ayyar taught his children English and arithmetic before departing for work in
the morning and upon returning in the evening. The reader
is referred to the excellent biography Chandra, A Biography
of S. Chandrasekhar (University of Chicago Press, 1991) by
Prof. Kameshwar C. Wali for an account of this remarkable
family and the course of the third child through his distinguished career in science. Chandra is the name by which S.
Chandrasekhar is universally known throughout the scien-
30 BIOGRAPHICAL MEMOIRS
tific world. Chandra’s life was guided by a dedication to
science that carried him out of his native culture to the
alien culture of foreign shores. The crosscurrents that he
navigated successfully, if not always happily, provide a fascinating tale. He was the foremost theoretical astrophysicist
of his time, to paraphrase his own accounting of Sir Arthur
Eddington.
The family moved to Madras in 1918 as C. S. Ayyar rose
to deputy accountant general. Chandra and his brothers
had private tutors then, with Chandra going to a regular
school in 1921. His second year in school introduced algebra and geometry, which so attracted him that he worked
his way through the textbooks the summer before the start
of school.
Chandra entered Presidency College in Madras in 1925,
studying physics, mathematics, chemistry, Sanskrit, and English. He found a growing liking for physics and mathematics and an ongoing attraction for English literature. One
can assume that his fascination with English literature contributed to his own lucid and impeccable writing style.
Chandra was inspired by the mathematical accomplishments of S. Ramanujan, who had gone to England and distinguished himself among the distinguished Cambridge mathematicians until his early death in 1920. Chandra aspired
to take mathematics honors, whereas his father saw the Indian Civil Service as the outstanding opportunity for a bright
young man. Mathematics seemed poor preparation for the
Civil Service. Sitalakshmi supported Chandra with the philosophy that one does best what one really likes to do.
Chandra compromised with physics honors, which placated
his father in view of the outstanding success of Sir C. V.
Raman.
On his own initiative Chandra read Arnold Sommerfeld’s
book Atomic Structures and Spectral Lines and attended lec-
SUBRAHMANYAN CHANDRASEKHAR 31
tures in mathematics. His physics professors noticed that
he was learning physics largely through independent reading and provided him with the freedom to attend mathematics lectures. In the autumn of 1928 Sommerfeld lectured at Presidency College. Chandra made it a point to
meet Sommerfeld and was taken aback to learn that the
old Bohr quantum mechanics, on which Sommerfeld’s book
was based, was superseded by the wave mechanics of
Schroedinger, Heisenberg, Dirac, Pauli, et al., and that the
Pauli exclusion principle replaced Boltzmann statistics with
Fermi-Dirac statistics. Sommerfeld had already applied the
new theory to electrons in metals and kindly provided
Chandra with galley proofs of his paper. Chandra launched
into an intensive study of the new quantum mechanics and
statistics and wrote his first professional research paper “The
Compton scattering and the new statistics” (1929). In January 1929 he communicated this work to Prof. R. H. Fowler
at Cambridge for publication in the Proceedings of the Royal
Society of London. The name Fowler suggested itself because
Fowler had applied the new statistics to collapsed stars (i.
e., white dwarfs). Fowler was an open-minded and generous individual who perceived the merit of Chandra’s paper,
which he duly communicated to the Royal Society. This
contact was to play a crucial role a year later when Chandra
arrived in England.
Heisenberg lectured at Presidency College in October
1929 and Chandra had the opportunity to carry on extensive discussion with him at the time. Then Meghnad Saha
at Allahabad, known for the statistical mechanics that provided the interpretation of stellar spectra, invited Chandra
for discussions of Chandra’s paper in the Proceedings of the
Royal Society of London. Wali, in his biography, contrasts this
early appreciation of Chandra’s work by the scientific com-
32 BIOGRAPHICAL MEMOIRS
munity with the class snobbery of the British Raj on the
personal level.
Final examinations at Presidency College came in March
1930 and Chandra established a record score. In February
Chandra was informed that a special Government of India
scholarship was to be offered to him to pursue study and
research in England for three years. When the scholarship
was announced publicly, Chandra experienced resentment
from fellow Indians who perceived him as abandoning his
country and his legacy. Worse, it was becoming clear that
Sitalakshmi was terminally ill and, if Chandra went to England, he would not see her again. True to form Sitalakshmi
decided the issue by declaring that Chandra was born for
the world and not just for her.
Chandra informed the authorities that he wished to use
his government scholarship to study and carry on research
with R. H. Fowler at Cambridge. The Office of the High
Commissioner of India proceeded with the arrangements.
Chandra departed Bombay on July 31, 1930, bound for
Venice, from where he traveled by rail to London, arriving
August 19. He undertook the journey in his personal pursuit of science, and that journey was culturally irreversible,
a departure from home from which he never really returned.
It is well known that Chandra spent his time on shipboard working out the statistical mechanics of the degenerate electron gas in white dwarf stars, appreciating, as Fowler
had not, that the upper levels of the degenerate electron
gas are relativistic. Since it is the upper levels that are affected by changes in density and temperature, it follows
that a density change ∆ρ and pressure change ∆p are related by ρ∆p/p∆ρ = 4/3 rather than the nonrelativistic value
5/3 employed earlier by Fowler. The value 4/3 meant that
the pressure supporting the star against gravity grows no
faster than the increasing gravitational force as the star con-
SUBRAHMANYAN CHANDRASEKHAR 33
tracts, with the result that there is a limiting mass above
which the internal pressure of the white dwarf cannot support the star against collapse. This is in contrast with the
familiar nonrelativistic situation where the pressure increases
more rapidly than the gravitational forces so that sufficient
contraction must ultimately provide a sufficient pressure to
block further contraction. The limiting mass was clearly of
the order of the mass M. of the Sun (2 × 1033 g). A precise
value would require detailed calculations of the interior
structure of the star with the precise value of ρ∆p/p∆ρ for
intermediate levels as well as the upper fully relativistic levels at each radius in the star. But the implication was clear.
A massive star, of which there are many, cannot fade out as
a white dwarf once its internal energy source is exhausted.
Instead it shrinks without limit, always too hot to become
completely degenerate, and disappears when the gravitational field above its surface becomes so strong that light
cannot escape. In modern language, the massive star eventually becomes a black hole. The reasoning was straightforward and the conclusion was startling. The repercussions
that ultimately followed his discovery served to push Chandra
farther into the obscure and lonely byways of science in a
foreign Western society and ever more distant from his cultural origins.
Upon arrival in London Chandra discovered that the Office of the Director of Public Instruction in Madras and the
High Commissioner of India in London had thoroughly
bungled his admission to Cambridge. What was more, the
secretary for the high commissioner’s office had not the
least interest in correcting the mistake and was openly rude
in his assertion of that fact. Chandra was saved only by the
eventual firm intervention of Fowler, who was vacationing
in Ireland at the time of Chandra’s arrival in London. The
34 BIOGRAPHICAL MEMOIRS
consequences of Chandra’s first research paper were more
far reaching than anyone could have imagined.
Chandra took up his studies at Cambridge and spent a
lonely but productive year in intensive study and research.
Sitalakshmi died on May 21, 1931, adding grief to his loneliness. Chandra was introduced to the monthly meetings of
the Royal Astronomical Society and became acquainted with
E. A. Milne and P. A. M. Dirac. Chandra devoted his research efforts to calculating opacities and applying his results to the construction of an improved model for the limiting mass of the degenerate star. Milne was enthusiastic
about the work, but it turned out later that his enthusiasm
was based more on his rivalry with A. S. Eddington than on
an appreciation of the scientific merits.
The year of intensive study at Cambridge moved Chandra
to look for a change of scenery, and at the invitation of
Max Born he spent the summer of 1931 at Born’s institute
at Gottingen. There he became acquainted with Ludwig
Biermann, Edward Teller, Leon Brillouin, and Werner
Heisenberg. Back at Cambridge in the autumn Chandra
continued his work on atomic absorption coefficients and
mean opacities, but with a growing sense of frustration from
his feeling that he was abandoning mathematics through
his pursuit of physics and abandoning pure physics through
his pursuit of astrophysics. Chandra was invited to present
his results on model stellar photospheres at the January
1932 meeting of the Royal Astronomical Society (RAS) and
was complimented by both Milne and Eddington following
the presentation.
Chandra’s feeling of frustration with his “peripheral science” led to his spending his third year at Bohr’s institute
in Copenhagen. He adapted readily to the informal atmosphere and became acquainted with Victor Weisskopf, Leon
Rosenfeld, M. Debrueck, H. Kopferman, and others. Dur-
SUBRAHMANYAN CHANDRASEKHAR 35
ing the time in Copenhagen Chandra succeeded in convincing himself that his real strength lay in developing and
expounding the implications of the basic physical laws of
nature as distinct from the pursuit of new laws of nature.
He found an interested and appreciative audience in the
physics community for his work on degenerate stars. Chandra
was invited to the University of Liege to lecture on his work,
following which he was presented with a bronze medal. The
overall experience of the year was to ease his mind and set
him firmly on a path in theoretical astrophysics.
Chandra finished the year with four papers on rotating
self-gravitating polytropes, which became his Ph.D. thesis.
His government scholarship ran out in August 1933 and
the question was what to do next. It was clear that there
were no opportunities in India unless he rode on the coattails of his uncle Raman, which he was loathe to do. Fortunately he won one of the highly competitive appointments
as a fellow of Trinity College, which ran for four years.
Milne nominated Chandra for fellow of the RAS, and the
future was clear for the immediate years at Cambridge. At
the monthly meetings in Burlington House Chandra and
such contemporaries as William McCrea generally sat in
the back row, but became acquainted with some of the denizens of the front row (e.g., Sir James Jeans, Sir Arthur
Eddington, Sir Frank Dyson, and such international visitors
as Henry Norris Russell and Harlow Shapley).
Chandra spent four weeks in the Soviet Union in the
summer of 1934 at the invitation of B. P. Gerasimovic, meeting
L. D. Landau and V. A. Ambartsumian, along with many
other enthusiastic young men. Unhappily only Landau and
Ambartsumian survived the massive purges that were soon
to follow. Ambartsumian grasped the significance of Chandra’s
work on dwarf stars and suggested that it was worth working out exactly (i.e., by direct radial integration of the ex-
36 BIOGRAPHICAL MEMOIRS
act equations, using the complete pressure-density relation).
This moved Chandra to tackle that immense problem upon
his return to Cambridge.
The work was accomplished with the aid of a hand calculator and was completed by the end of 1934. He submitted
his results for presentation at the January 1935 meeting of
the RAS. Eddington had taken an interest in the work through
the autumn, often dropping by Chandra’s room to see how
things were progressing, but never saying a word to Chandra
about his own private thoughts. Eddington suggested to the
secretary of the RAS that Chandra’s work merited double
the usual fifteen minutes for presentation and then set himself
up to present a paper with the title “Relativistic degeneracy”
immediately following. Eddington refused to divulge the
nature of his presentation beforehand. McCrea notes in his
obituary for Chandra that Eddington began by pointing
out that Chandra’s calculations were entirely correct based
on the relativistic degenerate electron gas. Eddington then
noted that the result predicted that a white dwarf with mass
in excess of the critical value (≈ 1.4 M.) would continue to
radiate and shrink until it disappeared. Then Eddington
went on to declare that stars do not behave in that way, and
Chandra’s calculations showed only that the theory of relativistic degeneracy is incorrect. Later he asserted that the
Pauli exclusion principle does not apply to relativistic electrons. One might have asked Eddington how he knew that
stars do not behave in that way, but Eddington was so formidable and influential a person that no one did, apparently. Egos were the same then as now, and one has only to
read Eddington’s remarkable monograph Fundamental Theory
(Cambridge University Press, 1944) to realize that he was
coming around to the idea that he could deduce the physical nature of the universe from his own personal declarations.
SUBRAHMANYAN CHANDRASEKHAR 37
The physicists, Chandra’s young contemporaries (e.g.,
Pauli, Rosenfeld, Dirac, and others), considered Eddington’s
assertions to be nonsense, but Eddington moved in a different world. R. H. Fowler and H. N. Russell did not voice the
essential points in opposition to Eddington’s assertions, evidently intimidated by Eddington’s preeminence. Russell, for
instance, refused to allow Chandra to say a few words in
response to Eddington’s hour long exposition of his personal views at the meeting of the International Astronomical Union (IAU) in Paris in July 1935. Chandra managed a
brief comment at the “International Colloquium on Astrophysics: Novae and White Dwarfs” in Paris in July 1939, but
Russell quickly closed the session before a discussion could
proceed.
The question of returning to India was raised by C. S.
Ayyar, but Chandra found himself increasingly out of sympathy with the political nature of academia in India. Then
Harlow Shapley invited Chandra to visit the Harvard Observatory. Chandra arrived in Boston on December 8, 1935.
