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Author name code: chandrasekhar
ADS astronomy entries on 2022-09-14
author:"Chandrasekhar, Subrahamanyan" year:1930-1990
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Title: Book-Review - Selected Papers - Stellar Structure and Stellar
Atmospheres - Radiative Transfer and Negative Ion of Hydrogen -
Stochastic Statistical and Hydrodynamic Problems in Physics and
Astronomy - Plasma Physics Hydrodynamic and Hydromagnetic Stability
and Application of the Tensor / Virial Theorem - Relativistic
Astrophysics
Authors: Chandrasekhar, S.
1990JHA....21..383C Altcode:
No abstract at ADS
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Title: Book-Review - Selected Papers
Authors: Chandrasekhar, S.
1990Sci...250..306C Altcode:
No abstract at ADS
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Title: Book-Review - Selected Papers - V.2 - Radiative Transfer and
Negative Ion of Hydrogen
Authors: Chandrasekhar, S.; Pagel, B.
1990Obs...110..136C Altcode:
No abstract at ADS
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Title: Book-Review - Selected Papers - V.3 - Stochastic Statistical
and Hydromagnetic Problems in Physics and Astronomy - V.4 - Plasma
Physics Hydrodynamics and Hydromagnetic Stability and Applications
of the Tensor Virial Theorem
Authors: Chandrasekhar, S.
1990Sci...248.1142C Altcode:
No abstract at ADS
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Title: Science and scientific attitudes
Authors: Chandrasekhar, S.
1990Natur.344..285C Altcode:
On Apollonius, Kepler and Einstein, Newton and Shakespeare, and Madonna
and Mrs Pelham.
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Title: Book-Review - Selected Papers - Plasma Physics Hydrodynamic and
Hydromagnetic Stability and Applications of the Tensor Virial Theorem
Authors: Chandrasekhar, S.
1990AstQ....7R.188C Altcode:
No abstract at ADS
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Title: Book-Review - Selected Papers - Stochastic Statistical and
Hydromagnetic Problems in Physics and Astronomy
Authors: Chandrasekhar, S.
1990AstQ....7Q.188C Altcode:
No abstract at ADS
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Title: Selected papers of S. Chandrasekhar. Volume 5: Relativistic
astrophysics.
Authors: Chandrasekhar, S.
1990spc5.book.....C Altcode:
This volume presents a collection of papers of S. Chandrasekhar
published during the years 1963 - 1984 and covers all of Chandrasekhar's
contributions to the general theory of relativity and to relativity's
astrophysical applications, except his research on black holes and
on colliding gravitational waves. The papers included are grouped
into three parts. Contents: Part I. Relativistic instabilities
and post-Newtonian approximations. Part II. The stability and
the equilibrium of rotating systems in general relativity. Part
III. Miscellaneous papers in general relativity.
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Title: Discotic liquid crystals
Authors: Chandrasekhar, S.; Ranganath, G. S.
1990RPPh...53...57C Altcode:
No abstract at ADS
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Title: Book-Review - Selected Papers - V.5 - Relativistic Astrophysics
Authors: Chandrasekhar, S.
1990AstQ....7..252C Altcode:
No abstract at ADS
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Title: Book-Review - Selected Papers - V.2 - Radiative Transfer and
Negative Ion of Hydrogen
Authors: Chandrasekhar, S.
1989S&T....78..491C Altcode:
No abstract at ADS
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Title: Book-Review - Selected Papers - V.1 - Stellar Structure and
Stellar Atmospheres
Authors: Chandrasekhar, S.
1989S&T....78..382C Altcode:
No abstract at ADS
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Title: Book-Review - Selected Papers - VOL.1 - Stellar Structure
and Stellar Atmospheres
Authors: Chandrasekhar, S.
1989Sci...245.1517C Altcode:
No abstract at ADS
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Title: Selected papers of S. Chandrasekhar. Volume 1: Stellar
structure and stellar atmospheres.
Authors: Chandrasekhar, S.
1989spc1.book.....C Altcode:
This volume presents a collection of papers of S. Chandrasekhar
published during the years 1929 - 1945. The criteria for selection are
essentially two: first, the papers have not been included in any of
the author's books, and second, the papers treat matters of possible
historical interest not treated in sufficient detail elsewhere. The
papers included are grouped into five sections. Contents: Part I. The
theory of white dwarf stars. Part II. The equilibrium of distorted
polytropes. Part III. Stellar evolution. Part IV. Integral theorems
on the equilibrium of a star. Part V. Theory of stellar atmospheres.
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Title: Selected papers of S. Chandrasekhar. Volume 4: Plasma physics,
hydrodynamic and hydromagnetic stability, and applications of the
tensor-virial theorem.
Authors: Chandrasekhar, S.
1989spc4.book.....C Altcode:
This volume presents a collection of papers of S. Chandrasekhar
published during the years 1954 - 1971. The papers included are
grouped into three parts. Contents: Part I. Plasma physics. Part
II. Hydrodynamic and hydromagnetic stability. Part III. Tensor virial
theorem and its applications.
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Title: Selected papers of S. Chandrasekhar. Volume 2: Radiative
transfer and negative ion of hydrogen.
Authors: Chandrasekhar, S.
1989spc2.book.....C Altcode:
This volume contains much of S. Chandrasekhar's analysis of
various transfer problems done primarily in the period 1944 -
1950. Contents: Part I. Radiative transfer. 1. The method of
discrete ordinates. 2. Radiative transfer allowing for polarization
of scattering light. 3. Principles of invariance: the H- and the X-
and Y-functions. 4. Miscellaneous problems. 5. Review articles. Part
II. Investigation on the negative ion of hydrogen and of two-electron
atoms.
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Title: Selected papers of S. Chandrasekhar. Volume 3: Stochastic,
statistical, and hydromagnetic problems in physics and astronomy.
Authors: Chandrasekhar, S.
1989spc3.book.....C Altcode:
This volume presents a collection of papers of S. Chandrasekhar
published during the years 1941 - 1958. The papers included are
grouped into four parts. Contents: Part I. Dynamical friction and
Brownian motion. Part II. Statistical problems in astronomy. Part
III. Statistical theory of hydrodynamic and hydromagnetic
turbulence. Part IV. Hydromagnetic problems in astrophysics.
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Title: Radiative transfer - a personal account.
Authors: Chandrasekhar, S.
1989fapi.conf...19C Altcode:
Contents: 1. Preliminaries. 2. The solutions for some typical problems
by the method of discrete ordinates. 3. The equation of transfer
incorporating the polarization of the radiation field - the problem with
a constant flux. 4. The general vector equation of transfer in terms
of Stokes parameters. 5. The impact of Ambartsumian's principles of
invariance. 6. Solutions of the coupled systems of integral equations
which follow from the principles of invariance. 7. The polarization
of the sunlit sky.
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Title: To Victor Ambartsumian on his 80th birthday
Authors: Chandrasekhar, S.
1988Ap.....29..408C Altcode: 1989Ap.....29..408C
No abstract at ADS
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Title: Book-Review - Truth and Beauty - Aesthetics and Motivations
in Science
Authors: Chandrasekhar, S.
1988S&T....76R..46C Altcode:
No abstract at ADS
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Title: Massless particles from a perfect fluid
Authors: Chandrasekhar, S.
1988Natur.333..506C Altcode:
No abstract at ADS
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Title: Book-Review - Truth and Beauty - Aesthetics and Motivations
in Science
Authors: Chandrasekhar, S.
1988Sci...240.1807C Altcode:
No abstract at ADS
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Title: Truth and Beauty - Aesthetics and Motivations in Science
Authors: Chandrasekhar, S.; Bondi, H.
1988Natur.331..668C Altcode:
No abstract at ADS
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Title: To Victor Ambartsumian on his 80TH Birthday
Authors: Chandrasekhar, S.
1988Afz....29....7C Altcode:
No abstract at ADS
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Title: Book-Review - Eddington - the Most Distinguished Astrophysicist
of his Time
Authors: Chandrasekhar, S.; Woolley, R.
1987JHA....18..143C Altcode:
No abstract at ADS
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Title: Book-Review - Eddington - the Most Distinguished Astrophysicist
of his Time
Authors: Chandrasekhar, S.
1987AstQ....5..255C Altcode:
No abstract at ADS
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Title: On the Collision of Impulsive Gravitational Waves when Coupled
with Null Dust
Authors: Chandrasekhar, S.; Xanthopoulos, B. C.
1986RSPSA.403..189C Altcode: 1986RSLPS.403..189C
The problem of colliding impulsive gravitational waves is considered
when the region of space-time, after the instant of collision, is
filled with a mixture of null dusts moving in opposite directions. The
extension of the resulting space-time, to regions before the instant
of collision, shows that null dust follows the leading edges of the
impulsive waves, and, further, that one can arrange that the space-time
in these regions is identical with what prevails when a perfect fluid
with ɛ = p fills the region after the instant of collision. This
ambiguity in the space-time, after the instant of collision, must be
traced to an inherent ambiguity in the nature of null dust and its
relation with a perfect fluid with ɛ = p.
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Title: Karl Schwarzschild Lecture: The Aesthetic Base of the General
Theory of Relativity
Authors: Chandrasekhar, S.
1986MitAG..67...19C Altcode:
Karl Schwarzschild is one of the towering physical scientists of this
century. The breadth and range of his contributions are staggering:
they cover the entire range of physics, astronomy, and astrophysics of
his time. The author considers Schwarzschild's attitude and approach to
scientific problems, as he can discern them from his published papers;
especially his solution of the equations of general relativity.
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Title: Some Exact Solutions of Gravitational Waves Coupled with
Fluid Motions
Authors: Chandrasekhar, S.; Xanthopoulos, B. C.
1985RSPSA.402..205C Altcode: 1985RSLPS.402..205C
Some exact solutions of Einstein's equations are found which represent
the interaction of gravitational waves with a perfect fluid in which the
velocity of sound equals the velocity of light. These solutions, unlike
the solutions representing the collision of impulsive gravitational
waves, are bounded by a space-time singularity and have some resemblance
to cosmological solutions: every time-like trajectory, extended into
the past, encounters the singularity. Moreover, in the generic case,
matter may be considered as being created at the singularity.
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Title: On the Collision of Impulsive Gravitational Waves when Coupled
with Fluid Motions
Authors: Chandrasekhar, S.; Xanthopoulos, B. C.
1985RSPSA.402...37C Altcode: 1985RSLPS.402...37C
An exact solution of Einstein's equations, with a source derived from a
perfect fluid in which the energy density, ɛ, is equal to the pressure,
p, is obtained. The solution describes the space-time following the
collision of plane impulsive gravitational waves and is the natural
generalization of the Nutku-Halil solution of the vacuum equations, in
the region of interaction under similar basic conditions. A consistent
extension of the solution, prior to the instant of collision,
requires that the fluid in the region of interaction is the direct
result of a transformation of incident null-dust (i.e. of massless
particles describing null trajectories). The ultimate result of the
collision is the development of a space-time singularity, the nature
of which is strongly dependent on the amplitude and the character of
the sound waves that are present. The distribution of ɛ that follows
the collision has many intriguing features. The solution obtained in
this paper provides the first example of an induced transformation of
a massless into a massive particle.
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Title: Book-Review - the Mathematical Theory of Black-Holes
Authors: Chandrasekhar, S.; Thorne, K. S.
1985AmJPh..53R1013C Altcode:
No abstract at ADS
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Title: Book-Review - the Mathematical Theory of Black-Holes
Authors: Chandrasekhar, S.; Thorne, K. S.
