Editing Apps/RotatingPolytropes (section)

==Example Equilibrium Configurations==
===Reviews===
* [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)], ARAA, 5, 465

===Uniform Rotation===

* [https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract E. A. Milne (1923)], MNRAS, 83, 118: ''The Equilibrium of a Rotating Star''
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Apparently, only n = 3 polytropic configurations are considered.
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* [https://ui.adsabs.harvard.edu/abs/1924MNRAS..84..665V/abstract H. von Zeipel (1924)], MNRAS, 84, 665: ''The radiative equilibrium of a rotating system of gaseous masses
* [https://ui.adsabs.harvard.edu/abs/1924MNRAS..84..684V/abstract H. von Zeipel (1924)], MNRAS, 84, 684:  ''The radiative equilibrium of a slightly oblate rotating star''
* [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..267C/abstract S. Chandrasekhar &amp; N. R. Lebovitz (1968)], ApJ, 152, 267:  ''The Pulsations and the Dynamical Stability of Gaseous Masses in Uniform Rotation''
* <font color="maroon"><b>Article in French!</b></font> [https://ui.adsabs.harvard.edu/abs/1970A%26A.....4..423T/abstract J. - L. Tassoul &amp; J. P. Ostriker (1970)], Astron. Ap., 4, 423:  ''Secular Stability of Uniformly Rotating Polytropes''
* [https://ui.adsabs.harvard.edu/abs/1972ApJ...171..103L/abstract N. R. Lebovitz &amp; G. W. Russell (1972)], ApJ, 171, 103:  ''The Pulsations of Polytropic Masses in Rapid, Uniform Rotation''
* [https://ui.adsabs.harvard.edu/abs/1981ApJ...249..746C/abstract M. J. Clement (1981)], ApJ, 249, 746:  ''Normal modes of oscillation for rotating stars.  I &#8212; The effect of rigid rotation on four low-order pulsations''
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<font color="green">In this paper, the effects of rigid rotation on four axisymmetric modes are found for several equilibrium systems including polytopes and a 15 solar-mass stellar model.</font>  Normal modes are determined <font color="green">by solving directly on a two-dimensional grid the linearized dynamical equations governing adiabatic oscillations &hellip; This brute force approach has many obvious dangers, all of which are realized in practice.</font>
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* [https://ui.adsabs.harvard.edu/abs/1985Ap%26SS.113..125C/abstract R. Caimmi (1985)], Astrophysics and Space Science, 113, 125:  ''Emden-Chandrasekhar Axisymmetric, Rigidly Rotating Polytropes.  III. Determination of Equilibrium Configurations by an Improvement of Chandrasekhar's Method''

===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

===Hachisu and Various Collaborators (Before HSCF)===

====Focus on Incompressible Configurations====
* [https://ui.adsabs.harvard.edu/abs/1980PThPh..63.1957F/abstract Toshio Fukushima, Yoshiharu Eriguchi, Daiichir Sugimoto &amp; Gennadii S. Bisnovatyi-Kogan (1980)], Progress of Theoretical Physics, 63, 1957: ''Concave Hamburger Equilibrium of Rotating Bodies''
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<font color="green">&hellip; computed the structure of uniformly rotating polytropes with ''small but finite'' values of polytropic index.  In the case of high angular momentum there appeared a concave hamburger-like shape of equilibrium, and the sequence of shapes seemed to continue into a toroid.</font>

<font color="green">&hellip; the Maclaurin spheroid does not represent the incompressible limit of the rotaing [''sic''] polytropic gas because of its restriction of the figure.  The computed sequence of equilibria clarifies the relation between the Maclaurin spheroid and the [[Apps/DysonWongTori#Self-Gravitating.2C_Incompressible_.28Dyson-Wong.29_Tori|Dyson-Wong toroid]].  Moreover it is the sequence of minimum-energy configurations.</font>