He enjoyed the friendly atmosphere but was unhappy with
the informality after the tightly structured society at Cambridge. He became acquainted with Fred Whipple, Gerard
Kuiper, Jerry Mulders, and others. Shapley liked Chandra’s
lectures so well that he nominated Chandra for election to
the Harvard Society of Fellows. Then Otto Struve invited
Chandra to visit the Yerkes Observatory of the University of
Chicago, followed by an offer of a position as research associate for a year with the expectation that it would become a
tenure track appointment in a year. The formal offer came
from the office of Chancellor Robert Maynard Hutchins.
By the end of the month Chandra had returned to England.
The Eddington factor had the effect of closing the doors
in England, and India offered no acceptable situation. So
38 BIOGRAPHICAL MEMOIRS
Chandra accepted Struve’s offer, much to the disgust of his
father who saw his son receding farther into the mists of
foreign culture.
Since his departure from India in July 1930 Chandra had
corresponded occasionally with Lalitha Doraiswamy who had
been a fellow student in physics at Presidency College. She
was in Bangalore in 1935 working in Raman’s laboratory.
They were both aware that they did not know each other
very well, and Chandra had fretted over whether a marriage relationship might interfere with his pursuit of science. Chandra returned to India for a visit in August 1936
and wrote to Lalitha that he would be at Madras. She took
the train to Madras to meet him and his misgivings vanished when they met after six years of geographical separation. They were married September 11, 1936.
Chandra and Lalitha spent a month in Cambridge on
their way to Boston and then the Yerkes Observatory. Struve
contacted the legal counsel of the University of Chicago to
arrange a visa for Chandra as a missionary, for otherwise
there was no quota for Indians to enter the United States.
They arrived at the Yerkes Observatory on Williams Bay on
Lake Geneva in Wisconsin on December 21, 1936. They
stayed a few days with the Kuipers until their house was
ready, and the cold Wisconsin weather was offset by the
friendliness of the atmosphere at the observatory.
Lalitha recognized the importance of Chandra’s singleminded pursuit of science, and she supported him at the
expense of her own career. She was active in the American
Association of University Women and her outgoing sociability complemented Chandra’s more austere view of life so
that they got on very well in their new surroundings.
The University of Chicago provided Chandra with his scientific home for the next fifty-nine years, but there were
difficult moments. Chancellor Hutchins intervened on more
SUBRAHMANYAN CHANDRASEKHAR 39
than one occasion to smooth the way. For instance, in 1938
Struve organized a course in astronomy on the campus of
the university to be taught by members of the Yerkes Observatory. However Henry G. Gale, dean of physical sciences,
vetoed Chandra’s participation, evidently on grounds of skin
color. When the problem was referred to Hutchins he said,
“By all means have Mr. Chandrasekhar teach.” At that point
it became clear why the original offer of a position had
come from the chancellor’s office rather than through the
dean.
In 1946 Princeton honored Chandra by offering him the
office and position vacated by the retirement of Henry Norris
Russell with a salary approximately double Chandra’s salary
at Chicago. Chandra was inclined to accept. Hutchins
matched the Princeton salary and asked Chandra to come
by his office to discuss the matter. In the course of the
discussion Hutchins remarked that, if conditions for
Chandra’s research were better at Princeton, then he would
not attempt to dissuade Chandra from leaving. When Chandra
responded that he did not think so, Hutchins noted that
Chicago could not offer Chandra the honor of succeeding
a Henry Norris Russell because Chicago had no Russell.
Then he asked Chandra for the name of the person who
had succeeded to Kelvin’s chair at the University of Glasgow.
Chandra replied that he had no idea; to which Hutchins
replied, “Well, there you are.” Chandra declined the Princeton
offer and Hutchins remarked on more than one occasion
that acquiring Chandra for the University of Chicago was
one of his major accomplishments as chancellor.
The course of Chandra’s research is perhaps best summarized by the monographs that he wrote as he completed
each phase of his work. An Introduction to the Study of Stellar
Structure (1939) contains his development of the theory of
stellar structure, including his work on degenerate stars
40 BIOGRAPHICAL MEMOIRS
and the mass limit for white dwarfs, and still makes an
excellent textbook on the subject. The Principles of Stellar
Dynamics (1943) and “Stochastic problems in physics and
astronomy” (1943) outline his development of the theory
of the dynamics of the motions of stars in the presence of
many other stars, showing the frictional drag exerted by
neighboring stars and setting up the basic theory for the
evolution of clusters of stars. Radiative Transfer (1950) contains his systematic development of the radiative flow of
energy in stellar interiors and photospheres including his
work on the negative hydrogen ion that dominates the opacity
at the surface of a star.
In 1952 the Department of Astronomy revamped its graduate curriculum to keep up with the rapid development in
the fields of atomic physics, stellar atmospheres, and stellar
evolution. Chandra had been offering a repertoire of basic
courses in stellar structure and radiative transfer. These
courses, based in large part on his own fundamental work,
provided excellent background for the theoretical students,
but were heavy going for the observational students and
lacked up-to-date information needed by both groups of
students. Chandra was alienated by the revision and Enrico
Fermi seized the opportunity to invite Chandra to become
a member of the Department of Physics and the Institute
for Nuclear Studies (now the Enrico Fermi Institute). Chandra
accepted the invitation and henceforth confined his teaching principally to the Department of Physics, commuting
from Yerkes to Chicago two days a week to teach. In 1964
Chandra moved permanently to the Chicago campus, the
transition catalyzed by John Simpson’s offer of a spacious
corner office in the newly constructed Laboratory for Astrophysics and Space Research.
It is ironic that 1952 was also the year Chandra took up
the onerous task of managing editor of the Astrophysical
SUBRAHMANYAN CHANDRASEKHAR 41
Journal. He carried on the responsibilities in his own style,
personally attending to the problems of production, refereeing, and politics within the community. The editing was
managed with the help of a secretary and an editorial assistant at the University of Chicago Press. Under Chandra’s
leadership the journal developed into the leading international journal in astrophysics. The journal was in reality
privately owned by the University of Chicago. Chandra was
its heart and soul, and Chandra realized the unstable character of the situation. In 1967 he set in motion a reorganization that would transfer the primary responsibility to the
American Astronomical Society (AAS), although the actual
production was to continue at the University of Chicago
Press. The rapid expansion of the journal from six issues a
year to two large issues a month made it increasingly difficult for a single editor to handle, particularly with Chandra’s
establishment of the Astrophysical Journal Letters in 1967. So
Chandra proposed that there be associate editors to assist
the managing editor. To make a long story short, the new
order of things was approved by the American Astronomical Society, and Chandra was able to pass on his enormous
burden to the new team in 1971. It is remarkable that during his years as editor Chandra carried on his scientific
research at a rate not noticeably diminished at the same
time that he taught his quota of courses in the Department
of Physics. It is an example of the extraordinary feats that
can be accomplished through dedication and self-discipline
to the exclusion of nearly everything else in one’s life. His
retirement from the position as editor was a great relief to
Chandra. He had never intended that the burden should
have continued for so long.
Chandra and Lalitha were faced with the question of U.S.
citizenship, and after thinking about it for a time came to
the conclusion that it was the only realistic choice. It was a
42 BIOGRAPHICAL MEMOIRS
big step away from their origins, but to do otherwise would
have ignored the fact of their permanent commitment to a
life in the United States. So in 1953 they became naturalized citizens. Lalitha’s careful explanation of the evolution
of their thinking did little to assuage the bitter feelings of
C. S. Ayyar who saw the move only as a betrayal of their
cultural origins rather than an inevitable evolution in their
circumstances. Following citizenship Chandra was elected
to the National Academy of Sciences in 1955.
During Chandra’s early years as editor, the field of plasma
physics and the confinement of ionized gas in magnetic
fields in the laboratory was coming into prominence, with
the hope, still unrealized today, of producing available power
through the fusion of hydrogen into helium. At the same
time it was being appreciated that the physics of fully ionized gases (i.e., plasmas) is the basis for the dynamical behavior of stellar interiors, atmospheres, and the interstellar
gas. Plasma conditions range all the way from the tenuous,
essentially collisionless gases in space to the incredibly dense
plasma in the central regions of a star. Chandra was attracted by the challenge of the unknown. He expounded
the existing theory of collisionless plasma in a course on
the foundations of plasma physics based on the standard
free-particle approach and the collisionless Boltzmann equation. S. K. Trehan put together a book Plasma Physics (University of Chicago Press, 1960) based on the notes from that
course. In collaboration with A. N. Kaufman and K. M.
Watson Chandra carried through the immense calculation
of the dynamical stability of the collisionless plasma confined in an axial magnetic field. At the same time Chandra
entered into an extensive study of the dynamical stability of
fluids in various configurations, including the presence of
magnetic fields and rotation of the entire system. His con-
SUBRAHMANYAN CHANDRASEKHAR 43
tributions are summarized in his monograph Hydrodynamic
and Hydromagnetic Stability (1961).
From there Chandra took up the classical and unfinished
problem of the dynamics of rotating, self-gravitating spheroids of homogeneous incompressible fluids. The problem
had been initiated by Newton in connection with the oblateness of Earth and carried on from there by such great
names as Maclaurin, Reimann, Dedekind, Jacobi, Dirichlet,
et al. Chandra reopened the unfinished problems with the
tensor virial equations whose great power had not been
appreciated up to that time. The results of that work appear in his monograph Ellipsoidal Figures of Equilibrium (1969).
The work on selfgravitating objects soon brought Chandra
to the doorstep of general relativity as the basic theory of
gravity. His efforts in that field led to development of the
Chandrasekhar-Friedman-Schultz instability, which became
a source of gravitational radiation from black holes. Extensive investigation of the Kerr metric and the rotating black
hole led to the monograph The Mathematical Theory of Black
Holes (1983). Chandra also developed the post-Newtonian
approximation for treating the field equations of general
relativity. It is now the means for calculating the gravitational radiation from multiple star systems, etc. He went on
to work out a variety of exact solutions to the equations of
general relativity in collaboration with B. C. Xanthopoulos
and V. Ferrari, showing some of the remarkable singularities
that turn up in the interaction of gravitational waves and at
the apex of the conical space solutions. One of the more
curious discoveries was that the radial pulsations of a star,
which are known from Newtonian gravitation to exhibit
overstability in the presence of dissipation (e.g., viscosity)
become unstable in general relativity through the energy
loss represented by the emission of gravitational waves. Thus
the star without internal dissipation is stable according to
44 BIOGRAPHICAL MEMOIRS
Newtonian theory, but unstable in the context of general
relativity.
As a brief aside it is interesting to note that in 1982 Chandra
was invited to lecture on Sir Arthur Eddington at the celebration at Cambridge of the hundredth anniversary of his
birth. The lectures are published in the small book Eddington,
the Most Distinguished Astrophysicist of His Time (1983). The
lectures emphasize the remarkable insights of Eddington
into stellar structure and his early recognition of the implications of Einstein’s general relativity. Chandra’s reflections on Eddington’s assertions on electron degeneracy and
the Pauli exclusion principle are of particular interest.
By 1990 Chandra had developed a growing interest and
admiration for the work of Sir Isaac Newton, and over the
next several years he constructed a detailed and critical
review of Newton’s Principia. The results of this effort are
published as Newton’s Principia for the Common Reader (1995).
This was the first time that a world class physicist undertook a thorough reading and critical commentary of the
Principia, dispelling such perpetuated notions that Newton’s
theory of the perturbations of the orbit of the Moon is in
error, or that some of his diagrams were incorrectly drawn.
Chandra’s book Truth and Beauty (1987) shows an entirely different side of his thinking. It includes his Ryerson
Lecture “Shakespeare, Newton, and Beethoven” in which
he explored and compared the motivations and feelings
involved in the creation of science and art.
Chandra’s scientific papers are collected in seven volumes
under the title Selected Papers, S. Chandrasekhar (1989-96).
They complement the monographs listed above and provide a more detailed historical picture of the day-by-day
development of his thinking.
Chandra attached great importance to training Ph.D. students. He saw them clearly as the future of astrophysics
SUBRAHMANYAN CHANDRASEKHAR 45
when the present generation of working scientists has passed
into retirement and beyond. Struve had assigned him the
responsibility for the weekly colloquium, held on Monday
afternoons, and Chandra saw to it that the graduate students were in regular attendance. The Yerkes faculty, graduate
students, and visitors presented their work at appropriate
times, and Chandra gave each hundredth colloquium himself, as well as many in between. The count of weekly colloquia
passed 500 before Chandra moved to the campus. He also
conducted seminars on Monday evenings for the edification of the graduate students, who took turns reporting on
interesting papers that had appeared in the literature.
