Editing Apps/RotatingPolytropes (section)

==Efforts to Construct Equilibrium Configurations Prior to 1968==
The results of the following, chronologically listed research efforts have largely been summarized in the review by [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)].  Text colored green has been taken directly from the (immediately preceding) cited paper, often from its abstract. 

* R. Dedekind (1860), J. Reine Angew. Math., 58, 217
* P. G. Lejeune. Dirichlet (1860), J. Reine Angel. Math., 58, 181
* G. Riemann (1860), ''Abhandl. Konigl. Ges. Wis. Gottingen'', 9, 3 (Verlag von B. G. Teubner, Leipzig; reprinted 1953, Dover Publ., New York)
* Lord Rayleigh (1880), ''Proc. London Math. Soc.'', 9, 57
* H. Poincar&eacute; (1885), ''Acta Math.'', 7, 259
* H. Poincar&eacute; (1903), ''Figures d'Equilibre d'une Masse Fluide'' (C. Naud, Paris)
* V. Volterra (1903), ''Acta Math.'', 27, 105
* W. Thomson &amp; P. G. Tait (1912), ''Treatise on Natural Philosophy'' (Cambridge Univ. Press)
* J. H. Jeans (1919) Phil. Trans. Roy. Soc., 218, 157
* R. Wavre (1932), ''Figures Planetaires et Geodesie'' (Gauthier-Villars, Paris)
* [https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract S. Chandrasekhar (1933)], MNRAS, 93, 390:  ''The equilibrium of distorted polytropes.  I.  The rotational problem''
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The purpose of this paper is <font color="green">&hellip; to extend Emden's [work] to the case of rotating gas spheres which in their non-rotating states have polytropic distributions described by the so-called Emden functions.  &hellip; the gas sphere is set rotating at a constant small angular velocity <math>~\omega</math>. &hellip; we shall assume that the rotation is so slow that the configurations are only slightly oblate.</font>
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* L. Lichtenstein (1933), ''Gleichgewichtsfiguren Rotierinder Fl&uuml;ssigkeiten (Verlag von Julius Springer, Berlin)
* V. C. A. Ferraro (1937), MNRAS, 97, 458
* Cowling (1941), MNRAS, 101, 367
* G. Randers (1942), ApJ, 95, 454
* P. Ledoux (1945), ApJ, 102, 143
* Cowling &amp; Newing (1949), ApJ, 109, 149
* S. Rosseland (1949), ''The Pulsation Theory of Variable Stars'' (Clarendon Press, Oxford)
* P. A. Sweet (1950), MNRAS, 110, 548
* Cowling (1951), ApJ, 114, 272
* P. Ledoux (1951), ApJ, 117, 373
* Dive, P. (1952), Bull. Sci. Math., 76, 38
* R. A. Lyttleton (1953), ''The Stability of Rotating Liquid Masses'' (Cambridge Univ. Press)
* W. S. Jardetzky (1958) ''Theories of Figures of Celestial Bodies'' (Interscience, New York)
* [http://adsabs.harvard.edu/abs/1958HDP....51..353L P. Ledoux (1958)], ''Handbuch der Physik'', 51, 605 (Fl&uuml;gge, S., Ed., Springer-Verlag, Berlin)
* [http://adsabs.harvard.edu/abs/1958HDP....51..353L P. Ledoux &amp; Th. Walraven (1958)], ''Handbuch der Physik'', 51, 353 (Fl&uuml;gge, S., Ed., Springer-Verlag, Berlin)
* [ [[Appendix/References#Appendix_of_EFE|EFE]] Publication I ] [https://www.sciencedirect.com/science/article/pii/0022247X60900251 S. Chandrasekhar (1960)], J. Mathematical Analysis and Applications, 1, 240:  ''The virial theorem in hydromagnetics''
* [ [[Appendix/References#Appendix_of_EFE|EFE]]Publication II ] [https://ui.adsabs.harvard.edu/abs/1961ApJ...134..500L/abstract N. R. Lebovitz (1961)], ApJ, 134, 500:  ''The virial tensor and its application to self-gravitating fluids''
* C. Pekeris, Z. Alterman &amp; H. Jarosch (1961), ''Proc. Natl. Acad. Sci. U. S., 47, 91
* [ [[Appendix/References#Appendix_of_EFE|EFE]] Publication V ] [https://ui.adsabs.harvard.edu/abs/1962ApJ...135..248C/abstract S. Chandrasekhar &amp; N. R. Lebovitz (1962a)], ApJ, 135, 248:  ''On the oscillations and the stability of rotating gaseous masses''
