Subrahmanyan Chandrasekhar INDEX CARD SIGNED Nobel Prize physics autograph BOX

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Seller: memorabilia111 ✉️ (808) 100%, Location: Ann Arbor, Michigan, US, Ships to: US & many other countries, Item: 176305800221 Subrahmanyan Chandrasekhar INDEX CARD SIGNED Nobel Prize physics autograph BOX. Important Subrahmanyan Chandrasekhar INDEX CARD SIGNED Nobel Prize physics autograph The Chandrasekhar limit (/tʃʌndrəˈseɪkər/) is the maximum mass of a stable white dwarf star. The currently accepted value of the Chandrasekhar limit is about 1.4 M☉ (2.765×1030 kg). White dwarfs resist gravitational collapse primarily through electron degeneracy pressure (compare main sequence stars, which resist collapse through thermal pressure). The Chandrasekhar limit is the mass above which electron degeneracy pressure in the star's core is insufficient to balance the star's own gravitational self-attraction. Consequently, a white dwarf with a mass greater than the limit is subject to further gravitational collapse, evolving into a different type of stellar remnant, such as a neutron star or black hole. Those with masses up to the limit remain stable as white dwarfs The limit was named after Subrahmanyan Chandrasekhar, an Indian astrophysicist who improved upon the accuracy of the calculation in 1930, at the age of 20, in India by calculating the limit for a polytrope model of a star in hydrostatic equilibrium, and comparing his limit to the earlier limit found by E. C. Stoner for a uniform density star. Importantly, the existence of a limit, based on the conceptual breakthrough of combining relativity with Fermi degeneracy, was indeed first established in separate papers published by Wilhelm Anderson and E. C. Stoner in 1929. The limit was initially ignored by the community of scientists because such a limit would logically require the existence of black holes, which were considered a scientific impossibility at the time. The fact that the roles of Stoner and Anderson are often forgotten in the astronomy community has been noted. Chandrasekhar decided to use special relativity to refine the calculations of his these director. Thus he finds the maximum mass for a white dwarf, the famous limit beyond which a star, having exhausted its fuel, collapses under gravity. Eddington is opposed for a long time to Chandra and in 1939 he wrote a book about the structure of the stars who closed definitively the question to the community of obedient scientists. Yet the conclusion of Chandra was a harbinger of the existence of black holes. In 1939 Chandrasekhar publishes the first of its treaties, "Introduction to the Study of Stellar Structure." In 1943 he published another treatise "Principles of stellar dynamics". In 1950, "Radiative Transfer" deals with the theory of stellar atmospheres. In 1961 he published Hydrodynamic and hydromagnetic Stability. After the discovery of quasars and CMB, Chandrasekhar published several articles bright, on the theory of relativistic stars and their stabilities. Neutron Stars, Black Holes and Quasar are his favorite subjects of study. In 1983, he sums up all his work in "The Mathematical Theory of Black Holes."
Subrahmanyan Chandrasekhar, (born October 19, 1910, Lahore, India [now in Pakistan]—died August 21, 1995, Chicago, Illinois, U.S.), Indian-born American astrophysicist who, with William A. Fowler, won the 1983 Nobel Prize for Physics for key discoveries that led to the currently accepted theory on the later evolutionary stages of massive stars. Chandrasekhar was the nephew of Sir Chandrasekhara Venkata Raman, who won the Nobel Prize for Physics in 1930. Chandrasekhar was educated at Presidency College, at the University of Madras, and at Trinity College, Cambridge. From 1933 to 1936 he held a position at Trinity. Chandrasekhar, Subrahmanyan; black hole Chandrasekhar, Subrahmanyan; black hole An overview of Subrahmanyan Chandrasekhar's contribution to the understanding of black holes. © Open University (A Britannica Publishing Partner) See all videos for this article By the early 1930s, scientists had concluded that, after converting all of their hydrogen to helium, stars lose energy and contract under the influence of their own gravity. These stars, known as white dwarf stars, contract to about the size of Earth, and the electrons and nuclei of their constituent atoms are compressed to a state of extremely high density. Chandrasekhar determined what is known as the Chandrasekhar limit—that a star having a mass more than 1.44 times that of the Sun does not form a white dwarf but instead continues to collapse, blows off its gaseous envelope in a supernova explosion, and becomes a neutron star. An even more massive star continues to collapse and becomes a black hole. These calculations contributed to the eventual understanding of supernovas, neutron stars, and black holes. Chandrasekhar came up with the idea for a limit on his voyage to England in 1930. However, his ideas met strong opposition, particularly from English astronomer Arthur Eddington, and took years to be generally accepted. Chandrasekhar joined the staff of the University of Chicago, rising from assistant professor of astrophysics (1938) to Morton D. Hull distinguished service professor of astrophysics (1952), and became a U.S. citizen in 1953. He did important work on energy transfer by radiation in stellar atmospheres and convection on the solar surface. He also attempted to develop the mathematical theory of black holes, describing his work in The Mathematical Theory of Black Holes (1983). Get exclusive access to content from our 1768 First Edition with your subscription. Subscribe today Chandrasekhar was awarded the Gold Medal of the Royal Astronomical Society in 1953, the Royal Medal of the Royal Society in 1962, and the Copley Medal of the Royal Society in 1984. His other books included An Introduction to the Study of Stellar Structure (1939), Principles of Stellar Dynamics (1942), Radiative Transfer (1950), Hydrodynamic and Hydromagnetic Stability (1961), Truth and Beauty: Aesthetics and Motivations in Science (1987), and Newton’s Principia for the Common Reader (1995). Subrahmanyan Chandrasekhar FRS[1] (19 October 1910 – 21 August 1995)[3] was an American astrophysicist of Indian origin who spent his professional life in the United States.[4] He was awarded the 1983 Nobel Prize for Physics with William A. Fowler for "...theoretical studies of the physical processes of importance to the structure and evolution of the stars". His mathematical treatment of stellar evolution yielded many of the current theoretical models of the later evolutionary stages of massive stars and black holes.[5][6] The Chandrasekhar limit is named after him. Chandrasekhar worked on a wide variety of physical problems in his lifetime, contributing to the contemporary understanding of stellar structure, white dwarfs, stellar dynamics, stochastic process, radiative transfer, the quantum theory of the hydrogen anion, hydrodynamic and hydromagnetic stability, turbulence, equilibrium and the stability of ellipsoidal figures of equilibrium, general relativity, mathematical theory of black holes and theory of colliding gravitational waves.[7] At the University of Cambridge, he developed a theoretical model explaining the structure of white dwarf stars that took into account the relativistic variation of mass with the velocities of electrons that comprise their degenerate matter. He showed that the mass of a white dwarf could not exceed 1.44 times that of the Sun – the Chandrasekhar limit. Chandrasekhar revised the models of stellar dynamics first outlined by Jan Oort and others by considering the effects of fluctuating gravitational fields within the Milky Way on stars rotating about the galactic centre. His solution to this complex dynamical problem involved a set of twenty partial differential equations, describing a new quantity he termed 'dynamical friction', which has the dual effects of decelerating the star and helping to stabilize clusters of stars. Chandrasekhar extended this analysis to the interstellar medium, showing that clouds of galactic gas and dust are distributed very unevenly. Chandrasekhar studied at Presidency College, Madras (now Chennai) and the University of Cambridge. A long-time professor at the University of Chicago, he did some of his studies at the Yerkes Observatory, and served as editor of The Astrophysical Journal from 1952 to 1971. He was on the faculty at Chicago from 1937 until his death in 1995 at the age of 84, and was the Morton D. Hull Distinguished Service Professor of Theoretical Astrophysics.[8] Contents 1 Early life and education 2 At Cambridge University 3 Career and research 3.1 Early career 3.2 World War II 3.3 Philosophy of systematization 3.4 Work with students 3.5 Other activities 4 Personal life 5 Awards, honours and legacy 5.1 Nobel prize 5.2 Other awards 5.3 Legacy 6 Publications 6.1 Books 6.2 Notes 6.3 Journals 6.4 Books and articles about Chandrasekhar 7 References 8 External links Early life and education Chandrasekhar was born on 19 October 1910 in Lahore, Punjab, British India (now Pakistan) in a Tamil Iyer family,[9] to Sita Balakrishnan (1891–1931) and Chandrasekhara Subrahmanya Ayyar (1885–1960)[10] who was stationed in Lahore as Deputy Auditor General of the Northwestern Railways at the time of Chandrasekhar's birth. He had two elder sisters, Rajalakshmi and Balaparvathi, three younger brothers, Vishwanathan, Balakrishnan, and Ramanathan and four younger sisters, Sarada, Vidya, Savitri, and Sundari. His paternal uncle was the Indian physicist and Nobel laureate C. V. Raman. His mother was devoted to intellectual pursuits, had translated Henrik Ibsen's A Doll's House into Tamil and is credited with arousing Chandra's intellectual curiosity at an early age.