Chandra supervised forty-six known Ph.D. research students,
many of whom have become prominent in the field of astrophysics, and not a few of whom are members of the
National Academy of Sciences. Chandra was a stern taskmaster who insisted on rigorous training and research. The
graduate courses in theoretical astrophysics taught at Yerkes
by Chandra were the usual preparation, until the early fifties. After that most of Chandra’s students came through
the Department of Physics. Once a student successfully completed the Ph.D., Chandra gave his full support in getting
the student established in the scientific community. In fact
Chandra’s support was not limited to his students alone.
He appeared at critical moments in the career of this writer,
as with others as well.
It is no surprise, of course, to learn that Chandra was
awarded many honorary degrees and medals. He was elected
a fellow of the Royal Society in 1944, which awarded him
the Bruce Medal in 1952. The Royal Astronomical Society
awarded him its Gold Medal in 1953. He was awarded the
National Medal of Science by President Lyndon Johnson in
1967. The fundamental nature of Chandra’s mass limit for
degenerate stars has come to be appreciated in the astronomy
46 BIOGRAPHICAL MEMOIRS
and physics communities, recognizing that it is perhaps the
most direct and striking example of the effect of quantum
physics on macroscopic bodies. Chandra was awarded a Nobel
Prize by King Carl Gustav in 1983 in recognition of his
work of fifty years before. On the other hand it must be
appreciated that Chandra’s work on radiative transfer, stellar dynamics, dynamical stability of fluids, plasmas and
selfgravitating bodies, and gravitational theory collectively
represent a much larger contribution to physics and astrophysics than the more spectacular mass limit.
Chandra’s death in 1995 heralded the end of the era that
developed the basic physics of the star. He was the most
prolific and wide ranging of those who applied hard physics to astronomical problems.
I EXPRESS MY APPRECIATION to D. E. Osterbrock for his careful reading
of the manuscript and several important suggestio
SUBRAHMANYAN CHANDRASEKHAR
The University of Chicago, Chicago, Illinois 60637, USA
1. Introduction
When we think of atoms, we have a clear picture in our minds: a central nucleus
and a swarm of electrons surrounding it. We conceive them as small objects of
sizes measured in Angstroms (~l0-8 cm); and we know that some hundred
different species of them exist. This picture is, of course, quantified and made
precise in modern quantum theory. And the success of the entire theory may be
traced to two basic facts: first, the Bohr radius of the ground state of the hydrogen
atom, namely,
(1)
where h is Planck’s constant, m is the mass of the electron and e is its charge,
provides a correct measure of atomic dimensions; and second, the reciprocal of
Sommerfeld’s fine-structure constant,
(2)
gives the maximum positive charge of the central nucleus that will allow a stable
electron-orbit around it. This maximum charge for the central nucleus arises
from the effects of special relativity on the motions of the orbiting electrons.
We now ask: can we understand the basic facts concerning stars as simply as
we understand atoms in terms of the two combinations of natural constants (1)
and (2). In this lecture, I shall attempt to show that in a limited sense we can.
The most important fact concerning a star is its mass. It is measured in units
of the mass of the sun, which is 2 x 1033 gm: stars with masses very much less
than, or very much more than, the mass of the sun are relatively infrequent. The
current theories of stellar structure and stellar evolution derive their successes
largely from the fact that the following combination of the dimensions of a mass
provides a correct measure of stellar masses:
where G is the constant of gravitation and H is the mass of the hydrogen atom.
In the first half of the lecture, I shall essentially be concerned with the question:
how does this come about?
S. Chandrasekhar 143
2. The role of radiation pressure
A central fact concerning normal stars is the role which radiation pressure plays
as a factor in their hydrostatic equilibrium. Precisely the equation governing
the hydrostatic equilibrium of a star is
where P denotes the total pressure, p the density, and M (r) is the mass interior
to a sphere of radius r. There are two contributions to the total pressure P:
that due to the material and that due to the radiation. On the assumption that
the matter is in the state of a perfect gas in the classical Maxwellian sense,
the material or the gas pressure is given by
where T is the absolute temperature, k is the Boltzmann constant, and µ is
the mean molecular weight (which under normal stellar conditions is 1.0).
The pressure due to radiation is given by
where α denotes Stefan’s radiation-constant. Consequently, if radiation
contributes a fraction (1−β ) to the total pressure, we may write
1 1
l - j 3 3 =
To bring out explicitly the role of the radiation pressure in the equilibrium
of a star, we may eliminate the temperature, T, from the foregoing equations
and express P in terms of p and β instead of in terms of p and T. We find:
and
(9)
The importance of this ratio, (1−β), for the theory of stellar structure was
first emphasized by Eddington. Indeed, he related it, in a famous passage in
his book on The Internal Constitution of the Stars, to the ‘happening of the stars’.1
A more rational version of Eddington’s argument which, at the same time,
isolates the combination (3) of the natural constants is the following:
There is a general theorem2 which states that the pressure, at the centre
of a star of a mass M in hydrostatic equilibrium in which the density, p (r), at
a point at a radial distance, r, from the centre does not exceed the mean density,
(r), interior to the same point r, must satisfy the inequality,
144 Physics 1983
Fig. 1. A comparison of an inhomogeneous distribution of density in a star (b) with the two
homogeneous configurations with the constant density equal to the mean density (a) and equal to
the density at the centre (c).
where denotes the mean density of the star and its density at the centre.
The content of the theorem is no more than the assertion that the actual pressure
at the centre of a star must be intermediate between those at the centres of the
two configurations ofuniform density, one at a density equal to the mean density
of the star, and the other at a density equal to the density pC at the centre (see
Fig. 1). If the inequality (10) should be violated then there must, in general,
be some regions in which adverse density gradients must prevail; and this implies
instability. In other words, we may consider conformity with the inequality (10)
as equivalent to the condition for the stable existence of stars.
The right-hand side of the inequality (10) together with P given by equation
(9), yields, for the stable existence of stars, the condition,
or, equivalently,
(12)
where in the foregoing inequalities, is a value of β at the centre of the star.
Now Stefan’s constant, a, by virtue of Planck’s law, has the value
(13)
Inserting this value a in the inequality (12) we obtain
(14)
We observe that the inequality (14) has isolated the combination (3) of
S. Chandrasekhar 145
natural constants of the dimensions of a mass; by inserting its numerical value
given in equation (3) we obtain the inequality,
≥ (15)
This inequality provides an upper limit to (1 for a star of a given mass. Thus,
(16)
where (1 is uniquely determined by the mass M of the star and the mean
molecular weight, µ, by the quartic equation,
= 5.48 (17)
In Table 1, we list the values of 1 for several values of µ2 M. From this
table it follows in particular, that for a star of solar mass with a mean molecular
weight equal to 1, the radiation pressure at the centre cannot exceed 3 percent
of the total pressure.
Table 1
The maximum radiation pressure, (1
at the centre of a star of a given mass, M.
1 1
0.01 0.56 0.50 15.49
.03 1.01 .60 26.52
.10 2.14 .70 50.92
.20 3.83 .80 122.5
.30 6.12 .85 224.4
0.40 9.62 0.90 519.6
What do we conclude from the foregoing calculation? We conclude that to the
extent equation (17) is at the base of the equilibrium of actual stars, to that
extent the combination of natural constants (3), providing a mass of proper
magnitude for the measurement of stellar masses, is at the base of a physical
theory of stellar structure.
3. Do stars have enough energy to cool?
The same combination of natural constants (3) emerged soon afterward in a
much more fundamental context of resolving a paradox Eddington had
formulated in the form of an aphorism: ‘a star will need energy to cool.’ The
paradox arose while considering the ultimate fate of a gaseous star in the light
of the then new knowledge that white-dwarf stars, such as the companion of
Sirius, exist, which have mean densities in the range l05
-l07
gm cm-3. AS
Eddington stated3
I do not see how a star which has once got into
this compressed state is ever going to get out of it...
It would seem that the star will be in an awkward predicament when its supply of subatomic energy fails.
The paradox posed by Eddington was reformulated in clearer physical terms
by R. H. Fowler.4 His formulation was the following:
The stellar material, in the white-dwarf state,
will have radiated so much energy that it has less energy than the same matter in normal atoms expanded at
the absolute zero of temperature. If part of it were
removed from the star and the pressure taken off, what
could it do?
Quantitatively, Fowler’s question arises in this way.
An estimate of the electrostatic energy, per unit volume of an assembly
of atoms, of atomic number Z, ionized down to bare nuclei, is given by
E v = l . 3 2 x 1 01 1Z 2
p 4 / 3 , (18)
while the kinetic energy ofthermal motions, per unit volume of free particles
in the form of a perfect gas of density, p, and temperature, T, is given by
T (19)
Now if such matter were released of the pressure to which it is subject, it can
resume a state of ordinary normal atoms only if
(20)
or, according to equations ( 18) and (19), only if
p <
This inequality will be clearly violated if the density is sufficiently high. This
is the essence of Eddington’s paradox as formulated by Fowler. And Fowler
resolved this paradox in 1926 in a paper’ entitled ‘Dense Matter’ - one of the
great landmark papers in the realm ofstellar structure: in it the notions of Fermi
statistics and of electron degeneracy are introduced for the first time.
4. Fowler’s resolution of Eddington’s paradox; the degeneracy of the electrons in whitedwarf stars
In a completely degenerate electron gas all available parts of the phase space,
with momenta less than a certain ‘threshold’ - the Fermi ‘threshold’ -
are occupied consistently with the Pauli exclusion-principle i.e., with two
electrons per ‘cell’ of volume h3 of the six-dimensional phase space. Therefore,
S. Chandrasekhar 147
denotes the number of electrons, per unit volume,
then the assumption-of complete degeneracy is equivalent to the assertion,
(22)
The value of the threshold momentum is determined by the normalization
condition
(23)
where n denotes the total number of electrons per unit volume.
For the distribution given by (22), the pressure P and the kinetic energy
of the electrons (per unit volume), are given by
and
(24)
where and are the velocity and the kinetic energy of an electron having a
momentum
If we set
(26)
appropriate for non-relativistic mechanics, in equations (24) and (25), we find
and
(27)
(28)
Fowler’s resolution of Eddington’s paradox consists in this: at the temperatures
and densities that may be expected to prevail in the interiors of the white-dwarf
stars, the electrons will be highly degenerate and must be evaluated in
accordance with equation (28) and not in accordance with equation (19); and
equation (28) gives,
(29)
Comparing now the two estimates (18) and (29), we see that, for matter of the
density occurring in the white dwarfs, namely 105
gm cm-3, the total kinetic
energy is about two to four times the negative potential-energy; and Eddington’s
148 Physics 1983
paradox does not arise. Fowler concluded his paper with the following highly
perceptive statement:
The black-dwarf material is best likened to a single gigantic molecule in
its lowest quantum state. On the Fermi-Dirac statistics, its high density
can be achieved in one and only one way, in virtue of a correspondingly
great energy content. But this energy can no more be expended in radiation
than the energy of a normal atom or molecule. The only difference between
black-dwarf matter and a normal molecule is that the molecule can exist in
a free state while the black-dwarf matter can only so exist under very high
external pressure.
5. The theory of the white-dwarf stars; the limiting mass
The internal energy (= 3 P/2) of a degenerate electron gas that is associated with
a pressure P is zero-point energy; and the essential content of Fowler’s paper is that
this zero-point energy is so great that we may expect a star to eventually settle
down to a state in which all of its energy is of this kind. Fowler’s argument can
be more explicitly formulated in the following manner.5
According to the expression for the pressure given by equation (27), we
have the relation,
(30)
where is the mean molecular weight per electron. An equilibrium configuration in which the pressure, P, and the density are related in the manner,
(31)
is an Emden polytrope of index n. The degenerate configurations built on the
equation of state (30) are therefore polytropes of index 3/2; and the theory of
polytropes immediately provides the relation,
(32)
or, numerically, for K1 given by equation (30),
For a mass equal to the solar mass and = 2, the relation (33) predicts R
= 1.26 x and a mean density of 7.0 x l05 g m / c m3
. These values are
precisely of the order of the radii and mean densities encountered in whitedwarf stars. Moreover, according to equations (32) and (33), the radius of the
white-dwarf configuration is inversely proportional to the cube root of the mass.
On this account, finite equilibrium configurations are predicted for all masses.
And it came to be accepted that the white-dwarfs represent the last stages in
the evolution of all stars.
