Subrahmanyan Chandrasekhar

How to use this course

Subrahmanyan Chandrasekhar, known everywhere as Chandra, worked in an unusual rhythm. He would choose a field, study it for five to ten years, solve its central problems, and close the period with a book that set out the whole subject from the beginning. Then he moved on. He described his career as a quest after perspectives. This course follows those periods, from white dwarfs in 1930 to black holes in the 1980s.

The first period contains the most famous episode of his life. In 1935 Arthur Eddington, the leading astrophysicist of the day, publicly rejected Chandrasekhar’s upper mass limit for white dwarfs. Lessons 03 and 04 tell that story with the physics on both sides. The companion course on Eddington treats the same events with the same facts.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. SI units are used, except that \(G=c=1\) in Lesson 10.

Timeline

Lahore and Madras, 1910–1930

Subrahmanyan Chandrasekhar was born on 19 October 1910 in Lahore, then in British India and now in Pakistan, where his father, C. Subrahmanyan Ayyar, was an auditor for the North Western Railway. The family was Tamil and later settled in Madras (now Chennai). Chandrasekhar was the eldest son in a family of ten children. His father’s younger brother was C. V. Raman, who won the 1930 Nobel Prize in Physics for the effect that bears his name.

He was taught at home by his parents and private tutors until he was eleven, then attended the Hindu High School in Triplicane, Madras, from 1922 to 1925. At Presidency College, Madras, he began in physics in 1925. He read widely in the new quantum mechanics, much of it in original papers that reached Madras with a delay of months.

Sommerfeld and Fowler

In 1928 Arnold Sommerfeld visited Presidency College on a lecture tour. Chandrasekhar met him and learned that the old quantum theory he had been studying had been replaced by wave mechanics and Fermi–Dirac statistics. He later called the meeting the most important event of his life. Within months Chandrasekhar had applied the new statistics to the Compton scattering of radiation in stellar interiors. Ralph H. Fowler, in Cambridge, communicated the paper to the Proceedings of the Royal Society in 1929. Chandrasekhar also read Fowler’s 1926 paper on dense matter, which explained white dwarfs as stars supported by degenerate electrons. That paper set the direction of his first research.

He graduated with a BSc (Hons) in physics in June 1930. In July the Government of India awarded him a scholarship for research in England, and he chose Cambridge, to work with Fowler.

The voyage and Cambridge, 1930–1933

Chandrasekhar left India on 31 July 1930. During the sea voyage to England he returned to Fowler’s theory of white dwarfs. Fowler had treated the electrons as non-relativistic. Chandrasekhar computed the electron momenta at white-dwarf densities and saw that they approach \(m_ec\), so special relativity must be included. When he did so, he found that the equilibrium had a maximum mass. He was nineteen.

At Cambridge he was a research student of Fowler at Trinity College. His short note on “The Maximum Mass of Ideal White Dwarfs” appeared in the Astrophysical Journal in 1931, followed by a longer paper in the Monthly Notices. He found that Edmund Stoner in Leeds and Wilhelm Anderson in Tartu had reached related results in 1929–30, using stars of uniform density. Lev Landau published the limit independently in 1932. Chandrasekhar’s contribution was to treat the star properly as a self-gravitating polytrope, which gave the exact value and the full structure of the models (Lesson 04).

Copenhagen and the doctorate

On the advice of Paul Dirac, he spent 1932–33 at the Institute for Theoretical Physics in Copenhagen, where he met Niels Bohr and Léon Rosenfeld. He took his PhD at Cambridge in the summer of 1933, with a thesis on rotating self-gravitating polytropes. Edward Arthur Milne and Arthur Eddington examined it. In October 1933 he was elected to a Prize Fellowship at Trinity College for 1933–37, an unusual honour for an Indian student of that time.

Trinity and Eddington, 1933–1937

At Trinity, Chandrasekhar dined at the same high table as Eddington, who was a fellow of the college, and the two became friends. Eddington had hoped Chandrasekhar’s work would help settle his own disagreements with Milne about stellar models. In 1934, after a visit to the Soviet Union where Victor Ambartsumian encouraged him to finish the theory, Chandrasekhar returned to white dwarfs. He computed exact models for the full relativistic equation of state, integrating the equations numerically with a desk calculating machine. Eddington took a close interest in the work as it progressed.

