User:Tohline/Apps/MaclaurinSpheroids

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Whitworth's (1981) Isothermal Free-Energy Surface
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Maclaurin Spheroids (axisymmetric structure)

LSU Structure still.gif

There is no particular reason why one should guess ahead of time that the equilibrium properties of any rotating, self-gravitating configuration should be describable in terms of analytic functions. As luck would have it, however, the gravitational potential at the surface of and inside an homogeneous spheroid is expressible in terms of analytic functions (The potential is constant on concentric spheroidal surfaces that generally have a different axis ratio from the spheroidal mass distribution.) Furthermore, the gradient of the gravitational potential is separable in cylindrical coordinates, proving to be a simple linear function of both <math>\varpi</math> and <math>z</math>.

If the spheroid is uniformly rotating, this behavior conspires nicely with the behavior of the centrifugal acceleration — which also will be a linear function of <math>\varpi</math> — to permit an analytic (and integrable) prescription of the pressure gradient. Not surprisingly, it resembles the functional form of the pressure gradient that is required to balance the gravitational force in uniform-density spheres.

As a consequence of this good fortune, the equilibrium structure of a uniformly rotating, uniform-density (n = 0), axisymmetric configuration can be shown to be precisely an oblate spheroid whose internal properties are describable in terms of analytic expressions. These expressions were first derived by Colin Maclaurin (1742) in A Treatise of Fluxions, and have been enumerated in many subsequent publications (e.g., Tassoul 1978; Chandrasekhar 1987).

Properties of Uniform-Density Spheroids

Surface Definition

Let <math>a_1</math> be the equatorial radius and <math>a_3</math> the polar radius of a uniform-density object whose surface is defined precisely by an oblate spheroid. The degree of flattening of the object may be parameterized in terms of the axis ratio <math>a_3/a_1</math>, or in terms of the object's eccentricity,

<math>

e \equiv \biggl[ 1 - \biggl(\frac{a_3}{a_1}\biggr)^2 \biggr]^{1/2} .

</math>

(For an oblate spheroid, <math>a_3 \leq a_1</math>; hence, the eccentricity is restricted to the range <math>0 \leq e \leq 1</math>.) The meridional cross-section of such a spheroid is an ellipse with the same eccentricity. The foci of this ellipse lie in the equatorial plane of the spheroid at a distance <math>\varpi = ea_1</math> form the minor (<math>z</math>) axis.

Mean Radius

For purposes of normalization, it will be useful to define the mean radius of the spheroid as,

<math>

a_\mathrm{mean} \equiv \biggl[a_1^2 a_3 \biggr]^{1/3} = a_1 (1 - e^2)^{1/6} ,

</math>

which is equivalent to the radius of a sphere in the limit <math>a_3 = a_1</math> (<math>e=0</math>).

Mass

The total mass of such a spheroid is,

<math>

M = \frac{4\pi}{3}~a_1^2 a_3 \rho = \frac{4\pi}{3}~a_1^3 \rho (1 - e^2)^{1/2} .

</math>

Gravitational Potential

In an accompanying discussion (not yet typed!) entitled, Properties of Homogeneous Ellipsoids, an expression is given for the gravitational potential <math>\Phi(\vec{x})</math> at an internal point or on the surface of an homogeneous ellipsoid with semi-axes <math>(x,y,z) = (a_1,a_2,a_3)</math>. For an homogeneous, oblate spheroid in which <math>a_1 = a_2 \geq a_3</math>, this analytic expression defining the potential reduces to the form,

<math>

\Phi(\varpi,z) = -\pi G \rho \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 \varpi^2 + A_3 z^2 \biggr) \biggr],

</math>

where, as defined elsewhere (not yet typed!), the coefficients <math>A_1</math>, <math>A_3</math>, and <math>I_\mathrm{BT}</math> are functions only of the spheroid's eccentricity. Specifically,

   <math>
   A_1
   </math>
   <math>
   =
   </math>
   <math>
   \frac{1}{e^2} \biggl[\frac{\sin^{-1}e}{e} - (1-e^2)^{1/2}  \biggr](1-e^2)^{1/2}
   </math>
   <math>
   A_3
   </math>
   <math>
   =
   </math>
   <math>
   \frac{2}{e^2} \biggl[(1-e^2)^{-1/2} -\frac{\sin^{-1}e}{e} \biggr](1-e^2)^{1/2}
   </math>
   <math>
   I_\mathrm{BT}
   </math>
   <math>
   =
   </math>
   <math>
   2A_1 + A_3(1-e^2) = 2 (1-e^2)^{1/2} \biggl[ \frac{\sin^{-1}e}{e}\biggr]
   </math>

Governing Relations

<math> (n+1) P = H\rho</math> .