* [ [[Appendix/References#Appendix_of_EFE|EFE]] Publication X ] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1069C/abstract S. Chandrasekhar &amp; N. R. Lebovitz (1962b)], ApJ, 136, 1069:  ''On the oscillations and the stability of rotating gaseous masses.  II. The homogeneous, compressible model''
* [ [[Appendix/References#Appendix_of_EFE|EFE]] Publication XI ] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1082C/abstract S. Chandrasekhar &amp; N. R. Lebovitz (1962c)], ApJ, 136, 1082:  ''On the oscillations and the stability of rotating gaseous masses.  III. The distorted polytropes''
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<font color="green">If one ''assumes'' that the mass is distributed uniformly, the equilibrium configurations are the well-known Maclaurin spheroids.  This paper will be devoted to finding the oscillation frequencies of the Maclaurin spheroids.</font>
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* [ [[Appendix/References#Appendix_of_EFE|EFE]] Publication XII ] [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1105C/abstract S. Chandrasekhar &amp; N. R. Lebovitz (1962d)], ApJ, 136, 1105:  ''On the occurrence of multiple frequencies and beats in the &beta; Canis Majoris stars''
* P. H. Roberts &amp; K. Stewartson (1963), ApJ, 137, 777
* [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1129R/abstract P. H. Roberts (1963a)], ApJ, 137, 1129:  ''On Highly Rotating Polytropes. I.''
* [ [[Appendix/References#Appendix_of_EFE|EFE]] Publication XIV ] [https://ui.adsabs.harvard.edu/abs/1963ApJ...137.1162C/abstract S. Chandrasekhar &amp; N. R. Lebovitz (1963)], ApJ, 137, 1162:  ''On the oscillations of the Maclaurin spheroid belonging to the third harmonics''
* [https://ui.adsabs.harvard.edu/abs/1963ApJ...138..809R/abstract P. H. Roberts (1963b)], ApJ, 138, 809:  ''On Highly Rotating Polytropes. II.''
* [ [[Appendix/References#Appendix_of_EFE|EFE]] Publication XIX ] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1182C/abstract S. Chandrasekhar (1963)], ApJ, 138, 1182:  ''The equilibrium and stability of the Roche ellipsoids''
* [ [[Appendix/References#Appendix_of_EFE|EFE]] Publication XX ] [https://ui.adsabs.harvard.edu/abs/1963ApJ...138.1214L/abstract N. R. Lebovitz (1963)], ApJ, 138, 1214:  ''On the principle of the exchange of stabilities.  I. The Roche ellipsoids''
* Chandrasekhar (1964a), ApJ, 139, 664
* Chandrasekhar (1964b), ApJ, 140, 417
* Chandrasekhar (1964c), ApJ, 140, 599
* [ [[Appendix/References#Appendix_of_EFE|EFE]] Publication XXIII ] [https://ui.adsabs.harvard.edu/abs/1964ApNr....9..323C/abstract Chandrasekhar &amp; N. R. Lebovitz (1964)], Astrophysica Norvegica, 9, 232:  ''On the ellipsoidal figures of equilibrium of homogeneous masses'' &#8212; <font color="maroon"><b>Excellent Review!</b></font>
* [https://ui.adsabs.harvard.edu/abs/1964ApJ...140..583H/abstract M. Hurley &amp; P. H. Roberts (1964)], ApJ, 140, 583:  ''On Highly Rotating Polytropes.  III.''
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<font color="green">This paper gives numerical results deduced from the theory developed in the two previous papers of this series.  Particular attention is devolted to the models proposed in the second of these, which are based on the assumption that, to an adequate approximation, the equidensity surfaces within the polytropes are spheroids whose eccentricity increases from center to surface &hellip; A comparsion is made with the investigations of [https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract S. Chandrasekhar (1933)] and James</font> (1962, private communication prior to its 1964 publication); see also [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1082C/abstract S. Chandrasekhar &amp; N. R. Lebovitz (1962c)].