The meeting of 11 January 1935

Chandrasekhar presented the results to the Royal Astronomical Society in London on 11 January 1935. Eddington spoke immediately afterwards, on a paper whose contents he had not disclosed to Chandrasekhar in advance. He did not dispute the mathematics. He argued that the relativistic formula for degeneracy pressure was wrong, because in his view it combined special relativity with a non-relativistic quantum theory in an illegitimate way. With his corrected formula every star could end as a stable white dwarf, and there would be no maximum mass. In print he called Chandrasekhar’s result almost a reductio ad absurdum of the relativistic formula. Milne also opposed Chandrasekhar’s conclusions. Most astronomers present, deferring to Eddington, assumed the young man was wrong.

Eddington repeated his criticism at the International Astronomical Union in Paris in July 1935, where Chandrasekhar was not given the chance to reply, and at Harvard in 1936. Bohr, Rosenfeld and Wolfgang Pauli told Chandrasekhar privately that Eddington’s argument had no basis, but they declined to say so in print. Christian Møller and Chandrasekhar (1935), Rudolf Peierls (1936), and Dirac, Peierls and Maurice Pryce (1942) published refutations of Eddington’s physics. Eddington never withdrew his view and restated it as late as 1940.

Chandrasekhar decided that he would not spend his career defending a correct result. In 1939 he wrote the theory into An Introduction to the Study of Stellar Structure and moved to other subjects. He and Eddington stayed on friendly terms and corresponded until Eddington’s death in 1944. Chandrasekhar admired him for the rest of his life, ranked him with Karl Schwarzschild as the greatest astronomer of his generation, and gave the Eddington centenary lectures in 1982, published as Eddington: The Most Distinguished Astrophysicist of His Time (1983).

Leaving England

At the invitation of Harlow Shapley he lectured at Harvard from January to March 1936. Otto Struve and Robert Hutchins then offered him a post at the University of Chicago’s Yerkes Observatory. He travelled to India and in September 1936 married Lalitha Doraiswamy, a fellow physics student from Presidency College. In January 1937 they arrived in Williams Bay, Wisconsin.

White dwarfs and the limiting mass

For the general reader. When a star like the Sun runs out of fuel, it shrinks into a white dwarf, a ball the size of the Earth. What holds it up is not heat but quantum mechanics: electrons cannot be squeezed into the same state. Chandrasekhar showed that this support has a limit. In a heavy enough star the electrons move near the speed of light, their pressure grows too slowly to resist gravity, and no balance is possible. The dividing line, about 1.4 times the mass of the Sun, is the Chandrasekhar limit.

The equation of state

At zero temperature, electrons fill all momentum states up to the Fermi momentum \(p_F\). With two spin states, the number density is \(n_e=8\pi p_F^3/(3h^3)\). The pressure is the momentum flux, \(P=\frac{8\pi}{3h^3}\int_0^{p_F}v(p)\,p^3\,dp\) with \(v=pc^2/E\). Writing \(x=p_F/(m_ec)\),

Here \(\mu_e\) is the number of nucleons per electron: 2 for helium, carbon or oxygen. The ions supply the mass; the electrons supply the pressure.

Polytropes

For \(P=K\rho^{1+1/n}\), hydrostatic equilibrium reduces to the Lane–Emden equation \(\xi^{-2}(\xi^2\theta')'=-\theta^n\), with \(\rho=\rho_c\theta^n\) and \(r=\alpha\xi\), \(\alpha^2=(n+1)K\rho_c^{1/n-1}/(4\pi G)\). The mass is \(M=4\pi\alpha^3\rho_c\,\omega_n\) with \(\omega_n=-\xi_1^2\theta'(\xi_1)\), and the radius is \(R=\alpha\xi_1\). Eliminating \(\rho_c\) gives

For \(n=3/2\) this is \(R\propto M^{-1/3}\): more massive white dwarfs are smaller. For \(n=3\), \(\alpha\propto\rho_c^{-1/3}\) and \(M\) does not depend on \(\rho_c\) at all.

In 1935 Chandrasekhar solved the problem exactly with the full equation of state. Setting \(y^2=1+x^2\) and \(\phi=y/y_0\) with \(y_0\) its central value, equilibrium becomes

a one-parameter family of equations interpolating between \(n=3/2\) (small \(y_0\)) and \(n=3\) (\(y_0\to\infty\)). Its solutions give the mass–radius curve in the figure at the top of this page.

