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# Subramanyan Chandrasekhar: A short Biography

Subrahmanyan Chandrasekhar was an Indian astrophysicist who was awarded the 1983 Nobel Prize in Physics along with William A. Fowler for the theoretical studies of the physical processes of importance to the structure and evolution of the stars.

Chandrasekhar was born on 19 October 1910 in Lahore, Punjab, British India (now in Pakistan). His parents were Chandrasekhara Subrahmanya Ayyar and Sita Balakrishnan. His father was a Deputy Auditor General of the Northwestern Railways stationed at Lahore at the time of Chandrasekhar’s birth.

The famous Indian physicist and Nobel laureate C.V. Raman was his paternal uncle. He was tutored at home until he was 12 years old. From 1922-to 25, he attended the Hindu High School, Triplicane, Madras. From 1925-to 30, he studied at Presidency College, Madras.

In 1930, Chandrasekhar obtained a Government of India scholarship to pursue graduate studies at the University of Cambridge. He was admitted to Trinity College, Cambridge. At the invitation of the German physicist Max Born, he spent the summer of 1931 at Born’s institute at Gottingen.

In 1935, Chandrasekhar moved to the United States after he was invited by the Director of the Harvard Observatory, Harlow Shapley to be a visiting lecturer in theoretical astrophysics for a three-month period. Through a recommendation from the famous Dutch astrophysicist Gerard Kuiper, Otto Struve, the Director of the Yerkes Observatory in Williams Bay, Wisconsin invited Chandrasekhar to his observatory in March 1936 and offered him the job.

Chandrasekhar initially declined the offer but finally accepted and returned to Yerkes as an Assistant Professor of Theoretical Astrophysics in December 1936. The Yerkes Observatory was run by the University of Chicago.

He was promoted to Associate Professor in 1941 and to Professor in 1943, at just 33 years of age. During World War 2, he worked at the Ballistic Research Laboratory at the Aberdeen Proving Ground in Maryland. In 1946, Princeton University offered Chandrasekhar a position vacated by the famous astronomer Henry Norris Russel with a salary double that of Chicago’s.

However, Robert Maynard Hutchins, the President of the University of Chicago, increased his salary to match that of Princeton’s and persuaded him to stay in Chicago. In 1952, he became the Morton D. Hull Distinguished Service Professor of Theoretical Astrophysics at The Institute for Nuclear Studies (now called The Enrico Fermi Institute (EFI)) at the invitation of the famous physicist Enrico Fermi.

Chandrasekhar remained at the University of Chicago for his entire career. In 1953, he and his wife took American citizenship. Chandrasekhar declined many offers from other universities, including one to succeed Henry Norris Russel as the director of the Princeton University Observatory. Chandrasekhar died of a sudden heart attack at the University of Chicago Hospital in 1995.

**References**

- C, Wali., K. C. (Kameshwar) (1991). Chandra: a biography of S. Chandrasekhar. Chicago: University of Chicago Press. pp. 9. ISBN 978-0226870540. OCLC 21297960.
- Subramanyan Chandrasekhar Biographical. NobelPrize.org. Retrieved 24 September 2019.
- Trehan, Surindar Kumar (1995). Subrahmanyan Chandrasekhar 1910–1995 (PDF). Biographical Memoirs of Fellows of the Indian National Science Academy, 23, 101–119.
- S Chandrashekhar, India’s great astrophysicist: Why Google Doodle is celebrating the Nobel prize winner. The Financial Express. 19 October 2017. Retrieved 19 October 2017.

# What are the scientific contributions of Dr. Subramanyan Chandrasekhar?

# The first paper

While studying at the Presidency College, Madras, Chandrasekhar wrote his first paper titled “The Compton Scattering and the New Statistics” in 1929.

**Reference**

Chandrasekhar, S. (1929). The Compton scattering and the new statistics. Proceedings of the Royal Society London A, (125), 231–237. https://doi.org/10.1098/rspa.1929.0163

# What is the Chandrasekhar Limit?

Starting from a sea voyage to England in 1930, Chandrasekhar published a series of papers between 1931 and 1935 in which he worked on the calculation of the statistics of a degenerate Fermi gas. Chandrasekhar solved the hydrostatic equation together with the nonrelativistic Fermi gas equation of state, and also treated the case of a relativistic Fermi gas and gave his famous limit.

In 1932, this value was also computed by the famous Soviet physicist Lev Davidovich Landau, who, however, did not apply it to white dwarfs and concluded that the quantum laws might be invalid for stars heavier than 1.5 solar masses. The Chandrasekhar limit is the maximum mass of a stable white dwarf star.

