Editing SSC/Structure/Polytropes/Analytic (section)

=====Example Interval=====
As an example, let's set <math>A=1</math> and examine the oscillation interval between <math>~m=0</math> and <math>m=1</math>, that is, over the range, <math>0 \le \Delta \le \pi</math> which corresponds to the parameter interval <math>\xi = [1, e^{2\pi}]</math>.  The denominator of <math>\theta_{5F}</math> is positive for all values of <math>\xi</math> and, over this specified interval, the numerator of <math>\theta_{5F}</math> is also always positive.  The blue curve in the following figure presents a plot of <math>\theta_{5F}(x)</math> and the green curve presents a plot of the first derivative (the slope) of the function <math>d\theta_{5F}(x)/d\xi</math> over the desired interval, where <math>x \equiv \xi/e^{2\pi}</math>; note that the horizontal axis is shown in logarithmic units.

<div align="center" id="Fig3">
<table border="2" cellpadding="10">
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  <td align="center">
Figure 3: &nbsp; Our Determination and Presentation of <br />a Segment of the <math>\theta_{5F}</math> Function as originally derived by<br />{{ Srivastava62 }} </td>
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[[File:PlotTheta5F.png|450px|center|Srivastava's Lane-Emden function for n = 5]]
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</div>

At both ends of the chosen parameter interval &#8212; that is, at <math>~\Delta = 0</math> and at <math>~\Delta = \pi</math> &#8212; the function <math>~\theta_{5F} = 0</math> and, correspondingly as depicted in the figure, the blue curve touches the horizontal axis.  At the beginning of the interval (<math>~\Delta =0</math>), the slope of the function and, correspondingly, the green curve, has the (positive) value,

<div align="center">
<table border="0" cellpadding="5" align="center">
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  <td align="right">
<math>~\frac{d\theta_{5F}}{d\xi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3\cos(0)-3\sin(0) + 2\sin^3(0) }{2\xi^{3/2}[3-2\sin^2(0)]^{3/2}}  
= \frac{3}{2(3^{3/2})} = (2^2 \cdot 3)^{-1/2} \approx 0.28868 
\, .
</math>
  </td>
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</table>
</div>
At the end of the interval (<math>~\Delta=\pi</math>), the slope of the function as well as the green curve, has the (negative) value,

<div align="center">
<table border="0" cellpadding="5" align="center">
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<math>~\frac{d\theta_{5F}}{d\xi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3\cos(\pi)-3\sin(\pi) + 2\sin^3(\pi) }{2\xi^{3/2}[3-2\sin^2(\pi)]^{3/2}}  
= \frac{-3}{2e^{3\pi}(3^{3/2})} = -e^{-3\pi} (2^2 \cdot 3)^{-1/2} \approx  -2.3296 \times 10^{-5} 
\, .
</math>
  </td>
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</table>
</div>

Over this interval, <math>~\theta_{5F}</math> reaches its maximum when the slope of the function is zero, that is, at the value of <math>~\Delta</math> where,

<div align="center">
<table border="0" cellpadding="5" align="center">
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<math>~
0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3\cos\Delta -3\sin\Delta +2(1-\cos^2\Delta)\sin\Delta</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
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<math>~3\cos\Delta -\sin\Delta -2\cos^2\Delta \sin\Delta</math>
  </td>
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<math>~\Rightarrow ~~~~1</math>
  </td>
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<math>~=</math>
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<math>~3\cot\Delta -2\cos^2\Delta \, .</math>
  </td>
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</table>
</div>
Rewriting both of these trigonometric functions in terms of the tangent function and adopting the shorthand notation,
<div align="center">
<math>~y \equiv \tan\Delta \, ,</math>
</div>
this condition becomes,

<div align="center">
<table border="0" cellpadding="5" align="center">

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<math>~1</math>
  </td>
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<math>~=</math>
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<math>~\frac{3}{y} -\frac{2}{1+y^2} </math>
  </td>
</tr>

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<math>~\Rightarrow ~~~~ y(y^2 + 1)</math>
  </td>
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<math>~=</math>
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<math>~3(y^2+1) - 2y </math>
  </td>
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<math>~\Rightarrow ~~~~ y^3 - 3y^2 + 3y - 3</math>
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<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
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</table>
</div>

<div align="center" id="CubicRoot">
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<font color="red">'''ASIDE:'''</font>  As is well known and documented &#8212; see, for example [http://mathworld.wolfram.com/CubicFormula.html Wolfram MathWorld] or [http://en.wikipedia.org/wiki/Cubic_function Wikipedia's discussion] of the topic &#8212; the roots of any cubic equation can be determined analytically.  In order to evaluate the root(s) of our particular cubic equation, we have drawn from the utilitarian [http://www.math.vanderbilt.edu/~schectex/courses/cubic/ online summary provided by Eric Schechter at Vanderbilt University].  For a cubic equation of the general form,
<div align="center">
<math>~ay^3 + by^2 + cy + d = 0 \, ,</math>
</div> 
a real root is given by the expression,
<div align="center">
<math>~
y = p + \{q + [q^2 + (r-p^2)^3]^{1/2}\}^{1/3} + \{q - [q^2 + (r-p^2)^3]^{1/2}\}^{1/3}
\, ,</math>
</div> 
where,
<div align="center">
<math>~p \equiv -\frac{b}{3a} \, ,</math> &nbsp;&nbsp;&nbsp;&nbsp;
<math>~q \equiv \biggl[p^3 + \frac{bc-3ad}{6a^2} \biggr] \, ,</math>
&nbsp;&nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp; 
<math>~r=\frac{c}{3a} \, .</math>
</div>
In our particular case,
<div align="center">
<math>~a =1\, ,</math> &nbsp;&nbsp;&nbsp;&nbsp;
<math>~b =-3\, ,</math> &nbsp;&nbsp;&nbsp;&nbsp;
<math>~c = +3 \, ,</math>
&nbsp;&nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp; 
<math>~d = - 3 \, .</math>
</div>
[[File:WolframAlphaCubicSolver.png|thumbnail|right|100px|WolframAlpha]]Hence, interestingly enough,
<div align="center">
<math>~p = q = r = + 1 \, ,</math>
</div>
which implies that the real root is,
<div align="center">
<table border="0" cellpadding="5" align="center">

