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

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<font color="green">The oscillations of slowly rotating polytopes are treated in</font> this paper.  The initial equilibrium configurations are constructed as in Chandrasekhar (1933).
<font color="green">The oscillations of slowly rotating polytopes are treated in</font> this paper.  The initial equilibrium configurations are constructed as in Chandrasekhar (1933).
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* [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..208S/abstract R. Stoeckly (1965)], ApJ, 142, 208:  ''Polytropic Models with Fast, Non-Uniform Rotation'' <font color="red">[NOTE: Article not available via SAO/NASA ADS.]</font>
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Models with polytropic index n = 1.5.<font color="green">&hellip; for the case of non-uniform rotation, no meridional currents, and axial symmetry.  The angular velocity assigned &hellip; 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).</font>
<font color="green">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.</font>
<font color="green">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 &hellip;  Non-uniformity of rotation would then be an element bearing on star formation and could be a factor in double-star formation.</font>
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* [https://ui.adsabs.harvard.edu/abs/1973ApJ...180..171O/abstract J. P. Ostriker &amp; P. Bodenheimer (1973)], ApJ, 155, 987 [Part III]
* [https://ui.adsabs.harvard.edu/abs/1973ApJ...180..171O/abstract J. P. Ostriker &amp; P. Bodenheimer (1973)], ApJ, 155, 987 [Part III]

Revision as of 22:29, 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).

  • 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.

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

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