1985AmJPh..53Q1013C Altcode:
No abstract at ADS
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Title: Book-Review - Eddington - the Most Distinguished Astrophysicist
of his Time
Authors: Chandrasekhar, S.; Whitrow, G. J.
1985Obs...105...16C Altcode:
No abstract at ADS
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Title: Book-Review - the Mathematical Theory of Black-Holes
Authors: Chandrasekhar, S.; Gottlober, S.
1985AN....306..128C Altcode:
No abstract at ADS
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Title: On stars, their evolution and their stability.
Authors: Chandrasekhar, S.
1985UsFiN.145..489C Altcode:
No abstract at ADS
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Title: Book-Review - Eddington - the Most Distinguished Astrophysicist
of his Time
Authors: Chandrasekhar, S.
1985S&T....69...34C Altcode:
No abstract at ADS
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Title: Book-Review - Eddington
Authors: Chandrasekhar, S.; Ronan, C. A.
1984JBAA...95Q..42C Altcode:
No abstract at ADS
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Title: Obituary - Hopkins, Jeanne 1923-1984
Authors: Chandrasekhar, S.
1984S&T....68..492C Altcode:
No abstract at ADS
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Title: On Stars, Their Evolution and Their Stability
Authors: Chandrasekhar, S.
1984Sci...226..497C Altcode:
The present investigation represents an attempt to show that, in a
limited sense, the basic facts concerning stars can be understood as
simply as atoms, taking into account an understanding of atoms in
terms of two combinations of natural constants. It is pointed out
that the most important fact concerning a star is its mass, which
is measured in units of the mass of the sun. The current theories of
stellar structure and stellar evolution derive their successes largely
from the fact that a certain combination of the dimensions of a mass
provides a correct measure of stellar masses. Attention is given to
the role of radiation pressure, the degeneracy of the electrons in
white-dwarf stars, the theory of the white-dwarf stars and the limiting
mass, the conditions under which normal stars develop degenerate cores,
the evolution of massive stars and the onset of gravitational collapse,
and the mathematical theory of black holes.
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Title: On the Nutku-Halil Solution for Colliding Impulsive
Gravitational Waves
Authors: Chandrasekhar, S.; Ferrari, V.
1984RSPSA.396...55C Altcode: 1984RSLPS.396...55C
The equations appropriate for space-times with two space-like
Killing-vectors are set up, ab initio, and explicit expressions for
the components of the Riemann, the Ricci, and the Einstein tensors in
a suitable tetrad-frame are written. The equations for the vacuum are
reduced to a single equation of the Ernst type. It is then shown that
the simplest linear solution of the Ernst equation leads directly to
the Nutku-Halil solution for two colliding impulsive gravitational
waves with uncorrelated polarizations. Thus, in some sense, the
Nutku-Halil solution occupies the same place in space-times with two
space-like Killing-vectors as the Kerr solution does in space-times
with one time-like and one space-like Killing-vector. The Nutku-Halil
solution is further described in a Newman-Penrose formalism; and the
expressions for the Weyl scalars, in particular, make the development
of curvature singularities manifest. Finally, a theorem analogous to
Robinson's theorem (but much less strong) is proved for space-times
with two space-like Killing-vectors.
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Title: Book-Review - Eddington - the Most Distinguished Astrophysicist
of his Time
Authors: Chandrasekhar, S.
1984Sci...224.1364C Altcode:
No abstract at ADS
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Title: On stars, their evolution and their stability
Authors: Chandrasekhar, S.
1984RvMP...56..137C Altcode:
No abstract at ADS
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Title: The general theory of relativity - Why 'It is probably the
most beautiful of all existing theories'
Authors: Chandrasekhar, S.
1984JApA....5....3C Altcode:
An attempt is made to objectively evaluate the frequent judgment
of Einstein's general theory of relativity, by such distinguished
physicists as Pauli (1921), Dirac, Born, and Rutherford, as 'beautiful'
and 'a work of art'. The criteria applied are that of Francis Bacon
('There is no excellent beauty that hath not some strangeness in the
proportions') and that of Heisenberg ('Beauty is the proper conformity
of the parts to one another and to the whole'). The strangeness in
the proportions of the theory of general relativity consists in its
relating, through juxtaposition, the concepts of space and time and
those of matter and motion; these had previously been considered
entirely independent. The criterion of 'conformity' is illustrated
through the directness with which the theory allows the description
of black holes.
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Title: On the onset of relativistic instability in highly centrally
condensed stars
Authors: Chandrasekhar, S.; Lebovitz, N. R.
1984MNRAS.207P..13C Altcode:
It is shown that, along the polytropic sequence, the criterion for
relativistic instability due to radial perturbation can be written to
less than 3 percent accuracy in where the adiabatic constant is close
to 4/3. This criterion is expected to indicate when the relativistic
instability will become relevant for highly centralized condensed stars
like those in the late stages of stellar evolution when elements of
higher atomic number are being synthesized.
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Title: On Algebraically Special Perturbations of Black Holes
Authors: Chandrasekhar, S.
1984RSPSA.392....1C Altcode: 1984RSLPS.392....1C
Algebraically special perturbations of black holes excite gravitational
waves that are either purely ingoing or purely outgoing. Solutions,
appropriate to such perturbations of the Kerr, the Schwarzschild,
and the Reissner-Nordstrom black-holes, are obtained in explicit forms
by different methods. The different methods illustrate the remarkable
inner relations among different facets of the mathematical theory. In
the context of the Kerr black-hole they derive from the different
ways in which the explicit value of the Starobinsky constant emerges,
and in the context of the Schwarzschild and the Reissner-Nordstrom
black-holes they derive from the potential barriers surrounding them
belonging to a special class.
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Title: Book-Review - Eddington - the Most Distinguished Astrophysicist
of his Time
Authors: Chandrasekhar, S.
1984JBAA...94...93C Altcode:
No abstract at ADS
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Title: Book-Review - the Mathematical Theory of Black-Holes
Authors: Chandrasekhar, S.; King, A.
1984Obs...104...39C Altcode:
No abstract at ADS
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Title: Book-Review - Eddington - the Most Distinguished Astrophysicist
of his Time
Authors: Chandrasekhar, S.; Hoyle, F.
1984Natur.307Q.575C Altcode:
No abstract at ADS
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Title: Book-Review - the Mathematical Theory of Black-Holes
Authors: Chandrasekhar, S.
1984S&T....67Q.428C Altcode:
No abstract at ADS
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Title: The mathematical theory of black holes.
Authors: Chandrasekhar, S.
1984grg..conf....5C Altcode:
This discussion of the mathematical theory of black holes is restricted
to features of the theory that reveal its richness. The mathematical
properties that are examined derive from a consideration of the
perturbations of black holes. Particular attention is given to the
problem of the reflection and transmission of incident gravitational
waves by the Schwarzschild black hole; the Kerr space-time; and the
Reissner-Nordstrom solution.
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Title: Book-Review - Eddington
Authors: Chandrasekhar, S.
1984AExpr...1...42C Altcode:
No abstract at ADS
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Title: Book-Review - Eddington
Authors: Chandrasekhar, S.
1984AExpr...1S..39C Altcode:
No abstract at ADS
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Title: Book-Review - the Mathematical Theory of Black-Holes
Authors: Chandrasekhar, S.; Penrose, R.
1983Natur.306..404C Altcode:
No abstract at ADS
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Title: The Mathematical Theory of Black Holes
Authors: Chandrasekhar, S.
1983grg1.conf....6C Altcode:
No abstract at ADS
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Title: The mathematical theory of black holes
Authors: Chandrasekhar, S.
1983mtbh.book.....C Altcode: 1983QB843.B55C48...
A detailed treatment of the mathematical theory of black holes is
presented. The analytical methods on which the theory is based are
reviewed, and a space-time of sufficient generality to encompass
the different situations arising in the study of black holes is
developed. The Schwarzschild space-time and the perturbations of the
Schwarzschild black hole are addressed. The Reissner-Nordstrom solution,
the Kerr metric, geodesics in Kerr space-time, electromagnetic waves
in Kerr geometry, gravitational perturbations of the Kerr black hole,
and spin-1/2 particles in Kerr geometry are discussed. Other solution
and methods are examined.
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Title: The Highly Collapsed Configurations of a Stellar Mass
Authors: Chandrasekhar, S.
1982blho.book...25C Altcode:
No abstract at ADS
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Title: On Crossing the Cauchy Horizon of a Reissner-Nordstrom
Black-Hole
Authors: Chandrasekhar, S.; Hartle, J. B.
1982RSPSA.384..301C Altcode: 1982RSLPS.384..301C
The behaviour, on the Cauchy horizon, of a flux of gravitational
and/or electromagnetic radiation crossing the event horizon of a
Reissner-Nordstrom black-hole is investigated as a problem in the theory
of one-dimensional potential-scattering. It is shown that the flux of
radiation received by an observer crossing the Cauchy horizon, along
a radial time-like geodesic, diverges for all physically reasonable
perturbations crossing the event horizon, even including those with
compact support.
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Title: On the Potential Barriers Surrounding the Schwarzschild
Black-Hole
Authors: Chandrasekhar, S.
1982sag..conf..120C Altcode:
No abstract at ADS
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Title: Achievements in Astronomy. (Book Reviews: Oort and the
Universe)
Authors: Chandrasekhar, S.
1981Sci...211..272V Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Achievements in Astronomy. (Book Reviews: Oort and the
Universe)
Authors: Chandrasekhar, S.
1981Sci...211..272C Altcode:
No abstract at ADS
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Title: The Gravitational Perturbations of the Kerr Black Hole. IV. The
Completion of the Solution
Authors: Chandrasekhar, S.
1980RSPSA.372..475C Altcode:
This paper eliminates the last remaining lacuna in the information
that was needed to make the solution for the perturbations in the
metric coefficients of the Kerr space-time fully explicit. The
requisite information is obtained from a pair of equations which is
complementary to the one considered in paper III; and the solution
of the Newman-Penrose equations governing the perturbations is, thus,
completed.
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Title: The general theory of relativity - The first thirty years
Authors: Chandrasekhar, S.
1980ConPh..21..429C Altcode:
The principal landmarks in the development of general relativity
(exclusive of cosmology) during the first 30 years after its founding
are reviewed. Consideration is given to experimental and observational
verifications of the predictions of the general theory of relativity,
the post-Newtonian approximation, and Schwarzschild's solution
for the space-time metric outside a central spherically symmetric
distribution of mass (the Schwarszchild black hole). Other problems
discussed include the relativistic equations of hydrostatic equilibrium
for spherically symmetric distribution of masses and the problem of
gravitational collapse.
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Title: Evidence for a Large Population of Shocked Interstellar Clouds
Authors: Chandrasekhar, S.
1980JApA....1...33C Altcode:
Reprinting of an invited discourse delivered by the author at the
Seventeenth General Assembly of the International Astronomical Union
held at Montreal, Canada.
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Title: The 1979 Milne Lecture - Milne, Edward-Arthur and the
Development of Modern Astrophysics
Authors: Chandrasekhar, S.
1980QJRAS..21...93C Altcode:
No abstract at ADS
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Title: On One-Dimensional Potential Barriers Having Equal Reflexion
and Transmission Coefficients
Authors: Chandrasekhar, S.