TECHNIQUE:  <font color="green">&hellip; method developed by Eriguchi, in which the boundary value problem of gravitational equilibrium is transformed into the Cauchy problem by making the analytic continuation into the complex plane.</font>
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* [https://ui.adsabs.harvard.edu/abs/1981PThPh..65.1870E/abstract Yoshiharu Eriguchi &amp; Daiichiro Sugimoto (1981)], Progress of Theoretical Physics, 65, 1870:  ''Another Equilibrium Sequence of Self-Gravitating and Rotating Incompressible Fluid''
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<font color="green">It has been said that there are only two axisymmetric equilibrium sequences in the case of self-gravitating, uniformly rotating ''incompressible'' fluids &#8212; Maclaurin spheroids and [[Apps/DysonWongTori#Self-Gravitating.2C_Incompressible_.28Dyson-Wong.29_Tori|Dyson-Wong toroids]] &hellip; We have computed &hellip; an intermediate sequence which branches off the spheroids and extends to toroids.</font>

TECHNIQUE:  ''Guess'' the location of the configuration's ''surface'' in the meridional plane then, assuming the density is uniform everywhere inside this surface, determine the corresponding gravitational potential using the integral form of the Poisson equation and a Green's function written in terms of Legendre polynomials. Iterate on this guess until hydrostatic balance is achieved.
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* [https://ui.adsabs.harvard.edu/abs/1982PThPh..67..844E/abstract Y. Eriguchi &amp; I. Hachisu (1982)], Progress of Theoretical Physics, 67, 844:  ''New Equilibrium Sequences Bifurcating from Maclaurin Sequence''
* [https://ui.adsabs.harvard.edu/abs/1982PThPh..67.1068E/abstract Y. Eriguchi, I. Hachisu &amp; D. Sugimoto (1982)], Progress of Theoretical Physics, 67, 1068:  ''Dumb-Bell-Shape Equilibria and Mass-Shedding Pear-Shape of Selfgravitating Incompressible Fluid''
* [https://ui.adsabs.harvard.edu/abs/1984PASJ...36..497H/abstract I. Hachisu &amp; Y. Eriguchi (1984)], PASJapan, 36, 497:  ''Bifurcation points on the Maclaurin sequence''
* [https://ui.adsabs.harvard.edu/abs/1985A%26A...148..289E/abstract Y. Eriguchi &amp; I. Hachisu (1985)], Astronomy &amp; Astrophysics, 148, 289:  ''Maclaurin hamburger sequence''
* [https://ui.adsabs.harvard.edu/abs/1986A%26A...168..130E/abstract Y. Eriguchi, E. Mueller &amp; I. Hachisu (1986)], Astronomy &amp; Astrophysics, 168, 130:  ''Meridional flow in a self-gravitating body.  I. Mechanical flow in a barotropic star with constant specific angular momentum''

====Focus on Compressible Configurations====
* [https://ui.adsabs.harvard.edu/abs/1978PASJ...30..507E/abstract Y. Eriguchi (1978)], PASJapan, 30, 507:  ''Hydrostatic Equilibria of Rotating Polytropes''
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<font color="green">This paper is based on the author's dissertation, submitted to the Univerrsity of Tokyo, in partial fulfillment of the requirements for the doctorate.</font>

Results 4a:  n = 1.5, 4.0, and 4.9, all with uniform rotation; compared to published results of James and of 

Results 4b:  n = 1.5 only, with a <math>~\dot\varphi(\varpi)</math> rotation law &#8212; obtained from combining eqs. (30) and (7) &#8212; that ''resembles'' the so-called j-constant [[AxisymmetricConfigurations/SolutionStrategies#SRPtable|simple rotation profile]], namely,
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<math>~{\dot\varphi}^2</math>
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<math>~=</math>
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<math>~
\frac{A}{[e^{2t}\sin^2\theta + \alpha^2]^{3/2}} 
= 
\frac{A}{[(r/R_0)^2\sin^2\theta + \alpha^2]^{3/2}}
= 
\frac{A}{[(\varpi/R_0)^2 + \alpha^2]^{3/2}} \, .
</math>
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After employing this "equation (30)" rotation law, <font color="green">&hellip; Rapid rotation near the central region results in density inversion and a "ring"-like structure appears in figure 7.  No other author has used the rotation law (30), and therefore a comparison cannot be made.  The structure in figure 6 resembles the results of [https://ui.adsabs.harvard.edu/abs/1968ApJ...154..627M/abstract Mark (1968)], and density inversion appears also in [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..208S/abstract Stoeckly's (1965)] results.</font>

