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* [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''
* [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''
* [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''
* <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/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''
* [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''

Revision as of 20:38, 26 June 2019

Rotationally Flattened Polytropes

Whitworth's (1981) Isothermal Free-Energy Surface
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Earliest Efforts to Construct Equilibrium Configurations

The results of the following identified 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
  • 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.

 

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
  • [ EFE Publication XIV ] S. Chandrasekhar & N. R. Lebovitz (1963), ApJ, 137, 1162: On the oscillations of the Maclaurin spheroid belonging to the third harmonics
  • [ EFE Publication XIX ] S. Chandrasekhar (1963), ApJ, 138, 1182: The equilibrium and stability of 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!
  • Clement (1965a), ApJ, 140, 1045
  • Clement (1965b), ApJ, 142, 243
  • [ EFE Publication XXV ] S. Chandrasekhar (1965), ApJ, 142, 890: The equilibrium and the stability of the Riemann ellipsoids. I
  • [ 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

Example Equilibrium Configurations

Reviews

Uniform Rotation

 

Apparently, only n = 3 polytropic configurations are considered.

 

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.

 

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.

 

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.

 

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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