Its currently accepted value is about 1.4 M☉ (2.765×1030 kg). In 1930, Chandrasekhar improved the accuracy of his calculation 1930 by calculating the limit for a polytrope model of a star in hydrostatic equilibrium and compared his limit to the earlier limit found by E. C. Stoner for a uniform density star. White dwarfs resist gravitational collapse primarily through electron degeneracy 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. A white dwarf with a mass greater than the limit is subject to further gravitational collapse converting it into a different type of stellar remnants, such as a neutron star or a black hole.

Those with masses up to the limit remain stable as white dwarfs. The value for the limit varies depending on the nuclear composition of the star. Chandrasekhar gave the following equation for the limit based on the equation of state for an ideal Fermi gas.

where ℏ is the reduced Planck’s constant, c is the speed of light, G is the Universal Gravitational Constant,

is the average molecular weight per electron which depends upon the chemical composition of the star, m_{H} is the mass of the hydrogen atom, ** ω^{0}_{3}** ≈ 2.018236 is a constant connection with the solution to the Lane–Emden equation.

A more accurate value of the limit than that given by this simple model requires adjusting the various factors like the electrostatic interactions between the electrons and nuclei and effects caused by nonzero temperature. In 1987, Elliott H. Lieb and Horng-Tzer Yau gave a rigorous derivation of the limit from a relativistic many-particle Schrödinger equation.

**References**

- Timmes, F. X., Woosley, S. E., Weaver, Thomas A. (1996). The Neutron Star and Black Hole Initial Mass Function. Astrophysical Journal. 457, 834–843. arXiv:astro-ph/9510136. Bibcode:1996ApJ…457..834T. doi:10.1086/176778.
- Chandrasekhar, S., Milne, E. A. (1931). The Highly Collapsed Configurations of a Stellar Mass. Monthly Notices of the Royal Astronomical Society. 91 (5): 456-466. Bibcode:1931 MNRAS..91..456C. Doi:10.1093/mnras/91.5.456.
- Chandrasekhar, S. (1935). The Highly Collapsed Configurations of a Stellar Mass (second paper). Monthly Notices of the Royal Astronomical Society. 95 (3): 207–225. Bibcode:1935MNRAS..95..207C. doi:10.1093/mnras/95.3.207.
- On Stars, Their Evolution and Their Stability, Nobel Prize lecture, Subrahmanyan Chandrasekhar, December 8, 1983.
- Lieb, Elliott H., Yau, Horng-Tzer. (1987). A rigorous examination of the Chandrasekhar theory of stellar collapse (PDF). Astrophysical Journal. (323), 140–144. Bibcode:1987ApJ…323..140L. doi:10.1086/165813.

# Research at Cambridge

In his first year at Cambridge, he calculated the mean opacities and applied those results to the construction of an improved model for the limiting mass of the degenerate star. He 1933, he was awarded his Ph.D. degree at Cambridge with a thesis among his four papers on rotating self-gravitating polytropes.

**References**

- Chandrasekhar, S. (1932). The stellar coefficients of absorption and opacity. Part II. Proceedings of the Royal Society A: Mathematical, Physical & Engineering Sciences, 135 (827). pp. 472-490. ISSN 1364-5021
- Chandrasekhar, S. (1932). Model stellar photospheres. Monthly Notices of the Royal Astronomical Society, 92. pp. 186-195. ISSN 0035-8711
- Chandrasekhar, S. (1933). The equilibrium of distorted polytropes. I. The rotational problem. Monthly Notices of the Royal Astronomical Society, 93. pp. 390-406. ISSN 0035-8711

# Research at Gottingen

In 1931, at the Born’s institute at Gottingen, he worked on the opacities, atomic absorption coefficients, and model stellar photospheres.

**References**

- Chandrasekhar, S. (1931). The stellar coefficients of absorption arid opacity. Proceedings of the Royal Society A: Mathematical, Physical & Engineering Sciences, 133 (821). pp. 241-254. ISSN 1364-5021
- Chandrasekhar, S. (1931). The highly collapsed configurations of a stellar mass. Monthly Notices of the Royal Astronomical Society, 91. pp. 456-466. ISSN 0035-8711

# World War 2

During World War 2, Chandrasekhar worked at the Ballistic Research Laboratory at the Aberdeen Proving Ground in Maryland, U. S. A., where he worked on problems of ballistics such as the decay of plane shock waves, optimum height for the bursting of the shell, conditions for the existence of shock waves and the normal reflection of a blast wave.

**References**

- Chandrasekhar, S. (1943). On the conditions for the existence of three shock waves. Ballistic Research Laboratory, Aberdeen Proving Ground, Report, (367).
- Chandrasekhar, S. (1943). Optimum Height for the Bursting of a 105mm Shell (PDF) (No. BRL-MR-139). Army Ballistic Research Lab Aberdeen Proving Ground MD, U. S. A.
- Chandrasekhar, S. (1943). On the decay of plane shock waves. Ballistic Research Laboratories, Aberdeen Proving Ground, Maryland, USA.