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* [https://ui.adsabs.harvard.edu/abs/1982PThPh..68..191H/abstract I. Hachisu, Y. Eriguchi &amp; D. Sugimoto (1982)], Progress of Theoretical Physics, 68, 191:  ''Rapidly Rotating Polytropes and Concave Hamburger Equilibrium''
* [https://ui.adsabs.harvard.edu/abs/1982PThPh..68..206H/abstract I. Hachisu &amp; Y. Eriguchi (1982)], Progress of Theoretical Physics, 68, 206:  ''Bifurcation and Fission of Three Dimensional, Rigidly Rotating and Self-Gravitating Polytropes''
* [https://ui.adsabs.harvard.edu/abs/1983PThPh..69.1131E/abstract Y. Eriguchi &amp; I. Hachisu (1983)], Progress of Theoretical Physics, 69, 1131: ''Two Kinds of Axially Symmetric Equilibrium Sequences of Self-Gravitating and Rotating Incompressible Fluid:  Two-Ring Sequence and Core-Ring Sequence''
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<font color="green">The computational scheme is much the same as that used in the computation of one-ring equilibrium sequence</font> &hellip; see [https://ui.adsabs.harvard.edu/abs/1981PThPh..65.1870E/abstract Eriguchi &amp; Sugimoto (1981)], above.
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* [https://ui.adsabs.harvard.edu/abs/1983MNRAS.204..583H/abstract I. Hachisu &amp; Y. Eriguchi (1983)], MNRAS, 204, 583:  ''Bifurcations and phase transitions of self-gravitating and uniformly rotating fluid''
* [https://ui.adsabs.harvard.edu/abs/1983PThPh..70.1534E/abstract Y. Eriguchi &amp; I. Hachisu (1983)], Progress of Theoretical Physics, 70, 1534:  ''Gravitational Equilibrium of a Multi-Body Fluid System''
* [https://ui.adsabs.harvard.edu/abs/1984Ap%26SS..99...71H/abstract I. Hachisu &amp; Y. Eriguchi (1984)], Astrophysics &amp; Space Sciences, 99, 71:  ''Fission Sequence and Equilibrium Models of Rigidity [sic] Rotating Polytropes''
* [https://ui.adsabs.harvard.edu/abs/1988ApJS...66..315H/abstract I. Hachisu, J. E. Tohline &amp; Y. Eriguchi (1988)], ApJS, 66, 315:  ''Fragmentation of Rapidly Rotating Gas Clouds.  II. Polytropes &#8212; Clues to the Outcome of Adiabatic Collapse''
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<font color="green">We find a fission sequence from an ellipsoidal configuration to a binary by way of dumb-bell equilibrium.</font>
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====Ellipsoidal and Binary Systems====

* [https://ui.adsabs.harvard.edu/abs/1984PASJ...36..239H/abstract I. Hachisu &amp; Y. Eriguchi (1984)], PASJapan, 36, 239: ''Fission of dumbbell equilibrium and binary state of rapidly rotating polytropes''
* [https://ui.adsabs.harvard.edu/abs/1984PASJ...36..491E/abstract Y. Eriguchi &amp; I. Hachisu (1984)], PASJapan, 36, 491:  ''Bifurcation points on the one-ring sequence of uniformly rotating and self-gravitating fluid''
* [https://ui.adsabs.harvard.edu/abs/1984Ap%26SS..99...71H/abstract I. Hachisu &amp; Y. Eriguchi (1984)], in <b>Double Stars, Physical Properties and Generic Relations.</b> Proceedings of IAU Colloquium No. 80, held at Lembang, Java, June 3-7, 1983.  Editors, Bambang Hidayat, Zdenek Kopal, Jurgen Rahe; Publisher, D. Reidel Pub. Co., Dordrecht, Holland; Boston, pp. 71-74:  ''Fission Sequence and Equilibrium Models of Rigidity [sic] Rotating Polytropes'' <font color="maroon"><b>&#8212; Excellent figure illustrating fission!</b></font>
* [https://ui.adsabs.harvard.edu/abs/1985A%26A...142..256E/abstract Y. Eriguchi &amp; I. Hachisu (1985)], Astronomy and Astrophysics, 142, 256:  ''Fission sequences of self-gravitating and rotating fluid with internal motion''