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Rotationally Flattened Polytropes

Whitworth's (1981) Isothermal Free-Energy Surface
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Here we review what has been learned over the past century+ regarding the structural properties of rotationally flattened polytropes having indexes in the range, ~0 < n < \infty. Separate chapters are devoted to configurations having the "bookend" index values of ~n = 0 (incompressible) and ~n = \infty (isothermal), and to configurations obeying the white dwarf equation of state. Our review will be divided chronologically into three parts with, generally speaking, the following two publications serving to identify the approximate location of segmentation boundaries:

  • R. A. James (1964), which serves as the first notable example of an astrophysicist drawing upon the capabilities of a digital computer to solve the set of equations that define — without approximation — the equilibrium structure of axisymmetric, rotationally flattened polytropes. (Rotating white dwarf structures are also examined.)
  • I. Hachisu (1986a), in which a versatile, efficient, and accurate technique for constructing rotating equilibrium configurations — hereafter referred to as the Hachisu Self-Consistent Field (HSCF) technique — was detailed. In retrospect, we recognize that this numerical tool was used by many groups to construct excellent initial equilibrium models for numerical stability studies.

Because equilibrium models along the incompressible (n = 0) Maclaurin sequence can be fully described analytically, we should not be surprised to find that Maclaruin spheroids are often used as a base of comparison when the properties of rotationally flattened, compressible configurations are discussed. Drawing from the review by N. R. Lebovitz (1967), here are a few relevant features to keep in mind:

  • According to Lichtenstein's theorem, each configuration has a plane of symmetry perpendicular to the axis of rotation; this is the equatorial plane.
  • Maclaurin spheroids are a subset of the broader class of equilibrium configurations referred to as Riemann S-type ellipsoids; specifically, the subset containing only axisymmetric configurations.
  • Along the Maclaurin equilibrium sequence, each configuration is an oblate spheroid whose position along the sequence can be uniquely specified by the spheroid's axis ratio, b/a, or alternatively by the spheroid's eccentricity, ~e \equiv \sqrt{1 - a_3^2/a_1^2}.
  • When the angular momentum of the equilibrium configuration, ~J_\mathrm{Mac} = \tfrac{2}{5}Ma_1^2 \Omega , is zero, the Maclaurin figure is … a sphere. As ~J_\mathrm{Mac} increases, the figure flattens progressively, approaching an infinitely thin circular disk as ~J_\mathrm{Mac} approaches infinity.

Prior to the work of James (1964)

N. R. Lebovitz (1967) offers a concise review of efforts that were made to construct models of rotationally flattened polytropic structures prior to work of R. A. James (1964). We will draw heavily from this review.

Distortions Introduced by Uniform Rotation

Let's take the following steps …

  • Detail the results from 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 ~\omega. … we shall assume that the rotation is so slow that the configurations are only slightly oblate.

  • Compare with the similar analysis for n = 3 conducted earlier by E. A. Milne (1923), MNRAS, 83, 118: The Equilibrium of a Rotating Star
  • Use as a point of comparison for the more recent work by 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

Principle Governing Equations

Spherical Coordinate Base

Poisson Equation

~
\frac{1}{r^2} \frac{\partial }{\partial r} \biggl[ r^2 \frac{\partial \Phi }{\partial r} \biggr] 
+ \frac{1}{r^2 \sin\theta} \frac{\partial }{\partial \theta}\biggl(\sin\theta ~ \frac{\partial \Phi}{\partial\theta}\biggr)

~=

~4\pi G\rho

The Two Relevant Components of the
Euler Equation

~{\hat{e}}_r:    


~0

=


- \biggl[ \frac{1}{\rho} \frac{\partial P}{\partial r}+  \frac{\partial \Phi }{\partial r} \biggr]  + \biggl[ \frac{j^2}{r^3 \sin^2\theta} \biggr]

~{\hat{e}}_\theta:    


~0

=


- \biggl[ \frac{1}{\rho r}  \frac{\partial P}{\partial\theta} +  \frac{1}{r} \frac{\partial \Phi}{\partial\theta}  \biggr] +  \biggl[ \frac{j^2}{r^3 \sin^3\theta} \biggr] \cos\theta

Choices and Adaptations

Among the set of simple rotation profiles that have been adopted by various research groups over the years, Milne (1923), Chandrasekhar (1933) and James (1964) all choose what would generally be considered the simplest, which is the assumption of uniform rotation ~(\dot\varphi = \omega_0). This means that,

~j^2 = r^4 \sin^4\theta \omega_0^2 \, .