1980RSPSA.369..425C Altcode: 1980RSLPS.369..425C
Based on the results of earlier studies on the perturbations of the
Schwarzschild and the Reissner-Nordstrom black holes, it is shown that
there exists a very general class of potential pairs (V^(+) and V^(-))
which yield the same reflexion and transmission coefficients. It
is further shown that these potentials, V^(+) and V^(-), satisfy an
infinite hierarchy of integral equalities which are, formally, the same
as the conserved quantities allowed by the Korteweg-deVries equation.
---------------------------------------------------------
Title: The role of general relativity in astronomy - Retrospect
and prospect
Authors: Chandrasekhar, S.
1980HiA.....5...45C Altcode:
Past confirmations of the general theory of relativity by the three
classical tests are reviewed, and attention is given to the role which
general relativity plays in our understanding of the large-scale
structure of the universe (i.e., in the realm of cosmology). The
nature of gravitational collapse as a central problem in astronomy
is discussed along with the author's considered prospects for general
relativity in astronomy.
---------------------------------------------------------
Title: On the Metric Perturbations of the Reissner-Nordstrom
Black Hole
Authors: Chandrasekhar, S.; Xanthopoulos, B. C.
1979RSPSA.367....1C Altcode: 1979RSLPS.367....1C
The two pairs of one-dimensional wave equations which govern the odd
and the even-parity perturbations of the Reissner-Nordstrom black hole
are derived directly from a treatment of its metric perturbations. The
treatment closely parallels the corresponding treatment in the context
of the Schwarzschild black hole.
---------------------------------------------------------
Title: Beauty and the quest for beauty in science
Authors: Chandrasekhar, S.
1979PhT....32g..25C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: The Gravitational Perturbations of the Kerr Black
Hole. III. Further Amplifications
Authors: Chandrasekhar, S.
1979RSPSA.365..425C Altcode: 1979RSLPS.365..425C
The present paper is devoted to an amplification of the solution of
the Newman-Penrose equations considered in the two earlier papers
of this series. The principal amplification consists in showing that
the function Psi , in terms of which the metric perturbations are most
simply expressed (and which was thought to require quadratures), besides
being separable in its variables, is expressible directly in terms of
the Teukolsky functions (and eliminates the need for quadratures). It
is further shown that the completion of the solution for the metric
perturbations requires the consideration of four additional equations
which follow from four Ricci identities (not hitherto considered); and
the solution of these equations is found. It is also pointed out that
while the perturbation in the Weyl scalar, Psi <SUB>2</SUB>, can be set
equal to zero, it cannot be deduced to be zero. (The contrary statement
in the earlier paper arose from an error of a factor 2 in one of the
equations.) Numerical verification of some of the principal equations
and identities of the theory is provided. An important aspect of the
analysis contained in this paper is the emergence of several crucial
identities among the Teukolsky functions which one might despair of
verifying directly.
---------------------------------------------------------
Title: On the Equations Governing the Perturbations of the
Reissner-Nordstrom Black Hole
Authors: Chandrasekhar, S.
1979RSPSA.365..453C Altcode: 1979RSLPS.365..453C
By considering suitable combinations of the Weyl scalars and the spin
coefficients, the basic equations governing the perturbations of the
Reissner-Nordstrom black hole, in the Newman-Penrose formalism, are
decoupled; a fundamental pair of decoupled equations are obtained. It is
then shown how this pair of decoupled equations can be transformed into
one dimensional wave equations which are appropriate for describing
the perturbations of odd and of even parity. A simple relation is
obtained which will allow derivation of a solution belonging to one
parity from a solution belonging to the opposite parity. Finally,
equations are derived in terms of which one can readily ascertain how
an arbitrary superposition of gravitational and electromagnetic waves,
incident on the black hole, will be reflected and absorbed.
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Title: An introduction to the theory of the Kerr metric and its
perturbations.
Authors: Chandrasekhar, S.
1979grec.conf..370C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: The deformed figures of the Dedekind ellipsoids in the
post-Newtonian approximation to general relativity: corrections
and amplifications.
Authors: Chandrasekhar, S.; Elbert, D. D.
1978ApJ...220..303C Altcode:
Two errors in the analysis of an earlier paper on the same subject are
corrected. It is found that as a consequence of the corrections, the
solution to the post-Newtonian equations (appropriate to determining
the deformed figures of the Dedekind ellipsoid) now diverges at a point
where the axes of the ellipsoid are in the ratios 1:0.3370:0.2850. In
addition, the fourth-harmonic oscillations of the Dedekind ellipsoid
are considered. It is found that it becomes dynamically unstable when
its axes are in the ratios 1:0.3121:0.2680.
---------------------------------------------------------
Title: On the linear perturbations of the Schwarzschild and the
Kerr geometries
Authors: Chandrasekhar, S.
1978pans.proc..528C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: The Gravitational Perturbations of the Kerr Black Hole. I. The
Perturbations in the Quantities which Vanish in the Stationary State
Authors: Chandrasekhar, S.
1978RSPSA.358..421C Altcode: 1978RSLPS.358..421C
As a preliminary towards a complete integration of the Newman-Penrose
equations governing the gravitational perturbations of the Kerr
black hole, the perturbations in the spin coefficients and in
the components of the Weyl tensor, which vanish in the stationary
state, are considered. The manner of treatment of the basic equations
yields Teukolsky's equations expressed directly in terms of the basic
derivative operators of the theory and, further, suggests a preferred
gauge in which two of the components of the Weyl tensor are governed
by the same equations as a Maxwell field. Various identities and
relations that are needed in subsequent work are assembled. In two
appendixes, the solution of Maxwell's equations in Kerr geometry and
the perturbations of the charged Kerr-Newman black hole are considered.
---------------------------------------------------------
Title: The Kerr metric and stationary axisymmetric gravitational
fields
Authors: Chandrasekhar, S.
1978RSPSA.358..405C Altcode: 1978RSLPS.358..405C
A treatment of Einstein's equations governing vacuum gravitational
fields which are stationary and axisymmetric is shown to divide itself
into three parts: a part essentially concerned with a choice of gauge
(which can be chosen to ensure the occurrence of an event horizon
exactly as in the Kerr metric); a part concerned with two of the basic
metric functions which in two combinations satisfy a complex equation
(Ernst's equation) and in one combination satisfies a symmetric pair of
real equations; and a third part which completes the solution in terms
of a single ordinary differential equation of the first order. The
treatment along these lines reveals many of the inner relations which
characterize the general solutions, provides a derivation of the Kerr
metric which is direct and verifiable at all stages, and opens an
avenue towards the generation of explicit classes of exact solutions
(an example of which is given).
---------------------------------------------------------
Title: The Gravitational Perturbations of the Kerr Black Hole. II. The
Perturbations in the Quantities which are Finite in the Stationary
State
Authors: Chandrasekhar, S.
1978RSPSA.358..441C Altcode:
The present paper completes the integration of the linearized
Newman-Penrose equations governing the gravitational perturbations
of the Kerr black hole. The equations which determine the solutions
are the four (complex) Bianchi identities (not used in part I) and
the 24 equations which follow from the commutation relations. The
principal results are (1) the demonstration that the perturbation in
the Weyl scalar Psi <SUB>2</SUB> must vanish in a gauge in which the
scalars Psi <SUB>1</SUB> and Psi <SUB>3</SUB> are assumed to vanish
identically; (2) the determination of the relative normalization of the
radial functions (left unspecified in part I) through an integrability
condition. Further, the solution to the integrability condition defines
a function involving quadratures over Teukolsky's radial and angular
functions; and it is in terms of this function that the perturbations
in the metric coefficients are determined.
---------------------------------------------------------
Title: Why are the stars as they are
Authors: Chandrasekhar, S.
1978pans.proc....1C Altcode:
Astronomical theories under consideration 40 years ago are surveyed with
reference to the question of why stars possess the characteristics that
they have. Eddington's (1926) parable of a physicist on a cloudbound
planet is examined, and its relation to the (1 - beta-asterisk)
theorem and to the combination (hc/G) to the 3/2 power divided by H
squared is discussed. Fowler's resolution to Eddington's paradox is
described with reference to the question of whether the stars have
enough energy to cool. The theory of degenerate configurations is
surveyed with attention to the limiting mass. Other topics include a
criterion for the stage at which stars can develop degenerate cores
and the minimum mass for gravitational collapse to be possible.
---------------------------------------------------------
Title: The gravitational perturbations of the Kerr black hole. I. The
perturbations in the quantities which vanish in the stationary state.
Authors: Chandrasekhar, S.
1977RSPSA.358..421C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: The gravitational perturbations of the Kerr black hole. II. The
perturbations in the quantities which are finite in the stationary
state.
Authors: Chandrasekhar, S.
1977RSPSA.358..441C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: The Kerr metric and stationary axisymmetric gravitational
fields.
Authors: Chandrasekhar, S.
1977RSPSA.358..405C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: The "Black Hole" in Astrophysics
Authors: Chandrasekhar, S.
1977nass.book..208C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: On the Reflexion and Transmission of Neutrino Waves by a Kerr
Black Hole
Authors: Chandrasekhar, S.; Detweiler, S.
1977RSPSA.352..325C Altcode: 1976RSPSA.352..325C; 1977RSLPS.352..325C
The equations governing the two-component neutrino are reduced to
the form of a one-dimensional wave equation. And it is shown how the
absence of super-radiance (i.e. a reflexion coefficient in excess of
one) for incident neutrino waves and its manifestation for incident
electromagnetic and gravitational waves (of suitable frequencies)
emerge very naturally from the character of the respective potential
barriers that surround the Kerr black hole.
---------------------------------------------------------
Title: On the Equations Governing the Gravitational Perturbations
of the Kerr Black Hole
Authors: Chandrasekhar, S.; Detweiler, S.
1976RSPSA.350..165C Altcode: 1976RSLPS.350..165C
Teukolsky's radial equation governing the general, non-axisymmetric,
gravitational perturbations of the Kerr black hole is reduced to
the form of a one-dimensional wave equation by making use of the
transformation which enables the treatment of the non-axisymmetric
modes in exactly the same way as the axisymmetric modes.
---------------------------------------------------------
Title: REVIEW: Liquid crystals
Authors: Chandrasekhar, S.
1976RPPh...39..613C Altcode:
An extended tutorial review article on the physics and properties
of thermotropic liquid crystals (LX) as examples of mesomorphic
states. Thermotropic LX (nematic, cholesteric, and smectic LX) are
contrasted to lyotropic LX in an introductory section. Statistical
theories of nematic order, the continuum theory of the nematic
state, properties of cholesteric LX, and properties of smectic LX
are the main headings. Topics discussed include: electrical and
magnetic birefringence, Maier-Saupe theory, deformation modes of
thermotropic LX (splay, twist, bend modes), Frank disclinations,
electrohydrodynamic (EHD) instabilities, fast turnoff EHD mode, chevron
patterns of oscillating domains, backflow effects, light scattering,
thermomechanical coupling, Freedericksz effect, memory effect, and
transitions between LX states.
---------------------------------------------------------
Title: Liquid crystals
Authors: Chandrasekhar, S.
1976RPPh...39..615C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: The Solution of Dirac's Equation in Kerr Geometry
Authors: Chandrasekhar, S.
1976RSPSA.349..571C Altcode: 1976RSLPS.349..571C
Dirac's equation for the electron in Kerr geometry is separated;
and the general solution is expressed as a superposition of solutions
derived from a purely radial and a purely angular equation.
---------------------------------------------------------
Title: The Solution of Maxwell's Equations in Kerr Geometry
Authors: Chandrasekhar, S.
