Editing Apps/RotatingPolytropes (section)

===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''