And in place of the co-latitude, ~\theta, they all adopt the coordinate,

~\mu \equiv \cos\theta ~~~\Rightarrow ~~~ \frac{\partial}{\partial\mu} = - \frac{\partial}{\sin\theta \partial\theta} \, .

As a result, the set of three remaining scalar governing equations becomes,

Poisson Equation

~
\frac{1}{r^2} \frac{\partial }{\partial r} \biggl[ r^2 \frac{\partial \Phi }{\partial r} \biggr] 
+ \frac{1}{r^2} \frac{\partial }{\partial \mu}\biggl[ (1-\mu^2) \frac{\partial \Phi}{\partial\mu}\biggr]

~=

~4\pi G\rho

Chandrasekhar (1933), p. 391, Eq. (4')
James (1964), p. 553, Eq. (2.1)

The Two Relevant Components of the Euler Equation

~{\hat{e}}_r:    


\rho \omega_0^2 r (1-\mu^2)

=


+ \biggl[ \frac{\partial P}{\partial r} +  \rho \frac{\partial \Phi }{\partial r} \biggr]

~{\hat{e}}_\theta:    


\rho \omega_0^2 r^2 \mu

=


- \biggl[ \frac{\partial P}{\partial\mu} +  \rho\frac{\partial \Phi}{\partial\mu}  \biggr]

Milne (1923), top of p. 126
Chandrasekhar (1933), p. 391, Eq. (3')
James (1964), p. 553, Eqs. (2.2) & (2.3)

NOTES:

  • In place of ~P, Milne uses ~W;
  • For the gravitational potential, Milne and Chandrasekhar both adopt the convention, ~V \equiv -\Phi , while James adopts the notation, ~\Psi \equiv - \Phi;
  • James includes terms with azimuthal derivatives in his equation set; these terms are set to zero (as reflected here) when seeking axisymmetric structures.


Chandrasekhar's Approach

Using the two components of the Euler equation to express spatial derivatives of the gravitational potential in terms of spatial derivatives of the gas pressure, that is, rewriting them in the form,

~{\hat{e}}_r:    


\frac{\partial \Phi }{\partial r}

=


-  \frac{1}{\rho} ~\frac{\partial P}{\partial r} +  \omega_0^2 r (1-\mu^2)

~{\hat{e}}_\theta:    


\frac{\partial \Phi}{\partial\mu}

=


- \frac{1}{\rho} ~\frac{\partial P}{\partial\mu} - \omega_0^2 r^2 \mu

we can fold them into the Poisson equation to obtain a 2nd-order PDE that relates ~\rho(r, \theta) to ~P(r, \theta), namely,

~4\pi G\rho

~=

~
\frac{1}{r^2} \frac{\partial }{\partial r} \biggl\{ r^2 \biggl[-  \frac{1}{\rho} ~\frac{\partial P}{\partial r} +  \omega_0^2 r (1-\mu^2)\biggr]  \biggr\} 
+ \frac{1}{r^2} \frac{\partial }{\partial \mu}\biggl\{ (1-\mu^2) \biggl[ - \frac{1}{\rho} ~\frac{\partial P}{\partial\mu} - \omega_0^2 r^2 \mu  \biggr] \biggr\}

 

~=

~
- \frac{1}{r^2} \frac{\partial }{\partial r} \biggl[\frac{r^2}{\rho} ~\frac{\partial P}{\partial r} \biggr]  
+\frac{1}{r^2} \frac{\partial }{\partial r} \biggl[\omega_0^2 r^3 (1-\mu^2)\biggr]  
- \frac{1}{r^2} \frac{\partial }{\partial \mu} \biggl[ \frac{(1-\mu^2)}{\rho} ~\frac{\partial P}{\partial\mu}  \biggr] 
- \frac{1}{r^2} \frac{\partial }{\partial \mu} \biggl[ \omega_0^2 r^2 \mu (1-\mu^2)   \biggr]

 