[11] The family moved from Lahore to Allahabad in 1916, and finally settled in Madras in 1918. Chandrasekhar was tutored at home until the age of 12.[11] In middle school his father would teach him Mathematics and Physics and his mother would teach him Tamil. He later attended the Hindu High School, Triplicane, Madras during the years 1922–25. Subsequently, he studied at Presidency College, Madras from 1925 to 1930, writing his first paper, "The Compton Scattering and the New Statistics", in 1929 after being inspired by a lecture by Arnold Sommerfeld.[12] He obtained his bachelor's degree, BSc (Hon.), in physics, in June 1930. In July 1930, Chandrasekhar was awarded a Government of India scholarship to pursue graduate studies at the University of Cambridge, where he was admitted to Trinity College, Cambridge, secured by R. H. Fowler with whom he communicated his first paper. During his travels to England, Chandrasekhar spent his time working out the statistical mechanics of the degenerate electron gas in white dwarf stars, providing relativistic corrections to Fowler's previous work (see Legacy below). At Cambridge University In his first year at Cambridge, as a research student of Fowler, Chandrasekhar spent his time calculating mean opacities and applying his results to the construction of an improved model for the limiting mass of the degenerate star. At the meetings of the Royal Astronomical Society, he met E. A. Milne. At the invitation of Max Born he spent the summer of 1931, his second year of post-graduate studies, at Born's institute at Göttingen, working on opacities, atomic absorption coefficients, and model stellar photospheres. On the advice of P. A. M. Dirac, he spent his final year of graduate studies at the Institute for Theoretical Physics in Copenhagen, where he met Niels Bohr. After receiving a bronze medal for his work on degenerate stars, in the summer of 1933, Chandrasekhar was awarded his PhD degree at Cambridge with a thesis among his four papers on rotating self-gravitating polytropes. On 9 October, he was elected to a Prize Fellowship at Trinity College for the period 1933–1937, becoming only the second Indian to receive a Trinity Fellowship after Srinivasa Ramanujan 16 years earlier. He had been so certain of failing to obtain the fellowship that he had already made arrangements to study under Milne that autumn at Oxford, even going to the extent of renting a flat there.[12] During this time, Chandrasekhar became acquainted with British physicist Sir Arthur Eddington. In an infamous encounter at the Royal Astronomical Society in London in 1935, Eddington publicly ridiculed the concept of the Chandrasekhar limit.[11] Although Eddington would later be proved wrong by computers and the first positive identification of a black hole in 1972, this encounter caused Chandrasekhar to contemplate employment outside the UK. Later in life, on multiple occasions, Chandrasekhar expressed the view that Eddington's behaviour was in part racially motivated.[13] Career and research Early career In 1935, Chandrasekhar was invited by the Director of the Harvard Observatory, Harlow Shapley, to be a visiting lecturer in theoretical astrophysics for a three-month period. He travelled to the United States in December. During his visit to Harvard, Chandrasekhar greatly impressed Shapley, but declined his offer of a Harvard research fellowship. At the same time, Chandrasekhar met Gerald Kuiper, a noted Dutch astrophysical observationalist who was then a leading authority on white dwarfs. Kuiper had recently been recruited by Otto Struve, the Director of the Yerkes Observatory in Williams Bay, Wisconsin, which was run by the University of Chicago, and the university's President Robert Maynard Hutchins. Having known of Chandrasekhar, Struve was then considering him for one of three faculty posts in astrophysics, along with Kuiper; the other opening had been filled by Bengt Stromgren, a Danish theorist.[12] Following a praise-filled recommendation from Kuiper, Struve invited Chandrasekhar to Yerkes in March 1936 and offered him the job. Though Chandrasekhar was keenly interested, he initially declined the offer and left for England; after Hutchins sent a radiogram to Chandrasekhar during the voyage, he finally accepted, returning to Yerkes as an Assistant Professor of Theoretical Astrophysics in December 1936.[12] Chandrasekhar remained at the University of Chicago for his entire career. He was promoted to associate professor in 1941 and to full professor two years later at just 33 years of age.[12] In 1952, he became Morton D. Hull Distinguished Service Professor of Theoretical Astrophysics. In 1953, he and his wife, Lalitha Chandrasekhar, took American citizenship.[14] Famously, Chandrasekhar declined many offers from other universities, including one to succeed Henry Norris Russell, the preeminent American astronomer, as director of the Princeton University Observatory. After the Laboratory for Astrophysics and Space Research (LASR) was built by NASA in 1966 at the University, Chandrasekhar occupied one of the four corner offices on the second floor. (The other corners housed John A. Simpson, Peter Meyer, and Eugene N. Parker.) Chandrasekhar lived at 4800 Lake Shore Drive after the high-rise apartment complex was built in the late 1960s, and later at 5550 Dorchester Building. World War II During World War II, Chandrasekhar worked at the Ballistic Research Laboratory at the Aberdeen Proving Ground in Maryland. While there, he worked on problems of ballistics, resulting in reports such as 1943's On the decay of plane shock waves, Optimum height for the bursting of a 105mm shell, On the Conditions for the Existence of Three Shock Waves,[15] and The normal reflection of a blast wave.[16][7] Chandrasekhar's expertise in hydrodynamics led Robert Oppenheimer to invite him to join the Manhattan Project at Los Alamos, but delays in the processing of his security clearance prevented him from contributing to the project. It has been rumoured that he visited the Calutron project. Philosophy of systematization He wrote that his scientific research was motivated by his desire to participate in the progress of different subjects in science to the best of his ability, and that the prime motive underlying his work was systematization. "What a scientist tries to do essentially is to select a certain domain, a certain aspect, or a certain detail, and see if that takes its appropriate place in a general scheme which has form and coherence; and, if not, to seek further information which would help him to do that".[17] Chandrasekhar developed a unique style of mastering several fields of physics and astrophysics; consequently, his working life can be divided into distinct periods. He would exhaustively study a specific area, publish several papers in it and then write a book summarizing the major concepts in the field. He would then move on to another field for the next decade and repeat the pattern. Thus he studied stellar structure, including the theory of white dwarfs, during the years 1929 to 1939, and subsequently focused on stellar dynamics, theory of Brownian motion from 1939 to 1943. Next, he concentrated on the theory of radiative transfer and the quantum theory of the negative ion of hydrogen from 1943 to 1950. This was followed by sustained work on turbulence and hydrodynamic and hydromagnetic stability from 1950 to 1961. In the 1960s, he studied the equilibrium and the stability of ellipsoidal figures of equilibrium, and also general relativity. During the period, 1971 to 1983 he studied the mathematical theory of black holes, and, finally, during the late 80s, he worked on the theory of colliding gravitational waves.[7] Work with students Chandra worked closely with his students and expressed pride in the fact that over a 50-year period (from roughly 1930 to 1980), the average age of his co-author collaborators had remained the same, at around 30. He insisted that students address him as "Chandrasekhar" until they received their PhD degree, after which time they (as other colleagues) were encouraged to address him as "Chandra". When Chandrasekhar was working at the Yerkes Observatory in 1940s, he would drive 150 miles (240 km) to and fro every weekend to teach a course at the University of Chicago. Two of the students who took the course, Tsung-Dao Lee and Chen-Ning Yang, won the Nobel prize before he could get one for himself. Regarding classroom interactions during his lectures, noted astrophysicist Carl Sagan stated from firsthand experience that "frivolous questions" from unprepared students were "dealt with in the manner of a summary execution", while questions of merit "were given serious attention and response".[4] Other activities From 1952 to 1971 Chandrasekhar was editor of The Astrophysical Journal.[18] When Eugene Parker submitted a paper on his discovery of solar wind in 1957, two eminent reviewers rejected the paper. However, since Chandra as an editor could not find any mathematical flaws in Parker's work, he went ahead and published the paper in 1958.