Summary

From the above derivations, we can describe the properties of a spherical <math>~n</math> = 1 polytrope as follows:

  • Mass:
Given the density, <math>\rho_c</math>, and the radius, <math>R</math>, of the configuration, the total mass is,

<math>M = \frac{4}{\pi} \rho_c R^3 </math> ;

and, expressed as a function of <math>M</math>, the mass that lies interior to radius <math>r</math> is,

<math>\frac{M_r}{M} = \frac{1}{\pi} \biggl[ \sin\biggl(\frac{\pi r}{R} \biggr) - \biggl(\frac{\pi r}{R} \biggr)\cos\biggl(\frac{\pi r}{R} \biggr) \biggr]</math> .

  • Pressure:
Given values for the pair of model parameters <math>( \rho_c , R )</math>, or <math>( M , R )</math>, or <math>( \rho_c , M )</math>, the central pressure of the configuration is,

<math>P_c = \frac{2 G}{\pi} \rho_c^2 R^2 = \frac{\pi G}{8}\biggl( \frac{M^2}{R^4} \biggr) = \biggl[ \frac{1}{2\pi} G^3 \rho_c^4 M^2 \biggr]^{1/3}</math> ;

and, expressed in terms of the central pressure <math>P_c</math>, the variation with radius of the pressure is,

<math>P(r)= P_c \biggl[\frac{R}{\pi r} \sin\biggl(\frac{\pi r}{R}\biggr) \biggr]^2</math> .

  • Enthalpy:
Throughout the configuration, the enthalpy is given by the relation,

<math>H(r) = \frac{2 P(r)}{ \rho(r)} = \frac{GM}{R} \biggl[\frac{R}{\pi r} \sin\biggl(\frac{\pi r}{R}\biggr) \biggr]</math> .

  • Gravitational potential:
Throughout the configuration — that is, for all <math>r \leq R</math> — the gravitational potential is given by the relation,

<math>\Phi_\mathrm{surf} - \Phi(r) = H(r) = \frac{GM}{R} \biggl[\frac{R}{\pi r} \sin\biggl(\frac{\pi r}{R}\biggr) \biggr] </math> .

Outside of this spherical configuration— that is, for all <math>r \geq R</math> — the potential should behave like a point mass potential, that is,

<math>\Phi(r) = - \frac{GM}{r} </math> .

Matching these two expressions at the surface of the configuration, that is, setting <math>\Phi_\mathrm{surf} = - GM/R</math>, we have what is generally considered the properly normalized prescription for the gravitational potential inside a spherically symmetric, <math>~n</math> = 1 polytropic configuration:

<math>\Phi(r) = - \frac{G M}{R} \biggl\{ 1 + \biggl[\frac{R}{\pi r} \sin\biggl(\frac{\pi r}{R}\biggr) \biggr] \biggr\} </math> .

  • Mass-Radius relationship:
We see that, for a given value of <math>\rho_c</math>, the relationship between the configuration's total mass and radius is,

<math>M \propto R^3 ~~~~~\mathrm{or}~~~~~R \propto M^{1/3} </math> .

  • Central- to Mean-Density Ratio:
The ratio of the configuration's central density to its mean density is,

<math>\frac{\rho_c}{\bar{\rho}} = \biggl(\frac{\pi M}{4 R^3} \biggr)\biggl(\frac{3 M}{4 \pi R^3} \biggr) = \frac{\pi^2}{3} </math> .

Related Wikipedia Discussions


Whitworth's (1981) Isothermal Free-Energy Surface

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