~=

~
- \frac{1}{r^2} \frac{\partial }{\partial r} \biggl[\frac{r^2}{\rho} ~\frac{\partial P}{\partial r} \biggr]  
- \frac{1}{r^2} \frac{\partial }{\partial \mu} \biggl[ \frac{(1-\mu^2)}{\rho} ~\frac{\partial P}{\partial\mu}  \biggr] 
+3\omega_0^2 (1-\mu^2) - \omega_0^2  (1-3\mu^2)

~\Rightarrow ~~~ 2\omega_0^2 -4\pi G\rho

~=

~
\frac{1}{r^2} \frac{\partial }{\partial r} \biggl[\frac{r^2}{\rho} ~\frac{\partial P}{\partial r} \biggr]  
+ \frac{1}{r^2} \frac{\partial }{\partial \mu} \biggl[ \frac{(1-\mu^2)}{\rho} ~\frac{\partial P}{\partial\mu}  \biggr]

Chandrasekhar (1933), p. 391, Eq. (5)

Chandrasekhar (1933) refers to this last expression as "the fundamental equation of the problem." If, following Chandrasekhar's lead, we adopt a polytropic equation of state,

~P

~=

~K\rho^{1+1/n} \, ,

Chandrasekhar (1933), p. 392, Eq. (6)

this "fundamental equation" can be rewritten strictly in terms of the configuration's axisymmetric density distribution, ~\rho(r,\theta). Chandrasekhar first adopts a dimensionless function, ~\Theta(\xi, \mu), that is related to the normalized density via the expression,

~\Theta

~\equiv

~\biggl( \frac{\rho}{\rho_c}  \biggr)^{1/n}

~\Rightarrow ~~~ P = K \rho^{1+1/n}

~=

~\rho_c^{1+1/n} K \Theta^{n+1} \, .

(In doing this, Chandrasekhar is obviously adopting an a path toward solution that parallel's the familiar method used to determine the structure of isolated, nonrotating polytropes.) Adopting the additional pair of dimensionless variables,

~\xi

~\equiv

~r \biggl[ \frac{(n+1)K}{4\pi G} ~\rho_c^{1/n - 1} \biggr]^{-1 / 2}

      and      

~v

~\equiv

~\frac{\omega_0^2}{2\pi G \rho_c}

Chandrasekhar's "fundamental equation" becomes,

~
\frac{1}{\xi^2} \frac{\partial }{\partial \xi} \biggl[\xi^2 ~\frac{\partial \Theta}{\partial \xi} \biggr]  
+ \frac{1}{\xi^2} \frac{\partial }{\partial \mu} \biggl[ (1-\mu^2) ~\frac{\partial \Theta}{\partial\mu}  \biggr]

~=

~v -\Theta^n \, .

Chandrasekhar (1933), p. 392, Eq. (11)

Milne's Additional Specificity

Selected Equation of State

In his examination of the effects of (uniform) rotation on the equilibrium structure of an otherwise spherically symmetric star, Milne (1923) focused on models in which the pressure,

~P_\mathrm{tot}

~=

~P_\mathrm{gas} + P_\mathrm{rad} \, .

When a dimensionless parameter, ~\beta, is used to quantify the ratio of the gas pressure to the total pressure — that is, if we set

~\beta \equiv \frac{P_\mathrm{gas}}{P_\mathrm{tot}} ~~~~\Rightarrow ~~~~ \frac{P_\mathrm{rad}}{P_\mathrm{tot}} = (1-\beta) \, ,

then, as we have detailed in our separate discussion of Milne's (1930) early work on bipolytropic stellar models, the pressure-density relation and the temperature-density relation become, respectively,

~P

~=

~ 
\biggl[ \biggl( \frac{\Re}{\mu_e}\biggr)^4 \biggl(\frac{1-\beta}{\beta^4}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]^{1/3} \rho^{1 + 1/3}

    and,    

~\frac{T^3}{\rho}

~=

\biggl[ \biggl( \frac{\Re}{\mu_e}\biggr) \biggl(\frac{1-\beta}{\beta}\biggr) \frac{3}{a_\mathrm{rad}} \biggr] \, .

Now, when building a realistic stellar model, one must expect that, in general, the parameter ~\beta will vary with position throughout the model. But if the assumption is made that ~\beta has the same value throughout the equilibrium configuration, then we are effectively adopting a polytropic equation of state with,

~n

~=

~3 \, ,

~K

~=

~\biggl[ \biggl( \frac{\Re}{\mu_e}\biggr)^4 \biggl(\frac{1-\beta}{\beta^4}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]^{1/3} \, .