1976RSPSA.349....1C Altcode: 1976RSLPS.349....1C
Explicit solutions for the vector-potential of a vacuum Maxwell-field
in Kerr geometry are obtained in terms of Teukolsky's radial and
angular functions.
---------------------------------------------------------
Title: Astronomers' Petition
Authors: Schramm, David N.; Anders, Edward; Chandrasekhar, S.; Palmer,
Patrick; Parker, Eugene; Simpson, J. A.
1976Sci...191.1124S Altcode:
No abstract at ADS
---------------------------------------------------------
Title: On a Transformation of Teukolsky's Equation and the
Electromagnetic Perturbations of the Kerr Black Hole
Authors: Chandrasekhar, S.
1976RSPSA.348...39C Altcode: 1976RSLPS.348...39C
Teukolsky's equation, governing the perturbations (scalar,
electromagnetic, and gravitational) of the Kerr black hole, is
transformed, by a simple change of variables, in a manner such that
there is formally no difference in the treatments of the axisymmetric
and the non-axisymmetric modes: the role of 'm' is effectively
eliminated. By considering in detail the case of electromagnetic
perturbations we show how, in all cases, the problems can be reduced
to problems in the theory of penetration of one dimensional potential
barriers with, however, certain novel features. The phenomenon of
super-radiance, peculiar to the Kerr metric, emerges, for example,
in an unexpected guise. The case of scalar perturbations is considered
briefly in an appendix.
---------------------------------------------------------
Title: On coupled second harmonic oscillations of the congruent
Darwin ellipsoids.
Authors: Chandrasekhar, S.
1975ApJ...202..809C Altcode:
The paper develops a general theory of coupled second harmonic
oscillations (with synchronous oscillations as a special degenerate
case) of the congruent Darwin ellipsoids. The congruent Darwin
ellipsoids, in equilibrium, consist of two identical ellipsoids facing
each other and rotating in circular orbits about their common center
of mass with a constant angular velocity. The relationship of this
model to the tidal interactions of a binary system is considered.
---------------------------------------------------------
Title: On the Equations Governing the Axisymmetric Perturbations of
the Kerr Black Hole
Authors: Chandrasekhar, S.; Detweiler, S.
1975RSPSA.345..145C Altcode: 1975RSLPS.345..145C
It is shown how Teukolsky's equation, governing the perturbations
of the Kerr black hole, can be reduced, in the axisymmetric case,
to a one-dimensional wave equation with four possible potentials. The
potentials are implicitly, dependent on the frequency; and besides,
depending on circumstances, they can be complex. In all cases
(i.e. whether or not the potentials are real or complex), the problem
of the reflexion and the transmission of gravitational waves by the
potential barriers can be formulated, consistently, with the known
conservation laws. It is, further, shown that all four potentials lead
to the same reflexion and transmission coefficients.
---------------------------------------------------------
Title: The Quasi-Normal Modes of the Schwarzschild Black Hole
Authors: Chandrasekhar, S.; Detweiler, S.
1975RSPSA.344..441C Altcode: 1975RSLPS.344..441C
The quasi-normal modes of a black hole represent solutions of the
relevant perturbation equations which satisfy the boundary conditions
appropriate for purely outgoing (gravitational) waves at infinity
and purely ingoing waves at the horizon. For the Schwarzschild
black hole the problem reduces to one of finding such solutions for a
one-dimensional wave equation (Zerilli's equation) for a potential which
is positive everywhere and is of short-range. The notion of quasi-normal
modes of such one-dimensional potential barriers is examined with two
illustrative examples; and numerical solutions for Zerilli's potential
are obtained by integrating the associated Riccati equation.
---------------------------------------------------------
Title: On the Equations Governing the Perturbations of the
Schwarzschild Black Hole
Authors: Chandrasekhar, S.
1975RSPSA.343..289C Altcode:
A coherent self-contained account of the equations governing the
perturbations of the Schwarzschild black hole is given. In particular,
the relations between the equations of Bardeen & Press, of Zerilli
and of Regge & Wheeler are explicitly established.
---------------------------------------------------------
Title: Development of general relativity
Authors: Chandrasekhar, S.
1974Natur.252...15C Altcode:
This article is based on an invited talk given at the 1974 annual
meeting of the American Physical Society in Chicago on the occasion
of the award of the Dannie Heineman Prize for Mathematical Physics,
February 5, 1974. (The substance of this lecture has been published
for private circulation in the Summer 1974 issue of The University of
Chicago Magazine.)
---------------------------------------------------------
Title: The deformed figures of the Dedekind ellipsoids in the
post-Newtonian approximation to general relativity.
Authors: Chandrasekhar, S.; Elbert, D. D.
1974ApJ...192..731C Altcode:
The effects of general relativity, in the post-Newtonian approximation,
on the Dedekind figures of equilibrium of homogeneous masses are
determined. It is shown how the post-Newtonian figures can be obtained
by first altering the velocity field in the Dedekind ellipsoid
appropriately, and then subjecting it to a suitable Lagrangian
displacement cubic in the coordinates. The solution exhibits a
singularity at a point where the axes of the Dedekind ellipsoid are in
the ratios 1:0.6158:0.4412. However, in contrast to what happens along
the Jacobian sequence, the occurrence of the singularity along the
Dedekind sequence is not associated with the onset of any instability
at that point by a strict Newtonian-like dynamic perturbation.
---------------------------------------------------------
Title: The Increasing Role of General Relativity in Astronomy
(Halley Lecture for 1972) [Observatory 92(990), 160, October 1972)]
Authors: Chandrasekhar, S.
1974SvPhU..17..309C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: On slowly rotating homogeneous masses in general relativity
Authors: Chandrasekhar, S.; Miller, John C.
1974MNRAS.167...63C Altcode:
The present paper is devoted to a study of slowly rotating homogeneous
masses in which the energy density E is a constant. The structure
of such configurations is determined with the aid of equations
derived by Hartle in the exact framework of general relativity. These
configurations have a natural limit in that the static, non-rotating,
configurations must have radii (R) exceeding 9/8 times the Schwarzschild
radius (R5). The derived structures, for varying RiR5, are illustrated
by a series of graphs. A result of particular interest which emerges
is that the ellipticity of the configuration, for varying radius but
constant mass and angular momentum, exhibits a very pronounced maximum
at RiR5 .
---------------------------------------------------------
Title: The Stability of Relativistic Systems (invited Paper)
Authors: Chandrasekhar, S.
1974IAUS...64...63C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: On a Criterion for the Onset of Dynamical Instability by a
Nonaxisymmetric Mode of Oscillation Along a Sequence of Differentially
Rotating Configurations
Authors: Chandrasekhar, S.
1974ApJ...187..169C Altcode:
In this paper, a variational base is derived for locating the point
of onset of dynamical instability along a sequence of differentially
rotating configurations. The underlying idea is to seek the conditions
for the existence of quasi-stationary, nonaxisymmetric modes of
deformation in frames of reference rotating with various preassigned
angular velocities. Subject headings: hydrodynamics - instabilities
- rotation
---------------------------------------------------------
Title: The black hole in astrophysics: the origin of the concept
and its role.
Authors: Chandrasekhar, S.
1974ConPh..15....1C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: The Stability of Stellar Masses in General Relativity
Authors: Chandrasekhar, S.
1974gra..conf..162C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: On a Criterion for the Occurrence of a Dedekind-Like Point of
Bifurcation Along a Sequence of Axisymmetric Systems. Relativistic
Theory of Uniformly Rotating Configurations
Authors: Chandrasekhar, S.; Friedman, John L.
1973ApJ...185....1C Altcode:
Stationary nonaxisymmetric systems in general relativity are
considered. It is shown that the theory of such systems can
be developed along lines which closely parallel the theory of
nonstationary axisymmetric systems. Equations are derived which govern
small nonaxisymmetric departures from equilibrium of axisymmetric
configurations of perfect fluid in uniform rotation. In terms of these
equations, the condition that a uniformly rotating configuration will
allow a quasi-stationary nonaxisymmetric deformation with a -dependence
of the form e (where m is an integer greater than or equal to one)
is obtained. A variational principle expressing this condition is also
derived. Subject headings: relativity - rotation
---------------------------------------------------------
Title: On a Criterion for the Occurrence of a Dedekind-Like Point of
Bifurcation Along a Sequence of Axisymmetric Systems. 11. Newtonian
Theory for Differentially Rotating Configurations
Authors: Chandrasekhar, S.; Lebovitz, Norman R.
1973ApJ...185...19C Altcode:
The equations in the Newtonian theory, which govern quasi-stationary
nonaxisymmetric deformations of axisymmetric configurations in
nonuniform rotation, are derived; and the condition for the existence
of such deformations with a -dependence of the form e (where m is
an integer greater than or equal to 1) is expressed in terms of
a variational principle. The condition for the case m = 2 applies
for the occurrence of a Dedekind-like point of bifurcation. In an
appendix the variational principle governing the axisymmetric modes
of oscillation of differentially rotating systems is reformulated in a
manner that avoids the solution of a secondorder partial differential
equation. Subject heading: rotation
---------------------------------------------------------
Title: On the Stability of Axisymmetric Systems to Axisymmetric
Perturbations in General Relativity.IV. Allowance for Gravitational
Radiation in an Odd-Parity Mode
Authors: Chandrasekhar, S.; Friedman, John L.
1973ApJ...181..481C Altcode:
In the present paper, the variational principle derved in Paper II
is clarified; and it is shown how it may be used to treat the damping
of the axisymmetric oscillations of a uniformly rotating star, by the
emission of gravitational radiation in an odd-parity mode. It is further
shown that the expression, for the imaginary part of the frequency as
a surface integral (at infinity), which follows from the variational
principle, is consistent with the requirements of the conservation of
energy. Subject headings: gravitation - relativity - rotation
---------------------------------------------------------
Title: Ellipsoidal figures of equilibrium.
Authors: Chandrasekhar, S.
1973efe..book.....C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: On the Stability of Axisymmetric Systems to Axisymmetric
Perturbations in General Relativity. III. Vacuum Metrics and Carter's
Theorem
Authors: Chandrasekhar, S.; Friedman, John L.
1972ApJ...177..745C Altcode:
The analysis of Paper II is specialized to vacuum metrics appropriately
for a discussion of their stability. And Carter's theorem, that
asymptotically fiat axisymmetric vacuum metrics, external to black
holes, cannot allow nontrivial axisymmetric neutral deformations,
is deduced.
---------------------------------------------------------
Title: The increasing role of general relativity in astronomy
Authors: Chandrasekhar, S.
1972Obs....92..160C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: On the Stability of Axisymmetric Systems to Axisymmetric
Perturbations in General Relativity. II. a Criterion for the Onset
of Instability in Uniformly Rotating Configurations and the Frequency
of the Fundamental Mode in Case of Slow Rotation
Authors: Chandrasekhar, S.; Friedman, John L.
1972ApJ...176..745C Altcode:
The theory developed in Paper I is applied to solve two problems in
general relativity: to obtain a criterion for the onset of instability
in a uniformly rotating configuration via a neutral mode of axisymmetric
oscillation; and to obtain an exact and an explicit formula for
the square of the frequency of the fundamental axisymmetric mode of
oscillation of a configuration rotating uniformly but slowly.
---------------------------------------------------------
Title: On the Stability of Axisymmetric Systems to Axisymmetric
Perturbations in General Relativity, I. The Equations Governing
Nonstationary, and Perturbed Systems
Authors: Chandrasekhar, S.; Friedman, John L.