# What is Chandrasekhar Number?

Chandrasekhar number is a dimensionless quantity used in magnetic convection to represent the ratio of the Lorentz force to the viscosity. It is defined as

where **μ _{0}** is the magnetic permeability,

**ρ**is the density of the fluid,

**ν**is the kinematic viscosity and

**λ**is the magnetic diffusivity.

**B**and

_{0}**d**are the characteristic magnetic field and a length scale of the system respectively. The number gives a measure of the magnetic field which is proportional to the square of a characteristic magnetic field in a system.

**Reference**

- Hurlburt, N.E., Matthews, P.C. and Rucklidge, A.M. (2000). Solar Magnetoconvection. Solar Physics, 192, 109-118. https://doi.org/10.1023/A:1005239617458

# What is Chandrasekhar Friction?

When a massive object moves through a cloud of smaller lighter objects, the effect of gravity causes the light bodies to accelerate and gain momentum and kinetic energy. Since there is a loss of momentum and kinetic energy for the body under consideration through the gravitational interactions with surrounding matter in space.

The effect is called dynamical friction or Chandrasekhar friction. It was discovered by Chandrasekhar in 1943. During the formation of planetary systems, dynamical friction between the protoplanet and the protoplanetary disk causes energy to be transferred from the protoplanet to the disk which results in the inward migration of the protoplanet.

When colliding, the dynamical friction between stars causes matter to sink toward the center of the galaxy and causes the orbits of the stars to be randomized. This process is called violent relaxation and can change two spiral galaxies into one larger elliptical galaxy.

The more massive galaxies tend to be found near the center of a galaxy cluster because the effect of the two-body collisions slows down the galaxy and the drag effect is greater if the galaxy mass is larger. The Chandrasekhar dynamical friction formula for the change in velocity of the object involves integrating over the phase space density of the field of matter. The Chandrasekhar friction formula is given by

Where **G** is the Gravitational constant, **M** is the mass under consideration, **m** is the mass of each star in the star distribution, **v _{M}** is the velocity of the object under consideration, in a frame where the center of gravity of the matter field is initially at rest,

**ln(Λ)**is the Coulomb logarithm,

**f(v)**is the number density distribution of the stars.

**References**

- Chandrasekhar, S. (1943). Dynamical friction. I. General considerations: the coefficient of dynamical friction. Astrophysical Journal, 97. pp. 255-262. ISSN 0004-637X https://doi.org/10.1086%2F144517
- Chandrasekhar, S. (1943). Dynamical friction. II. The rate of escape of stars from clusters and the evidence for the operation of dynamical friction. Astrophysical Journal, 97. pp. 263-273. ISSN 0004-637X https://doi.org/10.1086%2F144518
- Chandrasekhar, S. (1943). Dynamical Friction. III. A more exact theory of the rate of escape of stars from clusters. Astrophysical Journal, 98. pp. 54-60. ISSN 0004-637X https://doi.org/10.1086%2F144544

# What are Chandrasekhar-Kendall Functions?

In 1957, when attempting to solve the force-free magnetic fields, Chandrasekhar and P. C. Kendall derived the axisymmetric eigenfunctions of the curl operator which are called as Chandrasekhar-Kendall functions.

The two scientists independently derived the results but agreed to publish the paper together. If the force-free magnetic field equation is written as **∇×H=λH** with the assumption of divergence-free field **∇.H=0**, then the most general solution for the axisymmetric case is **H=1/λ ∇×(∇×ψn ̂)+∇×ψn ̂**, where** n ̂ **is a unit vector and the scalar function ψ satisfies the Helmholtz equation **∇^2 ψ+λ^2 ψ=0**.

**References**

- Chandrasekhar, Subrahmanyan (1956). On force-free magnetic fields. Proceedings of the National Academy of Sciences. 42 (1), 1–5. doi:10.1073/pnas.42.1.1. ISSN 0027-8424.
- Chandrasekhar, Subrahmanyan; Kendall, P. C. (September 1957). On Force-Free Magnetic Fields. The Astrophysical Journal. 126, 457. Bibcode:1957ApJ…126..457C. doi:10.1086/146413. ISSN 0004-637X. PMC 534220.

## What is Chandrasekhar’s H-function?

Chandrasekhar’s H-function is the solution to the problems involving scattering in atmospheric radiation. H-functions play a very important role in the wide class of problems of analytical radiative transfer theory. The H-function is the solution of the well-known integral equations, both nonlinear and linear. H-function satisfies two integral equations, one of them is nonlinear;

defined in the interval **0≤μ≤1**, where the Characteristic function **ψ(μ**) is an even polynomial in μ satisfying the following condition

If the equality is satisfied in this condition, it is called the conservative case, otherwise non-conservative.