With this realization, Chandrasekhar's (1933) work should be considered a generalization of Milne's (1923) modeling effort. Also, the results of the latter's work should match Chandrasekhar's results for the specific case of a rotating ~n=3 polytrope.

Radiative Equilibrium

Milne (1923) made an effort to ensure that his equilibrium models were not only in hydrostatic balance but that they also were in "radiative equilibrium;" see especially his §II.9. Drawing on Example B from our introductory discussion of nonadiabatic environments, Milne accomplished this by, effectively, adopting the steady-state specific-entropy expression,

~\rho T \cancelto{0}{\frac{ds_\mathrm{tot}}{dt}}

~=

~
\rho \epsilon_\mathrm{nuc} - \nabla \cdot \vec{F}_\mathrm{rad} \, .

In this expression, ~\epsilon_\mathrm{nuc}(\rho,T) specifies the rate at which (specific) energy is released via thermonuclear reactions, and

~\vec{F}_\mathrm{rad}

~=

~- \frac{c}{3\rho\kappa_R} \nabla (a_\mathrm{rad}T^4) \, .

Given that the energy per unit volume in the radiation field is, ~E_\mathrm{rad} = a_\mathrm{rad} T^4, this "radiative equilibrium" condition may be rewritten as,

~
\nabla \cdot \biggl[ \frac{1}{3\rho\kappa_R} \nabla E_\mathrm{rad} \biggr]

~=

~- \frac{\rho \epsilon_\mathrm{nuc} }{c} \, .

This is identical to equation (7) of Milne (1923) except: (a) His divergence and gradient operators appear as they would in Cartesian coordinates; and (b) an extra factor of ~4\pi appears in the term on the right-hand side of his expression. We attribute the extra factor of ~4\pi to slightly different definitions of the energy derived from nuclear reactions; specifically, we suspect that, ~\epsilon_\mathrm{nuc} = 4\pi \epsilon_\mathrm{Milne}, because immediately following his equation (5) — near the top of his p. 123 — we find the sentence … "Now suppose that ~4\pi\epsilon is the energy evolved per unit mass per second at [a given location]."

Compare Radiative Equilibrium with Mechanical Equilibrium

Given that,

~\tfrac{1}{3} E_\mathrm{rad}

~=

~P_\mathrm{rad} = (1-\beta) P_\mathrm{tot} \, ,

the radiative equilibrium condition can be rewritten as,

~
\nabla \cdot \biggl\{ \frac{1}{\rho\kappa_R} \nabla \biggl[(1-\beta)P_\mathrm{tot}\biggr] \biggr\}

~=

~- \frac{\rho \epsilon_\mathrm{nuc} }{c} \, .

If we now express the differential operators in terms of spherical coordinates and (for the time being) assume that ~\kappa_R and ~\beta are both independent of position, this becomes,

~- \frac{\rho \epsilon_\mathrm{nuc}\kappa_R }{c~(1-\beta)}

~=

~
\nabla \cdot \biggl\{  \biggl[{\hat{e}}_r \biggl[ \frac{1}{\rho } \frac{\partial P_\mathrm{tot}}{\partial r}\biggr] + {\hat{e}}_\theta \biggl[\frac{1}{\rho r} \frac{\partial P_\mathrm{tot}}{\partial \theta}\biggr] \biggr\}

 

~=

~
\frac{1}{r^2}\frac{\partial}{\partial r}\biggl[ \frac{r^2}{\rho } \frac{\partial P_\mathrm{tot}}{\partial r}\biggr] 
+ \frac{1}{r\sin\theta} \frac{\partial}{\partial\theta}\biggl[\frac{\sin\theta}{\rho r} \frac{\partial P_\mathrm{tot}}{\partial \theta}\biggr]

 

~=

~
\frac{1}{r^2}\frac{\partial}{\partial r}\biggl[ \frac{r^2}{\rho } \frac{\partial P_\mathrm{tot}}{\partial r}\biggr] 
+ \frac{1}{r} \frac{\partial}{\partial\mu}\biggl[\frac{(1-\mu^2)}{\rho r} \frac{\partial P_\mathrm{tot}}{\partial \mu}\biggr] \, .