1972ApJ...175..379C Altcode:
Axisymmetric systems in general relativity are considered. The field
and the fluid equations that are appropriate to general nonstationary
(but axisymmetric) systems are first derived. They are then specialized
to yield the equations which govern stationary equilibrium. The
equations whieh determine the evolution of small departures from
equilibrium are also obtained. Related matters that are considered
include the Landau-Lifshitz complex, the conserved quantities, and the
constancy of the baryon number and the angular momentum (per baryon)
of a fluid element as it moves. The theory is developed with a view
toward establishing criteria for the stability of rotating systems to
axisymmetric perturbations.
---------------------------------------------------------
Title: INVITED PAPER The Stability of Stellar Configurations in
General Relativity.
Authors: Chandrasekhar, S.
1972BAAS....4Q.315C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: On the “Derivation” of Einstein's Field Equations
Authors: Chandrasekhar, S.
1972AmJPh..40..224C Altcode:
An attempt is made to clarify the physical and the mathematical
reasonings that underlie Einstein's laws of gravitation. It appears
that only by a mixture of physical reasonableness, mathematical
simplicity, and aesthetic sensibility can one arrive at Einstein's field
equations. The general theory of relativity is in fact an example of
“the power of speculative thought.” The topics considered include a
discussion of the principle of equivalence and the view of space-time
as a geometric manifold. Two “derivations” of Einstein's equations
are given: one based on physical reasonableness and the other based
on a variational principle and mathematical simplicity.
---------------------------------------------------------
Title: A limiting case of relativistic equilibrium.
Authors: Chandrasekhar, S.
1972grec.conf..185C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Some elementary applications of thevirial theorem to stellar
dynamics
Authors: Chandrasekhar, S.; Elbert, Donna D.
1972MNRAS.155..435C Altcode:
The dynamical evolution of spherical and spheroidal systems of mass
points is examined with the aid of the scalar and the tensor forms
of the virial theorem. Spherical systems with positive total energy
tend to disperse to infinity while those with negative total energy
execute periodic oscillations of finite amplitude. Spheroidal systems
with positive total energy exhibit expansion like the spherical systems;
but they also become less oblate (and sometimes actually become prolate)
if they are initially oblate, and less prolate if they are initially
prolate. Spheroidal systems with negative total energy collapse to
smaller volumes while enhancing their initial oblateness or prolateness.
---------------------------------------------------------
Title: The Post-Newtonian Effects of General Relativity on the
Equilibrium of Uniformaly Rotating Bodies.VI. The Deformed Figures
of the Jacobi Ellipsoids (Continued)
Authors: Chandrasekhar, S.
1971ApJ...167..455C Altcode:
The theory of the (deformed) post-Newtonian Jacobi ellipsoid developed
in an earlier paper is specialized suitably to make a comparison
with the Newtonian ellipsoid having the same angular momentum and
baryon number.
---------------------------------------------------------
Title: The Post-Newtonian Effects of General Relativity on the
Equilibrium of Uniformly Rotating Bodies. V. The Deformed Figures
of the Maclaurin Spheroids (Continued)
Authors: Chandrasekhar, S.
1971ApJ...167..447C Altcode:
The theory of the (deformed) post-Newtonian Maclaurin spheroid developed
in an earlier paper is specialized suitably to make a comparison
with the Newtonian spheroid having the same angular momentum and
baryon number.
---------------------------------------------------------
Title: Criterion for the Instability of a Uniformly Rotating
Configuration in General Relativity
Authors: Chandrasekhar, S.; Friedman, John L.
1971PhRvL..26.1047C Altcode:
Uniformly rotating configurations in general relativity are considered,
and a condition is obtained that they can be quasistatically deformed
without violating any of the requirements for equilibrium. This
condition extends, into the domain of the rotating stars, the
criterion for the onset of dynamical instability (via a neutral mode
of oscillation) that occurs by radial pulsations in nonrotating stars.
---------------------------------------------------------
Title: An Announcement for Our Authors
Authors: Chandrasekhar, S.
1971ApJ...164..397C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: An Announcement for Our Authors
Authors: Chandrasekhar, S.
1971ApJ...164L..79C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Editorial
Authors: Chandrasekhar, S.
1970ApJ...162....1C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: The Evolution of the Jacobi Ellipsoid by Gravitational
Radiation
Authors: Chandrasekhar, S.
1970ApJ...161..571C Altcode:
The quasi-static evolution of the Jacobi ellipsoid by gravitational
radiation is determined by integrating the equation that gives the
rate of dissipation of the angular momentum with the constraint that
the ellipsoid remains Jacobian at all times. It is found that the
evolution is in the direction of increasing angular velocity toward
a non-radiating state at the point of bifurcation with the Maclaurin
sequence.
---------------------------------------------------------
Title: The Effect of Gravitational Radiation on the Secular Stability
of the Maclaurin Spheroid
Authors: Chandrasekhar, S.
1970ApJ...161..561C Altcode:
It is shown that the dissipation of energy by gravitational radiation
induces (in the manner of viscosity) a secular instability of the
Maclaurin spheroid at the point of bifurcation where the Jacobian
sequence branches off. But the mode of oscillation that is made unstable
by radiation reaction is not the same one that is made unstable by
viscosity.
---------------------------------------------------------
Title: The Instability of the Congruent DARWIN Ellipsoids. II
Authors: Chandrasekhar, S.
1970ApJ...160.1043C Altcode:
The resonant oscillations of one of two congruent Darwin ellipsoids,
forced by the natural oscillations of the other, are considered; and
the instability of the ellipsoids to synchronous coupled oscillations
is traced to this resonance.
---------------------------------------------------------
Title: Editorial
Authors: Chandrasekhar, S.
1970ApJ...160....1C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: The 2½-POST-NEWTONIAN Equations of Hydrodynamics and Radiation
Reaction in General Relativity
Authors: Chandrasekhar, S.; Esposito, F. Paul
1970ApJ...160..153C Altcode:
In this paper the equations of hydrodynamics in the 2 '-post-Newtonian
approximation to general relativity are derived. In this approximation
all terms of O(c-9) are retained consistently with Einstein's field
equations; it is also the approximation in which terms representing
the reaction of the fluid to the emission of gravitational radiation
by the system first make their appearance. The paper is in four
parts. In Part I (by S. C.) the lowest-order terms in the metric
coefficients are derived which are consequences of the imposition
of the Sommerfeld radiation-condition at infinity. It is shown
(following an early investigation of Trautman) that these terms
are of O(c-5) in gss, of O(c-5) in gs , and of O(c-0) in g . Unique
expressions are bbtained for these terms. They are found to be purely
of Newtonian origin. In Part II (by S. C. and F. P. E.) the equations
of motion governing the fluid in the -post-Newtonian approximation are
derived. In addition to the coefficients already determined, these
equations depend on a knowledge of the term of O(c-7) in goo. This
term is determined by an explicit appeal to the field equation. It is
further shown that this approximation brings no change to the density
(c2pn5 V - g) and the linear momentum ( a) that are conserved in the
second post-Newtonian approximation. In Part III (by S. C.) it is
shown that the terms of O( ) in the equations of motion contribute
principally to the dissipation of the energy and the angular momentum
conserved in the second post-Newtonian approximatioP. The rates of
dissipation of energy and of angular momentum that are predicted
are in exact agreement with the expectations of the linearized
theory of gravitational radiation. Finally, in Part IV (by S. C. and
F. P. E.) the energy, 009 - C9 pit0 V - g, to be associated with the
221 -post-Newtonian approximation is derived by evaluating the (0,
0)-component of the Landau-Lifshitz complex and the conserved density
in the 3v1tost-Newtonian approximation.
---------------------------------------------------------
Title: Solutions of Two Problems in the Theory of Gravitational
Radiation
Authors: Chandrasekhar, S.
1970PhRvL..24..611C Altcode:
The evolution of an elongated rotating configuration by gravitational
radiation and the possibility of a secular instability being induced by
it are considered in the context of the classical homogeneous figures
of Maclaurin and Jacobi. The triaxial Jacobian ellipsoid evolves in the
direction of increasing angular velocity and approaches (exponentially)
the point of bifurcation where it ceases to radiate. Further, radiation
reaction does not make the Maclaurin spheroid secularly unstable past
the point of bifurcation.
---------------------------------------------------------
Title: Post-Newtonian Methods and Conservation Laws
Authors: Chandrasekhar, S.
1970rela.conf...81C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Editorial
Authors: Chandrasekhar, S.
1969ApJ...158..847C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Conservation Laws in General Relativity and in the
Post-Newtonian Approximations
Authors: Chandrasekhar, S.
1969ApJ...158...45C Altcode:
It is shown how the exact conservation laws of general relativity,
expressed in terms of the symmetric energy-momentum complex of Landau
and Lifshitz, can be used to determine the various conserved quantities
in the different post-Newtonian approximations. Particular attention
is given to the conserved energy which emerges as the integral over
the whole of space of the difference between the (0,0)-com- ponent
of the Landau-Lifshitz complex and the energy of the conserved mass
present. The method is illustrated in the framework of the first
post-Newtonian equations of hydrodynamics
---------------------------------------------------------
Title: The Second Post-Newtonian Equations of Hydrodynamics in
General Relativity
Authors: Chandrasekhar, S.; Nutku, Yavuz
1969ApJ...158...55C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: The Instability of the Congruent Darwin Ellipsoids
Authors: Chandrasekhar, S.
1969ApJ...157.1419C Altcode:
In this paper a class of synchronous coupled oscillations of the
congruent Darwin ellipsoids is con- sidered; and it is shown that
two of the five modes of oscillation belonging to this class excite
instabilities along the entire Darwin sequence
---------------------------------------------------------
Title: Editorial
Authors: Chandrasekhar, S.
1969ApJ...155..371C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Editorial
Authors: Chandrasekhar, S.
1968ApJ...154....1C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: The Pulsations and the Dynamical Stability of Gaseous Masses
in Uniform Rotation
Authors: Chandrasekhar, S.; Lebovitz, N. R.
1968ApJ...152..267C Altcode:
A variational principle, applicable to axisymmetric oscillations
of uniformly rotating axisymmetric configurations, is established
On the assumption that the Lagrangian displacement (describing
the oscillation) at any point is normal to the level surface (of
constant total potential) through that point, it is shown how the
variational expression, for the frequencies of oscillation, can be
reduced to simple quadratures. The reduction is explicitly carried
out for certain stratifications of special interest. Some new results
on the oscillations of slowly rotating configurations are included;
and a number of related observations on their stability are also made
---------------------------------------------------------
Title: The Virial Equations of the Fourth Order
Authors: Chandrasekhar, S.
1968ApJ...152..293C Altcode:
The virial equations of the fourth order are derived; and the results of
certain formal developments needed for their practical usefulness are
also given The equations are then used to locate the neutral points,
along the Maclaurin and the Jacobian sequences, that belong to the
fourth harmonics
---------------------------------------------------------
Title: Editorial
Authors: Chandrasekhar, S.
1968ApJ...152L...1C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: A tensor virial-equation for stellar dynamics
Authors: Chandrasekhar, S.; Lee, Edward S.
1968MNRAS.139..135C Altcode:
A tensor virial-equation is derived for a system consisting of equal
mass-points, appropriately for stellar dynamics, by starting with the
Liouville equation governing an ensemble of such systems in the six
N-dimensional phase space.