[19] During the years 1990 to 1995, Chandrasekhar worked on a project devoted to explaining the detailed geometric arguments in Sir Isaac Newton's Philosophiae Naturalis Principia Mathematica using the language and methods of ordinary calculus. The effort resulted in the book Newton's Principia for the Common Reader, published in 1995. Chandrasekhar was an honorary member of the International Academy of Science.[citation needed] Personal life Chandrasekhar married Lalitha Doraiswamy in September 1936. He had met her as a fellow student at Presidency College. Chandrasekhar was the nephew of C.V. Raman, who was awarded the Nobel Prize for Physics in 1930. He became a naturalized citizen of the U.S. in 1953. Many considered him as warm, positive, generous, unassuming, meticulous, and open to debate, while some others as private, intimidating, impatient and stubborn regarding non-scientific matters,[4] and unforgiving to those who ridiculed his work.[20] Chandrasekhar died of a sudden heart attack at the University of Chicago Hospital in 1995, having survived a prior heart attack in 1975.[4] He was survived by his wife, who died on 2 September 2013 at the age of 102.[21] She was a serious student of literature and western classical music.[20] Once when involved in a discussion about the Gita, Chandrasekhar said, "I should like to preface my remarks with a personal statement in order that my later remarks will not be misunderstood. I consider myself an atheist".[22] This was also confirmed many times in his other talks.[23] In an interview with Kevin Krisciunas at the University of Chicago, on 6 October 1987, Chandrasekhar commented: "Of course, he (Otto Struve) knew I was an atheist, and he never brought up the subject with me".[24] Awards, honours and legacy Nobel prize Chandrasekhar was awarded the Nobel Prize in Physics in 1983 for his studies on the physical processes important to the structure and evolution of stars. Chandrasekhar accepted this honor, but was upset the citation mentioned only his earliest work, seeing it as a denigration of a lifetime's achievement. He shared it with William A. Fowler. Other awards An exhibition on life and works of Subrahmanyan Chandrasekhar was held at Science City, Kolkata, on January, 2011. Elected a Fellow of the Royal Society (FRS) in 1944[1] Henry Norris Russell Lectureship (1949)[25] Bruce Medal (1952)[26] Gold Medal of the Royal Astronomical Society (1953)[27] Rumford Prize of the American Academy of Arts and Sciences (1957)[28] National Medal of Science, USA (1966)[29] Padma Vibhushan (1968) Henry Draper Medal of the National Academy of Sciences (1971)[30] Copley Medal of the Royal Society (1984) Honorary Fellow of the International Academy of Science (1988)[citation needed] Gordon J. Laing Award (1989) Jansky Lectureship before the National Radio Astronomy Observatory Humboldt Prize[when?] Legacy Chandrasekhar's most notable work is on the astrophysical Chandrasekhar limit. The limit gives the maximum mass of a white dwarf star, ~1.44 solar masses, or equivalently, the minimum mass that must be exceeded for a star to collapse into a neutron star or black hole (following a supernova). The limit was first calculated by Chandrasekhar in 1930 during his maiden voyage from India to Cambridge, England for his graduate studies. In 1979, NASA named the third of its four "Great Observatories" after Chandrasekhar. This followed a naming contest which attracted 6,000 entries from fifty states and sixty-one countries. The Chandra X-ray Observatory was launched and deployed by Space Shuttle Columbia on 23 July 1999. The Chandrasekhar number, an important dimensionless number of magnetohydrodynamics, is named after him. The asteroid 1958 Chandra is also named after Chandrasekhar. The Himalayan Chandra Telescope is named after him. In the Biographical Memoirs of Fellows of the Royal Society of London, R. J. Tayler wrote: "Chandrasekhar was a classical applied mathematician whose research was primarily applied in astronomy and whose like will probably never be seen again."[1] Chandrasekhar guided 45 students to their PhDs.[31] After his death, his widow Lalitha Chandrasekhar made a gift of his Nobel Prize money to the University of Chicago towards the establishment of the Subrahmanyan Chandrasekhar Memorial Fellowship. First awarded in the year 2000, this fellowship is given annually to an outstanding applicant to graduate school in the PhD programs of the Department of Physics or the Department of Astronomy and Astrophysics.[32] S. Chandrasekhar Prize of Plasma Physics is an award given by Association of Asia Pacific Physical Societies (AAPS) to outstanding plasma physicists, started in the year 2014.[33] The Chandra Astrophysics Institute (CAI) is a program offered for high school students who are interested in astrophysics mentored by MIT scientists[34] and sponsored by the Chandra X-ray Observatory.[35] American astronomer Carl Sagan, who studied mathematics under Chandrasekhar at the University of Chicago, praised him in the book The Demon-Haunted World: "I discovered what true mathematical elegance is from Subrahmanyan Chandrasekhar." On 19 October 2017, Google showed a Google Doodle in 28 countries honouring Chandrasekhar's 107th birthday and the Chandrasekhar limit.[36][37] In 2010, on account of Chandra's 100th birthday, University of Chicago conducted a symposium titled Chandrasekhar Centennial Symposium 2010 which was attended by leading astrophysicists such as Roger Penrose, Kip Thorne, Freeman Dyson, Jayant V. Narlikar, Rashid Sunyaev, G. Srinivasan, and Clifford Will. Its research talks were published in 2011 as a book titled Fluid flows to Black Holes: A tribute to S Chandrasekhar on his birth centenary.[38][39][40] ON STARS, THEIR EVOLUTION AND THEIR STABILITY Nobel lecture, 8 December, 1983 by 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. 164 Physics 1983 REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 13a 14. 15. 16. 17. 18. 19. 20. 21. 22. Eddington, A. S., The Internal Constitution of the Stars (Cambridge University Press, England, 1926), p. 16. Chandrasekhar, S., Mon. Not. Roy. Astr. Soc., 96, 644 (1936). Eddington, A. S., The Internal Constitution of the Stars (Cambridge University Press, England, 1926), p. 172. Fowler, R. H., Mon. Not. Roy. Astr. Soc., 87, 114 (1926). Chandrasekhar, S., Phil. Mag., 11, 592 (1931). Chandrasekhar, S., Astrophys.]., 74, 81 (1931). Chandrasekhar, S., Mon. Not. Roy. Astr. Soc., 91, 456 (1931). Chandrasekhar, S., Observatory, 57, 373 (1934). Chandrasekhar, S., Mon. Not. Roy. Astr. Soc., 95, 207 (1935). Chandrasekhar, S., Z.f. Astrophysik, 5, 321 (1932). Chandrasekhar, S., Observatory, 57, 93 (1934). Chandrasekhar, S., Astrophys. J., 140, 417 (1964); see also Phys. Rev. Lett., 12, 114 and 437 (1964). Chandrasekhar, S., Astrophys.J., 142, 1519 (1965). Chandrasekhar, S. and Lebovitz, N. R., Mon. Not. Roy. Astr. Soc., 207, 13 P (1984). Chandrasekhar, S. and Tooper, R. F., Astrophys .J., 139, 1396 (1964). Detweiler, S. and Lindblom, L., Astrophys.J. Supp., 53, 93 (1983). Chandrasekhar, S., Astrophys. J., 161, 561 (1970); see also Phys. Rev. Lett., 24, 611 I and 762 (1970). For an account of these matters pertaining to the classical ellipsoids see Chandrasekhar, S., Ellipsoidal Figures of Equilibrium (Yale University Press, New Haven, 1968). Chandrasekhar, S., Ellipsoidal Figures of Equilibrium (Yale University Press, New Haven, 1968), Chap. 5, 37. Friedman, J. L., Comm. Math. Phys., 62, 247 (1978); see also Friedman, J. L. and Schutz, B. F., Astrophys.J., 222, 281 (1977). Comins, N., Mon. Not. Roy. Astr. Soc., 189, 233 and 255 (1979). Friedman, J. L., Phys. Rev. Lett., 51, 11 (1983). The author’s investigations on the mathematical theory of black holes, continued over the years 1974- 1983, are summarized in his last book The Mathematica/ Theory of Black Holes (Clarendon Press, Oxford, 1983). The reader may wish to consult the following additional references: 1. Chandrasekhar, S., ‘Edward Arthur Milnc: his part in the development of modern astrophysics,’ Quuart.J. Roy. Astr. Soc., 21, 93-107 (1980). 2. Chandrasekhar, S., Eddington; The Most Distinguished Astrophysicist of His Time (Cambridge University Press, 1983). A native of Tamil Nadu, Chandrasekhar was born in Lahore (British India), October 19, 1910. Fellow in Chennai (formerly Madras) India, Chandra obtained his doctorate in 1933 at the age of 22 years at Trinity College, Cambridge University in the UK, under the direction of Ralph Howard Fowler (1889-1944). He got a job at the University of Chicago in 1937 and U.S. citizenship in 1953. This is one of the major theoretical astrophysicists of the 20th century. His research had oriented on the evolution of stars. It is he who found the limit (Chandrasekhar limit), from which a white dwarf becomes unstable. Above, she collapsed in neutron star and explodes as a supernova. In 1930, Chandrasekhar, who was 20 years, calculates the limit while traveling by boat to Bombay. He discovers that the famous Eddington and Fowler had forgotten to consider the effects of relativity in their calculations. His awards are numerous: - Price Rumford in 1957 (work on radiative transfer), - Bruce Medal in 1952, - Gold Medal of the Royal Astronomical Society in 1953, - National Medal of Science in 1966 - Henry Draper Medal in 1971, - Nobel Prize in Physics in 1983 with William Fowler for his theoretical studies of physical processes governing the structure and evolution of stars. - Copley Medal in 1984, Passion for astrophysics and quantum theory, he attended the age of 17 at a conference of Arnold Johannes Wilhelm Sommerfeld (1868-1951). The young Chandra discovered quantum mechanics of Heisenberg and Schrödinger. Sommerfeld has a decisive influence on his career as an astrophysicist. Cambridge is the best university for Theoretical Astrophysics, thanks to the astrophysicist Arthur Stanley Eddington (1882 - 1944) known for his work on Einstein's general relativity and the structure of stars. This is Eddington who piloted the expedition of the solar eclipse of 1919, which verified the predictions of Einstein's theory. Ralph Howard Fowler had studied the density of white dwarfs.   