User:Tohline/Apps/RotatingPolytropes

Rotationally Flattened Polytropes

Earliest Efforts to Construct Equilibrium Configurations

The results of the following, chronologically listed research efforts have largely been summarized in the review by N. R. Lebovitz (1967).

• 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. Poincaré (1885), Acta Math., 7, 259
• H. Poincaré (1903), Figures d'Equilibre d'une Masse Fluide (C. Naud, Paris)
• V. Volterra (1903), Acta Math., 27, 105
• W. Thomson & 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)
• S. Chandrasekhar (1933), MNRAS, 93, 390: The equilibrium of distorted polytropes. I. The rotational problem

The purpose of this paper is … 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. … the gas sphere is set rotating at a constant small angular velocity <math>~\omega</math>. … we shall assume that the rotation is so slow that the configurations are only slightly oblate.

• L. Lichtenstein (1933), Gleichgewichtsfiguren Rotierinder Flüssigkeiten (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 & 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)
• P. Ledoux (1958), Handbuch der Physik, 51, 605 (Flügge, S., Ed., Springer-Verlag, Berlin)
• P. Ledoux & Th. Walraven (1958), Handbuch der Physik, 51, 353 (Flügge, S., Ed., Springer-Verlag, Berlin)
• [ EFE Publication I ] S. Chandrasekhar (1960), J. Mathematical Analysis and Applications, 1, 240: The virial theorem in hydromagnetics
• [ EFEPublication II ] N. R. Lebovitz (1961), ApJ, 134, 500: The virial tensor and its application to self-gravitating fluids
• C. Pekeris, Z. Alterman & H. Jarosch (1961), Proc. Natl. Acad. Sci. U. S., 47, 91
• [ EFE Publication V ] S. Chandrasekhar & N. R. Lebovitz (1962a), ApJ, 135, 248: On the oscillations and the stability of rotating gaseous masses
• [ EFE Publication X ] S. Chandrasekhar & N. R. Lebovitz (1962b), ApJ, 136, 1069: On the oscillations and the stability of rotating gaseous masses. II. The homogeneous, compressible model
• [ EFE Publication XI ] S. Chandrasekhar & N. R. Lebovitz (1962c), ApJ, 136, 1082: On the oscillations and the stability of rotating gaseous masses. III. The distorted polytropes

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.

• [ EFE Publication XII ] S. Chandrasekhar & N. R. Lebovitz (1962d), ApJ, 136, 1105: On the occurrence of multiple frequencies and beats in the β Canis Majoris stars
• P. H. Roberts & K. Stewartson (1963), ApJ, 137, 777
• P. H. Roberts (1963a), ApJ, 137, 1129: On Highly Rotating Polytropes. I.
• [ EFE Publication XIV ] S. Chandrasekhar & N. R. Lebovitz (1963), ApJ, 137, 1162: On the oscillations of the Maclaurin spheroid belonging to the third harmonics
• P. H. Roberts (1963b), ApJ, 138, 809: On Highly Rotating Polytropes. II.
• [ EFE Publication XIX ] S. Chandrasekhar (1963), ApJ, 138, 1182: The equilibrium and stability of the Roche ellipsoids
• [ EFE Publication XX ] 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
• [ EFE Publication XXIII ] Chandrasekhar & N. R. Lebovitz (1964), Astrophysica Norvegica, 9, 232: On the ellipsoidal figures of equilibrium of homogeneous masses — Excellent Review!
• M. Hurley & P. H. Roberts (1964), ApJ, 140, 583: On Highly Rotating Polytropes. III.
• R. A. James (1964), 140, 552: The Structure and Stability of Rotating Gas Masses

Structures have been determined for axially symmetric [uniformly] rotating gas masses, in the polytropic and white-dwarf cases … 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.