---------------------------------------------------------
Title: The Post-Newtonian Effects of General Relativity on the
Equilibrium of Uniformly Rotating Bodies.IV. The Roche Model
Authors: Chandrasekhar, S.
1967ApJ...148..645C Altcode:
In this paper, the Roche model, consisting of a tenuous envelope in
the field of a central rigidly rotating central mass, is considered
in the framework of the post-Newtonian equations of hydrodynamics;
and the effect, in particular, on the critical isobaric surface with
the lenticular shape is determined.
---------------------------------------------------------
Title: The Post-Newtonian Effects of General Relativity on the
Equilibrium of Uniformly Rotating Bodies. III. The Deformed Figures
of the Jacobi Ellipsoids
Authors: Chandrasekhar, S.
1967ApJ...148..621C Altcode:
The effects of general relativity, in the post-Newtonian approximation,
on the Jacobian figures of equilibrium of uniformly rotating homogeneous
masses are determined. It is shown, for example, that the post-Newtonian
figure is obtained by a deformation of the Jacobi ellipsoid by
a suitable Lagrangian displacement cubic in the coordinates. The
solution of the post-Newtonian equations exhibits an indeterminacy at
the point of bifurcation M2, where the Jacobian sequence branches off
from the Maclaurin sequence, and a singularity at a point J4, where
the axes of the Jacobi ellipsoid are in the ratios 1:0 2972:0.2575. The
indeterminacy in the solution at M2 arises from the fact that at this
point the Maclaurin spheroid is neutral to an infinitesimal deformation
proportional to (s , -x2, 0); and the singularity at J4 arises from
the fact that at this point the Jacobi ellipsoid is unstable to the
deformation induced by the effects of general relativity.
---------------------------------------------------------
Title: Editorial Notice
Authors: Chandrasekhar, S.; Clemence, G. M.
1967ApJ...148..319C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Editorial
Authors: Chandrasekhar, S.
1967ApJ...148....1C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: On a Post-Galilean Transformation Appropriate to the
Post-Newtonian Theory of Einstein, Infeld and Hoffmann
Authors: Chandrasekhar, S.; Contopoulos, G.
1967RSPSA.298..123C Altcode:
It is shown that the line element, which forms the basis of the
post-Newtonian theory of Einstein, Infeld and Hoffmann for the motion of
mass points under their mutual gravitational attractions, is invariant
in form to a certain post-Galilean transformation. It is necessary that
the transformation, expressed as an expansion in inverse powers of c^2
(c is the velocity of light), include terms of O(c<SUP>-2</SUP>) in the
transformation for the spatial coordinate and terms of O(c<SUP>-4</SUP>)
in the transformation for the time coordinate. Comparison with the
Lorentz transformation (between two frames in uniform relative motion
with a velocity V), expanded similarly in inverse powers of c^2,
shows (1) that the spatial part of the transformation includes the
Lorentzian terms (up to O(c<SUP>-2</SUP>)) and allows, in addition, an
arbitrary rotation, a uniform motion, and a shift of the origin (all of
O(c<SUP>-2</SUP>)) and (2) that the temporal part of the transformation
includes the Lorentzian terms (up to O(c<SUP>-4</SUP>)) and an
additional term of purely gravitational origin. It is the presence
of this last gravitational term that gives to the transformation its
non-Lorentzian character. For a suitable choice of the constants in
the post-Galilean transformation, the parameter V can be interpreted
as a velocity of relative motion between the two frames, even as in the
Lorentz transformation. The invariance of the form of the line element
to the transformation ensures that the equations of motion which follow
are similarly invariant to the transformation. This fact is further
verified by showing that the Lagrangians in the two frames differ
by the total derivative of a function. The relations between the ten
constants of the motion in the two frames are found. And the special
case when the transformation can be regarded as one appropriate to a
'centre of mass system' is briefly considered.
---------------------------------------------------------
Title: Virial Relations for Uniformly Rotating Fluid Masses in
General Relativity
Authors: Chandrasekhar, S.
1967ApJ...147..383C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: The Stability of Gaseous Masses in the Post-Newtonian
Approximation
Authors: Chandrasekhar, S.
1967rta3.book...74C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: The Post-Newtonian Effects on the Equilibrium of the Maclaurin
Spheroids
Authors: Chandrasekhar, S.
1967rta3.book...68C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: The Post-Newtonian Effects of General Relativity on the
Equilibrium of Uniformly Rotating Bodies. II. The Deformed Figures
of the Maclaurin Spheroids
Authors: Chandrasekhar, S.
1967ApJ...147..334C Altcode:
The equations of post-Newtonian hydrodynamics are solved appropriately
for a uniformly rotating homogeneous mass with symmetry about the axis
of rotation. The post-Newtonian figure is obtained as a deformation
of the Newtonian Maclaurin spheroid (with semi-axes and , say) by
a Lagrangian displacement proportional to (2) =a'12( 2Xl 2x2, - ),
where w denotes the distance from the axis of rotation and x1 and x2
are the Cartesian coordinates in the equatorial plane. It is shown that
the equation defining the boundary of the post-Newtonian configuration
is of the form 2 2 R 5 4 2+-2-t-2S2t(e)--4-22 =0, ai a3 a1 a1 a1 a3
(e) is a determinate function of the eccentricity e of the Maclaurin
spheroid and R5 (= 2CM/c2) is the Schwarzschild radius. The function
S t(e) is tabulated in the paper. Further, the angular velocity of
rotation of the post-Newtonian configuration differs from that of the
Maclaurin spheroid by an amount which is also tabulated. The solution
of the post-Newtonian equations exhibits a singularity at a certain
eccentricity e*(= 0.985226) of the Maclaurin spheroid. The origin
of this singularity is that at e* the Maclaurin spheroid allows an
infinitesimal neutral deformation by a displacement proportional to
(2); and the Newtonian instability of the Maclaurin spheroid at e*
is excited by the post-Newtonian effects of general relativity.
---------------------------------------------------------
Title: The Post-Newtonian Equations of Hydrodynamics in General
Relativity
Authors: Chandrasekhar, S.
1967rta3.book...43C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Editorial Notice
Authors: Chandrasekhar, S.
1967AJ.....72..452C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: The Equilibrium and the Stability of the Riemann Ellipsoids. II
Authors: Chandrasekhar, S.
1966ApJ...145..842C Altcode:
In this paper we consider ellipsoidal figures of eqmlibnum (of
semi-axes a,, a,, and a,) of homogeneous masses rotating uniformly
vith an angular velocity and with internal motions having a uniform
vorticity (in the rotating frame) in the case that the directions of
and do not coincide. Riemaun `5 theorem, that in this case and must
lie in a principal plane of the ellipsoid, is shown to follow from
a consideiation of the non-diagonal components of the second-order
tensor-virial theorem. The conditions for equilibrium are also derived;
and the domains of occupancy of these Riemaun ellipsoids in the (a,/a,,
a,/a,)-plane (on the assumpt'ons, which entail no loss of generality,
that fl and have no camponents in the x,-direction and that a, > a,)
are explicitly specified. It is shown that the eqmlibrium ellipsoids are
of three types: ellipsoids of type I which occupy the domain 2a1 >
(a, + a,) and a, > a, > a,; ellipsoids of type II for which a,
> 2a, and a,/a, (<1) are limited by a locus along which fpdx =
0; and ellipsoids of type III which occupy the domain limited by 2a,
< (a, - a,) and a locus along which P, = = 0 and a, > a,. And
quite generally, it is shown that an ellipsoid, represented by a point
in the allowed domain of occupancy, is a figure of equilihrium for
two different states of motion ( , ) and ( t, t); and that the two
resulting configurations are adjoints of one another in the sense of
Dedekind's theorem. Ellipsoids of type I may he considered as hranching
off from the Maclaurin sequence with an odd mode of oscillation
neutralized at the point of bifurcation by the choice of P, and ,
( l, and being zero). And ellipsoids of type III may be nmilarly
considered as hranching off from the ellipsoids of type S (for which
the directions of and coincide with the x,-axis) along the curve where
they are marginally unstable. The stability of the Riemaun ellipsoids
with respect to oscillations helonging to the second harmonics is also
investigated. It is first shown that the characteristic frequencies
of oscillation of an ellipsoid and its adjoint are the same; and
further that 11 and are allowed proper frequencies. The loci along
which instability sets in, in the different domains of occupancy, are
determined. Of particular interest are the facts that all ellipsoids
of type II are unstable; that along the curve where the ellipsoids of
type III branch off from ellipsoids of type S, the stability passes
from the latter to the former; and that among the ellipsoids of type
I there are some very highly flattened ones that are stable. Several
statements of Riemaun concerning the stability of these ellipsoids are
not suhstantiated hy the present detailed investigation. The origin
of Riemann s errors is clarified in the paper by Lebovitz following
this one.
---------------------------------------------------------
Title: Editorial
Authors: Chandrasekhar, S.
1966ApJ...143....1C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: The Stability of Gaseous Masses for Radial and Non-Radial
Oscillations in the Post-Newtonian Approximation of General
Relativity.
Authors: Chandrasekhar, S.
1965ApJ...142.1519C Altcode:
The stability of gaseous masses with respect to radial as well
as non-radial oscillations is considered in the framework of the
post-Newtonian equations of hydrodynamics The onset of dynamical
instability at a radius R determined by a formula of the type 2CM
K c2 (where K is a constant) is confirmed in case the "ratio of the
specific heats" = (a log p/a log p) (where the subscript S denotes that
the derivative is with respect to constant entropy) is a constant. An
expression for K is derived which does not involve any knowledge of the
equilibrium configuration beyond the Newtonian framework; and the values
of K appropriate to the polytropes are also listed. With respect to the
onset of instability for non-radial oscillations, it is shown that the
classical criterion of Schwarzschild based on the discriminant S(r)=dP
pdp dr dr is replaced by one based on the discriminant (r) =S(r)+ (F +
is;;:;) where His the internal energy (per unit volume) and F is a ratio
defined by the relation p11 = p/(F - 1). An alternative form for (r),
namely, (r) =S(r)[1+{' d {} } d } where F3 = 1 + (a log T/a log p),,
shows that the condition for the occurrence of convective instability
is unaltered in the post-Newtonian approximation.
---------------------------------------------------------
Title: The Post-Newtonian Equations of Hydrodynamics in General
Relativity.
Authors: Chandrasekhar, S.
1965ApJ...142.1488C Altcode:
The standard Eulerian equations of hydrodynamics are generalized to
take into account, consistently with Einstein's field equations, all
effects of order 1/c2. It is further shown that these post-Newtonian
equations allow integrals of motion which are entirely analogous to
the Newtonian integrals that express the conservation of mass, linear
momentum, angular momentum, and energy. The continued validity of
these conservation laws enables a consistent definition of "mass,"
"momentum," and "energy" in the framework of the post-Newtonian
theory. Besides the equations of motion, an appropriate tensor form
of the virial theorem is also derived.
---------------------------------------------------------
Title: The Post-Newtonian Effects of General Relativity on the
Equilibrium of Uniformly Rotating Bodies. I. The Maclaurin Spheroids
and the Virial Theorem.
Authors: Chandrasekhar, S.
1965ApJ...142.1513C Altcode:
The post-Newtonian effects of general relativity on the equilibrium of
uniformly rotating bodies are considered with the aid of a suitably
generalized version of the classical tensor virial theorem. An exact
relation exhibiting the relativistic effects is obtained; and it shows
that, if the figure of equilibrium is approximated by a spheroid,
the effect of general relativity is to attribute to the spheroid a
larger angular velocity than the Newtonian value.