Chandrasekhar decided to use special relativity to refine the calculations of his these director. Thus he finds the maximum mass for a white dwarf, the famous limit beyond which a star, having exhausted its fuel, collapses under gravity. Eddington is opposed for a long time to Chandra and in 1939 he wrote a book about the structure of the stars who closed definitively the question to the community of obedient scientists. Yet the conclusion of Chandra was a harbinger of the existence of black holes. In 1939 Chandrasekhar publishes the first of its treaties, "Introduction to the Study of Stellar Structure." In 1943 he published another treatise "Principles of stellar dynamics". In 1950, "Radiative Transfer" deals with the theory of stellar atmospheres. In 1961 he published Hydrodynamic and hydromagnetic Stability. After the discovery of quasars and CMB, Chandrasekhar published several articles bright, on the theory of relativistic stars and their stabilities. Neutron Stars, Black Holes and Quasar are his favorite subjects of study. In 1983, he sums up all his work in "The Mathematical Theory of Black Holes." size of Earth compared to a white dwarf Image: size of Earth compared to a white dwarf.   Subrahmanyan Chandrasekhar Image: Subrahmanyan Chandrasekhar (1910 - 1995), called Chandra, is the nephew of Nobel laureate Chandrasekhara Venkata Raman physics laureate in 1930. Nobel Prize himself in 1983, his name was given to the asteroid Chandra (1958) and X-ray Space Telescope put into orbit by the Space Shuttle Columbia July 23, 1999. He was a member of the Royal Society from 1944 until his death. He died at 84, August 21, 1995 in Chicago in the United States. The Nobel Prize (/ˈnoʊbɛl/, NOH-bel; Swedish: Nobelpriset, [nʊˈbɛ̂lːˌpriːsɛt]; Norwegian: Nobelprisen) is a set of annual international awards bestowed in several categories by Swedish and Norwegian institutions in recognition of academic, cultural, or scientific advances. The will of the Swedish chemist, engineer and industrialist Alfred Nobel established the five Nobel prizes in 1895. The prizes in Chemistry, Literature, Peace, Physics, and Physiology or Medicine were first awarded in 1901.[1][3][4] The prizes are widely regarded as the most prestigious awards available in their respective fields.[5][6][7] In 1968, Sveriges Riksbank, Sweden's central bank, established the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel. The award is based on a donation received by the Nobel Foundation in 1968 from Sveriges Riksbank on the occasion of the bank's 300th anniversary. The first Prize in Economic Sciences was awarded to Ragnar Frisch and Jan Tinbergen in 1969. The Prize in Economic Sciences is awarded by the Royal Swedish Academy of Sciences, Stockholm, Sweden, according to the same principles as for the Nobel Prizes that have been awarded since 1901.[8] The Royal Swedish Academy of Sciences awards the Nobel Prize in Chemistry, the Nobel Prize in Physics, and the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel; the Nobel Assembly at the Karolinska Institute awards the Nobel Prize in Physiology or Medicine; the Swedish Academy grants the Nobel Prize in Literature; and the Norwegian Nobel Committee awards the Nobel Peace Prize. Between 1901 and 2018, the Nobel Prizes (and the Prizes in Economic Sciences, from 1969 on) were awarded 590 times to 935 people and organizations.[1] With some receiving the Nobel Prize more than once, this makes a total of 27 organizations and 908 individuals.[1][9] The prize ceremonies take place annually in Stockholm, Sweden (with the exception of the Peace Prize ceremony, which is held in Oslo, Norway). Each recipient (known as a "laureate") receives a gold medal, a diploma, and a sum of money that has been decided by the Nobel Foundation. (As of 2020, each prize is worth 9,000,000 SEK, or about US$935,366, €848,678, or £716,224.[10]) Medals made before 1980 were struck in 23-carat gold, and later in 18-carat green gold plated with a 24-carat gold coating. The prize is not awarded posthumously; however, if a person is awarded a prize and dies before receiving it, the prize may still be presented.[11] A prize may not be shared among more than three individuals, although the Nobel Peace Prize can be awarded to organizations of more than three people.[12] Contents 1 History 1.1 Nobel Foundation 1.1.1 Formation of Foundation 1.1.2 Foundation capital and cost 1.2 First prizes 1.3 Second World War 1.4 Prize in Economic Sciences 2 Award process 2.1 Nominations 2.2 Selection 2.3 Posthumous nominations 2.4 Recognition time lag 3 Award ceremonies 3.1 Nobel Banquet 3.2 Nobel lecture 4 Prizes 4.1 Medals 4.2 Diplomas 4.3 Award money 5 Controversies and criticisms 5.1 Controversial recipients 5.2 Overlooked achievements 5.3 Emphasis on discoveries over inventions 5.4 Gender disparity 6 Specially distinguished laureates 6.1 Multiple laureates 6.2 Family laureates 7 Refusals and constraints 8 Cultural impact 9 See also 10 References 10.1 Citations 10.2 Sources 11 Further reading 12 External links History A black and white photo of a bearded man in his fifties sitting in a chair. Alfred Nobel had the unpleasant surprise of reading his own obituary, which was titled The merchant of death is dead, in a French newspaper. Alfred Nobel (About this soundlisten (help·info)) was born on 21 October 1833 in Stockholm, Sweden, into a family of engineers.[13] He was a chemist, engineer, and inventor. In 1894, Nobel purchased the Bofors iron and steel mill, which he made into a major armaments manufacturer. Nobel also invented ballistite. This invention was a precursor to many smokeless military explosives, especially the British smokeless powder cordite. As a consequence of his patent claims, Nobel was eventually involved in a patent infringement lawsuit over cordite. Nobel amassed a fortune during his lifetime, with most of his wealth coming from his 355 inventions, of which dynamite is the most famous.[14] In 1888, Nobel was astonished to read his own obituary, titled The merchant of death is dead, in a French newspaper. It was Alfred's brother Ludvig who had died; the obituary was eight years premature. The article disconcerted Nobel and made him apprehensive about how he would be remembered. This inspired him to change his will.[15] On 10 December 1896, Alfred Nobel died in his villa in San Remo, Italy, from a cerebral haemorrhage. He was 63 years old.[16] Nobel wrote several wills during his lifetime. He composed the last over a year before he died, signing it at the Swedish–Norwegian Club in Paris on 27 November 1895.[17][18] To widespread astonishment, Nobel's last will specified that his fortune be used to create a series of prizes for those who confer the "greatest benefit on mankind" in physics, chemistry, physiology or medicine, literature, and peace.[19] Nobel bequeathed 94% of his total assets, 31 million SEK (c. US$186 million, €150 million in 2008), to establish the five Nobel Prizes.[20][21] Owing to skepticism surrounding the will, it was not approved by the Storting in Norway until 26 April 1897.[22] The executors of the will, Ragnar Sohlman and Rudolf Lilljequist, formed the Nobel Foundation to take care of the fortune and to organise the awarding of prizes.[23] Nobel's instructions named a Norwegian Nobel Committee to award the Peace Prize, the members of whom were appointed shortly after the will was approved in April 1897. Soon thereafter, the other prize-awarding organizations were designated. These were Karolinska Institute on 7 June, the Swedish Academy on 9 June, and the Royal Swedish Academy of Sciences on 11 June.[24] The Nobel Foundation reached an agreement on guidelines for how the prizes should be awarded; and, in 1900, the Nobel Foundation's newly created statutes were promulgated by King Oscar II.[19] In 1905, the personal union between Sweden and Norway was dissolved. Nobel Foundation Formation of Foundation Main article: Nobel Foundation A paper with stylish handwriting on it with the title "Testament" Alfred Nobel's will stated that 94% of his total assets should be used to establish the Nobel Prizes. According to his will and testament read in Stockholm on 30 December 1896, a foundation established by Alfred Nobel would reward those who serve humanity. The Nobel Prize was funded by Alfred Nobel's personal fortune. According to the official sources, Alfred Nobel bequeathed from the shares 94% of his fortune to the Nobel Foundation that now forms the economic base of the Nobel Prize.