S. Chandrasekhar 149
But it soon became clear that the foregoing simple theory based on Fowler’s
premises required modifications. For the electrons, at their threshold energies
at the centres of the degenerate stars, begin to have velocities comparable to that
of light as the mass increases. Thus, already for a degenerate star of solar mass
( w i t h pe = 2) the central density (which is about six times the mean density) is
4.19 x 106 g m / c m3
; and this density corresponds to a threshold momentum
= 1.29 mc and a velocity which is 0.63 c. Consequently, the equation of state
must be modified to take into account the effects of special relativity. And this
is easily done by inserting in equations (24) and (25) the relations,
(34)
in place of the non-relativistic relations (26). We find that the resulting equation
of state can be expressed, parametrically, in the form
(35)
(36)
and
(37)
And similarly
where
(38)
(39)
According to equations (35) and (36), the pressure approximates the relation
(30) for low enough electron concentrations but for increasing electron
concentrations the pressure tends to6
(40)
This limiting form of relation can be obtained very simply by setting = c
in equation (24); then
(41)
and the elimination with the aid of equation (23) directly leads to equation
(40).
While the modification of the equation of state required by the special
150 Physics 1983
theory of relativity appears harmless enough, it has, as we shall presently show,
a dramatic effect on the predicted mass-radius relation for degenerate
configurations.
The relation between P and corresponding to the limiting form (41) is
In this limit, the configuration is an Emden polytrope of index 3. And it is well
known that when the polytropic index is 3, the mass of the resulting equilibrium
configuration is uniquely determined by the constant of proportionality, in
the pressure-density relation. We have accordingly,
(43)
(In equation (43), 2.018 is a numerical constant derived from the explicit
solution of the Lane-Emden equation for n = 3.)
It is clear from general considerations’ that the exact mass-radius relation for the
degenerate configurations must provide an upper limit to the mass of such configurations given
by equation (43); and further, that the mean density of the configuration must tend to
infinity, while the radius tends to zero, and These conditions,
straightforward as they are, can be established directly by considering the
equilibrium of configurations built on the exact equation of state given by
equations (35) - (37). It is found that the equation governing the equilibrium of
such configurations can be reduced to the form8,9
(44)
where
(45)
and denotes the threshold momentum of the electrons at the centre of the
configuration and measures the radial distance in the unit
(46)
By integrating equation (44), with suitable boundary conditions and for
various initially prescribed values we can derive the exact mass-radius
relation, as well as the other equilibrium properties, of the degenerate
configurations. The principal results of such calculations are illustrated in
Figures 2 and 3.
The important conclusions which follow from the foregoing considerations
are: first, there is an upper limit, to the mass of stars which can become
degenerate configurations, as the last stage in their evolution; and second, that
s t a r s w i t h M > must have end states which cannot be predicted from
the considerations we have presented so far. And finally, we observe that the
i -
) -
i -
) -
I-
)-
Fi g. 2. T he f ull-li ne c ur ve re prese nts t he e xact ( mass-ra di us)-relati o n (l , is defi ne d i n e q uati o n ( 4 6)
a n d 3 de n otes t he li miti n g mass). T his c ur ve te n ds as y m pt oticall y t o t he ---- c ur ve a p pr o priate
t o t he l o w- mass de ge nerate c o nfi g urati o ns, a p pr o xi mate d b y p ol ytr o pes of i n de x 3/ 2. T he re gi o ns of
t he c o nfi g urati o ns w hic h ma y be c o nsi dere d as relati vistic are s h o w n s ha de d. ( Fr o m
C ha n drase k har, S., M o n. N ot. R oy. Astr. S oc., 9 5, 2 0 7 ( 1 9 3 5).)
c o m bi n ati o n of t h e n at ur al c o n st a nt ( 3) n o w e m er g e s i n t h e f u n d a m e nt al
c o nt e xt of M li mit gi v e n b y e q u ati o n ( 4 3): its si g nific a nc e f or t h e t h e or y of st ell ar
str uct ur e a n d st ell ar e v ol uti o n c a n n o l o n g er b e d o u bt e d.
6. U n der w h at c o n diti o ns c a n n or m al st ars de vel o p de ge ner ate c ores?
O nc e t h e u p p er li mit t o t h e m ass of c o m pl et el y d e g e n er at e c o nfi g ur ati o ns h a d
b e e n est a blis h e d, t h e q u esti o n t h at r e q uir e d t o b e r es ol v e d w as h o w t o r el at e its
e xist e nc e t o t h e e v ol uti o n of st ars fr o m t h eir g as e o us st at e. If a st ar h as a m ass l ess
t h a n Mli mit t h e ass u m pti o n t h at it will e v e nt u all y e v ol v et o w ar ds t h e c o m pl et el y
d e g e n er at e st at e a p p e ars r e as o n a bl e. B ut w h at if its m ass is gr e at er t h a n li mit
Physics 1983
Fig. 3. The full-line curve represents the exact (mass-density)-relation for the highly collapsed
configurations. This curve tends asymptotically to the dotted curve as (From Chandrasekhar, S., Mon. Not. Roy. Astr. Soc., 95, 207 (1935).)
Clues as to what might ensue were sought in terms of the equations and
inequalities of and 3.10,11
The first question that had to be resolved concerns the circumstances under
which a star, initially gaseous, will develop degenerate cores. From the physical
side, the question, when departures from the perfect-gas equation of state (5) will
set in and the effects of electron degeneracy will be manifested, can be readily
answered.
Suppose, for example, that we continually and steadily increase the density,
at constant temperature, of an assembly of free electrons and atomic nuclei,
in a highly ionized state and initially in the form of a perfect gas governed by the
equation of state (5). At first the electron pressure will increase linearly with p;
but soon departures will set in and eventually the density will increase in
accordance with the equation of state that describes the fully degenerate
electron-gas (see Fig. 4). The remarkable fact is that this limiting form of the
equation of state is independent of temperature.
However, to examine the circumstances when, during the course of evolution,
a star will develop degenerate cores, it is more convenient to express the
electron pressure (as given by the classical perfect-gas equation of state) in
terms of p and in the manner (cf. equation (7)).
where now denotes the electron pressure. Then, analogous to equation (9),
we can write
(48)
S. Chandrasekhar
Fig. 4. Illustrating how by increasing the density at constant temperature degeneracy always sets
in.
Comparing this with equation (42), we conclude that if
(49)
the given by the classical perfect-gas equation of state will be greater
than that given by the equation if degeneracy were to prevail, not only for the
prescribed and T, but for all and T having the same
Inserting for a its value given in equation (13), we find that the inequality
(49) reduces to
or equivalently,
(See Fig. 5)
(50)
(51)
154 Physics 1983
Fig. 5. Illustrating the onset of degeneracy for increasing density at constant β. Notice that there
are no intersections for β> 0.09212. In the figure, 1−β is converted into mass of a star built on the
standard model.
For our present purposes, the principal content of the inequality (51) is the
criterion that for a star to develop degeneracy, it is necessary that the radiation
pressure be less than 9.2 percent of This last inference is so central
to all current schemes of stellar evolution that the directness and the simplicity
of the early arguments are worth repeating.
The two principal elements of the early arguments were these: first, that
radiation pressure becomes increasingly dominant as the mass of the star
increases; and second , that the degeneracy of electrons is possible only so long as
the radiation pressure is not a significant fraction of the total pressure -
indeed, as we have seen, it must not exceed 9.2 percent of The
second of these elements in the arguments is a direct and an elementary consequence of the physics of degeneracy; but the first requires some amplification.
That radiation pressure must play an increasingly dominant role as the mass
of the star increases is one of the earliest results in the study of stellar structure
that was established by Eddington. A quantitative expression for this fact is
S . C h a n d r a s e k h a r 155
given by Eddington’s standard model which lay at the base of early studies
summarized in his The Internal Constitution of the Stars.
On the standard model, the fraction (= gas pressure/total pressure) is a
constant through a star. On this assumption, the star is a polytrope of index
3 as is apparent from equation (9); and, in consequence, we have the relation (cf.
equation (43))
(52)
where C is defined in equation (9). Equation (52) provides a quartic equation
for analogous to equation (17) for Equation (52) for = gives
(53)
On the standard model, then stars with masses exceeding M will have radiation
pressures which exceed 9.2 percent of the total pressure. Consequently stars with
M > M cannot, at any stage during the course of their evolution, develop
degeneracy in their interiors. Therefore, for such stars an eventual white-dwarf
state is not possible unless they are able to eject a substantial fraction of their
mass.
The standard model is, of course, only a model. Nevertheless, except under
special circumstances, briefly noted below, experience has confirmed the
standard model, namely that the evolution of stars of masses exceeding 7-8
must proceed along lines very different from those of less massive stars. These
conclusions, which were arrived at some fifty years ago, appeared then so
convincing that assertions such as these were made with confidence:
Given an enclosure containing electrons and atomic nuclei (total charge
zero) what happens if we go on compressing the material indefinitely?
( 1 9 3 2 ) 1 0
The life history of a star of small mass must be essentially different from
the life history of a star of large mass. For a star of small mass the natural
white-dwarf stage is an initial step towards complete extinction. A star of
large mass cannot pass into the white-dwarfstage and one is left speculating
on other possibilities. (1934) 8
And these statements have retained their validity.
While the evolution of the massive stars was thus left uncertain, there was no
such uncertainty regarding the final states of stars of sufficiently low mass.”
The reason is that by virtue, again, of the inequality (10), the maximum central
pressure attainable in a star must be less than that provided by the degenerate
equation of state, so long as
or, equivalently
(54)
(55)
156 Physics 1983
We conclude that there can be no surprises in the evolution of stars of mass
less than 0.43 = 2). The end stage in the evolution of such stars can
only be that of the white dwarfs. (Parenthetically, we may note here that the
inequality (55) implies that the so-called ‘mini’ black-holes of mass 1015
g m
cannot naturally be formed in the present astronomical universe.)
7. Some brief remarks on recent progress in the evolution of massive stars and the onset
of gravitational collapse
It became clear, already from the early considerations, that the inability of the
m a s s i v e s t a r s t o b e c a m e w h i t e d w a r f s m u s t r e s u l t i n t h e d e v e l o p m e n t o f
much more extreme conditions in their interiors and, eventually, in the onset of
gravitational collapse attended by the super-nova phenomenon. But the precise
manner in which all this will happen has been difficult to ascertain in spite of
great effort by several competent groups of investigators. The facts which must
be taken into account appear to be the following.*
In the first instance, the density and the temperature will steadily increase
without the inhibiting effect of degeneracy since for the massive stars considered
1 1 On this account, ‘nuclear ignition’ of carbon, say, will take
place which will be attended by the emission of neutrinos. This emission of
neutrinos will effect a cooling and a lowering of (1 but it will still be in
excess of The important point here is that the emission of neutrinos acts
selectively in the central regions and is the cause of the lowering of (1
in these regions. The density and the temperature will continue to increase till
the next ignition of neon takes place followed by further emission of neutrinos
and a further lowering of (1 T his succession of nuclear ignitions and
lowering of (1 w i l l c o n t i n u e t i l l 1 a n d a r e l a t i v i s t i c a l l y
degenerate core with a mass approximately that of the limiting mass
for = 2) forms at the centre. By this stage, or soon afterwards, instability of
some sort is expected to set in (see following $8) followed by gravitational
collapse and the phenomenon of the super-nova (of type II). In some instances,
what was originally the highly relativistic degenerate core of approximately 1.4
will be left behind as a neutron star. That this happens sometimes is confirmed by the fact that in those cases for which reliable estimates of the masses of
pulsars exist, they are consistently close to 1.4 However, in other instances -
perhaps, in the majority of the instances - what is left behind, after all ‘the
dust has settled’, will have masses in excess of that allowed for stable neutron
stars; and in these instances black holes will form.
In the case of less massive stars 6-8 the degenerate cores, which are
initially formed, are not highly relativistic. But the mass of core increases with
the further burning of the nuclear fuel at the interface of the core and the mantle;
and when the core reaches the limiting mass, an explosion occurs following
instability; and it is believed that this is the cause underlying super-nova
phenomenon of type I.
* I am grateful to Professor D. Arnett for guiding me through the recent literature and giving me
advice in the writing of this section.
From the foregoing brief description of what may happen during the late stages
in the evolution of massive stars, it is clear that the problems one encounters
are of exceptional complexity, in which a great variety of physical factors
compete. This is clearly not the occasion for me to enter into a detailed
discussion of these various questions. Besides, Professor Fowler may address
himself to some of these matters in his lecture that is to follow.
8. Instabilities of relativistic origin: (I) The vibrational instability of spherical stars
I now turn to the consideration of certain types of stellar instabilities which are
derived from the effects of general relativity and which have no counterparts
in the Newtonian framework. It will appear that these new types of instabilities
of relativistic origin may have essential roles to play in discussions pertaining to
gravitational collapse and the late stages in the evolution of massive stars.
We shall consider first the stability of spherical stars for purely radial
perturbations. The criterion for such stability follows directly from the linearized
equations governing the spherically symmetric radial oscillations of stars. In the
framework of the Newtonian theory of gravitation, the stability for radial
perturbations depends only on an average value of the adiabatic exponent,
which is the ratio of the fractional Lagrangian changes in the pressure and in the
density experienced by a fluid element following the motion; thus,
(56)
And the Newtonian criterion for stability is
< 4/3, dynamical instability of a global character will ensue with an e-folding
time measured by the time taken by a sound wave to travel from the centre to the
surface.