---------------------------------------------------------
Title: The Equilibrum and the Stability of the Riemann Ellipsoids. I.
Authors: Chandrasekhar, S.
1965ApJ...142..890C Altcode:
Riemann's problem is concerned with the ellipsoidal figures of
equilibrium of homogeneous masses rotating uniformly with an angular
velocity and with internal motions having a uniform vorticity in
the rotating frame. In this paper, the equilibrium and the stability
of these Riemann ellipsoids are considered in the special case the
axes of rotation and vorticity coincide with a principal axis of the
ellipsoid. It is shown that for the case considered (1) the equilibrium
figures can be delineated into sequences- the Riemann sequences-along
which the ratio f = / is a constant; (2) an ellipsoid which is a figure
of equilibrium for some given j is also a figure of equilibrium for
ft = (ai2 + a22)2/ai2ai2f, where ai and a2 are the semi-axes of the
ellipsoid in the equatorial plane; (3) the two states of internal
motion, corresponding to f and ft, lead to configurations which are
adjoint in the sense of a theorem due to Dedekind; (4) the first
member of a Riemann sequence is a Maclaurin spheroid which is stable
in the absence of any dissipative mechanism; (5) from each point
of the stable part of the Maclaurin sequence two Riemann sequences
bifurcate; (6) there exist two self-adjoint sequences along which f
= Jt = + (a12 + ai2) /aia and that limit the domain of the Riemann
sequences in the (a /ai, ai/ai) = plane; and (7) the bifurcation of
the Jacobian and the Dedekind sequences from, what is usually called,
the point of bifurcation is a special case of a much more general
phenomenon. The stability of the Riemann ellipsoids with respect to
modes of oscillation belonging to the second and the third harmonics
is also investigated. With respect to modes of oscillation belonging
to the second harmonics it is shown that (1) the Riemann ellipsoids
allow a non-trivial neutral mode of oscillation; (2) the characteristic
frequencies of oscillation of an ellipsoid and its adjoint are the same;
(3) the Riemann ellipsoids with f > -2 are stable with respect to
these modes; and (4) instability by one of these modes arises along the
sequences forj < -2. With respect to modes of oscillation belonging
to the third harmonics, it is shown that along all Riemann sequences
instability first arises by a mode which deforms the ellipsoid into
a pear-shaped configuration. The points at which instability sets in
along the different Riemann sequences and the loci, which separate the
regions of stability from the regions of instability in the domain of
the Riemann ellipsoids considered, are also determined.
---------------------------------------------------------
Title: The Equilibrium and the Stability of the Dedekind Ellipsoids.
Authors: Chandrasekhar, S.
1965ApJ...141.1043C Altcode:
Figures ot equilibrium of liquid masses with internal motions of uniform
vorticity define the Dedekind sequence. It has been known for a long
time that the Dedekind sequence of stationary ellipsoids is congruent
to the Jacobian sequence of uniformly rotating ellipsoids. It is shown
in this paper that the characteristic frequencies of oscillation of a
Dedekind ellipsoid, belonging to the second harmonics, are identical
with those of a Jacobi ellipsoid having the same figure; but that the
points at which instability sets in by a mode of oscillation belonging
to the third harmonics are different along the two sequences.
---------------------------------------------------------
Title: Non-Radial Oscillations of Gaseous Masses.
Authors: Chandrasekhar, S.; Lebovitz, Norman R.
1964ApJ...140.1517C Altcode:
In this paper the fundamental frequencies of non-radial oscillations of
polytropic gas spheres, belonging to spherical harmonics of orders 1 =
1 and 2, are evaluated, in a "second approximation," by a variational
method. Also, the value of the ratio of the specific heats is determined
for which an accidental degeneracy occurs between the fundamental
modes of radial oscillation and non-radial oscillation belonging to 1
= 2; it is found that this value of varies from 1 6 for a homogeneous
compressible sphere to 1 5719 for a polytrope of index n = 3.
---------------------------------------------------------
Title: Erratum: the Dynamical Instability of Gaseous Masses
Approaching the Schwarzschild Limit in General Relativity.
Authors: Chandrasekhar, S.
1964ApJ...140.1342C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: The Equilibrium and the Stability of the DARWIN Ellipsoids.
Authors: Chandrasekhar, S.
1964ApJ...140..599C Altcode:
Darwin's problem is concerned with the equilibrium and the
stability of synchronously rotating homogeneous masses under their
mutual gravitational and tidal interactions. The problem is solved
consistently, in a method of approximation due to Jeans, in two special
cases: the case when one of the two components is of infinitesimal
mass compared to the other and the case when the two components are
of equal mass and congruent. In the former case, the problem insofar
as the equilibrium and the stability of the infinitesimal mass is
concerned, is hardly distinguishable from Roche's simpler problem in
which the distorting mass is treated as a rigid sphere. However, in
Darwin's formulation, the distorting mass (in the case considered) is
a Maclaurin spheroid; and a principal result is that Darwin's problem
has no solution when the eccentricity of the spheroid exceeds a certain
maximum value (= 0.40504). In the case of the congruent components, the
maximum angular velocity of orbital rotation, the distance of closest
approach, and the Roche limit (where the equilibrium ellipsoid can be
deformed into a neighboring equilibrium ellipsoid by a quasi-static,
infinitesimal, solenoidal, ellipsoidal displacement), all occur at
different points along the sequence; and instability, by a mode of
natural oscillation of either component by itself, sets in at a still
different point. It appears, moreover, that of the two figures of
equilibrium one obtains (at each separation) those with the greater
elongations overlap; all the physically realizable equilibrium
ellipsoids are therefore stable with respect to their individual
natural oscillations. The bearing of these results on the concepts of
"limiting stability" and "partial stability" due to Darwin and Jeans
is briefly examined.
---------------------------------------------------------
Title: The Dynamical Instability of Gaseous Masses Approaching the
Schwarzschild Limit in General Relativity.
Authors: Chandrasekhar, S.
1964ApJ...140..417C Altcode:
In this paper the theory of the infinitesimal, baryon-number conserving,
adiabatic, radial oscillations of a gas sphere is developed in the
framework of general relativity. A variational base for determining
the characteristic frequencies of oscillation is established. It
provides a convenient method for obtaining sufficient conditions for
the occurrence of dynamical instability. The principal result of the
analysis is the demonstration that the Newtonian lower limit T4, for
the ratio of the specific heats , for insuring dynamical stability
is increased by effects arising from general relativity; indeed, is
increased to an extent that, so long as is finite, dynamical instability
will intervene before a mass contracts to the limiting radius (>2.25
GM/c2) compatible with hydrostatic equilibrium. Moreover, if should
exceed only by a small amount, then dynamical instability will occur
if the mass should contract to the radius Rc = K 4 C2 -3 where K is
a constant depending, principally, on the density distribution in the
configuration. The value of the constant K is explicitly evaluated for
the homogeneous sphere of constant energy density and the polytropes
of indices n = 1, 2, and 3.
---------------------------------------------------------
Title: The Dynamical Instability of the White-Dwarf Configurations
Approaching the Limiting Mass.
Authors: Chandrasekhar, S.; Tooper, Robert F.
1964ApJ...139.1396C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Otto Struve, 1897-1963.
Authors: Chandrasekhar, S.
1964ApJ...139..423C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: A General Variational Principle Governing the Radial and the
Non-Radial Oscillations of Gaseous Masses.
Authors: Chandrasekhar, S.
1964ApJ...139..664C Altcode:
In this paper a general variational principle, applicable to radial
as well as non-radial oscillations of gaseous masses, is formulated
and proved. And it is, further, shown that when the normal modes
are analyzed in vector spherical harmonics, the variational principle
requires that the square of the characteristic frequency of oscillation,
(r2, belonging to a particular spherical harmonic, is stationary
with respect to simultaneous variations of two independent radial
functions. A consequence of this result is that ff2 (belonging to
a particular harmonic) emerges as a characteristic root of a 2 X 2
matrix. Two simple illustrations of the variational principle are given.
---------------------------------------------------------
Title: On the ellipsoidal figures of equilibrium of homogeneous masses
Authors: Chandrasekhar, S.; Lebovitz, Norman R.
1964ApNr....9..323C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: The Equilibrium and the Stability of the Roche Ellipsoids.
Authors: Chandrasekhar, S.
1963ApJ...138.1182C Altcode:
Roche's problem is concerned with the equilibrium and the stability
of rotating homogeneous masses which are, further, distorted by
the constant tidal action of an attendant rigid spherical mass. This
ancient problem is reconsidered in this paper with the principal oblect
of determining the stability of the equilibrium configurations (the
ellipsoids of Roche) by a direct evaluation of their characteristic
frequencies of oscillation belonging to the second harmonics The
result of the evaluation is the demonstration that the Roche ellipsoid
becomes unstable at a point subsequent to the Roche limit where the
angular velocity of rotation, consistent with equilibrium, attains
its maximum value. This result requires a revision of the current
common view regarding the meaning that is to be attached to the Roche
limit Among related matters which are considered are the following:
the relationships that exist between the sequences of Roche and those
of Maclaurin, Jacobi, and Jeans; the exhibition and the isolation of
the second neutral point (belonging to the third harmonics) along the
Roche sequences; and the effect of compressibility on the stability of
the Roche ellipsoids. A result which emerges from these considerations
is the universal instability of the Jacobi ellipsoids under the least
tidal action. The methods used in this paper are those derived from
the virial theorem and its various extensions. The principal results
are summarized in Section X and are exhibited in Figures 1, 2, and 3.
---------------------------------------------------------
Title: The Ellipticity of a Slowly Rotating Configuration.
Authors: Chandrasekhar, S.; Roberts, P. H.
1963ApJ...138..801C Altcode:
The second-order virial theorem is used to set upper and lower bounds
for m/e for a slowly rotating confignration, where m is the ratio of the
centrifugal acceleration at the equator to the (average) gravitational
acceleration on its surface and ER is the ellipticity of its slightly
oblate figure of equilibrium. The bounds obtained are explicitly
evaluated for the polytropes, for a model consisting of a core and a
mantle of constant densities, and for a particular model for the earth.
---------------------------------------------------------
Title: Letter to the Editor: a General Variational Principle Governing
the Radial and the Non-Radial Oscillations of Gaseous Masses.
Authors: Chandrasekhar, S.
1963ApJ...138..896C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Non-Radial Oscillations and Convective Instability of Gaseous
Masses.
Authors: Chandrasekhar, S.; Lebovitz, Norman R.
1963ApJ...138..185C Altcode:
Modes of non-radial oscillation of gaseous masses belonging to spherical
harmonics of orders 1 = 1 and 3 are considered on the basis of the
first- and the third-order virial equations. For an assumed Lagrangian
displacement of the form = (Li; + Li) (where Li;jk and Lj represent a
total of twenty-one unspecified constants and cr is the characteristic
frequency to be determined), the theory predicts the occurrence of
modes of oscillation of two different types: modes (belonging to 1 =
3) which are analogous to the Kelvin modes of an incompressible sphere
and modes (belonging to 1 = 1) which are analogous to those discovered
by Pekeris for a homogeneous compressible sphere and which exhibit its
convective instability. For the latter modes, the virial equations lead
to a characteristic equation for cr2 of degree 2 whose coefficients
are integrals over the variables of the unperturbed configuration,
including its superpotential. The theory is applied to the polytropic
gas spheres, and it is shown that they are convectively unstable
(for the modes belonging to 1 = 1) if the ratio of the specific
heats is less than a certain critical value The critical values of
predicted by the (approximate) theory differ from 1 + 1/n (where n
is the polytropic index) by less than 1 per cent over the range of n
(<3.5) considered; the extent of this agreement is a measure of the
accuracy of the method based on the virial equations and the assumed
form of the Lagrangian displacement.