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* [https://ui.adsabs.harvard.edu/abs/1964ApJ...140..552J/abstract R. A. James (1964)], 140, 552:  ''The Structure and Stability of Rotating Gas Masses''
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<font color="green">Structures have been determined for axially symmetric</font> [uniformly] <font color="green">rotating gas masses, in the polytropic and white-dwarf cases &hellip; Physical parameters for the rotating configurations were obtained for values of n < 3, and for a range of white-dwarf configurations.  The existence of forms of bifurcation of the axially symmetric series of equilibrium forms was also investigated.  The white-dwarf series proved to lack such points of bifurcation, but they were found on the polytropic series for n < 0.808.</font>
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* D. Lynden-Bell (1964), ApJ, 139, 1195: ''On Large-Scale Instabilities during Gravitational Collapse and the Evolution of Shrinking Maclaurin Spheroids''
* I. W. Roxburgh (1964), MNRAS, 128, 157
* [https://archive.org/details/AllerStellarStructure P. Ledoux's (1965) Chapter 10, pp. 499-574 of ''Stellar Structure''] (Aller, L. H., McLaughlin, D. B., Eds., Univ. of Chicago Press, Chicago)
* [https://archive.org/details/AllerStellarStructure L. Mestel's (1965) Chapter xx, pp. 465-xxx of ''Stellar Structure''] (Aller, L. H., McLaughlin, D. B., Eds., Univ. of Chicago Press, Chicago)
* [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..208S/abstract R. Stoeckly (1965)], ApJ, 142, 208:  ''Polytropic Models with Fast, Non-Uniform Rotation'' <font color="maroon"><b>&#8212; The [[AxisymmetricConfigurations/SolutionStrategies#Uniform-Density_Initially_.28n.27_.3D_0.29|n' = 0 angular momentum distribution]] is first defined here!</b></font>
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Models with polytropic index n = 1.5.<font color="green">&hellip; for the case of non-uniform rotation, no meridional currents, and axial symmetry.  The angular velocity assigned &hellip; is a Gaussian function of distance from the axis. The exponential constant <math>~c</math> in this function is a parameter of non-uniformity of rotation, ranging from 0 (uniform rotation) to 1 (approximate spatial dependence of angular velocity that might arise during contraction from a uniformly rotating mass of initially homogeneous density).</font>

<font color="green">For <math>~c = 0</math>, a sequence of models having increasing angular momentum is known to terminate when centrifugal force balances gravitational force at the equator; this sequence contains no bifurcation point with non-axisymmetric models as does the sequence of Maclaurin spheroids with the Jacobi ellipsoids.</font>

<font color="green">For <math>~c \approx 1</math>, the distortion of interior equidensity contours of some models with fast rotation is shown to exceed that of the Maclaurin spheroids at their bifurcation point.  In the absence of a rigorous stability investigation, this result suggests that a star with sufficiently non-uniform rotation reaches a point of bifurcation &hellip;  Non-uniformity of rotation would then be an element bearing on star formation and could be a factor in double-star formation.</font>
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* M. J. Clement (1965a), ApJ, 140, 1045: ''A General Variational Principle Governing the Oscillations of a Rotating Gaseous Mass''
* M. J. Clement (1965b), ApJ, 141, 210:  ''The Radial and Non-Radial Oscillations of Slowly Rotating Gaseous Masses''
* M. J. Clement (1965c), ApJ, 142, 243: ''The Effect of a Small Rotation on the Convective Instability of Gaseous Masses''
* [ [[Appendix/References#Appendix_of_EFE|EFE]] Publication XXV ] [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract S. Chandrasekhar (1965)], ApJ, 142, 890:  ''The equilibrium and the stability of the Riemann ellipsoids.  I'' 
* [https://ui.adsabs.harvard.edu/abs/1965ApJS...11...95H/abstract M. Hurley &amp; P. H. Roberts (1965)], ApJSuppl, 11, 95:  ''On Highly Rotating Polytropes. IV.