When one examines the same problem in the framework of the general theory
of relativity, one finds 12 that, again, the stability depends on an average value
of but contrary to the Newtonian result, the stability now depends on the
radius of the star as well. Thus, one finds that no matter how high may be,
instability will set in provided the radius is less than a certain determinate
multiple of the Schwarzschild radius,
Rs = 2 GM/c2
. (58)
Thus, if for the sake of simplicity, we assume that is a constant through the
star and equal to 5/3, then the star will become dynamically unstable for radial
perturbations, if R1 < 2.4 R,. And further, if r1 instability will set in for
all R < (9/8) Rs
. The radius (9/8) Rs defines, in fact, the minimum radius which any
gravitating mass, in hydrostatic equilibrium, can have in the framework of general relativity.
This important result is implicit in a fundamental paper by Karl Schwarzschild
published in 1916. (Schwarzschild actually proved that for a star in which the
energy density is uniform, R > (9/8)Rs
.)
158 Physics 1983
In one sense, the most important consequence of this instability of relativistic
origin is that (again assumed to be a constant for the sake of simplicity)
differs from and is greater than 4/3 only by a small positive constant, then the
instability will set in for a radius R which is a large multiple of and, therefore,
under circumstances when the effects of general relativity, on the structure of
the equilibrium configuration itself, are hardly relevant. Indeed, it follows13 from
the equations governing radial oscillations of a star, in a first post-Newtonian
approximation to the general theory of relativity, that instability for radial
perturbations will set in for all
(59)
where K is a constant which depends on the entire march of density and pressure
in the equilibrium configuration in the Newtonian frame-work. Thus, for a
polytrope of index n, the value of the constant is given by
(60)
w h e r e is the Lane-Emden function in its standard normalization = 1
at = 0), is the dimensionless radial coordinate, defines the boundary
of the polytrope (where = 0) and is the derivative of at
Table 2
Values of the constant K in the inequality (59)
for various polytropic indices, n.
K n
In Table 2, we list the values of K for different polytropic indices. It should be
particularly noted that K increases without limit for 5 and the configuration
becomes increasingly centrally condensed.** Thus, already for n = 4.95 (for
which polytropic index = 8.09 x 106 K-46. In other words, for the highly
centrally condensed massive stars (for which differ from 4/3 by as little
* It is for this reason that we describe the instability as global.
** Since this was written, it has been possible to show (Chandrasekhar and Lebovitz 13a) that
for 5, the asymptotic behaviour of K is given by
and, further, that along the polytropic sequence, the criterion for instability (59) can be expressed
alternatively in the form
S. Chandrasekhar 159
as 0.01); the instability of relativistic origin will set in, already, when its radius
falls below 5 x 103 Clearly this relativistic instability must be considered
in the contexts of these problems.
A further application of the result described in the preceding paragraph is to
degenerate configurations near the limiting mass14. Since the electrons in
these highly relativistic configurations have velocities close to the velocity of
light, the effective value of will be very close to 4/3 and the post-Newtonian
relativistic instability will set in for a mass slightly less than that of the
limiting mass. On account of the instability for radial oscillations setting in for
a mass less than the period of oscillation, along the sequence of the
degenerate configurations, must have a minimum. This minimum can be
estimated to be about two seconds (see Fig. 6). Since pulsars, when they were
discovered, were known to have periods much less than this minimum value,
the possibility of their being degenerate configurations near the limiting mass
was ruled out; and this was one of the deciding factors in favour of the pulsars
being neutron stars. (But by a strange irony, for reasons we have briefly
explained in 7, pulsars which have resulted from super-nova explosions have
masses close to 1.4
Finally, we may note that the radial instability of relativistic origin is the
underlying cause for the existence of a maximum mass for stability: it is a direct
consequence of the equations governing hydrostatic equilibrium in general
relativity. (For a complete investigation on the periods of radial oscillation of
neutron stars for various admissible equations of state, see a recent paper by
Detweiler and Lindblom15.)
9. Instabilities of relativistic origin: (2) The secular instability of rotating stars derived
from the emission of gravitational radiation by non-axisymmetric modes of oscillation
I now turn to a different type of instability which the general theory of relativity
predicts for rotating configurations. This new type of instability16 has its origin
in the fact that the general theory of relativity builds into rotating masses a
dissipative mechanism derived from the possibility of the emission of gravitational radiation by non-axisymmetric modes of oscillation. It appears that this
instability limits the periods of rotation of pulsars. But first, I shall explain the
nature and the origin of this type of instability.
It is well known that a possible sequence of equilibrium figures of rotating
homogeneous masses is the Maclaurin sequence of oblate spheroids17. When
one examines the second harmonic oscillations of the Maclaurin spheroid, in a
frame of reference rotating with its angular velocity, one finds that for two of
these modes, whose dependence on the azimuthal angle is given by the
characteristic frequencies of oscillation, depend on the eccentricity in the
manner illustrated in Figure 7. It will be observed that one of these modes
becomes neutral (i.e., 0) when = 0.813 and that the two modes coalesce
when = 0.953 and become complex conjugates of one another beyond this
* By reason of the dominance of the radiation pressure in these massive stars and of being very
close to zero.
160
Fig. 6. The variation of the period of radial oscillation along the completely degenerate configurations. Notice that the period tends to infinity for a mass close to the limiting mass. There is
consequently a minimum period of oscillation along these configurations; and the minimum period
is approximately 2 seconds. (From J. Skilling, Pulsating Stars (Plenum Press, New York, 1968), p. 59.)
point. Accordingly, the Maclaurin spheroid becomes dynamically unstable at the
latter point (first isolated by Riemann). On the other hand, the origin of the
neutral mode at e = 0.813 is that at this point a new equilibrium sequence of
triaxial ellipsoids - the ellipsoids of Jacobi - bifurcate. On this latter account,
Lord Kelvin conjectured in 1883 that
if there be any viscosity, however slight . . . the equilibrium beyond
e = 0.81 cannot be secularly stable.
Kelvin’s reasoning was this: viscosity dissipates energy but not angular momentum. And since for equal angular momenta, the Jacobi ellipsoid has a
lower energy content than the Maclaurin spheroid, one may expect that the
action of viscosity will be to dissipate the excess energy of the Maclaurin
spheroid and transform it into the Jacobi ellipsoid with the lower energy. A
detailed calculation 18 of the effect of viscous dissipation on the two modes of
oscillation, illustrated in Figure 7, does confirm Lord Kelvin’s conjecture. It is
found that viscous dissipation makes the mode, which becomes neutral at
e= 0.813, unstable beyond this point with an e-folding time which depends
inversely on the magnitude of the kinematic viscosity and which further decreases monotonically to zero at the point, e = 0.953 where the dynamical
instability sets in.
Since the emission of gravitational radiation dissipates both energy and
angular momentum, it does not induce instability in the Jacobi mode; instead it
S. Chandrasekhar 161
Fig. 7. The characteristic frequencies (in the unit of the two even modes of secondharmonic oscillation of the Maclaurin sphcriod. The Jacobi sequence bifurcates from the Maclaurin sequence by the mode that is neutral = 0) at e = 0.813; and the Dcdekind sequence
bifurcates by the alternative mode at D. At O2, (e = 0.9529) the Maclaurin spheroid becomes
dynamically unstable. The real and the imaginary parts of the frequency, beyond O2 are shown by
the full line and the dashed curves, respectively. Viscous dissipation induces instability in the
branch of the Jacobi mode; and radiation-reaction induces instability in the branch DO, of the
Dedekind mode.
induces instability in the alternative mode at the same eccentricity. In the first
instance this may appear surprising; but the situation we encounter here
clarifies some important issues.
If instead of analyzing the normal modes in the rotating frame, we had
analyzed them in the inertial frame, we should have found that the mode which
becomes unstable by radiation reaction at e = 0.813, is in fact neutral at this
point. And the neutrality of this mode in the inertial frame corresponds to the
fact that the neutral deformation at this point is associated with the bifurcation
(at this point) of a new triaxial sequence-the sequence of the Dedekind
ellipsoids. These Dedekind ellipsoids, while they are congruent to the Jacobi
ellipsoids, they differ from them in that they are at rest in the inertial frame and
owe their triaxial figures to internal vortical motions. An important conclusion
that would appear to follow from these facts is that in the framework of general
relativity we can expect secular instability, derived from radiation reaction, to
arise from a Dedekind mode of deformation (which is quasi-stationary in the
inertial frame) rather than the Jacobi mode (which is quasi-stationary in the
rotating frame).
A further fact concerning the secular instability induced by radiation reaction, discovered subsequently by Friedman19 and by Comins20, is that the
162
modes belonging to higher values of m (= 3, 4, , ,) become unstable at smaller
eccentricities though the e-folding times for the instability becomes rapidly
longer. Nevertheless it appears from some preliminary calculations of Friedman21 that it is the secular instability derived from modes belonging to m = 3
(or 4) that limit the periods of rotation of the pulsars.
It is clear from the foregoing discussions that the two types of instabilities of
relativistic origin we have considered are destined to play significant roles in
the contexts we have considered.
10. The mathematical theory of black holes
So far, I have considered only the restrictions on the last stages of stellar
evolution that follow from the existence of an upper limit to the mass of
completely degenerate configurations and from the instabilities of relativistic
origin. From these and related considerations, the conclusion is inescapable
that black holes will form as one of the natural end products of stellar evolution
of massive stars; and further that they must exist in large numbers in the
present astronomical universe. In this last section I want to consider very
briefly what the general theory of relativity has to say about them. But first, I
must define precisely what a black hole is.
A black hole partitions the three-dimensional space into two regions: an
inner region which is bounded by a smooth two-dimensional surface called the
event horizon; and an outer region, external to the event horizon, which is
asymptotically flat; and it is required (as a part of the definition) that no point
in the inner region can communicate with any point of the outer region. This
incommunicability is guaranteed by the impossibility of any light signal, originating in the inner region, crossing the event horizon. The requirement of
asymptotic flatness of the outer region is equivalent to the requirement that the
black hole is isolated in space and that far from the event horizon the spacetime approaches the customary space-time of terrestrial physics.
In the general theory of relativity, we must seek solutions of Einstein’s
vacuum equations compatible with the two requirements I have stated. It is a
startling fact that compatible with these very simple and necessary requirements, the general theory of relativity allows for stationary (i.e., time-independent) black-holes exactly a single, unique, two-parameter family of solutions.
This is the Kerr family, in which the two parameters are the mass of the black
hole and the angular momentum of the black hole. What is even more remarkable, the metric describing these solutions is simple and can be explicitly
written down.
I do not know if the full import of what I have said is clear. Let me explain.
Black holes are macroscopic objects with masses varying from a few solar
masses to millions of solar masses. To the extent they may be considered as
stationary and isolated, to that extent, they are all, every single one of them,
described exactly by the Kerr solution. This is the only instance we have of an
exact description of a macroscopic object. Macroscopic objects, as we see them
all around us, are governed by a variety of forces, derived from a variety of
approximations to a variety of physical theories. In contrast, the only elements
S. Chandrasekhar 163
in the construction of black holes are our basic concepts of space and time.
They are, thus, almost by definition, the most perfect macroscopic objects there
are in the universe. And since the general theory of relativity provides a single
unique two-parameter family of solutions for their descriptions, they are the
simplest objects as well.
Turning to the physical properties of the black holes, we can study them best
by examining their reaction to external perturbations such as the incidence of
waves of different sorts. Such studies reveal an analytic richness of the Kerr
space-time which one could hardly have expected. This is not the occasion to
elaborate on these technical matters22. Let it suffice to say that contrary to
every prior expectation, all the standard equations of mathematical physics can
be solved exactly in the Kerr space-time. And the solutions predict a variety
and range of physical phenomena which black holes must exhibit in their
interaction with the world outside.
The mathematical theory of black holes is a subject of immense complexity;
but its study has convinced me of the basic truth of the ancient mottoes,
and
The simple is the seal of the true
Beauty is the splendour of truth
The subject is a fair field for the struggle to gain knowledge by scientific reasoning; and, win or lose, we find the joy
of contest.
—Sir Arthur Stanley Eddington, The
Internal Constitution of the Stars
(1926).
Stellar evolution has for many years
been one of the most exciting fields of
research in astronomy and astrophysics. In the early 1930s, a young
astrophysicist named Subrahmanyan
Chandrasekhar certainly felt this excitement when in his theoretical work he
found a fundamental parameter that
determines the destiny of stars. By
appling both relativity and the new
quantum mechanics, Chandrasekhar
found a critical mass, below which stars
end up as white dwarfs, and above
which, as later work would show, they
end up as neutron stars or black holes.