Next, following Milne's (1923) lead, let's assume that the pressure, ~P, that appears in Chandrasekhar's "fundamental equation" is only slightly different from ~P_\mathrm{tot} — that is, let's write,

~P = P_\mathrm{tot} + \delta P \, ,

then subtract the derived radiative equilibrium relation from Chandrasekhar's "fundamental equation." Doing this, we obtain,

~2\omega_0^2 -4\pi G\rho + \frac{\rho \epsilon_\mathrm{nuc}\kappa_R }{c~(1-\beta)}

~=

~
\frac{1}{r^2} \frac{\partial }{\partial r} \biggl[\frac{r^2}{\rho} ~\frac{\partial (\delta P)}{\partial r} \biggr]  
+ \frac{1}{r^2} \frac{\partial }{\partial \mu} \biggl[ \frac{(1-\mu^2)}{\rho} ~\frac{\partial (\delta P)}{\partial\mu}  \biggr] \, .

Notice that, if we specifically choose the value of ~\beta such that (see Milne's §I.6),

~\beta

~=

~1 - \frac{\kappa_R (\epsilon_\mathrm{nuc}/4\pi)}{c~G} \, ,

then the left-hand side of this relation simplifies considerably. Specifically, we end up with,

~2

~=

~
\frac{1}{r^2} \frac{\partial }{\partial r} \biggl[\frac{r^2}{\rho} ~\frac{\partial U}{\partial r} \biggr]  
+ \frac{1}{r^2} \frac{\partial }{\partial \mu} \biggl[ \frac{(1-\mu^2)}{\rho} ~\frac{\partial U}{\partial\mu}  \biggr] \, ,

Milne (1923), p. 125, Eq. (20)

where we have adopted the variable notation used by Milne (1923), viz., ~\delta P \rightarrow \omega_0^2 U \, . And, simultaneously, the condition for radiative equilibrium takes the form,

~- \frac{\rho \epsilon_\mathrm{nuc}\kappa_R }{c}

~=

~
\frac{1}{r^2}\frac{\partial}{\partial r}\biggl[ \frac{r^2}{3\rho } \frac{\partial E_\mathrm{rad}}{\partial r}\biggr] 
+ \frac{1}{r} \frac{\partial}{\partial\mu}\biggl[\frac{(1-\mu^2)}{3 \rho r} \frac{\partial E_\mathrm{rad}}{\partial \mu}\biggr]

 

~\Rightarrow ~~~ - 4\pi G \rho (1-\beta)

~=

~
\frac{1}{r^2}\frac{\partial}{\partial r}\biggl[ \frac{r^2}{3\rho } \frac{\partial (a_\mathrm{rad} T^4)}{\partial r}\biggr] 
+ \frac{1}{r} \frac{\partial}{\partial\mu}\biggl[\frac{(1-\mu^2)}{3 \rho r} \frac{\partial (a_\mathrm{rad} T^4)}{\partial \mu}\biggr]

 

~\Rightarrow ~~~ - \frac{3\pi G \rho (1-\beta)}{a_\mathrm{rad} }

~=

~
\frac{1}{r^2}\frac{\partial}{\partial r}\biggl[ \frac{r^2 T^3}{\rho } \frac{\partial T}{\partial r}\biggr] 
+ \frac{1}{r^2} \frac{\partial}{\partial\mu}\biggl[\frac{(1-\mu^2)T^3}{ \rho} \frac{\partial T}{\partial \mu}\biggr] \, .

Comment by J. E. Tohline on 25 July 2019:  Milne's Equation (19) does not include the factor of a_rad (the radiation constant) that is shown here in the denominator of the left-hand-side; this appears to be a typesetting error, as Milne's expression is even dimensionally incorrect without this factor.

Milne (1923), p. 125, Eq. (19)

 

Solutions

In determining the equilibrium configuration's axisymmetric density distribution, ~\rho(r,\theta), Chandrasekhar adopts a dimensionless function, ~\Theta(\xi, \mu), that is related to the normalized density via the expression,

~\Theta

~\equiv

~\biggl( \frac{\rho}{\rho_c}  \biggr)^{1/n}

~\Rightarrow ~~~ P = K \rho^{1+1/n}

~=

~\rho_c^{1+1/n} K \Theta^{n+1} \, .