---------------------------------------------------------
Title: The Points of Bifurcation Along the Maclaurin, the Jacobi,
and the Jeans Sequences.
Authors: Chandrasekhar, S.
1963ApJ...137.1185C Altcode:
The role which the second- and the third-order virial equations
governing equilibrium can play in isolating points of neutral stability
along equilibrium sequences is discussed and clarified. It is shown
that a necessary condition for the occurrence of a neutral point is
that a non-trivial Lagrangian displacement exists for which the first
variations of alt of the integral relations (five in the second order
and fifteen in the third order), provided by the virial equations,
vanish. By using this condition, it is possible, for example, to
isolate the point of bifurcation along the Jacobian sequence without
any prior specification of the nature of the sequence which follows
bifurcation. As further illustrations of the method, the known points
of neutral stability along the Maclaurin and the Jeans sequences are
also derived.
---------------------------------------------------------
Title: On the Stability of the Jacobi Ellipsoids.
Authors: Chandrasekhar, S.; Lebovitz, Norman R.
1963ApJ...137.1142C Altcode:
In this paper the problem of the small oscillations of the Jacobi
ellipsoids is solved, and all the characteristic frequencies belonging
to the second and the third harmonics are found. In particular,
the variation, along the Jacobian sequence, of the characteristic
frequency with respect to which the Jacobi ellipsoid becomes unstable
is exhibited.
---------------------------------------------------------
Title: The Equilibrium and the Stability of the Jeans Spheroids.
Authors: Chandrasekhar, S.; Lebovitz, Norman R.
1963ApJ...137.1172C Altcode:
The equilibrium and the stability of homogeneous masses distorted
by the tidal effects of a secondary (of mass M' at a distance R)
are re-examined on the basis of the second-order virial equations. In
agreement with known results,it is shown that, under circumstances when
the figure of equilibrium is a prolate spheroid, there is a maximum
value of p( - GM'/R3) which is compatible with equilibrium. The
problem of the small oscillations of these Jeans spheroids is next
considered. The characteristic frequencies of oscillation belonging to
the second harmonics are determined both in case the mass is considered
incompressible and in case it is considered compressible and subject
to the gas laws governing adiabatic changes. In the former case,
instability sets in when p attains its maximum value; and in the latter
case it sets in before that happens.
---------------------------------------------------------
Title: On the Oscillations of the Maclaurin Spheroid Belonging to
the Third Harmonics.
Authors: Chandrasekhar, S.; Lebovitz, Norman R.
1963ApJ...137.1162C Altcode:
The characteristic frequencies of oscillation of the Maclaurin spheroid
belonging to the third harmonics are found Two further points of
neutral stability, beyond the first at eccentricity e = 0.81267, are
isolated They occur at e = 0.89926 and e = 0.96937; it is the second
of these that is the analogue of the point of bifurcation along the
Jacobian sequence.
---------------------------------------------------------
Title: On the Oscillations and the Stability of Rotating Gaseous
Masses. III. The Distorted Polytropes.
Authors: Chandrasekhar, S.; Lebovitz, Norman R.
1962ApJ...136.1082C Altcode:
The theory of the oscillations of rotating gaseous masses, developed
in an earlier paper of this series, is here applied to determine the
effect of a small rotation ( on the fundamental modes of oscillation of
a polytrope. The basis for this application is provided by the theory of
rotationally distorted polytropes; this theory is reviewed and amplified
further to include a discussion of the superpotential. The various
tensors, in terms of which the characteristic frequencies of oscillation
are expressed, are evaluated for distorted polytropes, appropriately,
to the first order in . The final results on the effect of rotation
on the characteristic frequencies are presented in the form of tables.
---------------------------------------------------------
Title: On the Point of Bifurcation Along the Sequence of the Jacobi
Ellipsoids.
Authors: Chandrasekhar, S.
1962ApJ...136.1048C Altcode:
In this paper, the known point of bifurcation along the sequence of the
Jacobi ellipsoids is isolated by a new method based on equilibrium
considerations only. The method consists in finding an integral
property (or, more generally, a functional) of the configuration which
vanishes as a condition of equilibrium. The first variation of such a
functional will vanish at a point of bifurcation (and only at a point of
bifurcation) for a Lagrangian displacement which deforms the body from
the shape it has along an equilibrium sequence to the shape it will
have in the sequence following bifurcation. For finding a functional
j with the requisite properties, an equation for the third-order
virial (namely, fpi# ) is first established. And from an examination
of the conditions, which follow from this equation, for equilibrium,
it is found that J= [ +x2Q312+x1( 33 22) ] dx (where , is the tensor
potential of the gravitational field) has all the necessary properties
The first variation of j, for the Lagrangian displacement which deforms
a Jacobi ellipsoid into a pear-shaped object, is then evaluated, and
it is shown that its vanishing determines the point of bifurcation
along the Jacobian sequence, in agreement with Darwin's result.
---------------------------------------------------------
Title: The Potentials and the Superpotentials of Homogeneous
Ellipsoids.
Authors: Chandrasekhar, S.; Lebovitz, Norman R.
1962ApJ...136.1037C Altcode:
Explicit expressions are found for the various potentials and
superpotentials of homogeneous ellipsoids
---------------------------------------------------------
Title: On the Occurrence of Multiple Frequencies and Beats in the
β Canis Majoris Stars.
Authors: Chandrasekhar, S.; Lebovitz, Norman R.
1962ApJ...136.1105C Altcode:
An explanation is suggested for the occurrence of two nearly
equal frequencies and associated beats in the light- and in the
velocity-variations of the p Canis Majoris stars. It is shown that if
the ratio of the specific heats is 1 6 and the star is rotating, any
disturbance will excite two normal modes with nearly equal frequencies.
---------------------------------------------------------
Title: On the Oscillations and the Stability of Rotating Gaseous
Masses. II. The Homogeneous, Compressible Model.
Authors: Chandrasekhar, S.; Lebovitz, Norman R.
1962ApJ...136.1069C Altcode:
The pulsation frequencies of rotating, gaseous masses of uniform
density, i.e., of the Maclaurin spheroids, are found as functions of
the angular momentum H and the ratio of the specific heats . Numerical
calculations for the pulsation frequencies and normal modes are given
for = 1.3, 4,1.4, 1.5, 1.6, and . One finds that the value of at which
dynamical instability sets in is reduced from = 4 by rotation. One also
finds that, when = 1.6, the normal modes of oscillation one obtains
in the limit H 0 are both very far from being radial.
---------------------------------------------------------
Title: On Superpotentials in the Theory of Newtonian
Gravitation. II. Tensors of Higher Rank.
Authors: Chandrasekhar, S.; Lebovitz, Norman R.
1962ApJ...136.1032C Altcode:
In addition to the tensors considered in the earlier paper, the
following tensors are defined and studied: jj(x) =G#P(x') x½k'i dx',
ij;k(X) =Gj{P(x') (XffX'iL(Xi ½) Xk'dx , I x - x' 3 and jm1(x)= (x
)LX½ X% XJ$ )( 5Xk )( )dx These tensors are useful in problems (such
as the stability of the Jacobi ellipsoids) in which it is necessary
to examine the effects of perturbations belonging to the third harmonic.
---------------------------------------------------------
Title: On the Oscillations and the Stability of Rotating Gaseous
Masses.
Authors: Chandrasekhar, S.; Lebovitz, Norman R.
1962ApJ...135..248C Altcode:
In this paper the oscillations and the stability of a rotating
gaseous mass are considered on the basis of an appropriate tensor
form of the virial theorem. On the assumption that the Lagrangian
displacement can be expressed in the form = XjrXr (Xjr constants), a
characteristic equation for X (of order eighteen) is derived from the
nine integral relations provided by the virial theorem. An examination
of the roots of this characteristic equation enables the enumeration
of the properties of all the natural modes of oscillation belonging
essentially to harmonics not higher than the second. It is shown that
there are three principal groups among these modes: a group of three
modes, each of which exhibits a doublet character; a group of two
modes, one of which becomes neutral at a point where the condition
for the occurrence of a point of bifurcation is satisfied and both
of which become overstable at a higher angular velocity; and a group
which represents the coupling of two modes, one of which is purely
radial and the other of which is purely non-radial in the absence of
rotation. In addition to these modes, there are two "trivial" modes,
one of which is neutral and the other of which has a characteristic
frequency equal to the angular velocity.
---------------------------------------------------------
Title: An Interpretation of Double Periods in Beta Canis Majoris
Stars.
Authors: Chandrasekhar, S.; Lebovitz, N. R.
1962ApJ...135..305C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: On Super-Potentials in the Theory of Newtonian Gravitation.
Authors: Chandrasekhar, S.; Lebovitz, Norman R.
1962ApJ...135..238C Altcode:
The character of the gravitational equilibrium of bodies in rotation
and with prevalent magnetic fields depends on the tensor potential,
4=G})P(x') ( ')( ) dx', lx-x'13 and the associated tensors, ii= $
iidx and pq;ij= fvPXp% $dX This paper is devoted to a consideration of
these fundamental tensors It is shown, in particular, that the tensor
potential can be expressed in the form ii= where is the gravitational
potential as usually defined and x is a super-potential determined by
the equation V2x= -2 .
---------------------------------------------------------
Title: A Theorem on Rotating Polytropes.
Authors: Chandrasekhar, S.
1961ApJ...134..662C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Diffuse Reflection by a Semi-Infinite Atmosphere.
Authors: Horak, Henry G.; Chandrasekhar, S.
1961ApJ...134...45H Altcode:
Parallel light of flux density irF0 is incident on a plane-parallel,
atmosphere which scatters light in accordance with the phase function
p (c6s 0) = o + i#i (cos 0) + #2 (cos 0). The exact solution for the
emergent radiation field is found by using the invariance-principle
method.
---------------------------------------------------------
Title: The Geodesics in Godel's Universe
Authors: Chandrasekhar, S.; Wright, James P.
1961PNAS...47..341C Altcode:
No abstract at ADS
---------------------------------------------------------
Title: Diffuse Reflection by a Semi-Infinite Atmosphere.
Authors: Horak, Henry G.; Chandrasekhar, S.
1961AJ.....66S.286H Altcode:
The solution has been obtained for the problem of diffuse reflection
by a semi-infinite atmosphere which scatters light in accordance with
the phase function p(cos0) =w9 ~,P,(cos0) ~,P,(cos0), where ~~, W~,
~2 are constants and P,, P2 are Legendre polynomials. The albedo, W~,
can be less than or equal to unity. The solution was obtained by the
principle-of-invariance method.
---------------------------------------------------------
Title: Buchbesprechungen über: Three Copernican Treatises- The
Universe of Light- Radiative Transfer(Ref. A. UNSÖLD)
Authors: Rosen, E.; Bragg, W.; Chandrasekhar, S.
1961ZA.....51..148R Altcode:
No abstract at ADS
---------------------------------------------------------
Title: The Stability of Non-Dissipative Couette Flow in Hydromagnetics
Authors: Chandrasekhar, S.
1960PNAS...46..253C Altcode:
No abstract at ADS