Although we now recognize the
"Chandrasekhar limit" as a major discovery, its validity and importance remained in doubt among astronomers in
large part because a single individual
felt that all stars should become white
dwarfs in their terminal stages. A
dramatic and unanticipated confrontation took place at the January 1935
meeting of the Royal Astronomical Society of England. As we will see, the
brilliant but young Chandrasekhar,
armed with a fairly simple derivation
based on special relativity and the FerKameshwar C. Wali is an elementary-particle
theorist at Syracuse University, in New York.
He is writing a biography of Subrahmanyan
Chandrasekhar.
mi-Dirac quantum-statistical distribution laws, was no match for the ridicule
by Arthur Stanley Eddington, a renowned scientist with tremendous international stature, authority and influence. And because physicists failed
to counter Eddington publicly in their
own area of expertise, astronomers remained confused about stellar evolution until the 1950s.
Chandrasekhar has given1
a partial
account of the controversy. But a fuller
account of the 1935 meeting and the
circumstances leading to it demonstrates how discovery in science is often
beset by obstacles that do not arise
logically or objectively. Human factors, such as personal biases, prestige,
and authority play just as important a
role in science as in art or literature.
A glance at our current view of stellar evolution will help us appreciate
the importance of the issue at the
center of the controversy.
Our present understanding
A star is born, or so observational
evidence tells us, in the middle of a
condensing cloud of interstellar gas
and dust, composed mainly of hydrogen. All-pervading gravitational
forces, which are primarily responsible
for the condensation, compress the stellar material. In time, the interior becomes hot enough to initiate a nuclear
reaction in which the hydrogen nuclei
fuse to form helium. The star turns on.
This steady transformation of hydrogen into helium releases enormous
amounts of energy, creating enough
internal pressure to balance the crushing forces of gravity. The star reaches
equilibrium. During this "adult
stage," the star is situated on the socalled "main sequence" of the Hertzsprung-Russell diagram—a plot of a
star's spectral type, or surface temperature, against its absolute luminosity, or
total energy output. Eventually, all of
the central hydrogen is converted into
helium. The star leaves the main sequence as gravitational forces take
over and compress the central helium
core. As the temperature of the interior rises, the outer layers, where some
hydrogen may still be burning, expand;
the diameter of the star increases to ten
or a hundred times its main-sequence
value. At this stage, we call the star a
red giant.
Complex processes subsequently occur. In stars somewhat more massive
than the Sun, core temperatures rise
enough to burn the helium and to
create carbon and oxygen. This process
generates energy and pressure once
again, and the star reaches another
equilibrium stage, which last until all
the central helium is burnt and an
oxygen and carbon core remains.
Then, gravitational contraction begins
again. When the temperatures are
high enough to initiate carbon-oxygen
reactions, elements like neon, magnesium and silicon are created. Thus, in
its struggle for survival against the
forces of gravity, a star may synthesize
more and more complex elements in its
interior nuclear furnace. During any
of the stages, instabilities may develop,
producing nova or supernova explosions that eject large fragments of the
star into outer space.
The question arises: What happens
after a star finally exhausts all its
nuclear fuel and can no longer produce
the necessary energy and pressure to
withstand gravity? Current thought is
that the final mass of the star, or of the
stellar remnant of a nova or supernova,
determines the nature of its terminal
stage: a white dwarf, a neutron star or
a black hole. Only stars of fairly low
final mass become white dwarfs or
neutron stars. For a star to become a
white dwarf, its mass must not exceed
the critical value of 1.44 solar masses.
Neutron stars are remnants with
masses three or four times that of the
Sun. More massive stars do not become
white dwarfs or neutron stars unless
they lose substantial amounts of their
mass. They cannot win against gravity; their ultimate fate is surrender—
they become black holes.
Such is our present understanding of
the life cycle of stars. Let us now go
back a half century and follow the
scientific developments that led to
Chandrasekhar's discovery of the criti0031-9228 / 82 / 1000 33-08 / $01.00 i& 1982 American Institute of Physics PHYSICS TODAY / OCTOBER 1982 33
cal mass that divides stars of different
destinies.
Dense-matter puzzle
White dwarfs comprise a distinct
class of very faint stars with very high
surface temperatures. Low in luminosity but white in color, they lie off the
main sequence of the HertzsprungRussell diagram. Eddington himself
dramatized their existence and properties, describing them as
Strange objects, which persist in
showing a type of spectrum entirely out of keeping with their luminosity, [that] may ultimately teach
us more than a host which radiate
according to rule.
In the early twenties, three such
objects were known, the Companion of
Sirius being the most illustrious.
Known as Sirius comes, its mass was
reliably determined from the fact that
it formed a double star with Sirius, the
brightest star in the Galaxy. Each
component of this binary system has
approximately the same mass as the
Sun. The white color of Sirius comes, or
more precisely, its spectral type and
luminosity, indicated a radius of approximately 20 000 kilometers, astonishingly small for a star whose mass is
as great as that of the Sun. This radius
also implied a density of 61 000 g/cm3
,
or just about a ton per cubic inch!
Obviously, such dense matter is not to
be found terrestrially. According to
Eddington, when the laws of physics
are extrapolated to stellar conditions,
such exotic matter is not only allowed
to exist, but exists in such a way that it
has the compressibility of a perfect gas.
Walter Sydney Adams, the astrophysicist who discovered the spectral
characteristics that indicated the small
size of Sirius comes, tested some of the
bizarre expectations. At Eddington's
suggestion, Adams measured the shift
of the spectral lines to the red, an effect
predicted by Einstein's theory. This
shift depends on density, and if the
densities were as high as predicted, the
red shifts would correspond to a velocity of nearly 20 km/sec. Adams did
find such a shift, and, in Eddington's
words,
. . . killed two birds with one stone;
he has carried out a new test of
Einstein's general theory of relativity and he has confirmed our
suspicion that matter 2000 times
denser than platinum is not only
possible, but is actually present in
the universe.
The existence of such compact objects, however, presented a major puzzle that the astrophysical theory of the
early 1920s could not explain. Eddington himself identified the problem of
matter existing under such extraordinary conditions: Very-high-density material with the compressibility of a
perfect gas is possible if and only if the
temperatures in the interior of the star
are high enough to ionize the atoms
completely. The bare nuclei and electrons together form an aggregate of
matter very different from that found
on Earth. It is an assemblage of particles that are millions of times smaller
than normal atoms. This reduction in
size makes the high densities possible.
However, white dwarfs radiate energy into space. As the energy is radiated, the temperatures should fall, and
the electrons should recombine to form
ordinary atoms and ordinary matter.
But according to the standard astrophysical models of the 1920s, the
"dense" stage had a lower energy than
Eddington In the
1930s. (Photo courtesy
of the Bettman
Archive, Inc.)
the normal state. This implied that,
even after the extraordinary constraints of pressure were removed,
white-dwarf matter would continue to
exist in the dense state. If a spoon of
stellar matter were scooped out of the
white dwarf and brought down to
Earth, it would continue to be dense
matter!
Eddington highlighted this problem
both in his lectures before the Royal
Society and in his book, The Internal
Constitution of the Stars. Writing in
his usual charming and somewhat
playful style, he said:
I do not see how a star which has
once got into this compressed condition is ever going to get out of
it
When the star cools down and
regains the normal density ordinarily associated with solids, it must
expand and do work against gravity. The star will need energy in
order to cool. . . .
Imagine a body continually losing
heat but with insufficient energy
to grow cold.
The quantum solution
The puzzle remained unsolved until
the advent of the new quantum mechanics in the middle twenties. In
1926, almost immediately after the formulation of the laws of quantum statistics, Ralph Howard Fowler2
applied
them to the state of matter in white
dwarfs. If the dissociation of atoms
into free electrons and bare nuclei
caused the immense densities in white
dwarfs, Fowler argued, the classical
laws were not relevant, but the newly
discovered laws of quantum statistics
were.
The description of the electron distribution provided by Fermi-Dirac statistics gives rise to a relation between
pressure and density radically different from that which follows from classical statistics. The most important
characteristic of the new pressure-density relation was its lack of dependence
on temperature. Even at absolute zero
there is a finite pressure, whose value is
a function of density. Electrons under
the conditions found in white dwarfs—
confined, but free of particular nucleiwere said to be "degenerate." And the
"degeneracy pressure" associated with
the high densities encountered in white
dwarfs was found to be large enough to
sustain the star against the forces of
gravity.
Fowler also showed that Eddington's
worry could be discarded. Fowler's
evaluation of stellar matter at the temperatures and densities of white dwarfs
indicated that its energy is indeed
greater than that of ordinary matter
under terrestrial conditions. The
whole idea was beautifully summarized
in Fowler's paper at the 10 December
1926 meeting of the Royal Astronomical Society:
The black dwarf material is best
likened to a single gigantic molecule in its lowest quantum state.
On the Fermi-Dirac statistics, its
high density can be achieved in one
and only one way, in virtue of a
correspondingly great energy content. But this energy can no more
be expended in radiation than the
energy of a normal atom or molecule. The only difference between
black dwarf matter and a normal
molecule is that the molecule can
exist in a free state while the black
dwarf matter can only so exist
under very high external pressure.
Fowler preferred the phrase "black
dwarf material" to describe the stellar
contents of white dwarfs, because all
the energy that was not locked up in
the degenerate quantum state eventually would leak away to outer space.
Then, white dwarfs would become invisible. Fowler's pioneering discovery
was a brilliant extrapolation of the
very new quantum mechanics of the
atom. Most notably, his pressure-density relation allowed a star of any mass
to be a white dwarf in its final stage; it
would remain at peace, as it were, until
all its energy was radiated into space
and it became extinct. Eddington was
satisfied with such a course of evoluation, and matters concerning white
dwarfs appeared settled.
At this stage, Chandrasekhar entered the picture. As an undergraduate at Presidency College, in Madras,
India, he learned of the Fermi-Dirac
statistics from none other than Arnold
Sommerfeld during his visit to India in
1928. Chandrasekhar had read
Fowler's paper because it contained an
application of Fermi-Dirac statistics.
"My knowledge of physics, astronomy
and mathematics was rudimentary in
the extreme at the time," Chandrasekhar recalls. But his learning was adequate enough to enable him to draw
some far-reaching conclusions from
Fowler's discovery. He observed that
Fowler's pressure-density relation implied that stellar configurations consisting of degenerate electrons belonged to the class of polytropes of
index %. (A polytrope is stellar material in equilibrium under its own gravity,
obeying a general equation of state
P = Kp1 + 1/n
, where P is the pressure, p
is the density and n is the "polytropic
index.") Chandrasekhar concluded
that3
(i) the radius of a white dwarf is
inversely proportional to the cube
root of the mass—implying thereby that for every finite-mass star,
there is a finite radius, a conclusion which I have already alluded
to; (ii) the density is proportional to
the square of the mass; (iii) the
Chandrasekhar in the
1930s. (Photo courtesy
of S. Chandrasekhar.)
central density would be six times
the mean density p.
Chandrasekhar adds relativity
The last of Chandrasekhar's three
conclusions led him to ask a number of
key questions, which provided crucial
to the subsequent theory of white
dwarfs. If the central densities were so
high, would not the momenta of the
electrons, increasing as one moved
away from the center of the Fermi
sphere, reach magnitudes comparable
to their rest masses? If they did, the
special-relativistic variation of mass
with velocity would be important and
would have to be taken into account.
What would be the consequences?
These questions occupied Chandrasekhar's mind on his voyage to England
in the summer of 1930. He soon found
that the relativistic effects were indeed
important. In the extreme relativistic
limit, the pressure-density relation indicated a stellar configuration of polytropic index 3, instead of % as with
Fowler's nonrelativistic degenerate
configuration. Further, the total mass
of the configuration was uniquely determined by some fundamental atomic
constants and the mean molecular
weight p. of the stellar material. Using
the known values for the atomic constants, he found the mass to be 5.76/ju2
solar masses. In 1930, the canonical
value for p was 2.5, giving a mass of
approximately 0.91 solar masses
(which later became 1.44 solar masses
when fi was revised to 2). This was the
origin of the "critical mass," or Chandrasekhar limit.
Chandrasekhar brought two short
papers with him to Cambridge. One
dealt with the nonrelativistic degenerate configurations, the other with relativistic effects and the emergence of the
critical mass. Fowler had no difficulty
with the first one; he appreciated its
details and the progress Chandrasekhar had made. However, he was skeptical about the second. So was Edward
Arthur Milne, to whom Fowler sent the
paper for his opinion. Some months
later Milne did communicate a paper9
by Chandrasekhar giving a full derivation of the limiting mass.