• D. Lynden-Bell (1964), ApJ, 139, 1195
• I. W. Roxburgh (1964), MNRAS, 128, 157
• P. Ledoux's (1965) Chapter 10, pp. 499-574 of Stellar Structure (Aller, L. H., McLaughlin, D. B., Eds., Univ. of Chicago Press, Chicago)
• L. Mestel's (1965) Chapter xx, pp. 465-xxx of Stellar Structure (Aller, L. H., McLaughlin, D. B., Eds., Univ. of Chicago Press, Chicago)
• Clement (1965a), ApJ, 140, 1045
• R. Stoeckly (1965), ApJ, 142, 208: Polytropic Models with Fast, Non-Uniform Rotation [NOTE: Article not available via SAO/NASA ADS.]

Models with polytropic index n = 1.5.… for the case of non-uniform rotation, no meridional currents, and axial symmetry. The angular velocity assigned … 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). 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. 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 … Non-uniformity of rotation would then be an element bearing on star formation and could be a factor in double-star formation.

• Clement (1965b), ApJ, 142, 243
• [ EFE Publication XXV ] S. Chandrasekhar (1965), ApJ, 142, 890: The equilibrium and the stability of the Riemann ellipsoids. I [NOTE: Article not available via SAO/NASA ADS.]
• M. Hurley & P. H. Roberts (1965), ApJSuppl, 11, 95: On Highly Rotating Polytropes. IV.
• [ EFE Publication XXVII ] N. R. Lebovitz (1965), lecture notes. Inst. Ap., Cointe-Sclessin, Belgium, p. 29: The Riemann ellipsoids
• J. J. Monaghan & I. W. Roxburgh (1965), MNRAS, 131, 13: The structure of rapidly rotating polytropes

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. Here, results are presented for values of the polytropic index n = 1, 1.5, 2, 2.5, 3, 3.5, 4. (Apparently, uniform rotation is assumed.) … no discussion of stability is given, we assume that the polytropes become unstable at the equator before a point of bifurcation is reached.

• I. W. Roxburgh, J. S. Griffith & P. A. Sweet (1965), Z. Ap., 61, 203
• J.-L. Tassoul & M. Cretin (1965), Ann. Ap., 28, 982
• [ EFE Publication XXVIII ] S. Chandrasekhar (1966), ApJ, 145, 842: The equilibrium and the stability of the Riemann ellipsoids. II
• P. G. Drazin & L. N. Howard (1966), Advan. Appl. Mech., 9, 1
• W. A. Fowler (1966), ApJ, 144, 180
• M. Hurley, P. H. Roberts & K. Wright (1966), ApJ, 143, 535: The Oscillations of Gas Spheres
• [ EFE Publication XXIX ] N. R. Lebovitz (1966), ApJ, 145, 878: On Riemann's criterion for the stability of liquid ellipsoids
• J. P. Ostriker, P. Bodenheimer & D. Lynden-Bell (1966), Phys. Rev. Letters, 17, 816: Equilibrium Models of Differentially Rotating Zero-Temperature Stars

… 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 [Chandrasekhar (1931, ApJ, 74, 81)] mass limit <math>~M_3</math> is increased by only a few percent when uniform rotation is included in the models, … 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 … 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.

• 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 & J. P. Ostriker (1967), MNRAS (to appear)
• K. Rosenkilde (1967), ApJ (to appear)
• L. Rossner (1967), ApJ (to appear)

Example Equilibrium Configurations

Reviews

• N. R. Lebovitz (1967), ARAA, 5, 465

Uniform Rotation

• E. A. Milne (1923), MNRAS, 83, 118: The Equilibrium of a Rotating Star

Apparently, only n = 3 polytropic configurations are considered.

• H. von Zeipel (1924), MNRAS, 84, 665: The radiative equilibrium of a rotating system of gaseous masses
• H. von Zeipel (1924), MNRAS, 84, 684: The radiative equilibrium of a slightly oblate rotating star
• S. Chandrasekhar & N. R. Lebovitz (1968), ApJ, 152, 267: The Pulsations and the Dynamical Stability of Gaseous Masses in Uniform Rotation
• Article in French! J. - L. Tassoul & J. P. Ostriker (1970), Astron. Ap., 4, 423: Secular Stability of Uniformly Rotating Polytropes
• N. R. Lebovitz & G. W. Russell (1972), ApJ, 171, 103: The Pulsations of Polytropic Masses in Rapid, Uniform Rotation
• M. J. Clement (1981), ApJ, 249, 746: Normal modes of oscillation for rotating stars. I — The effect of rigid rotation on four low-order pulsations

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. Normal modes are determined by solving directly on a two-dimensional grid the linearized dynamical equations governing adiabatic oscillations … This brute force approach has many obvious dangers, all of which are realized in practice.

