Editing Apps/RotatingPolytropes (section)

===Differential Rotation===

* [https://ui.adsabs.harvard.edu/abs/1962ApJ...136.1082C/abstract S. Chandrasekhar &amp; N. R. Lebovitz (1962)], ApJ, 136, 1082
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<font color="green">The oscillations of slowly rotating polytopes are treated in</font> this paper.  The initial equilibrium configurations are constructed as in Chandrasekhar (1933).
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* <font color="maroon"><b>TORUS!</b></font> [https://ui.adsabs.harvard.edu/abs/1964ApJ...140.1067O/abstract J. P. Ostriker (1964)], ApJ, 140, 1067:  ''The Equilibrium of Self-Gravitating Rings''
* [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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* [https://ui.adsabs.harvard.edu/abs/1967MNRAS.136..293L/abstract D. Lynden-Bell &amp; J. P. Ostriker (1967)], MNRAS, 136, 293:  ''On the stability of differentially rotating bodies''
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<font color="green">A variational principle of great power is derived.  It is naturally adapted for computers, and may be used to determine the stability of any fluid flow including those in differentially-rotating, self-gravitating stars and galaxies.  The method also provides a powerful theoretical tool for studying general properties of eigenfunctions, and the relationships between secular and ordinary stability.  In particular we prove the anti-sprial theorem indicating that no stable (or steady( mode can have a spiral structure</font>.
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* [https://ui.adsabs.harvard.edu/abs/1971A%26A....15..329F/abstract K. J. Fricke &amp; R. C. Smith (1971)], Astronomy &amp; Astrophysics, 15, 329:  ''On Global Dynamical Stability of Rotating Stars''
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<font color="green">The local criteria for axisymmetric dynamical stabffity of rotating stars are shown to be globally valid by the use of a variational principle. These criteria are necessary and sufficient so long as the <b>perturbation of the gravitational potential can be neglected</b>.  In this note we restrict ourselves to the problem of dynamical instability using the variational principle of Lynden-Bell &amp; Ostriker (1967) in the form given to it by [https://ui.adsabs.harvard.edu/abs/1970ApJ...160..701L/abstract Lebovitz (1970)] to deduce global criteria &#8212;</font>
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* [https://ui.adsabs.harvard.edu/abs/1972ApJS...24..319S/abstract B. F. Schutz, Jr. (1972)], ApJSuppl., 24, 319:  ''Linear Pulsations and Starility of Differentially Rotating Stellar Models. I. Newtonian Analysis''
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<font color="green">A systematic method is presented for deriving the Lagrangian governing the evolution of small perturbations of arbitrary flows of a self-gravitating perfect fluid. The method is applied to a differentially rotating stellar model; the result is a Lagrangian equivalent to that of [https://ui.adsabs.harvard.edu/abs/1967MNRAS.136..293L/abstract D. Lynden-Bell &amp; J. P. Ostriker (1967)]. A sufficient condition for stability of rotating stars, derived from this Lagrangian, is simplified greatly by using as trial functions not the three components of the Lagrangian displacement vector, but three scalar functions &hellip; This change of variables saves one from integrating twice over the star to find the effect of the perturbed gravitational field.</font>
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* [https://ui.adsabs.harvard.edu/abs/1973ApJ...180..171O/abstract J. P. Ostriker &amp; P. Bodenheimer (1973)], ApJ, 180, 171 [Part III]:  ''On the Oscillations and Stability of Rapidly Rotating Stellar Models. III. Zero-Viscosity Polytropic Sequences''
* [https://ui.adsabs.harvard.edu/abs/1973ApJ...180..159B/abstract P. Bodenheimer &amp; J. P. Ostriker (1973)], ApJ, 180, 159 [Part VIII]
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An explanation is given regarding the specification of various so-called <math>~n'</math> angular momentum distributions.  Equilibrium models are built along the following <math>~(n, n')</math> sequences:&nbsp; <math>~(0, 0)</math>, <math>~(\tfrac{3}{2}, \tfrac{3}{2})</math>, <math>~(\tfrac{3}{2}, 1)</math>, <math>~(\tfrac{3}{2}, 0)</math>, <math>~(3, 0)</math>, and <math>~(3, \tfrac{3}{2})</math>. 