Despite the skepticism of the two
men, Chandrasekhar realized that the
existence of a critical mass was an
inevitable consequence of combining
special relativity and quantum statistics. By applying these two theories to
Eddington's standard model of a star,
one could see that a star of mass
greater than that given by Chandrasekhar's formula can not achieve the stable degenerate state of the white dwarf.
In a paper published in 1932, Chandrasekhar wrote4
For all stars of mass greater than
M [1.2 times the critical mass], the
perfect gas equation of state does
not break down, however high the
density may become and the matter does not become degenerate.
An appeal to Fermi-Dirac statistics to avoid the central singularity
cannot be made.
He continued,
Great progress in the analysis of
stellar structure is not possible
before we can answer the following
fundamental question: Given an
enclosure containing electrons and
atomic nuclei, what happens if we
go on compressing the material
indefinitely.
Not until much later did the farreaching implications of this question
became apparent. If the white-dwarf
stage was not the end stage of all stars,
did other stages exist? If they did, what
PHYSICS TODAY / OCTOBER 1982 35
were they? If not, and the collapse
continued indefinitely, what happened
to the stellar matter? Such questions
should have been asked immediately
and their answers should have been
sought. That this was not the case,
however, is due to the following incident.
In 1933, Chandrasekhar completed
his PhD in England and was elected a
Fellow of Trinity College. From his
earliest days in Cambridge he had developed a close working association and
friendship with Milne. He had also
come to know Eddington well. Eddington frequently visited Chandrasekhar
in his rooms at Trinity. They often
dined together at the high table (the
slightly elevated table at which faculty
and distinguished visitors eat in the
dining halls of the colleges in England).
Eddington knew Chandrasekhar's
work almost on a daily basis.
In the meantime, Chandrasekhar's
work had assumed a new significance,
because it touched upon a controversy
between Eddington and Milne. Milne
had suggested modifications in Eddington's perfect-gas model of the stars to
explain certain observed features of
opacity and temperature in the surface
layers of stellar matter. According to
Milne, every star should have a central
core of degenerate material surrounded by matter that obeyed the perfect
gas laws. Eddington did not approve of
such modifications of his model, and
made no secret of this. He had said, for
instance, "I have not read professor
Milne's paper, but I hardly think it is
necessary, for it would be absurd for me
to pretend that professor Milne has the
remotest chance of being right."
A meeting looms
Chandrasekhar, who immensely admired both these stalwarts of modern
astrophysics, saw the issue as a purely
scientific controversy that could be settled by deriving an exact equation of
state for stellar material under very
general conditions. His previous discovery of the value of the critical mass
had convinced him that not all stars
could have the degenerate core Milne
wished them to have. If the mass of the
star exceeded the critical mass, relativistic degeneracy restored the perfect
gas law for the pressure and density.
This situation gave Eddington a slight
edge in his controversy with Milne.
However, the existence of a critical
mass also meant that the white-dwarf
stage could not be the end stage for all
stars—a conclusion that distressed Eddington very much.
Chandrasekhar had assumed that
the exact equation of state would settle
the question. It would account for both
the relativistic effects and the possibility of inhomogeneous polytropes consisting of degenerate cores surrounded
Dinner of the Royal Astronomical Society Club, 12 June 1936. Chandrasekhar (seated third from
the left) attended as a guest of Eddington (second from left). W. M. Smart, then secretary of the
Royal Astronomical Society, and later the Regius professor of astronomy at Glasgow, Scotland,
is on Chandrasekhar's left. (Photograph courtesy of S. Chandrasekhar.)
by layers of ordinary matter. "I was
very pleased," Chandrasekhar recalls,
"because Eddington seemed to understand that this would certainly resolve
the controversy between Milne and
him."
Chandrasekhar thus proceeded to
work out a complete theory of white
dwarfs. Accomplishing his task by
1934, he submitted two papers to the
Royal Astronomical Society, and accepted their invitation to present a
brief account of his results at the January 1935 meeting. He was no stranger to these meetings, which were held
the second Friday of every month. He
had been introduced to the society by
Fowler in 1930 and had been elected a
fellow in 1933. Since then, he had
presented several papers and had already brought to the attention of Eddington, Milne and others his belief
that the fate of massive stars could not
be identical to that of less massive
stars. That conclusion, however, was
based on approximate models. Now he
had the exact solution to the problem,
reinforced by extensive numerical
analysis. Chandrasekhar did not doubt
the validity of his results and the profound challenge they presented to
those interested in stellar evolution. In
those days the society decided its program just a few days before the meeting, the list of papers to be presented
being distributed at the gathering. Papers had to be submitted one week
before the meeting. After 45 years,
Chandrasekhar recalls the events leading to that meeting5
I knew the assistant secretary, a
Miss Williams... rather well, and
she used to send me the program
ahead of the meeting. And on
Thursday evening I got the program and found that immediately
after my paper Eddington was giving a paper on "Relativistic Degeneracy." I was really very annoyed
because, here Eddington was coming to see me every day, and he
never told me he was giving a
paper.
Then I went to dine in College
and Eddington was there. Somehow I thought Eddington would
come to talk with me, but I did not
go talk with him. After dinner I
was standing by myself in the Combination Room where we used to
have coffee, and Eddington came
up to me and asked me, "I suppose
you are going to London tomorrow?" I said, "Yes." He said, "You
know your paper is very long. So I
have asked Smart [the secretary of
the RAS], to give you a half hour
for your presentation instead of
the customary 15 minutes." I said,
"That's very nice of you." And he
still did not tell me. So I was a
little nervous as to what the story
was.
The next day at Burlington
House there was a tea before the
meeting, and [William Hunter]
McCrea, who is a relativist, he and
I were standing together and Eddington came by. McCrea asked
Eddington, "Well, Professor Eddington, what are we to understand by Relativistic Degeneracy?"
Eddington turned to me and said,
"That's a surprise for you," and
walked away.
The controversy unfolds
The proceedings of the meeting, published in Observatory,6
contain the following account of Chandrasekhar's
paper:
Dr. Chandrasekhar read a paper
describing the research which he
has recently carried out, an ac36 PHYSICS TODAY / OCTOBER 1982
count of which has already appeared in the Observatory, 57.373,
1934, investigating the equilibrium of stellar configurations with
degenerate cores. He takes the
equation of state for degenerate
matter in its exact form, that is to
say, taking account of relativistic
degeneracy. An important result
of the work is that the life history
of a star of small mass must be
essentially different from that of a
star of large mass. There exists a
certain critical mass M. If the
star's mass is greater than M the
star cannot have a degenerate
core, but if the star's mass is less
than M it will tend, at the end of its
life history, towards a completely
collapsed state.
After a brief comment by Milne, the
President invited Eddington to speak
on "Relativistic Degeneracy." Eddington began by saying,6
Dr. Chandrasekhar has been referring to degeneracy. There are two
expressions commonly used in this
connection, "ordinary" degeneracy
and "relativistic" degeneracy, and
perhaps I had better begin by explaining the difference. They refer
to formulae expressing the electron pressure P in terms of the
electron density a. For ordinary
degeneracy Pe = Ko5
'
3
. But it is
generally supposed that this is
only the limiting form at low densities of a more complicated relativistic formula, which shows P varying as something between cr"3
and
<74/3
, approximating to <74/3
at the
highest densities. I do not know
whether I shall escape from this
meeting alive, but the point of my
paper is that there is no such thing
as relativistic degeneracy!
After making a few remarks about the
history of the problem, the difficulty he
had pointed out in 1924 and the way
Fowler's appeal to Fermi-Dirac statistics had solved the problem, Eddington
went on to say
But Dr. Chandrasekhar has revived it again. Fowler used the
ordinary formulae; Chandrasekhar, using the relativistic formula which has been accepted for
the last five years, shows that a
star of mass greater than a certain
limit M remains a perfect gas and
can never cool down. The star has
to go on radiating and radiating
and contracting and contracting
until, I suppose, it gets to a few km
radius, when gravity becomes
strong enough to hold in the radiation, and the star can at last find
peace.
Dr. Chandrasekhar had got this
result before, but he has rubbed it
in, in his last paper; and, when
discussing it with him, I felt driven
to the conclusion that this was
almost a reductio ad absurdum of
the relativistic degeneracy formula. Various accidents may intervene to save a star, but I want
more protection than that. I think
there should be a law of Nature to
prevent a star from behaving in
this absurd way!
If one takes the mathematical
derivation of the relativistic degeneracy formula as given in astronomical papers, no fault is to be
found. One has to look deeper into
its physical foundations, and these
are not above suspicion. The formula is based on a combination of
relativity mechanics and nonrelativity quantum theory, and I do not
regard the offspring of such a
union as born in lawful wedlock. I
feel satisfied that the current formula is based on a partial relativity theory, and that if the theory
is made complete the relativity
corrections are compensated, so
that we come back to the "ordinary" formula.
Eddington continued by making some
vague remarks about two kinds of
waves—progressive and standing—using a light-hearted analogy that made
several laugh:
I might compare the progressive
wave with Professor Stratton and
55
50
45
^4.0
535
en 2.5
§ 2 0
cr
15
1.0
05
0
0 0.1 0 2 0.3 0 4 0.5 0.6 0.7 0.8 0.9 10
MASS (1.44 M Q =1 )
Stellar mass versus radius. The dashed
curve follows from Fowler's nonrelativistic
degeneracy considerations. Here the radius of
equilibrium configurations decreases with increasing mass, but reaches zero only for an
infinite mass. Hence a star of any mass can
settle down comfortably to a white-dwarf
stage. In contrast, when special relativity is
taken into account, the solid curve results.
This curve represents the exact mass-radius
relation for completely degenerate configurations. As the mass approaches the limiting
mass of 1.44 solar masses, the radius becomes zero, indicating that only stars with
masses lower than 1.44 MQ can have an
equilibrium, white-dwarf stage. (From ref. 10.)
the standing wave with the President of the Royal Astronomical
Society; only, to make the analogy
a good one, the Society would have
to change its President gradually
and continuously, instead of suddenly every two years. The formulae which apply to such a President
would be different from the formulae which apply to an ordinary
individual.
Eddington's comparison implied that a
simple mistake in calculations had
been made. The effect on Chandrasekhar must have been chilling. Before he
could say anything, the president said:
the arguments of this paper will
need to be very carefully weighed
before we can discuss it. I ask you
to return [your] thanks to Sir Arthur Eddington.
He then called the next speaker to give
his paper.
After the meeting
It would be an understatement to say
that Chandrasekhar was left dumbfounded, shocked and depressed. Instead of gaining recognition for having
raised a challenging question, he found
his years of hard work summarily,
almost cavalierly, dismissed. Moreover, Chandrasekhar, barely twentyfour years old and almost a newcomer
to the research arena of astrophysics,
confronted in Eddington a figure whose
international prestige and authority
could destroy him. He must have been
overwhelmed by feelings of humiliation and helplessness when people
came by after the meeting, saying, "It
was too bad, too bad," expressing their
sorrow over what they now believed to
be Chandrasekhar's conceptual error.
In science, at times, one has the
inalienable right to be wrong. Certain
kinds of mistakes can even be a source
of pride. But on this occasion such was
not the case. Eddington had made
Chandrasekhar's work appear to be
based on a conceptual error, so that the
important conclusion Chandrasekhar
had stated with so much conviction was
wrong. If Eddington had felt so all
along, why hadn't he said so? Why, in
the privacy of Chandrasekhar's rooms,
did he encourage the young man to go
on with work that involved so much
tedious numerical labor? Was his motivation only to discredit Chandrasekhar
publicly, or could Eddington be right?
Chandrasekhar must have entertained
some of these thoughts at the conclusion of the meeting.
Chandrasekhar recalls having dinner later that day with Harry Hemley
Plaskett, an Oxford University astronomer. The dinner was quiet; neither of
them said a word. No hint of assurance
came from Plaskett; his silence gave
the impression that Eddington was
right. It was one of those things; misPHYSICS TODAY / OCTOBER 1982 37
takes happened. After the dinner,
Chandrasekhar went to see Plaskett off
at Paddington station. "Milne was
there," Chandrasekhar recalls, "and
Milne was absolutely in a state of
euphoria. Because Eddington's work
had shown that my limiting mass was
incorrect, his own idea that every star
had a degenerate core must be valid.
He told me that he felt it in his bones
that Eddington was right. I was really
angry at that.... Well, I wished he felt
it elsewhere."
Dejected and depressed, Chandrasekhar returned to Trinity at one o'clock in
the morning.
Later that morning, he saw Fowler
and told him what had transpired at
the meeting. Fowler offered some reassurrance. In private, so did some others. But Eddington continued his attacks.
During the International Astronomical Union meeting in Paris in 1935,
Eddington gave a talk in which he
called Chandrasekhar's work simple
heresy. Relativistic degeneracy did not
exist. Every star, no matter what its
mass, had the same final state; the idea
of a limiting mass was simply absurd.