**References**

- Chandrasekhar, Subrahmanyan. (2013). Radiative transfer. Courier Corporation.
- Hottel, H. C., Sarofim, A. F. (1967). Radiative transfer. McGraw-Hill.
- Nagirner, Dmitrij I., Ivanov, Vsevolod V. (2020). Chandrasekhar’s H-function revisited. Journal of Quantitative Spectroscopy and Radiative Transfer. 246, 106914. ISSN 0022-4073. https://doi.org/10.1016/j.jqsrt.2020.106914.

# What is Emden-Chandrasekhar Equation?

Emden–Chandrasekhar equation is a dimensionless form of the Poisson equation for the density distribution of a spherically symmetric isothermal gas sphere subjected to its own gravitational force. This equation was first discovered by Robert Emden in 1907. The equation is given by

where **ξ** is the dimensionless radius and **ψ** is related to the density of the gas sphere as

where **ρ _{c}** is the density of the gas at the center. The isothermal assumption is usually modeled to describe the core of a star.

Assuming an isothermal sphere has some disadvantages like if the density obtained as solution of this isothermal gas sphere decreases from the center, it decreases too slowly to give a well-defined surface and finite mass for the sphere.

This behavior of density causes increases in mass with an increase in radius. Thus, the model is usually valid to describe the core of a star, where the temperature is approximately constant.

**References**

- Chandrasekhar, S., Wares, Gordon W. (1949). The isothermal function. Astrophysical Journal, 109, 551-554. ISSN 0004-637X. http://dx.doi.org/10.1086/145167
- Chandrasekhar, Subrahmanyan. (1958). An introduction to the study of stellar structure. Vol. 2. Courier Corporation.
- Emden, R. (1907). Gaskugeln: Anwendungen der mechanischen warmetheorie auf kosmologische und meteorologische Probleme. B. Teubner, Leipzig.

# What are Chandrasekhar-Page Equations?

When seeking a separable solution to the Dirac equation in Kerr metric or Kerr–Newman metric, Chandrasekhar–Page equations were discovered by Chandrasekhar in 1976, which describe the wave function of the spin-1/2 massive particles.

He showed that a separable solution can be obtained from the Dirac equation in the Kerr metric. Don Page later extended this work to the Kerr-Newman metric, which is applicable to the charged black holes. By assuming a normal mode decomposition of the form

for the time and the azimuthal component of the spherical polar coordinates** (r,θ,ϕ)**, Chandrasekhar showed that the four bispinor components can be expressed as the product of radial and angular functions.

The two radial and angular functions, respectively, are denoted by **R _{(+1/2)} (r)**,

**R**and

_{(-1/2)}(r)**S**,

_{(+1/2) }(θ)**S**. The energy as measured at infinity is ω and the axial angular momentum is m which is a half-integer.

_{(-1/2)}(θ)**References**

- Chandrasekhar, S. (1976). The solution of Dirac’s equation in Kerr geometry. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. The Royal Society. 349 (1659): 571–575. Bibcode:1976RSPSA.349..571C. doi:10.1098/rspa.1976.0090. ISSN 2053-9169. S2CID 122791570.
- Page, Don N. (1976). Dirac equation around a charged, rotating black hole. Physical Review D. American Physical Society (APS). 14 (6): 1509-1510. Bibcode:1976PhRvD..14.1509P. doi:10.1103/physrevd.14.1509. ISSN 0556-2821.

# What is the Chandrasekhar Potential Energy Tensor?

Chandrasekhar potential energy tensor provides the gravitational potential of a body due to its own gravity created by the distribution of matter across the body. The Chandrasekhar tensor is a generalization of potential energy, in other words, the trace of the Chandrasekhar tensor provides the potential energy of the body. The Chandrasekhar potential energy tensor is defined as

Where

⇒

where **G** is the Gravitational constant, **ϕ(x)** is the self-gravitating potential from Newton’s law of gravity, **Φij **is the generalized version of **Φ**, **ρ(x) **is the matter density distribution, and **V** is the volume of the body. **W _{ij}** is a symmetric tensor. The trace of Chandrasekhar tensor

**W**is the potential energy

_{ij}**W**.

**References**

- Chandrasekhar, S., Lebovitz, N.R. (1962). The Potentials and the Superpotentials of Homogeneous Ellipsoids (PDF). Astrophysical Journal, 136, p.1037 –1047. Bibcode:1962ApJ…136.1037C. doi:10.1086/147456.
- Chandrasekhar, S., Fermi, E. (1953). Problems of Gravitational Stability in the Presence of a Magnetic Field (PDF). Astrophysical Journal, 118, p.116. Bibcode:1953ApJ…118..116C. doi:10.1086/145732.
- Chandrasekhar, Subrahmanyan. (1969). Ellipsoidal figures of equilibrium. Vol. 9. New Haven: Yale University Press.