[citation needed] The Nobel Foundation was founded as a private organization on 29 June 1900. Its function is to manage the finances and administration of the Nobel Prizes.[25] In accordance with Nobel's will, the primary task of the Foundation is to manage the fortune Nobel left. Robert and Ludvig Nobel were involved in the oil business in Azerbaijan, and according to Swedish historian E. Bargengren, who accessed the Nobel family archive, it was this "decision to allow withdrawal of Alfred's money from Baku that became the decisive factor that enabled the Nobel Prizes to be established".[26] Another important task of the Nobel Foundation is to market the prizes internationally and to oversee informal administration related to the prizes. The Foundation is not involved in the process of selecting the Nobel laureates.[27][28] In many ways, the Nobel Foundation is similar to an investment company, in that it invests Nobel's money to create a solid funding base for the prizes and the administrative activities. The Nobel Foundation is exempt from all taxes in Sweden (since 1946) and from investment taxes in the United States (since 1953).[29] Since the 1980s, the Foundation's investments have become more profitable and as of 31 December 2007, the assets controlled by the Nobel Foundation amounted to 3.628 billion Swedish kronor (c. US$560 million).[30] According to the statutes, the Foundation consists of a board of five Swedish or Norwegian citizens, with its seat in Stockholm. The Chairman of the Board is appointed by the Swedish King in Council, with the other four members appointed by the trustees of the prize-awarding institutions. An Executive Director is chosen from among the board members, a Deputy Director is appointed by the King in Council, and two deputies are appointed by the trustees. However, since 1995, all the members of the board have been chosen by the trustees, and the Executive Director and the Deputy Director appointed by the board itself. As well as the board, the Nobel Foundation is made up of the prize-awarding institutions (the Royal Swedish Academy of Sciences, the Nobel Assembly at Karolinska Institute, the Swedish Academy, and the Norwegian Nobel Committee), the trustees of these institutions, and auditors.[30] Foundation capital and cost The capital of the Nobel Foundation today is invested 50% in shares, 20% bonds and 30% other investments (e.g. hedge funds or real estate). The distribution can vary by 10 percent.[31] At the beginning of 2008, 64% of the funds were invested mainly in American and European stocks, 20% in bonds, plus 12% in real estate and hedge funds.[32] In 2011, the total annual cost was approximately 120 million krona, with 50 million krona as the prize money. Further costs to pay institutions and persons engaged in giving the prizes were 27.4 million krona. The events during the Nobel week in Stockholm and Oslo cost 20.2 million krona. The administration, Nobel symposium, and similar items had costs of 22.4 million krona. The cost of the Economic Sciences prize of 16.5 Million krona is paid by the Sveriges Riksbank.[31] First prizes A black and white photo of a bearded man in his fifties sitting in a chair. Wilhelm Röntgen received the first Physics Prize for his discovery of X-rays. Once the Nobel Foundation and its guidelines were in place, the Nobel Committees began collecting nominations for the inaugural prizes. Subsequently, they sent a list of preliminary candidates to the prize-awarding institutions. The Nobel Committee's Physics Prize shortlist cited Wilhelm Röntgen's discovery of X-rays and Philipp Lenard's work on cathode rays. The Academy of Sciences selected Röntgen for the prize.[33][34] In the last decades of the 19th century, many chemists had made significant contributions. Thus, with the Chemistry Prize, the Academy "was chiefly faced with merely deciding the order in which these scientists should be awarded the prize".[35] The Academy received 20 nominations, eleven of them for Jacobus van 't Hoff.[36] Van 't Hoff was awarded the prize for his contributions in chemical thermodynamics.[37][38] The Swedish Academy chose the poet Sully Prudhomme for the first Nobel Prize in Literature. A group including 42 Swedish writers, artists, and literary critics protested against this decision, having expected Leo Tolstoy to be awarded.[39] Some, including Burton Feldman, have criticised this prize because they consider Prudhomme a mediocre poet. Feldman's explanation is that most of the Academy members preferred Victorian literature and thus selected a Victorian poet.[40] The first Physiology or Medicine Prize went to the German physiologist and microbiologist Emil von Behring. During the 1890s, von Behring developed an antitoxin to treat diphtheria, which until then was causing thousands of deaths each year.[41][42] The first Nobel Peace Prize went to the Swiss Jean Henri Dunant for his role in founding the International Red Cross Movement and initiating the Geneva Convention, and jointly given to French pacifist Frédéric Passy, founder of the Peace League and active with Dunant in the Alliance for Order and Civilization. Second World War In 1938 and 1939, Adolf Hitler's Third Reich forbade three laureates from Germany (Richard Kuhn, Adolf Friedrich Johann Butenandt, and Gerhard Domagk) from accepting their prizes.[43] Each man was later able to receive the diploma and medal.[44] Even though Sweden was officially neutral during the Second World War, the prizes were awarded irregularly. In 1939, the Peace Prize was not awarded. No prize was awarded in any category from 1940 to 1942, due to the occupation of Norway by Germany. In the subsequent year, all prizes were awarded except those for literature and peace.[45] During the occupation of Norway, three members of the Norwegian Nobel Committee fled into exile. The remaining members escaped persecution from the Germans when the Nobel Foundation stated that the Committee building in Oslo was Swedish property. Thus it was a safe haven from the German military, which was not at war with Sweden.[46] These members kept the work of the Committee going, but did not award any prizes. In 1944, the Nobel Foundation, together with the three members in exile, made sure that nominations were submitted for the Peace Prize and that the prize could be awarded once again.[43] Prize in Economic Sciences Main article: Nobel Memorial Prize in Economic Sciences Map of Nobel laureates by country In 1968, Sweden's central bank Sveriges Riksbank celebrated its 300th anniversary by donating a large sum of money to the Nobel Foundation to be used to set up a prize in honour of Alfred Nobel. The following year, the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel was awarded for the first time. The Royal Swedish Academy of Sciences became responsible for selecting laureates. The first laureates for the Economics Prize were Jan Tinbergen and Ragnar Frisch "for having developed and applied dynamic models for the analysis of economic processes".[47][48] The Board of the Nobel Foundation decided that after this addition, it would allow no further new prizes.[49] Award process The award process is similar for all of the Nobel Prizes, the main difference being who can make nominations for each of them.[50] File:Announcement Nobelprize Chemistry 2009-3.ogv The announcement of the laureates in Nobel Prize in Chemistry 2009 by Gunnar Öquist, permanent secretary of the Royal Swedish Academy of Sciences File:Announcement Nobelprize Literature 2009-1.ogv 2009 Nobel Prize in Literature announcement by Peter Englund in Swedish, English, and German Nominations Nomination forms are sent by the Nobel Committee to about 3,000 individuals, usually in September the year before the prizes are awarded. These individuals are generally prominent academics working in a relevant area. Regarding the Peace Prize, inquiries are also sent to governments, former Peace Prize laureates, and current or former members of the Norwegian Nobel Committee. The deadline for the return of the nomination forms is 31 January of the year of the award.[50][51] The Nobel Committee nominates about 300 potential laureates from these forms and additional names.[52] The nominees are not publicly named, nor are they told that they are being considered for the prize. All nomination records for a prize are sealed for 50 years from the awarding of the prize.[53][54] Selection The Nobel Committee then prepares a report reflecting the advice of experts in the relevant fields. This, along with the list of preliminary candidates, is submitted to the prize-awarding institutions.[55] The institutions meet to choose the laureate or laureates in each field by a majority vote. Their decision, which cannot be appealed, is announced immediately after the vote.[56] A maximum of three laureates and two different works may be selected per award. Except for the Peace Prize, which can be awarded to institutions, the awards can only be given to individuals.[57] Posthumous nominations Although posthumous nominations are not presently permitted, individuals who died in the months between their nomination and the decision of the prize committee were originally eligible to receive the prize. This has occurred twice: the 1931 Literature Prize awarded to Erik Axel Karlfeldt, and the 1961 Peace Prize awarded to UN Secretary General Dag Hammarskjöld. Since 1974, laureates must be thought alive at the time of the October announcement. There has been one laureate, William Vickrey, who in 1996 died after the prize (in Economics) was announced but before it could be presented.