Editing Apps/Ostriker64 (section)

===The Dimensionless Radial Coordinate, &xi;, and Smallness Parameter, &beta;===

As we have [[SSC/Structure/Polytropes#Polytropic_Spheres|reviewed separately]], when researchers in the astrophysics community discuss the structure of ''spherical'' polytropes, the 
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<span id="LaneEmdenEquation"><font color="#770000">'''Lane-Emden Equation'''</font></span>
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{{Math/EQ_SSLaneEmden01}}
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invariably arises, as it is the governing 2<sup>nd</sup>-order ODE whose solution, <math>~\Theta_H(\xi)</math>, defines the internal structure of spherically symmetric equlibrium configurations.  Traditionally, as well, the dimensionless radial coordinate,
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<math>~\xi \equiv \frac{r}{a_n} \, ,</math>
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is defined in terms of <math>~a_n</math>, which is a natural length scale of the (spherical) problem.  Equation (42) of Ostriker's (1964) [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II] provides the traditional definition of <math>~a_n</math>.  It is therefore not surprising that, even though Ostriker's set of 1964 papers deal largely with the equilibrium and stability of ''ring-like'' configurations, he adopts a similar definition for the dimensionless radial coordinate; specifically, eq. (5) of [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II] states that,
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<math>~\alpha \xi \equiv r \, .</math>
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But, of course, in the context of Ostriker's presentation, <math>~r</math> is not a spherical radial coordinate but is, rather, as [[#Coordinate_System|defined above]]; and <math>~\alpha</math> is of the same order as the minor, cross-sectional radius of the torus.

[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 17 August 2018:  There appears to be a typographical error in the definition of &beta; that is provided by equation (6) in &sect;IIa of Ostriker's ''Paper II''.  The published equation defines &beta; as the ratio of &alpha; to ''r'' rather than, as we have indicated here, as the ratio of &alpha; to ''R''.  Equation (77) on p. 1078 of Paper II confirms this suspicion.]]In eq. (6) of [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], Ostriker also defines the dimensionless parameter,
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<math>~\beta \equiv \frac{\alpha}{R} \, ,</math>
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where <math>~R</math> is associated with the major radius of the ring.  Then he states that, <font color="darkgreen">"&hellip; since <math>~\alpha \ll R</math> (by hypothesis), we may be sure that <math>~\beta \ll 1</math> &hellip;"</font>

With the definitions of these two dimensionless parameters in hand &#8212; and, more specifically, after appreciating that,
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<math>~\frac{R}{r} = \frac{1}{\beta\xi}  ~~~\Rightarrow ~~~ \ln\frac{8R}{r} = \biggl[ \ln\frac{8}{\beta} - \ln\xi \biggr] </math>
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&#8212; we can follow Ostriker's lead and rewrite his derived expression for <math>~\Phi_\mathrm{TR}</math> in the form,

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<math>~\Phi_\mathrm{TR}(r,\phi)\biggr|_\mathrm{JPO}</math>
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<math>~=</math>
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<math>~ 
-\frac{GM}{\pi R}  \biggl\{ \ln\frac{8}{\beta} - \ln\xi 
+ \frac{\beta\xi}{2}\biggl[  - \biggl( \ln\frac{8}{\beta} -1 \biggr) + \ln\xi  \biggr]\cos\phi
</math>
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</tr>

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&nbsp;
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&nbsp;
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<math>~ 
+~ \frac{\beta^2\xi^2}{2^4} \biggl[ \biggl(2 \ln\frac{8}{\beta} - 3 - 2\ln\xi \biggr) + \biggl( 3 \ln\frac{8}{\beta} - 4 - 3\ln\xi \biggr)\cos 2\phi
\biggr] ~+ ~\cdots \biggr\} \, .
</math>
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Ostriker's (1964) [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], p. 1071, Eq. (26)
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