# What are Chandrasekhar Virial Equations?

Chandrasekhar virial equations are a hierarchy of moment equations of the Euler equations, developed by Chandrasekhar, Enrico Fermi, and Norman Lebovitz.

**References**

- Chandrasekhar, S., Lebovitz, N.R. (1962). The Potentials and the Superpotentials of Homogeneous Ellipsoids (PDF). Astrophysical Journal, 136, p.1037 –1047. Bibcode:1962ApJ…136.1037C. doi:10.1086/147456.
- Chandrasekhar, S., Fermi, E. (1953). Problems of Gravitational Stability in the Presence of a Magnetic Field (PDF). Astrophysical Journal, 118, p.116. Bibcode:1953ApJ…118..116C. doi:10.1086/145732.
- Chandrasekhar, Subrahmanyan. (1969). Ellipsoidal figures of equilibrium. Vol. 9. New Haven: Yale University Press.

# What is Batchelor-Chandrasekhar Equation?

The Batchelor–Chandrasekhar equation is the evolution equation for the scalar functions, defining the two-point velocity correlation tensor of homogeneous axisymmetric turbulence. It was discovered by George Batchelor and Subrahmanyan Chandrasekhar when they developed the theory of homogeneous axisymmetric turbulence based on Howard P. Robertson’s work on isotropic turbulence using an invariant principle.

This equation is an extension of Kármán–Howarth equation from isotropic to axisymmetric turbulence.

**References**

- Batchelor, G. K. (1946). The theory of axisymmetric turbulence. (PDF) Proceedings of the Royal Society A, 186(1007), 480–502. https://doi.org/10.1098/rspa.1946.0060
- Chandrasekhar, S. (1950). The theory of axisymmetric turbulence. (PDF). Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 242(855), 557–577. https://doi.org/10.1098/rsta.1950.0010
- Chandrasekhar, S. (1950). The decay of axisymmetric turbulence. (PDF). Proceedings of the Royal Society A, 203(1074), 358–364. https://doi.org/10.1098/rspa.1950.0143
- Davidson, P. (2004). Turbulence: An Introduction for Scientists and Engineers: 1st Edition. Oxford University Press.
- Robertson, H. P. (1940, April). The invariant theory of isotropic turbulence. Mathematical Proceedings of the Cambridge Philosophical Society, 36(2), pp. 209 – 223. DOI: https://doi.org/10.1017/S0305004100017199

# What is Schönberg–Chandrasekhar limit?

The Schönberg–Chandrasekhar limit is the maximum mass of a non-fusing, isothermal core that can support an enclosing envelope. It was discovered by Chandrasekhar and Mario Schonberg in 1942. It is the ratio of the core mass to the total mass of the core and envelope.

The values of the limit depend on the models used and the assumed chemical compositions of the core and envelope. Values are in the range of 0.10 to 0.15 (10% to 15% of the total stellar mass). This is the maximum to which a helium-filled core can grow, and if this limit is exceeded, as can only happen in massive stars, the core collapses, releasing energy that causes the outer layers of the star to expand into a red giant.

The Schönberg–Chandrasekhar limit comes into play when fusion in a main-sequence star exhausts the hydrogen at the center of the star. The star then contracts until hydrogen fuses in a shell surrounding a helium-rich core, both of which are surrounded by an envelope consisting primarily of hydrogen. The core increases in mass as the shell burns its way outwards through the star.

If the star’s mass is less than approximately 1.5 solar masses, the core will become degenerate before the Schönberg–Chandrasekhar limit is reached, and, on the other hand, if the mass is greater than approximately 6 solar masses, the star leaves the main sequence with a core mass already greater than the Schönberg–Chandrasekhar limit so its core is never isothermal before helium fusion.

In the remaining case, where the mass is between 1.5 and 6 solar masses, the core will grow until the limit is reached, at which point it will contract rapidly until helium starts to fuse in the core.

**References**

- Schönberg, M., Chandrasekhar, S. (1942). On the Evolution of the Main-Sequence Stars. Astrophysical Journal, 96(2), 161–172.
- Beech, M. (1988). The Schoenberg-Chandrasekhar limit: A polytropic approximation, Astrophysics and Space Science 147(2), 219-227. DOI 10.1007/BF00645666.
- David Darling. (n.d). Schönberg-Chandrasekhar limit. Retrieved December 4, 2020, from https://www.daviddarling.info/encyclopedia/S/Schonberg-Chandrasekhar_limit.html