(In doing this, Chandrasekhar is obviously adopting an a path toward solution that parallel's the familiar method used to determine the structure of isolated, nonrotating polytropes.) He then argues that, for slowly rotating configurations, the function, ~\Theta(\xi,\mu) can be effectively expressed as a small perturbation (in the two-dimensional meridional plane), ~v\Psi(\xi,\mu) \ll 1, added to the radially dependent "Emden's function," ~\theta(\xi), that defines the structure of non-rotating polytropic configurations; that is,

~\Theta(\xi,\mu)

~=

~
\theta(\xi) + v \Psi(\xi,\mu) + \mathcal{O}(v^2) \, ,

Chandrasekhar (1933), p. 392, Eq. (13)

where,

~v

~\equiv

~\frac{\omega_0^2}{2\pi G\rho_c} \ll 1 \, .

Chandrasekhar (1933), p. 392, Eq. (10)

He deduces that, to lowest order in a Legendre polynomial series,

~\Psi

~=

~
\psi_0(\xi) - \frac{5}{6} ~ \frac{\xi_1^2}{3\psi_2(\xi_1) + \xi_1 \psi_2^'(\xi_1)} \psi_2(\xi) P_2(\mu) \, ,

Chandrasekhar (1933), p. 395, Eq. (36)

where: ~P_2(\mu) = \tfrac{1}{2}(3\mu^2 - 1); ~\xi_1 is "the first zero of the Emden's function with index ~n"; and ~\psi_0 and ~\psi_2 satisfy the 2nd-order ODEs,

~\frac{1}{\xi^2} \frac{d}{d\xi} \biggl( \xi^2 \frac{d\psi_0}{d\xi} \biggr)

~=

~- n\theta^{n-1} \psi_0 + 1 \, ,

~\frac{1}{\xi^2} \frac{d}{d\xi} \biggl( \xi^2 \frac{d\psi_2}{d\xi} \biggr)

~=

~\biggl( - n\theta^{n-1} + \frac{6}{\xi^2} \biggr) \psi_2 \, .

Chandrasekhar (1933), p. 395, Eqs. (371 & 372)

Realizing that "the boundary ~\xi_0 is given by ~\Theta = 0," Chandrasekhar deduced as well that,

~\xi_0

~=

~\xi_1 + \frac{v}{|\theta_1^'|} \biggl[
\psi_0(\xi_1) - \frac{5}{6} ~ \frac{\xi_1^2 \psi_2(\xi_1) P_2(\mu)}{3\psi_2(\xi_1) + \xi_1 \psi_2^'(\xi_1)}  \biggr] \, ,

Chandrasekhar (1933), p. 395, Eq. (38)

Huge Reference List

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.

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  • 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 ~\omega. … 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
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  • S. Rosseland (1949), The Pulsation Theory of Variable Stars (Clarendon Press, Oxford)
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  • [ 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
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  • [ 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 ~c 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 ~c = 0, 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 ~c \approx 1, 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 ~n \ge 3. 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 ~M_3 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 ~M_3 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).

 

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 ~c 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 ~c = 0, 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 ~c \approx 1, 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 ~n' angular momentum distributions. Equilibrium models are built along the following ~(n, n') sequences:  ~(0, 0), ~(\tfrac{3}{2}, \tfrac{3}{2}), ~(\tfrac{3}{2}, 1), ~(\tfrac{3}{2}, 0), ~(3, 0), and ~(3, \tfrac{3}{2}).

 

Dynamical instability is shown to occur in differentially rotating polytropes with n = 3.33 and T/|W| > ∼ 0.14. This instability has a strong m = 1 mode, although the m = 2, 3, and 4 modes also appear.

Hachisu and Various Collaborators (Before HSCF)

Focus on Incompressible Configurations

 

… computed the structure of uniformly rotating polytropes with small but finite values of polytropic index. In the case of high angular momentum there appeared a concave hamburger-like shape of equilibrium, and the sequence of shapes seemed to continue into a toroid.

… the Maclaurin spheroid does not represent the incompressible limit of the rotaing [sic] polytropic gas because of its restriction of the figure. The computed sequence of equilibria clarifies the relation between the Maclaurin spheroid and the Dyson-Wong toroid. Moreover it is the sequence of minimum-energy configurations.

