Difference between revisions of "User:Tohline/Apps/RotatingPolytropes"

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<font color="green">Structures have been determined for axially symmetric</font> [uniformly] <font color="green">rotating gas masses, in the polytropic and white-dwarf cases &hellip; 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.</font>
<font color="green">Structures have been determined for axially symmetric</font> [uniformly] <font color="green">rotating gas masses, in the polytropic and white-dwarf cases &hellip; 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.</font>
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* [https://ui.adsabs.harvard.edu/abs/1965MNRAS.131...13M/abstract F. F. Monaghan &amp; I. W. Roxburgh (1965)], MNRAS, 131, 13:  ''The structure of rapidly rotating polytropes''
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<font color="green">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.</font>
Here, <font color="green">results are presented for values of the polytropic index n = 1, 1.5, 2, 2.5, 3, 3.5, 4.</font> (Apparently, uniform rotation is assumed.)  <font color="green">&hellip; no discussion of stability is given, we assume that the polytropes become unstable at the equator before a point of bifurcation is reached.</font>
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* [https://ui.adsabs.harvard.edu/abs/1970A%26A.....4..423T/abstract J. - L. Tassoul &amp; J. P. Ostriker (1970)], Astron. Ap., 4, 423
* [https://ui.adsabs.harvard.edu/abs/1970A%26A.....4..423T/abstract J. - L. Tassoul &amp; J. P. Ostriker (1970)], Astron. Ap., 4, 423

Revision as of 22:03, 16 June 2019

Rotationally Flattened Polytropes

Whitworth's (1981) Isothermal Free-Energy Surface
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Example Equilibrium Configurations

Reviews

Uniform Rotation

 

Apparently, only n = 3 polytropic configurations are considered.

 

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.

 

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.

Differential Rotation

 

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

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

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