The objection of 1935

Eddington accepted the mathematics but rejected the relativistic equation of state. He argued that the Fermi–Dirac counting of states was derived for a non-relativistic gas and could not legitimately be combined with the relativistic relation \(E^2=p^2c^2+m_e^2c^4\), and that the correct law was \(P\propto\rho^{5/3}\) at all densities. If so, \(n=3/2\) would hold for every white dwarf and any mass could reach equilibrium. The objection fails. The counting of states, \(2\cdot4\pi p^2\,dp\,V/h^3\), follows from quantizing a particle in a box and holds whatever the relation between energy and momentum. The Dirac equation for free electrons gives the same density of states. Peierls showed in 1936 that the result does not depend on the shape of the container, one of Eddington’s doubts. Elliott Lieb and Horng-Tzer Yau gave a rigorous derivation of the Chandrasekhar theory from quantum mechanics in 1987.

Observation confirms the theory. The masses of known white dwarfs cluster near \(0.6M_\odot\), none exceed the limit, and their radii follow the predicted mass–radius relation. Accreting white dwarfs that approach the limit explode as type Ia supernovae, and stellar cores that exceed it collapse to neutron stars or black holes.

Isothermal cores: the Schönberg–Chandrasekhar limit

For the general reader. When a star has burned the hydrogen at its centre, it is left with a core of helium that produces no energy and has the same temperature throughout. In 1942 Mario Schönberg and Chandrasekhar showed that such a core can support the layers above it only while it holds less than about a tenth of the star’s mass. Beyond that, the core contracts and the star swells into a red giant. The result explained why stars leave the main sequence and helped date star clusters.

Consider an isothermal core of mass \(M_c\), radius \(R_c\), temperature \(T_c\) and molecular weight \(\mu_c\), surrounded by an envelope that exerts pressure \(P_s\) on its surface. The virial theorem with surface pressure reads

where \(\beta\) is a constant of order unity fixed by the structure of the core. At fixed \(M_c\) and \(T_c\), the right side, divided by \(R_c^3\), has a maximum as a function of \(R_c\). So an isothermal core can withstand only a bounded surface pressure.

Sketch. Maximizing \(P_s(R_c)=\frac{3}{4\pi R_c^3}\big(\frac{M_ckT_c}{\mu_cm_H}-\frac{\beta GM_c^2}{3R_c}\big)\) gives \(P_{s,\max}\propto T_c^4/(\mu_c^4M_c^2)\). The envelope, matched at the core boundary where its temperature is \(T_c\), demands \(P_{\rm env}\propto T_c^4/(\mu_{\rm env}^4M^2)\) by homology relations for the envelope. Requiring \(P_{\rm env}\le P_{s,\max}\) gives \(M_c/M\lesssim C(\mu_{\rm env}/\mu_c)^2\). The constant \(0.37\) comes from the numerical models of Schönberg and Chandrasekhar.

A star whose helium core exceeds the limit cannot stay in thermal equilibrium. The core contracts on its thermal timescale, heats up, and the envelope expands: the star crosses rapidly to the red-giant branch. In low-mass stars the core becomes degenerate first and the limit does not apply.

Stellar dynamics and dynamical friction

For the general reader. In a galaxy, stars rarely collide, but each one feels the pull of many others. Chandrasekhar asked how these countless small tugs change a star’s motion over time. He found that a heavy body moving through a sea of lighter stars gathers a wake of them behind it, and the wake pulls it back. This drag, dynamical friction, makes globular clusters and satellite galaxies sink toward galactic centres and helps merge black holes.

Relaxation

A star of speed \(v\) passing another of mass \(m\) at impact parameter \(b\) receives a transverse kick \(\delta v\approx2Gm/(bv)\). In a system of \(N\) stars of radius \(R\), summing the squares of many random kicks over one crossing gives \(\Delta v^2/v^2\approx8\ln\Lambda/N\), with \(\Lambda\approx N\). The relaxation time is therefore about \(N/(8\ln N)\) crossing times, which is enormous for a galaxy and modest for a globular cluster. Chandrasekhar developed this theory in Principles of Stellar Dynamics (1942).

Dynamical friction

Only stars slower than \(M\) contribute, which gives the bracket. The drag is proportional to \(M\) itself, since a heavier body builds a heavier wake. In the same year Chandrasekhar wrote a long review, “Stochastic problems in physics and astronomy” (Reviews of Modern Physics, 1943), which treated dynamical friction and relaxation as a Fokker–Planck problem and became a standard reference on Brownian motion and random processes.