TECHNIQUE: … method developed by Eriguchi, in which the boundary value problem of gravitational equilibrium is transformed into the Cauchy problem by making the analytic continuation into the complex plane.

 

It has been said that there are only two axisymmetric equilibrium sequences in the case of self-gravitating, uniformly rotating incompressible fluids — Maclaurin spheroids and Dyson-Wong toroids … We have computed … an intermediate sequence which branches off the spheroids and extends to toroids.

TECHNIQUE: Guess the location of the configuration's surface in the meridional plane then, assuming the density is uniform everywhere inside this surface, determine the corresponding gravitational potential using the integral form of the Poisson equation and a Green's function written in terms of Legendre polynomials. Iterate on this guess until hydrostatic balance is achieved.

Focus on Compressible Configurations

 

This paper is based on the author's dissertation, submitted to the Univerrsity of Tokyo, in partial fulfillment of the requirements for the doctorate.

Results 4a: n = 1.5, 4.0, and 4.9, all with uniform rotation; compared to published results of James and of

Results 4b: n = 1.5 only, with a ~\dot\varphi(\varpi) rotation law — obtained from combining eqs. (30) and (7) — that resembles the so-called j-constant simple rotation profile, namely,

~{\dot\varphi}^2

~=

~
\frac{A}{[e^{2t}\sin^2\theta + \alpha^2]^{3/2}} 
= 
\frac{A}{[(r/R_0)^2\sin^2\theta + \alpha^2]^{3/2}}
= 
\frac{A}{[(\varpi/R_0)^2 + \alpha^2]^{3/2}} \, .

After employing this "equation (30)" rotation law, … Rapid rotation near the central region results in density inversion and a "ring"-like structure appears in figure 7. No other author has used the rotation law (30), and therefore a comparison cannot be made. The structure in figure 6 resembles the results of Mark (1968), and density inversion appears also in Stoeckly's (1965) results.

  • I. Hachisu, Y. Eriguchi & D. Sugimoto (1982), Progress of Theoretical Physics, 68, 191: Rapidly Rotating Polytropes and Concave Hamburger Equilibrium
  • I. Hachisu & Y. Eriguchi (1982), Progress of Theoretical Physics, 68, 206: Bifurcation and Fission of Three Dimensional, Rigidly Rotating and Self-Gravitating Polytropes
  • Y. Eriguchi & I. Hachisu (1983), Progress of Theoretical Physics, 69, 1131: Two Kinds of Axially Symmetric Equilibrium Sequences of Self-Gravitating and Rotating Incompressible Fluid: Two-Ring Sequence and Core-Ring Sequence
 

The computational scheme is much the same as that used in the computation of one-ring equilibrium sequence … see Eriguchi & Sugimoto (1981), above.

 

We find a fission sequence from an ellipsoidal configuration to a binary by way of dumb-bell equilibrium.

Ellipsoidal and Binary Systems

  • I. Hachisu & Y. Eriguchi (1984), PASJapan, 36, 239: Fission of dumbbell equilibrium and binary state of rapidly rotating polytropes
  • Y. Eriguchi & I. Hachisu (1984), PASJapan, 36, 491: Bifurcation points on the one-ring sequence of uniformly rotating and self-gravitating fluid
  • I. Hachisu & Y. Eriguchi (1984), in Double Stars, Physical Properties and Generic Relations. Proceedings of IAU Colloquium No. 80, held at Lembang, Java, June 3-7, 1983. Editors, Bambang Hidayat, Zdenek Kopal, Jurgen Rahe; Publisher, D. Reidel Pub. Co., Dordrecht, Holland; Boston, pp. 71-74: Fission Sequence and Equilibrium Models of Rigidity [sic] Rotating Polytropes — Excellent figure illustrating fission!
  • Y. Eriguchi & I. Hachisu (1985), Astronomy and Astrophysics, 142, 256: Fission sequences of self-gravitating and rotating fluid with internal motion

HSCF and Beyond

 

… The authors formulate a pair of criteria for rotating gas clouds that must be satisfied simultaneously if the final collapsed object is going to be unstable toward fragmentation … Centrally condensed objects become dynamically unstable to non-axisymmetric modes if T/|W| = 0.27. Self-gravitating toroidal systems, however, become dynamically unstable at T/|W| = 0.14.

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

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