[58] On 3 October 2011, the laureates for the Nobel Prize in Physiology or Medicine were announced; however, the committee was not aware that one of the laureates, Ralph M. Steinman, had died three days earlier. The committee was debating about Steinman's prize, since the rule is that the prize is not awarded posthumously.[11] The committee later decided that as the decision to award Steinman the prize "was made in good faith", it would remain unchanged.[59] Recognition time lag Nobel's will provided for prizes to be awarded in recognition of discoveries made "during the preceding year". Early on, the awards usually recognised recent discoveries.[60] However, some of those early discoveries were later discredited. For example, Johannes Fibiger was awarded the 1926 Prize in Physiology or Medicine for his purported discovery of a parasite that caused cancer.[61] To avoid repeating this embarrassment, the awards increasingly recognised scientific discoveries that had withstood the test of time.[62][63][64] According to Ralf Pettersson, former chairman of the Nobel Prize Committee for Physiology or Medicine, "the criterion 'the previous year' is interpreted by the Nobel Assembly as the year when the full impact of the discovery has become evident."[63] A room with pictures on the walls. In the middle of the room there is a wooden table with chairs around it. The committee room of the Norwegian Nobel Committee The interval between the award and the accomplishment it recognises varies from discipline to discipline. The Literature Prize is typically awarded to recognise a cumulative lifetime body of work rather than a single achievement.[65][66] The Peace Prize can also be awarded for a lifetime body of work. For example, 2008 laureate Martti Ahtisaari was awarded for his work to resolve international conflicts.[67][68] However, they can also be awarded for specific recent events.[69] For instance, Kofi Annan was awarded the 2001 Peace Prize just four years after becoming the Secretary-General of the United Nations.[70] Similarly Yasser Arafat, Yitzhak Rabin, and Shimon Peres received the 1994 award, about a year after they successfully concluded the Oslo Accords.[71] Awards for physics, chemistry, and medicine are typically awarded once the achievement has been widely accepted. Sometimes, this takes decades – for example, Subrahmanyan Chandrasekhar shared the 1983 Physics Prize for his 1930s work on stellar structure and evolution.[72][73] Not all scientists live long enough for their work to be recognised. Some discoveries can never be considered for a prize if their impact is realised after the discoverers have died.[74][75][76] Award ceremonies Two men standing on a stage. The man to the left is clapping his hands and looking towards the other man. The second man is smiling and showing two items to an audience not seen on the image. The items are a diploma which includes a painting and a box containing a gold medal. Behind them is a blue pillar clad in flowers. A man in his fifties standing behind a desk with computers on it. On the desk is a sign reading "Kungl. Vetensk. Akad. Sigil". Left: Barack Obama after receiving the Nobel Peace Prize in Oslo City Hall from the hands of Norwegian Nobel Committee Chairman Thorbjørn Jagland in 2009; Right: Giovanni Jona-Lasinio presenting Yoichiro Nambu's Nobel Lecture at Aula Magna, Stockholm in 2008 Except for the Peace Prize, the Nobel Prizes are presented in Stockholm, Sweden, at the annual Prize Award Ceremony on 10 December, the anniversary of Nobel's death. The recipients' lectures are normally held in the days prior to the award ceremony. The Peace Prize and its recipients' lectures are presented at the annual Prize Award Ceremony in Oslo, Norway, usually on 10 December. The award ceremonies and the associated banquets are typically major international events.[77][78] The Prizes awarded in Sweden's ceremonies' are held at the Stockholm Concert Hall, with the Nobel banquet following immediately at Stockholm City Hall. The Nobel Peace Prize ceremony has been held at the Norwegian Nobel Institute (1905–1946), at the auditorium of the University of Oslo (1947–1989), and at Oslo City Hall (1990–present).[79] The highlight of the Nobel Prize Award Ceremony in Stockholm occurs when each Nobel laureate steps forward to receive the prize from the hands of the King of Sweden. In Oslo, the Chairman of the Norwegian Nobel Committee presents the Nobel Peace Prize in the presence of the King of Norway.[78][80] At first, King Oscar II did not approve of awarding grand prizes to foreigners. It is said[by whom?] that he changed his mind once his attention had been drawn to the publicity value of the prizes for Sweden.[81] Nobel Banquet Main article: Nobel Banquet A set table with a white table cloth. There are many plates and glasses plus a menu visible on the table. Table at the 2005 Nobel Banquet in Stockholm After the award ceremony in Sweden, a banquet is held in the Blue Hall at the Stockholm City Hall, which is attended by the Swedish Royal Family and around 1,300 guests. The Nobel Peace Prize banquet is held in Norway at the Oslo Grand Hotel after the award ceremony. Apart from the laureate, guests include the President of the Storting, on occasion the Swedish prime minister, and, since 2006, the King and Queen of Norway. In total, about 250 guests attend. Nobel lecture According to the statutes of the Nobel Foundation, each laureate is required to give a public lecture on a subject related to the topic of their prize.[82] The Nobel lecture as a rhetorical genre took decades to reach its current format.[83] These lectures normally occur during Nobel Week (the week leading up to the award ceremony and banquet, which begins with the laureates arriving in Stockholm and normally ends with the Nobel banquet), but this is not mandatory. The laureate is only obliged to give the lecture within six months of receiving the prize, but some have happened even later. For example, US President Theodore Roosevelt received the Peace Prize in 1906 but gave his lecture in 1910, after his term in office.[84] The lectures are organized by the same association which selected the laureates.[85] Prizes Medals The Nobel Foundation announced on 30 May 2012 that it had awarded the contract for the production of the five (Swedish) Nobel Prize medals to Svenska Medalj AB. Between 1902 and 2010, the Nobel Prize medals were minted by Myntverket (the Swedish Mint), Sweden's oldest company, which ceased operations in 2011 after 107 years. In 2011, the Mint of Norway, located in Kongsberg, made the medals. The Nobel Prize medals are registered trademarks of the Nobel Foundation.[86] Each medal features an image of Alfred Nobel in left profile on the obverse. The medals for physics, chemistry, physiology or medicine, and literature have identical obverses, showing the image of Alfred Nobel and the years of his birth and death. Nobel's portrait also appears on the obverse of the Peace Prize medal and the medal for the Economics Prize, but with a slightly different design. For instance, the laureate's name is engraved on the rim of the Economics medal.[87] The image on the reverse of a medal varies according to the institution awarding the prize. The reverse sides of the medals for chemistry and physics share the same design.[88] A heavily decorated paper with the name "Fritz Haber" on it. Laureates receive a heavily decorated diploma together with a gold medal and the prize money. Here Fritz Haber's diploma is shown, which he received for the development of a method to synthesise ammonia. All medals made before 1980 were struck in 23 carat gold. Since then, they have been struck in 18 carat green gold plated with 24 carat gold. The weight of each medal varies with the value of gold, but averages about 175 grams (0.386 lb) for each medal. The diameter is 66 millimetres (2.6 in) and the thickness varies between 5.2 millimetres (0.20 in) and 2.4 millimetres (0.094 in).[89] Because of the high value of their gold content and tendency to be on public display, Nobel medals are subject to medal theft.[90][91][92] During World War II, the medals of German scientists Max von Laue and James Franck were sent to Copenhagen for safekeeping. When Germany invaded Denmark, Hungarian chemist (and Nobel laureate himself) George de Hevesy dissolved them in aqua regia (nitro-hydrochloric acid), to prevent confiscation by Nazi Germany and to prevent legal problems for the holders. After the war, the gold was recovered from solution, and the medals re-cast.[93] Diplomas Nobel laureates receive a diploma directly from the hands of the King of Sweden, or in the case of the peace prize, the Chairman of the Norwegian Nobel Committee. Each diploma is uniquely designed by the prize-awarding institutions for the laureates that receive them.[87] The diploma contains a picture and text in Swedish which states the name of the laureate and normally a citation of why they received the prize. None of the Nobel Peace Prize laureates has ever had a citation on their diplomas.[94][95] Award money The laureates are given a sum of money when they receive their prizes, in the form of a document confirming the amount awarded.