For a satellite on a circular orbit of radius \(r\) in a singular isothermal halo with circular speed \(v_c\), the formula gives the sinking time

Radiative transfer and the H-function

For the general reader. The light we receive from a star or a planet has been scattered many times on its way out. Calculating that journey exactly was a hard mathematical problem. In the 1940s Chandrasekhar found exact solutions by inventing new functions, now called Chandrasekhar’s H-functions. He also explained the polarization of light from the blue sky and showed which atom makes the Sun opaque.

In a plane-parallel atmosphere with optical depth \(\tau\) and direction cosine \(\mu\), the specific intensity obeys

for conservative isotropic scattering. The problem of a semi-infinite atmosphere with a constant outward flux, posed by Milne, is the basic model of a stellar atmosphere. Eddington’s approximation gives a source function linear in \(\tau\), \(S\propto\tau+\tfrac23\).

Chandrasekhar derived the H-equation from principles of invariance, an idea of Viktor Ambartsumian: the emergent radiation is unchanged if a thin layer is added to the top of a semi-infinite atmosphere. The same method, with matrix H-functions, gave him the exact solution for Rayleigh scattering with polarization, which explains the polarization pattern of the sunlit sky. His book Radiative Transfer (1950) collects this work. In the same period he computed the absorption coefficient of the negative hydrogen ion H⁻, which Rupert Wildt had proposed as the main source of opacity in the solar atmosphere.

Hydrodynamic and hydromagnetic stability

For the general reader. Heat a pan of liquid from below and, at some point, it starts to churn in regular cells. In the 1950s Chandrasekhar asked how rotation and magnetic fields change that threshold, questions that matter for the Sun, the Earth’s core and laboratory plasmas. He found that both inhibit convection, and his 1961 treatise became the standard reference on the stability of fluids.

A layer of fluid of depth \(d\), heated from below with temperature difference \(\Delta T\), has Rayleigh number \(Ra=g\alpha\Delta T\,d^3/(\nu\kappa)\), where \(\alpha\) is the expansion coefficient, \(\nu\) the viscosity and \(\kappa\) the thermal diffusivity. Linearizing the Boussinesq equations about the conducting state and seeking modes with horizontal wavenumber \(a/d\) gives, for stress-free boundaries, the marginal-stability curve \(Ra=(\pi^2+a^2)^3/a^2\). Rayleigh found in 1916 that its minimum is \(Ra_c=27\pi^4/4\approx657.5\) at \(a=\pi/\sqrt2\).

Rotation about the vertical with angular velocity \(\Omega\) has a similar effect, governed by the Taylor number \(Ta=4\Omega^2d^4/\nu^2\), with \(Ra=[(\pi^2+a^2)^3+\pi^2Ta]/a^2\). The 1961 book Hydrodynamic and Hydromagnetic Stability treats these problems, the stability of Couette flow between rotating cylinders, and the Rayleigh–Taylor and Kelvin–Helmholtz instabilities, all with the same analytic care. Many of its predictions were later confirmed in experiments, some carried out at Chicago at his suggestion.

Ellipsoids and relativistic instability

For the general reader. A spinning ball of fluid flattens. Spin it faster and, beyond a critical point, it prefers to become a three-axis ellipsoid. Newton, Maclaurin, Jacobi, Dedekind and Riemann studied these shapes. In the 1960s Chandrasekhar reworked the whole classical theory with a new tool, the tensor virial theorem. In the same decade he discovered that general relativity makes very compact stars unstable even when Newtonian theory says they are fine.

For a homogeneous spheroid of eccentricity \(e\) in uniform rotation, the equilibrium condition from the \(xx\) and \(zz\) components gives the Maclaurin relation

The Jacobi ellipsoids branch off at \(e=0.81267\), and \(\Omega^2/(\pi G\rho)\) reaches its maximum \(0.4493\) at \(e=0.92995\). Chandrasekhar and Norman Lebovitz used the virial equations of second and higher order to settle the stability of all these families, in work summarized in Ellipsoidal Figures of Equilibrium (1969). In 1970 Chandrasekhar showed that gravitational radiation makes Maclaurin spheroids unstable beyond the Jacobi bifurcation, the first case of what is now called the Chandrasekhar–Friedman–Schutz instability of rotating neutron stars.