# What is Chandrasekhar’s white dwarf equation?

Chandrasekhar’s white dwarf equation is an initial value ordinary differential equation introduced by Chandrasekhar, in his study of the gravitational potential of completely degenerate white dwarf stars. The equation is given

with the initial conditions **φ(0)=1,φ’ (0)=0 **where **φ** measures the density of the white dwarf, η is the nondimensional radial distance from the center and C is a constant which is related to the density of the white dwarf at the center. The boundary **η=η_{∞}**

*of the equation is defined by the condition*

**such that a range of φ becomes**

*φ(η*_{∞})=√C**√(C ) ≤φ≤1**. This condition is equivalent to saying that the density vanishes at

**η=η**.

_{∞}**References**

- Chandrasekhar, S. (2003). An introduction to the study of stellar structure. Dover Publications Inc.
- Davis, Harold Thayer (1962). Introduction to Nonlinear Differential and Integral Equations. Courier Corporation. ISBN 978-0-486-60971-3.

# What is known as Chandrasekhar Polarization?

Chandrasekhar Polarization is a partial polarization of emergent radiation at the limb of rapidly rotating early-type stars or binary star systems with a purely electron-scattering atmosphere. It was predicted theoretically by Chandrasekhar in 1946. He predicted that the purely electron stellar atmosphere emits a polarized light using Thomson’s law of scattering.

His theory predicted that 11 percent polarization could be observed at maximum, however, when applied to a spherical star, the net polarization effect was found to be zero because of the spherical symmetry.

J. Patrick Harrington and George W. Collins, II showed that this symmetry is broken if we consider a rapidly rotating star (or a binary star system), in which the star is not exactly spherical, but slightly oblate due to extreme rotation (or tidal distortion in the case of the binary system).

The symmetry is also broken in eclipsing binary star system. Attempts made to predict this polarization effect were initially unsuccessful, but rather led to the prediction of interstellar polarization. In 1983, scientists found the first evidence of this polarization effect on the star Algol, an eclipsing binary star system.

The polarization on a rapidly rotating star was not found until 2017 since it required a high-precision polarimeter. In September 2017, a team of scientists from Australia observed this polarization on the star Regulus, which rotates at 96.5 percent of its critical angular velocity for the breakup.

**References**

- Chandrasekhar, S. (1946). On the Radiative Equilibrium of a Stellar Atmosphere. X. The Astrophysical Journal, 103, p. 351. doi:10.1086/144816. Bibcode: 1946ApJ…103..351C
- Chandrasekhar, S. (1989). Selected Papers, Vol 2, Radiative transfer and negative ion of hydrogen. Chicago, University of Chicago Press. ISBN 9780226100920.
- Harrington, J. P., Collins, G. W. (1968). Intrinsic polarization of rapidly rotating early-type stars. The Astrophysical Journal, 151, p1051.
- Rucinski, S. M. (1970). An Upper Limit to the Chandrasekhar Polarization in Early Type Stars. Acta Astronomica, 20, p1.
- Kemp, J. C., Henson, G. D., Barbour, M. S., Kraus, D. J., Collins, G. W. (1983). Discovery of eclipse polarization in Algol. Astrophysical Journal, Part 2 – Letters to the Editor, 273, p.L85-L88.
- Cotton, D. V., Bailey, J., Howarth, I. D., Bott, K., Kedziora-Chudczer, L., Lucas, P. W., & Hough, J. H. (2017). Polarization due to rotational distortion in the bright star Regulus. Nature Astronomy, 1(10), 690.
- Hiltner, W. A. (1949). Polarization of radiation from distant stars by the interstellar medium. Nature, 163(4138), 283.

# What are Chandrasekhar’s X- and Y-functions?

In atmospheric radiation, Chandrasekhar’s X- and Y-function appear as the solutions to problems involving diffusive reflection and transmission, introduced by Chandrasekhar. In the theory of radiative transfer, of either thermal or neutron radiation, a position and direction-dependent intensity function are usually sought for the description of the radiation field. The intensity field can in principle be solved from the integrodifferential radiative transfer equation (RTE), but an exact solution is usually impossible and even in the case of geometrically simple systems can contain unusual special functions such as the Chandrasekhar’s H-function and Chandrasekhar’s X- and Y-functions. Chandrasekhar’s X- and Y- function **X(μ)**, **Y(μ)** in the interval **0≤μ≤1**, satisfies the pair of nonlinear integral equations,

where the characteristic function **Ψ(μ)** is an even polynomial in **μ** generally satisfying the condition,

**0 1<∞** is the optical thickness of the atmosphere. If the equality is satisfied in the above condition, it is called the conservative case, otherwise non-conservative. These functions are related to Chandrasekhar’s H-function as

**X(μ)→H(μ)**,

**Y(μ)→0**as

**r**and also

_{1}→∞**X(μ)→1**,

**Y(μ)→**e

^{r1/μ}as

**r**.

_{1}→0**References**

- Chandrasekhar, Subrahmanyan. (2013). Radiative transfer. Courier Corporation.
- Hottel, H. C., Sarofim, A. F. (1967). Radiative transfer. McGraw-Hill.
- Howell, J. R., Menguc, M. P, Siegel, R. (2010). Thermal radiation heat transfer, 5th Edition. CRC Press.
- Modest, M.0 F. (2013). Radiative heat transfer. Academic Press.