Chandrasekhar, who was present at
the meeting, was not even allowed to
respond. It was not until four years
later, at another Paris meeting, that
Chandrasekhar finally got an opportunity to say openly that Eddington was
wrong (see the photo on opposite page).
No simple method, no direct observational test that existed at the time,
could prove Eddington wrong, and Eddington's theoretical reasoning was
hard for most astronomers to follow.
Moreover, his style and prestige were
sufficient to convince most that he
might well be right. The following
excerpt from a letter by McCrea, the
relativist and astronomer mentioned
above by Chandrasekhar, illustrates
the situation with the most clarity.7
I remember one RAS meeting
when Eddington produced what
struck me then as an unanswerable argument. As I recall, it went
like this: Only special relativity is
involved. Basically in order to obtain the so-called 'relativistic' form
of any property, we require the
fundamental equations of the subject to be Lorentz-invariant, i.e. the
same for all inertial frames. If for
any reason some particular frame
of reference is uniquely defined by
the physics of a problem, then for
that problem it is absurd—said
Eddington—to require a lot of other frames to be of the same status.
In the case of a spherically symmetric star the rest frame of the
star is a uniquely preferred frame.
In discussing the physics of the
star we are not justified in claiming that any other frame is equivalent to that unique frame. So Eddington said that there can be no
requirement of Lorentz-invariance
in such a case. It was therefore not
surprising that insistence upon
that requirement led to a manifestly absurd result....
When I listened to Eddington on
this occasion I could not immediately weigh up all the implications
of what he said, but my instinct
seemed to tell me that he might be
right.
Later in the letter, McCrea admits,
What I am ashamed of is not having tried to get to the bottom of the
sort of argument Eddington produced. Had anyone other than
Eddington produced such arguments, I suppose I should have
done so. But they were superficially satisfying to me, and since they
satisfied Eddington, I confess that I
was content to let it go at that. In
any case I was not working at
stellar structure. However, I did
profess to know something about
special relativity and I ought to
have gone into the subject from
that side.
Appealing to physicists
The issue was mainly a question of
physics. Thus, Chandrasekhar thought
that the views of physicists were exceedingly important. A definitive verdict from Niels Bohr, Wolfgang Pauli
or Paul Dirac would quickly settle the
controversy and clear the cloud of
doubt that Eddington had cast over his
work. Chandrasekhar wrote a letter to
Leon Rosenfeld, a long-time friend, almost immediately after the January
1935 meeting8
... I have been spending months
on my stellar structure work with
the hope that for once there will be
no controversy. Now that my work
is completed, Eddington has started this "howler" and of course
Milne is happy. My work has
shown that his (Milne's) ideas in
many places are wrong, but my
work depends on the relativistic
degenerate formula and Milne can
now go ahead. The result is that
there is going to be a long period of
stress and confusion and if somebody like Bohr can authoritatively
make a pronouncement in the matter it will be of the greatest value
for further progress in the subject.
At the time, Rosenfeld was in Copenhagen working with Bohr. Chandrasekhar provided all the details of the
equations he had used and the objections Eddington has raised. Could Eddington be right? Rosenfeld replied
... I may say that your letter was
some surprise for me: for nobody
had ever dreamt of questioning the
equations, and Eddington's remark as reported in your letter is
utterly obscure. So I think you had
better cheer up and not let you
scare so much by high priests: for I
suppose you know enough Marxist
history to be aware of the fundamental identity of high priests and
mountebanks.
I submitted your letter to Bohr
immediately, and I can state as
follows the outcome of our examination of the question: In order to
apply Pauli's principle to an assembly of electrons without interactions, one has simply to express
that every non-degenerate stationary state of an electron in the
external field considered may be
occupied by zero or one electron.
Now in your case (no external
field), the stationary states are defined by the components of momentum and spin, and their energies by the relativistic formula (p. 2
of your letter). The eigenfunctions
are plane waves, which may be
defined either as stationary waves
by a condition of reflection at the
boundary of the volume V, or as
progressing waves by a periodicity
condition at the boundary; these
two cases become equivalent in the
limit, considered by you, of an (asymptotically) infinite volume, and
both yield for the asymptotic density distribution in the phase space
precisely the expression you have
used in your equations. Further,
this expression is relativistically
invariant. These quite obvious arguments would seem to settle the
question without any doubt.
In a follow-up letter, sent the very next
day, Rosenfeld wrote
Bohr and I are absolutely unable to
see any meaning in Eddington's
statements as reported in your second letter. The question, however,
seems quite simple and has certainly a unique solution. So, if
"Eddington's principle" had any
sense at all, it would be different
from Pauli's. Could you perhaps
induce Eddington to state his views
in terms intelligible to humble
mortals? What are the mysterious
reasons of relativistic invariance
which compel him to formulate a
natural law in what seems to ordinary human beings a non-relativistic manner.* That would be curious to know.
*It seems to us as if Eddington's
statement that several high speed
electrons might be in one cell of the
phase space would imply that to
another observer several slow
speed electrons, in contrast to Pauli's Principle, would be in the same
cell.
These private remarks reassured
Chandrasekhar about the correctness
38 PHYSICS TODAY / OCTOBER 1982
of his ideas, but they were not adequate
to quell the public doubt. In addition,
further discussions with Eddington did
not prove fruitful. He had his own
notions about elementary principles of
quantum mechanics and about progressive and standing waves:
We cannot combine the wave functions (progressive waves) to produce standing waves. They are
incoherent. If two functions are
written with a + symbol, we only
mean the both are present. We
cannot combine progressive plane
waves to produce standing waves
in the quantum theory.
Chandrasekhar struggled with alternate arguments and derivations that
would make Eddington understand.
He wrote to Rosenfeld again.
I am really sorry to trouble you,
but Eddington is reading a paper at
the Coloquium next Friday and I
want to have real missiles to throw
at him! Some really simple way of
demonstrating that any theory
which shows that P~pb/i is an
identical proportionality for all
densities [as Eddington believed]
must be necessarily self-contradictory.
And Rosenfeld replied,
I was very glad to read your preliminary note in the Observatory.
It seems to me that your new work
is very important indeed, and I
think everybody except Eddington
will admit it rests on a perfectly
sound basis.
As to the artillery fighting you are
planning against Eddington, I
could not imagine any missile
more devastating than the one contained in my last letter. I feel a
little dubious about the result of
such a fight, since I do not expect
Eddington, whatever the missiles,
to collapse like a star with
fiM = 1 — e; it wouldn't be dignified
enough for him to recant after he
has gone so far as denying the
existence of wave packets in quantum theory! Wouldn't it be a good
policy to leave him alone, instead
of losing one's time and temper in
fruitless arguments? Nevertheless
I wish you a great fun next Friday,
and I even regret not to be there to
enjoy the show.
Forced to wait
After more prolonged discussions
with Eddington, Chandrasekhar decided to send Eddington's manuscript (later published in the Monthly Notices of
the Royal Astronomical Society) to RoParls conference on white dwarfs and supernovae, August 1939. At
this conference, Chandrasekhar recalled later, "Eddington and I
exchanged our contrary views in no uncertain terms." The photograph
shows all participants in the conference. They are (front) Frederick J.
M. Stratton, Cecilia Helena Payne-Gaposchkin, Henry Norris Russell,
Amos J. Shaler, Eddington, Sergei Gaposchkin; (rear) Carlyle S. Beals,
Bengt Edlen, Pol F. Swings, Gerard P. Kuiper, Bengt G. D. StrOmgren,
Chandrasekhar, Walter Baade; Knut Lundmark is standing between
Chandrasekhar and Eddington. (Photograph courtesy of S.
Chandrasekhar.)
PHYSICS TODAY / OCTOBER 1982 39
senfeld and Bohr.
I have managed to get hold of
Eddington's manuscript. He gave
it to me and I am forwarding it to
you for you and Bohr alone to read.
I should be awfully glad if Bohr
could be persuaded to interest himself in the matter. ... It is terribly
important to settle that matter as
quickly as possible, otherwise intense confusion would result in
astrophysics.
Rosenfeld said:
I vividly realize your troubles and
feel very sorry for you. Bohr would
be quite willing to help you, but he
is very tired now and has to write
two articles due for February 15th;
after this is completed, he intends
to leave to some rest resort to
recover from a very strained semester. He therefore feels it difficult to concentrate himself on a
new subject just now; but he has a
proposal to you, which I think
would meet your wish in the best
possible way. Would you agree for
us to forward confidentially Eddington's manuscript to Pauli, together with a statement of the
circumstances and asking for an
"authoritative reply"?
Chandrasekhar readily agreed, and
the derivation of his relativistic formula, along with Eddington's manuscript and other relevant material,
were sent to Pauli. About Eddington's
manuscript, Rosenfeld remarked:
... After having courageously read
Eddington's paper twice, I have
nothing to change in my previous
statements; it is the wildest nonsense.
Pauli's response was equally conclusive. He did not think there was any
ambiguity on applying his exclusion
principle to relativistic systems. Eddington's principal error, according to
Pauli, was making the application of
the exclusion principle to the relativistic case depend upon the results of
astrophysical calculations. Unfortunately, he too was unwilling to enter into
the controversy. Astrophysics simply
was not at the center of interest of
many physicists. As a result, as Chandrasekhar anticipated in his letter to
Rosenfeld, confusion prevailed among
astronomers. Chandrasekhar wanted
an authoritative statement from a
physicist like Bohr, Pauli or Dirac, not
so much to convince physicists of the
correctness of his ideas, but to quell the
doubts among astronomers.
Faced with this situation, Chandrasekhar decided to ignore the whole
controversy and go on to other things.
"I had to make a decision," he recalls,
"Am I going to continue the rest of my
life fighting... or change to other
areas of interest. I said, well, I will
write a book and then change my interest. So I did." The book, An Introduction to the Study of Stellar Structure,
was published in 1939; in it, he gives a
full account of his theory of white
dwarfs.
More than two decades passed before
the theory was completely accepted.
Today, hundreds of white dwarfs are
known, and the masses of those that
are measured fall neatly without a
single exception on the curve Chandrasekhar predicted. The Chandrasekhar
limit now stands out as an important
discovery in astrophysics.
Before concluding, let me say that
this incident, strange as it may seem, in
no way affected the personal relationship between Chandrasekhar and Eddington. They continued to correspond, and Eddington's letters are full
of warmth, humor and affection as
these excerpts from letters written
between 1938 and 1943 show:
. . . My cycling x is still 75. I was
unlucky this Easter as I did two
rides of 743
/4 miles which do not
count; I still need four more rides
for the next quantum jump. However, I had marvelously fine
weather, and splendid country—
chiefly South Wales.
Tomorrow I have to put on weird
costume—knee breeches and silk
hose!—and get my Order from the
King. Then holiday—then the
I.A.U.—then the British Association at Cambridge which will
make my fourth conference this
year.. ..
As a lighter matter the following
incident may amuse you. The
Master recently had to give lunch
to the Dutchess of Kent. He applied to the local food office for
extra rations in order to entertain
H. R. H. suitably. They rose to the
occasion. Yes, he might have one
extra pennyworth of meat, and
some rice.
Chandrasekhar, in all my conversations with him, still has the highest
admiration for Eddington's extraordinary scientific achievements, his internationalism, his charm and wit and his
great influence in many fields of human endeavour. In the obituary
speech he gave for Eddington at the
University of Chicago, Chandrasekhar
says that posterity may rank Eddington, next to Karl Schwarzschild, as the
greatest astronomer of our time. Perhaps by reading what Chandrasekhar
says further along in the same tribute
we may understand why they were able
to keep such good relations.
I believe that anyone who has
known Eddington will agree that
he was a man of the highest
integrity and character. I do not
believe for example, that he ever
thought harshly of anyone. That
was why it was so easy to disagree
with him on scientific matters.
You can always be certain that he
would never misjudge you or think
ill of you on that account. This
cannot be said of others.
This year marks the centennial year of
Eddington's birth. Chandrasekhar has
been invited to deliver the centenary
lectures at Trinity College in Cambridge.
For us, the controversy forcefully
illustrates that science is a human
endeavor. Eddington, whose bold embrace of Einstein's general theory contributed so much to its early acceptance
in the English-speaking world, failed to
see the far-reaching consequence of a
very simple and straightforward application of the special theory. Why did
he, imaginative as he was, fail to
foresee the exciting possibilities of terminal states other than that of a white
dwarf, and instead discard them as
absurd? He came so close to being the
first to realize that black holes must
exist in the astronomical universe
(Note on page 37 the sentence of
Eddington's talk that I have italicized).
Had he only taken the opposite view
and dramatized in his characteristic
way the significance of Chandrasekhar's discovery, what would have been
the course of astrono