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Rotationally Flattened Polytropes

Whitworth's (1981) Isothermal Free-Energy Surface
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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 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. 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.

 

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 … A comparsion is made with the investigations of S. Chandrasekhar (1933) and James (1962, private communication prior to its 1964 publication); see also S. Chandrasekhar & N. R. Lebovitz (1962c).

 

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.

 

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.

  • 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
  • [ EFE Publication XXV ] S. Chandrasekhar (1965), ApJ, 142, 890: The equilibrium and the stability of the Riemann ellipsoids. I
  • M. Hurley & P. H. Roberts (1965), ApJSuppl, 11, 95: On Highly Rotating Polytropes. IV.
 

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).

  • [ 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.

 

… 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

Uniform Rotation

 

Apparently, only n = 3 polytropic configurations are considered.

 

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

 

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.

 

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.

 

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 perturbation of the gravitational potential can be neglected. In this note we restrict ourselves to the problem of dynamical instability using the variational principle of Lynden-Bell & Ostriker (1967) in the form given to it by Lebovitz (1970) to deduce global criteria —

  • B. F. Schutz, Jr. (1972), ApJSuppl., 24, 319: Linear Pulsations and Starility of Differentially Rotating Stellar Models. I. Newtonian Analysis
 

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 D. Lynden-Bell & 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 … This change of variables saves one from integrating twice over the star to find the effect of the perturbed gravitational field.

 

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>.

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

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