[87] The amount of prize money depends upon how much money the Nobel Foundation can award each year. The purse has increased since the 1980s, when the prize money was 880,000 SEK per prize (c. 2.6 million SEK altogether, US$350,000 today). In 2009, the monetary award was 10 million SEK (US$1.4 million).[96][97] In June 2012, it was lowered to 8 million SEK.[98] If two laureates share the prize in a category, the award grant is divided equally between the recipients. If there are three, the awarding committee has the option of dividing the grant equally, or awarding one-half to one recipient and one-quarter to each of the others.[99][100][101] It is common for recipients to donate prize money to benefit scientific, cultural, or humanitarian causes.[102][103] Controversies and criticisms Main article: Nobel Prize controversies Controversial recipients When it was announced that Henry Kissinger was to be awarded the Peace Prize, two of the Norwegian Nobel Committee members resigned in protest. Among other criticisms, the Nobel Committees have been accused of having a political agenda, and of omitting more deserving candidates. They have also been accused of Eurocentrism, especially for the Literature Prize.[104][105][106] Peace Prize Among the most criticised Nobel Peace Prizes was the one awarded to Henry Kissinger and Lê Đức Thọ. This led to the resignation of two Norwegian Nobel Committee members.[107] Kissinger and Thọ were awarded the prize for negotiating a ceasefire between North Vietnam and the United States in January 1973. However, when the award was announced, both sides were still engaging in hostilities.[108] Critics sympathetic to the North announced that Kissinger was not a peace-maker but the opposite, responsible for widening the war. Those hostile to the North and what they considered its deceptive practices during negotiations were deprived of a chance to criticise Lê Đức Thọ, as he declined the award.[53][109] Yasser Arafat, Shimon Peres, and Yitzhak Rabin received the Peace Prize in 1994 for their efforts in making peace between Israel and Palestine.[53][110] Immediately after the award was announced, one of the five Norwegian Nobel Committee members denounced Arafat as a terrorist and resigned.[111] Additional misgivings about Arafat were widely expressed in various newspapers.[112] Another controversial Peace Prize was that awarded to Barack Obama in 2009.[113] Nominations had closed only eleven days after Obama took office as President of the United States, but the actual evaluation occurred over the next eight months.[114] Obama himself stated that he did not feel deserving of the award, or worthy of the company in which it would place him.[115][116] Past Peace Prize laureates were divided, some saying that Obama deserved the award, and others saying he had not secured the achievements to yet merit such an accolade. Obama's award, along with the previous Peace Prizes for Jimmy Carter and Al Gore, also prompted accusations of a left-wing bias.[117] Literature Prize The award of the 2004 Literature Prize to Elfriede Jelinek drew a protest from a member of the Swedish Academy, Knut Ahnlund. Ahnlund resigned, alleging that the selection of Jelinek had caused "irreparable damage to all progressive forces, it has also confused the general view of literature as an art". He alleged that Jelinek's works were "a mass of text shovelled together without artistic structure".[118][119] The 2009 Literature Prize to Herta Müller also generated criticism. According to The Washington Post, many US literary critics and professors were ignorant of her work.[120] This made those critics feel the prizes were too Eurocentric.[121] Science prizes In 1949, the neurologist António Egas Moniz received the Physiology or Medicine Prize for his development of the prefrontal leucotomy. The previous year, Dr. Walter Freeman had developed a version of the procedure which was faster and easier to carry out. Due in part to the publicity surrounding the original procedure, Freeman's procedure was prescribed without due consideration or regard for modern medical ethics. Endorsed by such influential publications as The New England Journal of Medicine, leucotomy or "lobotomy" became so popular that about 5,000 lobotomies were performed in the United States in the three years immediately following Moniz's receipt of the Prize.[122][123] Overlooked achievements The Norwegian Nobel Committee declined to award a prize in 1948, the year of Gandhi's death, on the grounds that "there was no suitable living candidate." The Norwegian Nobel Committee confirmed that Mahatma Gandhi was nominated for the Peace Prize in 1937–39, 1947 and, a few days before he was assassinated in January 1948.[124] Later, members of the Norwegian Nobel Committee expressed regret that he was not given the prize.[125] Geir Lundestad, Secretary of Norwegian Nobel Committee in 2006, said, "The greatest omission in our 106 year history is undoubtedly that Mahatma Gandhi never received the Nobel Peace prize. Gandhi could do without the Nobel Peace prize. Whether Nobel committee can do without Gandhi is the question".[126] In 1948, the year of Gandhi's death, the Nobel Committee declined to award a prize on the grounds that "there was no suitable living candidate" that year.[125][127] Later, when the 14th Dalai Lama was awarded the Peace Prize in 1989, the chairman of the committee said that this was "in part a tribute to the memory of Mahatma Gandhi".[128] Other high-profile individuals with widely recognised contributions to peace have been missed out. Foreign Policy lists Eleanor Roosevelt, Václav Havel, Ken Saro-Wiwa, Sari Nusseibeh, and Corazon Aquino as people who "never won the prize, but should have".[129] In 1965, UN Secretary General U Thant was informed by the Norwegian Permanent Representative to the UN that he would be awarded that year's prize and asked whether or not he would accept. He consulted staff and later replied that he would. At the same time, Chairman Gunnar Jahn of the Nobel Peace prize committee, lobbied heavily against giving U Thant the prize and the prize was at the last minute awarded to UNICEF. The rest of the committee all wanted the prize to go to U Thant, for his work in defusing the Cuban Missile Crisis, ending the war in the Congo, and his ongoing work to mediate an end to the Vietnam War. The disagreement lasted three years and in 1966 and 1967 no prize was given, with Gunnar Jahn effectively vetoing an award to U Thant.[130][131] James Joyce, one of the controversial omissions of the Literature Prize The Literature Prize also has controversial omissions. Adam Kirsch has suggested that many notable writers have missed out on the award for political or extra-literary reasons. The heavy focus on European and Swedish authors has been a subject of criticism.[132][133] The Eurocentric nature of the award was acknowledged by Peter Englund, the 2009 Permanent Secretary of the Swedish Academy, as a problem with the award and was attributed to the tendency for the academy to relate more to European authors.[134] This tendency towards European authors still leaves some European writers on a list of notable writers that have been overlooked for the Literature Prize, including Europe's Leo Tolstoy, Anton Chekhov, J. R. R. Tolkien, Émile Zola, Marcel Proust, Vladimir Nabokov, James Joyce, August Strindberg, Simon Vestdijk, Karel Čapek, the New World's Jorge Luis Borges, Ezra Pound, John Updike, Arthur Miller, Mark Twain, and Africa's Chinua Achebe.[135] Candidates can receive multiple nominations the same year. Gaston Ramon received a total of 155[136] nominations in physiology or medicine from 1930 to 1953, the last year with public nomination data for that award as of 2016. He died in 1963 without being awarded. Pierre Paul Émile Roux received 115[137] nominations in physiology or medicine, and Arnold Sommerfeld received 84[138] in physics. These are the three most nominated scientists without awards in the data published as of 2016.[139] Otto Stern received 79[140] nominations in physics 1925–43 before being awarded in 1943.[141] The strict rule against awarding a prize to more than three people is also controversial.[142] When a prize is awarded to recognise an achievement by a team of more than three collaborators, one or more will miss out. For example, in 2002, the prize was awarded to Koichi Tanaka and John Fenn for the development of mass spectrometry in protein chemistry, an award that did not recognise the achievements of Franz Hillenkamp and Michael Karas of the Institute for Physical and Theoretical Chemistry at the University of Frankfurt.[143][144] According to one of the nominees for the prize in physics, the three person limit deprived him and two other members of his team of the honor in 2013: the team of Carl Hagen, Gerald Guralnik, and Tom Kibble published a paper in 1964 that gave answers to how the cosmos began, but did not share the 2013 Physics Prize awarded to Peter Higgs and François Englert, who had also published papers in 1964 concerning the subject. All five physicists arrived at the same conclusion, albeit from different angles. Hagen contends that an equitable solution is to either abandon the three limit restriction, or expand the time period of recognition for a given achievement to two years.