Relativistic instability

In Newtonian theory a star is dynamically stable against radial pulsation when its adiabatic index exceeds \(4/3\). In 1964 Chandrasekhar derived the variational principle for radial oscillations in general relativity and found that the critical index rises.

The result showed that supermassive stars, which have \(\Gamma_1\) only slightly above \(4/3\), collapse long before their radius reaches the Schwarzschild radius. It was one of the first results of relativistic astrophysics, just as quasars were discovered.

The mathematical theory of black holes

For the general reader. In his sixties Chandrasekhar turned to black holes, the objects whose possibility his white-dwarf limit had first suggested. He set out to understand how a black hole responds when disturbed: by waves of gravity, light or matter. Over a decade he worked out the whole theory for rotating black holes and wrote it into a single book. He regarded black holes as the most perfect macroscopic objects in nature, since a few numbers describe them exactly.

Units \(G=c=1\). A perturbation of the Schwarzschild metric with angular dependence of degree \(l\) and time dependence \(e^{i\sigma t}\) reduces to a one-dimensional wave equation in the tortoise coordinate \(r_*=r+2M\ln(r/2M-1)\):

for axial perturbations (Regge and Wheeler, 1957). Polar perturbations give a different potential, found by Frank Zerilli in 1970. Chandrasekhar showed that the two potentials are related by a transformation that gives them identical reflection and transmission coefficients, and he traced this to a hidden structure of the Einstein equations. With Steven Detweiler (1975) he computed the quasi-normal frequencies with which a perturbed black hole rings down, now measured in gravitational-wave signals from merging black holes.

For rotating black holes, Saul Teukolsky had shown in 1972–73 that the perturbation equations separate in the Kerr geometry. Chandrasekhar built from this a complete account of the Kerr black hole, including the reflection and transmission of waves and the transformation theory linking its different descriptions. The Mathematical Theory of Black Holes (1983) is the result. The Kerr metric of mass \(M\) and angular momentum \(J=aM\) has horizons at

In his last research period, with Valeria Ferrari and Basilis Xanthopoulos, he studied exact solutions describing colliding plane gravitational waves.

Chicago, Yerkes and the Astrophysical Journal, 1937–1980

Chandrasekhar joined the University of Chicago in January 1937 and worked at its Yerkes Observatory in Williams Bay, Wisconsin, where Otto Struve was building a strong department. He taught graduate courses there and, in later years, also on the main Chicago campus. He became a full professor in 1943 and in 1952 Morton D. Hull Distinguished Service Professor. During the Second World War he worked as a consultant for the Ballistic Research Laboratory at Aberdeen Proving Ground in Maryland, writing reports on shock waves and blast waves.

He supervised about fifty doctoral students, among them Donald Osterbrock, Guido Münch, Anne Underhill, Arthur Code, Jeremiah Ostriker and Roland Winston. From 1952 to 1971 he edited the Astrophysical Journal, for most of that time almost single-handed. He made it the leading journal of the field and insisted on strict, careful refereeing, often reading papers himself.

He and Lalitha became United States citizens in 1953. In 1964 he declined an offer to return to Cambridge. He was elected a Fellow of the Royal Society in 1944, and received the Bruce Medal (1952), the Gold Medal of the Royal Astronomical Society (1953), the Royal Medal (1962), the National Medal of Science (1966) and the Henry Draper Medal (1971).

Nobel, Newton, death and legacy

In 1983 Chandrasekhar received half of the Nobel Prize in Physics, for theoretical studies of the physical processes that govern the structure and evolution of stars. The other half went to William A. Fowler for nuclear reactions in stars. The Copley Medal of the Royal Society followed in 1984. In the same year as the Nobel Prize he published his book on Eddington, a generous portrait of the man who had opposed him.

In his last years he wrote about science and art in Truth and Beauty: Aesthetics and Motivations in Science (1987), and studied Isaac Newton’s Principia proposition by proposition, giving his own proofs before reading Newton’s. Newton’s Principia for the Common Reader appeared in 1995. He died of heart failure in Chicago on 21 August 1995, aged 84. Lalitha Chandrasekhar survived him.

Legacy

• Compact objects: the Chandrasekhar limit is the starting point for neutron stars, black holes and type Ia supernovae, the standard candles that revealed the accelerating expansion of the universe.
• Stellar evolution and dynamics: the Schönberg–Chandrasekhar limit, dynamical friction and the theory of stellar relaxation.
• Mathematical physics: the H-functions, the Chandrasekhar number, the tensor virial theorem and the perturbation theory of black holes.
• Institutions: the Astrophysical Journal and a generation of Chicago students.
• Honours: NASA’s Chandra X-ray Observatory, launched in 1999, and the asteroid 1958 Chandra bear his name.