• 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

• S. Chandrasekhar & N. R. Lebovitz (1962), ApJ, 136, 1082

The oscillations of slowly rotating polytopes are treated in this paper. The initial equilibrium configurations are constructed as in Chandrasekhar (1933).

• TORUS! J. P. Ostriker (1964), ApJ, 140, 1067: The Equilibrium of Self-Gravitating Rings
• R. Stoeckly (1965), ApJ, 142, 208: Polytropic Models with Fast, Non-Uniform Rotation [NOTE: Article not available via SAO/NASA ADS.]

Models with polytropic index n = 1.5.… for the case of non-uniform rotation, no meridional currents, and axial symmetry. The angular velocity assigned … 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). 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. 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 … Non-uniformity of rotation would then be an element bearing on star formation and could be a factor in double-star formation.

• J. P. Ostriker & P. Bodenheimer (1973), ApJ, 180, 171 [Part III]: On the Oscillations and Stability of Rapidly Rotating Stellar Models. III. Zero-Viscosity Polytropic Sequences
• P. Bodenheimer & J. P. Ostriker (1973), ApJ, 180, 159 [Part VIII]

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:  <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>.

• J. L. Friedman & B. F. Schutz (1978), ApJ, 222, 281
• R. H. Durisen & J. N. Imamura (1981), ApJ, 243, 612
• I. Hachisu, Y. Eriguchi & D. Sugimoto (1982), Progress of Theoretical Physics, 68, 191: Rapidly Rotating Polytropes and Concave Hamburger Equilibrium
• I. Hachisu & Y. Eriguchi (1984), Astrophysics & Space Sciences, 99, 71: Fission Sequence and Equilibrium Models of Rigidity [sic] Rotating Polytropes
• J. E. Tohline, R. H. Durisen & M. McCollough (1985), ApJ, 298, 220
• R. H. Durisen, R. A. Gingold, J. E. Tohline & A. P. Boss (1986), ApJ, 305, 281
• H. A. Williams & J. E. Tohline (1987), ApJ, 315, 594
• H. A. Williams & J. E. Tohline (1988), ApJ, 334, 449
• I. Hachisu, J. E. Tohline & Y. Eriguchi (1988), ApJS, 66, 315: Fragmentation of Rapidly Rotating Gas Clouds. II. Polytropes — Clues to the Outcome of Adiabatic Collapse
• P. J. Luyten (1990), MNRAS, 245, 614
• P. J. Luyten (1991), MNRAS, 248, 256
• A. G. Aksenov (1996), Astronomy Letters, 22, 634
• B. K. Pickett, R. H. Durisen & G. A. Davis (1996), ApJ, 458, 714
• B. K. Pickett, R. H. Durisen & R. Link (1997), Icarus, 126, 243
• J. Toman, J. N. Imamura, B. K. Pickett & R. H. Durisen (1998), ApJ, 497, 370
• J. N. Imamura, R. H. Durisen & B. K. Pickett (2000), ApJ, 528, 946
• J. M. Centrella, K. C. B. New, L. L. Lowe & J. D. Brown (2001), ApJL, 550, 193
• M. Shibata, S. Karino & Y. Eriguchi (2002), MNRAS, 334, 27
• M. Saijo, T. W. Baumgarte & S. L. Shapiro (2003), ApJ, 595, 352
• M. Saijo & S. Yoshida (2006), MNRAS, 368, 1429

See Also

• Our discussion of Rotating White Dwarfs: Example Equilibria
• Fission Theory of Binary Star Formation

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