[145] Similarly, the prohibition of posthumous awards fails to recognise achievements by an individual or collaborator who dies before the prize is awarded. The Economics Prize was not awarded to Fischer Black, who died in 1995, when his co-author Myron Scholes received the honor in 1997 for their landmark work on option pricing along with Robert C. Merton, another pioneer in the development of valuation of stock options. In the announcement of the award that year, the Nobel committee prominently mentioned Black's key role. Political subterfuge may also deny proper recognition. Lise Meitner and Fritz Strassmann, who co-discovered nuclear fission along with Otto Hahn, may have been denied a share of Hahn's 1944 Nobel Chemistry Award due to having fled Germany when the Nazis came to power.[146] The Meitner and Strassmann roles in the research was not fully recognised until years later, when they joined Hahn in receiving the 1966 Enrico Fermi Award. Emphasis on discoveries over inventions Alfred Nobel left his fortune to finance annual prizes to be awarded "to those who, during the preceding year, shall have conferred the greatest benefit on mankind".[147] He stated that the Nobel Prizes in Physics should be given "to the person who shall have made the most important 'discovery' or 'invention' within the field of physics". Nobel did not emphasise discoveries, but they have historically been held in higher respect by the Nobel Prize Committee than inventions: 77% of the Physics Prizes have been given to discoveries, compared with only 23% to inventions. Christoph Bartneck and Matthias Rauterberg, in papers published in Nature and Technoetic Arts, have argued this emphasis on discoveries has moved the Nobel Prize away from its original intention of rewarding the greatest contribution to society.[148][149] Gender disparity See also: List of female Nobel laureates In terms of the most prestigious awards in STEM fields, only a small proportion have been awarded to women. Out of 210 laureates in Physics, 181 in Chemistry and 216 in Medicine between 1901 and 2018, there were only three female laureates in physics, five in chemistry and 12 in medicine.[150][151][152][153] Factors proposed to contribute to the discrepancy between this and the roughly equal human sex ratio include biased nominations, fewer women than men being active in the relevant fields, Nobel Prizes typically being awarded decades after the research was done (reflecting a time when gender bias in the relevant fields was greater), a greater delay in awarding Nobel Prizes for women's achievements making longevity a more important factor for women (Nobel Prizes are not awarded posthumously), and a tendency to omit women from jointly awarded Nobel Prizes.[154][155][156][157][158][159] Despite these factors, Marie Curie is to date the only person awarded Nobel Prizes in two different sciences (Physics in 1903, Chemistry in 1911); she is one of only three people who have received two Nobel Prizes in sciences (see Multiple laureates below). Specially distinguished laureates Multiple laureates A black and white portrait of a woman in profile. Marie Curie, one of four people who have received the Nobel Prize twice (Physics and Chemistry) Four people have received two Nobel Prizes. Marie Curie received the Physics Prize in 1903 for her work on radioactivity and the Chemistry Prize in 1911 for the isolation of pure radium,[160] making her the only person to be awarded a Nobel Prize in two different sciences. Linus Pauling was awarded the 1954 Chemistry Prize for his research into the chemical bond and its application to the structure of complex substances. Pauling was also awarded the Peace Prize in 1962 for his activism against nuclear weapons, making him the only laureate of two unshared prizes. John Bardeen received the Physics Prize twice: in 1956 for the invention of the transistor and in 1972 for the theory of superconductivity.[161] Frederick Sanger received the prize twice in Chemistry: in 1958 for determining the structure of the insulin molecule and in 1980 for inventing a method of determining base sequences in DNA.[162][163] Two organizations have received the Peace Prize multiple times. The International Committee of the Red Cross received it three times: in 1917 and 1944 for its work during the world wars; and in 1963 during the year of its centenary.[164][165][166] The United Nations High Commissioner for Refugees has been awarded the Peace Prize twice for assisting refugees: in 1954 and 1981.[167] Family laureates The Curie family has received the most prizes, with four prizes awarded to five individual laureates. Marie Curie received the prizes in Physics (in 1903) and Chemistry (in 1911). Her husband, Pierre Curie, shared the 1903 Physics prize with her.[168] Their daughter, Irène Joliot-Curie, received the Chemistry Prize in 1935 together with her husband Frédéric Joliot-Curie. In addition, the husband of Marie Curie's second daughter, Henry Labouisse, was the director of UNICEF when he accepted the Nobel Peace Prize in 1965 on that organisation's behalf.[169] Although no family matches the Curie family's record, there have been several with two laureates. The husband-and-wife team of Gerty Cori and Carl Ferdinand Cori shared the 1947 Prize in Physiology or Medicine[170] as did the husband-and-wife team of May-Britt Moser and Edvard Moser in 2014 (along with John O'Keefe).[171] J. J. Thomson was awarded the Physics Prize in 1906 for showing that electrons are particles. His son, George Paget Thomson, received the same prize in 1937 for showing that they also have the properties of waves.[172] William Henry Bragg and his son, William Lawrence Bragg, shared the Physics Prize in 1915 for inventing the X-ray crystallography.[173] Niels Bohr was awarded the Physics prize in 1922, as was his son, Aage Bohr, in 1975.[169][174] Manne Siegbahn, who received the Physics Prize in 1924, was the father of Kai Siegbahn, who received the Physics Prize in 1981.[169][175] Hans von Euler-Chelpin, who received the Chemistry Prize in 1929, was the father of Ulf von Euler, who was awarded the Physiology or Medicine Prize in 1970.[169] C. V. Raman was awarded the Physics Prize in 1930 and was the uncle of Subrahmanyan Chandrasekhar, who was awarded the same prize in 1983.[176][177] Arthur Kornberg received the Physiology or Medicine Prize in 1959; Kornberg's son, Roger later received the Chemistry Prize in 2006.[178] Jan Tinbergen, who was awarded the first Economics Prize in 1969, was the brother of Nikolaas Tinbergen, who received the 1973 Physiology or Medicine Prize.[169] Alva Myrdal, Peace Prize laureate in 1982, was the wife of Gunnar Myrdal who was awarded the Economics Prize in 1974.[169] Economics laureates Paul Samuelson and Kenneth Arrow were brothers-in-law. Frits Zernike, who was awarded the 1953 Physics Prize, is the great-uncle of 1999 Physics laureate Gerard 't Hooft.[179] In 2019, married couple Abhijit Banerjee and Esther Duflo were awarded the Nobel prize for Economics[180]. Refusals and constraints A black and white portrait of a man in a suit and tie. Half of his face is in a shadow. Richard Kuhn, who was forced to decline his Nobel Prize in Chemistry Two laureates have voluntarily declined the Nobel Prize. In 1964, Jean-Paul Sartre was awarded the Literature Prize but refused, stating, "A writer must refuse to allow himself to be transformed into an institution, even if it takes place in the most honourable form."[181] Lê Đức Thọ, chosen for the 1973 Peace Prize for his role in the Paris Peace Accords, declined, stating that there was no actual peace in Vietnam.[182] During the Third Reich, Adolf Hitler hindered Richard Kuhn, Adolf Butenandt, and Gerhard Domagk from accepting their prizes. All of them were awarded their diplomas and gold medals after World War II. In 1958, Boris Pasternak declined his prize for literature due to fear of what the Soviet Union government might do if he travelled to Stockholm to accept his prize. In return, the Swedish Academy refused his refusal, saying "this refusal, of course, in no way alters the validity of the award."[182] The Academy announced with regret that the presentation of the Literature Prize could not take place that year, holding it back until 1989 when Pasternak's son accepted the prize on his behalf.[183][184] Aung San Suu Kyi was awarded the Nobel Peace Prize in 1991, but her children accepted the prize because she had been placed under house arrest in Burma; Suu Kyi delivered her speech two decades later, in 2012.[185] Liu Xiaobo was awarded the Nobel Peace Prize in 2010 while he and his wife were under house arrest in China as political prisoners, and he was unable to accept the prize in his lifetime. Cultural impact Being a symbol of scientific or literary achievement that is recognisable worldwide, the Nobel Prize is often depicted in fiction. This includes films like The Prize and Nobel Son about fictional Nobel laureates as well as fictionalised accounts of stories surrounding real prizes such as Nobel Chor, a film based on the theft of Rabindranath Tagore's prize.[186][187] The memorial symbol "Planet of Alfred Nobel" was opened in Dnipropetrovsk University of Economics and Law in 2008. On the globe, there are 802 Nobel laureates' reliefs made of a composite alloy obtained when disposing of military strategic missiles.[188][189] 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
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