The dispute with Eddington remains part of his public image. Historians stress that it did not stop his career, that Eddington’s position was rejected by physicists within a few years, and that the two men remained friends. Chandrasekhar himself drew the lesson that even great scientists can hold to preconceived ideas.

Connections

People linked to Subrahmanyan Chandrasekhar across this collection. A name without a link has no page here yet. See Subrahmanyan Chandrasekhar in the network of all 337 lives →

Teachers and advisors

• Ralph H. Fowler 1930-1933

Students

• Anne Underhill
• Arthur Code
• Donald Osterbrock
• Guido Münch
• Jeremiah Ostriker
• Roland Winston

Influenced by

• Arnold Sommerfeld 1928
• Isaac Newton
• Edmund Stoner

Influenced

• J. Robert Oppenheimer

Colleagues and collaborators

• Niels Bohr 1932-1935
• Paul Dirac
• Wolfgang Pauli
• Basilis Xanthopoulos
• Edward Arthur Milne
• Lev Landau
• Mario Schönberg 1942
• Norman Lebovitz
• Otto Struve 1937-1950
• Steven Detweiler 1975
• Valeria Ferrari

Rivals

• Arthur Eddington 1935-1944

Family

• C. V. Raman

Also mentioned in this life

• Arthur Compton
• Richard Dedekind
• Albert Einstein
• Enrico Fermi
• Hermann von Helmholtz
• Carl Gustav Jacob Jacobi
• William Thomson, Lord Kelvin
• Colin Maclaurin
• Max Planck
• Lord Rayleigh
• Bernhard Riemann

Fields

• astrophysics 6 lives
• general relativity 4 lives

On CoursesHub

• General Relativity

Sources and further reading

Biography

1. Subrahmanyan Chandrasekhar, Biographical, NobelPrize.org.
2. Subrahmanyan Chandrasekhar, MacTutor History of Mathematics.
3. Obituary, Quarterly Journal of the Royal Astronomical Society, via MacTutor.
4. The Chandrasekhar–Eddington dispute, MacTutor.
5. Subrahmanyan Chandrasekhar, Wikipedia.
6. K. C. Wali, Chandra: A Biography of S. Chandrasekhar, University of Chicago Press, 1991.
7. R. J. Tayler, Subrahmanyan Chandrasekhar, Biographical Memoirs of Fellows of the Royal Society 42 (1996).
8. A. I. Miller, Empire of the Stars, Houghton Mifflin, 2005.
9. S. Chandrasekhar, Eddington: The Most Distinguished Astrophysicist of His Time, Cambridge University Press, 1983.

Physics

1. R. H. Fowler, On dense matter, MNRAS 87 (1926) 114–122.
2. S. Chandrasekhar, The maximum mass of ideal white dwarfs, ApJ 74 (1931) 81–82.
3. S. Chandrasekhar, The highly collapsed configurations of a stellar mass (second paper), MNRAS 95 (1935) 207–225.
4. A. S. Eddington, On “relativistic degeneracy”, MNRAS 95 (1935) 194–206.
5. S. Chandrasekhar, An Introduction to the Study of Stellar Structure, University of Chicago Press, 1939.
6. M. Schönberg, S. Chandrasekhar, On the evolution of the main-sequence stars, ApJ 96 (1942) 161.
7. S. Chandrasekhar, Dynamical friction I, ApJ 97 (1943) 255.
8. S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys. 15 (1943) 1–89.
9. S. Chandrasekhar, Radiative Transfer, Oxford, 1950.
10. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Oxford, 1961.
11. S. Chandrasekhar, Ellipsoidal Figures of Equilibrium, Yale, 1969.
12. S. Chandrasekhar, The Mathematical Theory of Black Holes, Oxford, 1983.
13. E. H. Lieb, H.-T. Yau, A rigorous examination of the Chandrasekhar theory of stellar collapse, ApJ 323 (1987) 140–144.
14. Chandrasekhar limit, Wikipedia (history and primary references).
15. Stellar structure and compact objects before 1940, arXiv:1703.09991.
16. S. Chandrasekhar, On stars, their evolution and their stability, Nobel lecture, 1983.