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<font color="green">This concerns the structure of polytropes in solid-body rotation.   The underlying theory has been developed in two previous papers (Roberts 1963a, b) and has led to the numerical integrations tabulated herein.  An account of the properties of the polytropes deduced from the present results and a comparison with other studies of the problem are given elsewhere (Hurley and Roberts 1964).</font>
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* [ [[Appendix/References#Appendix_of_EFE|EFE]] Publication XXVII ] N. R. Lebovitz (1965), lecture notes. Inst. Ap., Cointe-Sclessin, Belgium, p. 29:  ''The Riemann ellipsoids''
* [https://ui.adsabs.harvard.edu/abs/1965MNRAS.131...13M/abstract J. J. Monaghan &amp; I. W. Roxburgh (1965)], MNRAS, 131, 13:  ''The structure of rapidly rotating polytropes''
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<font color="green">James attacked the problem by numerically solving the partial differential equations of the problem with the aid of an electronic computer, but even this method lead to difficulties for <math>~n \ge 3</math>.  Of all the methods used so far James' is undoubtedly the most accurate, but also the most laborious.</font>

Here, <font color="green">results are presented for values of the polytropic index n = 1, 1.5, 2, 2.5, 3, 3.5, 4.</font> (Apparently, uniform rotation is assumed.)  <font color="green">&hellip; no discussion of stability is given, we assume that the polytropes become unstable at the equator before a point of bifurcation is reached.</font>
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* I. W. Roxburgh, J. S. Griffith &amp; P. A. Sweet (1965), Z. Ap., 61, 203
* J.-L. Tassoul &amp; M. Cretin (1965), ''Ann. Ap.'', 28, 982
* [ [[Appendix/References#Appendix_of_EFE|EFE]] Publication XXVIII ] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..842C/abstract S. Chandrasekhar (1966)], ApJ, 145, 842:  ''The equilibrium and the stability of the Riemann ellipsoids.  II''
* P. G. Drazin &amp; L. N. Howard (1966), Advan. Appl. Mech., 9, 1
* W. A. Fowler (1966), ApJ, 144, 180
* [https://ui.adsabs.harvard.edu/abs/1966ApJ...143..535H/abstract M. Hurley, P. H. Roberts &amp; K. Wright (1966)], ApJ, 143, 535:  ''The Oscillations of Gas Spheres''
* [ [[Appendix/References#Appendix_of_EFE|EFE]] Publication XXIX ] [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..878L/abstract N. R. Lebovitz (1966)], ApJ, 145, 878:  ''On Riemann's criterion for the stability of liquid ellipsoids''
* [https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract J. P. Ostriker, P. Bodenheimer &amp; D. Lynden-Bell (1966)], Phys. Rev. Letters, 17, 816:  ''Equilibrium Models of Differentially Rotating Zero-Temperature Stars''
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<font color="green">&hellip; work by Roxburgh (1965, Z. Astrophys., 62, 134), Anand (1965, Proc. Natl. Acad. Sci. U.S., 54, 23), and James (1964, ApJ, 140, 552) shows that the</font> [Chandrasekhar (1931, ApJ, 74, 81)] <font color="green">mass limit <math>~M_3</math> is increased by only a few percent when uniform rotation is included in the models, &hellip;</font>

<font color="green">In this Letter we demonstrate that white-dwarf models with masses considerably greater than  <math>~M_3</math> are possible if differential rotation is allowed &hellip; models are based on the physical assumption of an axially symmetric, completely degenerate, self-gravitating fluid, in which the effects of viscosity, magnetic fields, meridional circulation, and relativistic terms in the hydrodynamical equations have been neglected.</font>
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* I. W. Roxburgh (1966a), ''Rotation and Magnetism in Stellar Structure and Evolution''. Symposium lecture (Goddard Space Flight Center, New York)
* I. W. Roxburgh (1966b), MNRAS, 132, 201
* D. Lynden-Bell &amp; J. P. Ostriker (1967), MNRAS (to appear)
* K. Rosenkilde (1967), ApJ (to appear)
* L. Rossner (1967), ApJ (to appear)