# What is the method of discrete ordinates?

In the theory of radiative transfer, of either thermal or neutron radiation, a position and direction-dependent intensity function are usually sought for the description of the radiation field.

The intensity field can in principle be solved from the integrodifferential radiative transfer equation (RTE), but an exact solution is usually impossible and even in the case of geometrically simple systems can contain unusual special functions such as the Chandrasekhar’s H-function and Chandrasekhar’s X- and Y-functions.

The method of discrete ordinates, or the Sn method, is one way to approximately solve the RTE by discretizing both the xyz-domain and the angular variables that specify the direction of radiation. These methods were developed by Chandrasekhar when he was working on radiative transfer.

**References**

- Modest, Michael F. (2013). Radiative Heat Transfer, 3rd edition. pp.542-543, Elsevier https://doi.org/10.1016/C2010-0-65874-3
- Roberts, Jeremy A. (2010). Direct Solution of the Discrete Ordinates Equations. DOI: 10.13140/RG.2.1.2694.4886
- Liou, Kuo-Nan. (1973). A Numerical Experiment on Chandrasekhar’s Discrete-Ordinate Method for Radiative Transfer: Applications to Cloudy and Hazy Atmospheres. Journal of Atmospheric Sciences, 30(7), 1303-1326. https://doi.org/10.1175/1520-0469(1973)030%3C1303:ANEOCD%3E2.0.CO;2

# What is Velikhov-Chandrasekhar Instability?

Magnetorotational instability (MRI) is a fluid instability that causes an accretion disk orbiting a massive central object to become turbulent. It arises when the angular velocity of a conducting fluid in a magnetic field decreases as the distance from the rotation center increases.

It is also known as the Velikhov-Chandrasekhar instability or Balbus–Hawley instability in the literature, not to be confused with the electrothermal Velikhov instability. The MRI is an important part of the dynamics in accretion disks.

The MRI was first noticed in a non-astrophysical context by Evgeny Velikhov in 1959 when considering the stability of the Couette flow of an ideal hydromagnetic fluid. His result was later generalized by Chandrasekhar in 1960.

**References**

- Velikhov, E. P. (1959), Stability of an Ideally Conducting Liquid Flowing Between Cylinders Rotating in a Magnetic Field, Journal of Experimental and Theoretical Physics, 36(5), pp. 1398–1404.
- Chandrasekhar, S. (1960), The stability of non-dissipative Couette flow in hydromagnetics, Proceedings of the National Academy of Sciences of the United States of America, 46 (2), pp. 253–257, Bibcode:1960PNAS…46..253C, doi:10.1073/pnas.46.2.253, PMC 222823, PMID 16590616

# What is Chandrasekhar–Wentzel Lemma?

Chandrasekhar–Wentzel lemma was derived by Chandrasekhar and Gregor Wentzel in 1965 while studying the stability of rotating liquid drops. The lemma states that if **S** is a surface bounded by a simple closed contour** C**, then

Here x is the position vector and n is the unit normal on the surface. An immediate consequence is that if S is a closed surface, then the line integral tends to zero, leading to the result

or in index notation we have

That is to say, the tensor

defined on a closed surface is always symmetric, i.e., T_{ij}=T_{ji}.

**References**

- Chandrasekhar, S. (1965). The Stability of a Rotating Liquid Drop. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. 286(1404), 1–26. doi:10.1098/rspa.1965.0127.
- Chandrasekhar, S., Wali, K. C. (2001). A Quest for Perspectives: Selected Works of S. Chandrasekhar: With Commentary. World Scientific.

# What is Chandrasekhar’s variational principle?

Chandrasekhar’s variational principle provides the stability criterion for a static barotropic star, subjected to radial perturbation. It was discovered by Chandrasekhar in 1960. A barotropic star with dρ/dr<0 and ρ(r)=0 is stable if the quantity

where

is nonnegative for all real functions ρ'(x) that conserves the total mass of the star

where x is the coordinate system fixed to the center of the star, R is the radius of the star, V is the volume of the star, ρ(x) is the unperturbed density, ρ'(x) is the small perturbed density such as that in the perturbed state, the total density is ρ+ρ’, Φ is the self-gravitating potential from Newton’s law of gravity and G is the Gravitational Constant.