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<math>~y</math>
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<math>~=</math>
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<math>~1 + \{2\}^{1/3} + \{0\}^{1/3} \, .</math>
  </td>
</tr>
</table>
</div>
(There is also a pair of imaginary roots, but they are irrelevant in the context of our overarching astrophysical discussion.)

Just for fun, we have also used WolframAlpha's online "cubic equation solver" widget to find the root(s) of our specific cubic equation.  Clicking on the thumbnail image provided here, on the right, displays the key result that was returned by this WolframAlpha widget.
  </td>
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</table>
</div>


The single, real root of this cubic equation is,
<div align="center">
<math>~y = 1 + 2^{1/3} \, ,</math>
</div>
which corresponds to,
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<math>~\Delta = \tan^{-1}(1 + 2^{1/3}) \, .</math>
</div>

[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 17 April 2015:  As far as I have been able to determine, this analytic prescription of xi_crit has not previously been derived, although, as is made clear in what follows, Murphy (1983) has assessed its value numerically to six significant digits.]]Hence, over this example interval, the maximum of Srivastava's <math>\theta_{5F}</math> function &#8212; and, hence also, the location at which the function's slope transitions from positive to negative values (denoted by the vertical red line in the above figure) &#8212; occurs at,
<div align="center">
<math>\xi_\mathrm{crit} \equiv e^{2\tan^{-1}(1+2^{1/3})} = 10.05836783\, .</math>
</div>
The corresponding value of the function at this critical radial location is,
<div align="center">
<math>\theta_{5F}|_\mathrm{max} \equiv \theta_{5F}(\xi_\mathrm{crit}) = (1+2^{1/3})[3 + (1+2^{1/3})^2]^{-1/2} e^{-\tan^{-1}(1+2^{1/3})} = 0.250260848 \, .</math>
</div>
This agrees precisely with the determination made by {{ Murphy83afull }} &#8212; see the excerpts from his paper displayed in the following boxed-in image &#8212; that the portion of the <math>~\theta_{5F}</math> function that falls in the interval <math>~1 \le (A\xi) < \xi_\mathrm{crit}</math> (the segment of the blue curve that lies to the left of the vertical red line in the above figure) is unphysical because the slope of the function is positive throughout that interval.  
<!--
<table border="1" cellpadding="10" align="center">
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  <th align="center">Excerpts (edited) from 
[http://adsabs.harvard.edu/abs/1983PASAu...5..175M J. O. Murphy (1983, Proc. Astr. Soc. Australia, 5, 175)]
  </th>
<tr>
  <td>
[[File:Murphy1983Extremum02.png|600px|center|Murphy's (1983) examination of Srivastava's function]]
[[Image:AAAwaiting01.png|600px|center|Murphy's (1983) examination of Srivastava's function]]
  </td>
</tr>
</table>
-->

<div align="center">
<table border="1" cellpadding="5" width="80%">
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Equation and text extracted<sup>&dagger;</sup> from p. 177 of &hellip;<br /> 
{{ Murphy83afigure }}
</td></tr>
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  <td align="center">
<!-- [[File:Murphy1983Extremum02.png|700px|center|Murphy's (1983) examination of Srivastava's function]] -->
<!-- [[Image:AAAwaiting01.png|400px|center|Norman &amp; Wilson (1978)]] -->
<table border="0" align="center" cellpadding="8" width="70%">
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on the interval <math>~[1, e^{2\pi} ]</math> &nbsp; &nbsp; &hellip; &nbsp; &nbsp; <math>~d\theta_{5F}/d\xi > 0</math> in the range [1, 10.0583]

<p></p>
----
<p></p>

<math>~\theta_{5F}(\zeta)_\mathrm{MAX} = 0.2503 \sqrt{A}</math> &nbsp; &nbsp; at &nbsp; &nbsp;  <math>~\zeta = 10.0583/A</math>
  </td>
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  </td>
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<tr><td align="left"><sup>&dagger;</sup>Equations and text displayed here,  with presentation order &amp; layout modified from the original publication.</td></tr>
</table>
</div>

On the other hand, the segment that falls in the interval, <math>~\xi_\mathrm{crit} \le (A\xi) \le e^{2\pi}</math>, whose function values lie in the range, <math>~\theta_{5F}|_\mathrm{max} \ge (A^{-1/2} \theta_{5F}) \ge 0</math> &#8212; that is, the segment of the blue curve that lies to the right of the vertical red line in the above figure &#8212; can be used to describe the <math>~n=5</math> "envelope" of a bipolytropic configuration because the function value is positive while it's first derivative is negative.