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* [https://ui.adsabs.harvard.edu/abs/1978ApJ...222..281F/abstract J. L. Friedman &amp; B. F. Schutz (1978)], ApJ, 222, 281
* [https://ui.adsabs.harvard.edu/abs/1981ApJ...243..612D/abstract R. H. Durisen &amp; J. N. Imamura (1981)], ApJ, 243, 612
* See [[#Hachisu_and_Various_Collaborators|Hachisu and Various Collaborators]], below.
* [https://ui.adsabs.harvard.edu/abs/1985ApJ...298..220T/abstract J. E. Tohline, R. H. Durisen &amp; M. McCollough (1985)], ApJ, 298, 220
* [https://ui.adsabs.harvard.edu/abs/1986ApJ...305..281D/abstract R. H. Durisen, R. A. Gingold, J. E. Tohline &amp; A. P. Boss (1986)], ApJ, 305, 281
* [https://ui.adsabs.harvard.edu/abs/1987ApJ...315..594W/abstract H. A. Williams &amp; J. E. Tohline (1987)], ApJ, 315, 594
* [https://ui.adsabs.harvard.edu/abs/1988ApJ...334..449W/abstract H. A. Williams &amp; J. E. Tohline (1988)], ApJ, 334, 449
* [https://ui.adsabs.harvard.edu/abs/1990MNRAS.245..614L/abstract P. J. Luyten (1990)], MNRAS, 245, 614
* [https://ui.adsabs.harvard.edu/abs/1991MNRAS.248..256L/abstract P. J. Luyten (1991)], MNRAS, 248, 256
* [https://ui.adsabs.harvard.edu/abs/1996AstL...22..634A/abstract A. G. Aksenov (1996)], Astronomy Letters, 22, 634
* [https://ui.adsabs.harvard.edu/abs/1996ApJ...458..714P/abstract B. K. Pickett, R. H. Durisen &amp; G. A. Davis (1996)], ApJ, 458, 714
* [https://ui.adsabs.harvard.edu/abs/1997Icar..126..243P/abstract B. K. Pickett, R. H. Durisen &amp; R. Link (1997)], Icarus, 126, 243
* [https://ui.adsabs.harvard.edu/abs/1998ApJ...497..370T/abstract J. Toman, J. N. Imamura, B. K. Pickett &amp; R. H. Durisen (1998)], ApJ, 497, 370
* [https://ui.adsabs.harvard.edu/abs/2000ApJ...528..946I/abstract J. N. Imamura, R. H. Durisen &amp; B. K. Pickett (2000)], ApJ, 528, 946
* [https://ui.adsabs.harvard.edu/abs/2001ApJ...550L.193C/abstract J. M. Centrella, K. C. B. New, L. L. Lowe &amp; J. D. Brown (2001)], ApJL, 550, 193 &#8212; see also, [https://ui.adsabs.harvard.edu/abs/1987ApJ...323..592H/abstract Hachisu, Tohline &amp; Eriguchi (1987)]
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<font color="green">Dynamical instability is shown to occur in differentially rotating polytropes with n = 3.33 and T/|W| > &sim; 0.14.  This instability has a strong m = 1 mode, although the m = 2, 3, and 4 modes also appear.</font>
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* [https://ui.adsabs.harvard.edu/abs/2002MNRAS.334L..27S/abstract M. Shibata, S. Karino &amp; Y. Eriguchi (2002)], MNRAS, 334, 27
* [https://ui.adsabs.harvard.edu/abs/2003ApJ...595..352S/abstract M. Saijo, T. W. Baumgarte &amp; S. L. Shapiro (2003)], ApJ, 595, 352
* [https://ui.adsabs.harvard.edu/abs/2006MNRAS.368.1429S/abstract M. Saijo &amp; S. Yoshida (2006)], MNRAS, 368, 1429