**References**

- Chandrasekhar, S. (1960). A general variational principle governing the radial and the non-radial oscillations of gaseous masses. Astrophysical Journal, 139, p. 664. Doi: 10.1086/147792, Bibcode:1964ApJ…139..664C
- Chandrasekhar, S. (1981). Hydrodynamic and hydromagnetic stability. Courier Corporation.
- Binney, James., Tremaine, Scott. (2011). Galactic dynamics: Second Edition. Princeton University Press.

# Awards, Honours conferred upon Dr. Chandrasekhar and his Legacy

The list of awards and honors Chandrasekhar received are as follows;

- Fellow of the Royal Society (FRS) (1944)
- Henry Norris Russell Lectureship (1949)
- Bruce Medal (1952)
- Gold Medal of the Royal Astronomical Society (1953)
- Rumford Prize of the American Academy of Arts and Sciences (1957)
- National Medal of Science, USA (1966)
- Padma Vibhushan (1968)
- Henry Draper Medal of the National Academy of Sciences (1971)
- Marian Smoluchowski Medal (1973)
- The Nobel Prize in Physics (1983)
- Copley Medal of the Royal Society (1984)
- Honorary Fellow of the International Academy of Science (1988)
- Gordon J. Laing Award (1989)
- Golden Plate Award of the American Academy of Achievement (1990)
- Jansky Lectureship before the National Radio Astronomy Observatory
- Humboldt Prize
- The Nobel Prize in Physics 1983 was divided equally between Subramanyan Chandrasekhar “for his theoretical studies of the physical processes of importance to the structure and evolution of the stars” and William Alfred Fowler “for his theoretical and experimental studies of the nuclear reactions of importance in the formation of the chemical elements in the universe.”

In 1979, NASA named the third of its four Great Space Observatories after Chandrasekhar called the Chandra X-ray Observatory which was launched and deployed by Space Shuttle Columbia on 23 July 1999.

The asteroid 1958 Chandra is named after Subramanyan Chandrasekhar. It was discovered on 24 September 1970, by Argentinian astronomer Carlos Cesco at the Yale–Columbia Southern Station of the Leoncito Astronomical Complex in San Juan, Argentina. It was named after Chandrasekhar by the Minor Planet Center on 1 November 1979.

Himalayan Chandra Telescope which is a 2.01-meter diameter optical-infrared telescope of the Indian Astronomical Observatory located at Hanle near Leh in the Union Territory of Ladakh is named after him.

**References**

- Tayler, R. J. (1996). Subrahmanyan Chandrasekhar. 19 October 1910 – 21 August 1995. Biographical Memoirs of Fellows of the Royal Society, 42, 80–94. doi:10.1098/rsbm.1996.0006. S2CID 58736242.
- Grants, Prizes and Awards│American Astronomical Society. (2010, January 24). Archived from the American Astronomical society. https://web.archive.org/web/20100124125308/http:/aas.org/grants/awards.php
- Astronomical Society of the Pacific: Past Winners of the Catherine Wolfe Bruce Gold Medal. Astronomical Society of the Pacific. Archived from the original on 21 July 2011. Retrieved 8 December 2020.
- Winners of the Gold Medal of the Royal Astronomical Society. Royal Astronomical Society. Archived from the original on 25 May 2011. Retrieved 8 December 2020.
- Past Recipients of the Rumford Prize. American Academy of Arts and Sciences. http://www.amacad.org/about/rumford.aspx. Retrieved 24 February 2011.
- The President’s National Medal of Science: Recipient Details – NSF – National Science Foundation. National Science Foundation. https://www.nsf.gov/od/nms/recip_details.cfm?recip_id=73
- Henry Draper Medal. National Academy of Sciences. Archived from the original on 26 January 2013. Retrieved 8 December 2020.
- Golden Plate Awardees of the American Academy of Achievement. American Academy of Achievement. https://achievement.org/our-history/golden-plate-awards/#science-exploration
- The Nobel Prize in Physics 1983. The Nobel Foundation. https://www.nobelprize.org/prizes/physics/1983/summary/
- MPC/MPO/MPS Archive. The International Astronomical Union Minor Planet Center. Retrieved 8 December 2020.
- 1958 Chandra (1970 SB). The International Astronomical Union Minor Planet Center. Retrieved 8 December 2020.
- Schmadel, Lutz D. (2007). (1958) Chandra. Dictionary of Minor Planet Names. Springer Berlin Heidelberg. p. 157. doi:10.1007/978-3-540-29925-7_1959. ISBN 978-3-540-00238-3.
- Name NASA’s Next Great Observatory. (1998). Harvard-Smithsonian Center for Astrophysics. Retrieved December 8, 2020.
- Bagla, Pallava. (7 January 2002), India Unveils World’s Highest Observatory. National Geographic News. http://news.nationalgeographic.com/news/2001/12/1227_020104indiaobs.html

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