Editing SSC/Structure/Polytropes/Analytic (section)

====Summary====

From the above derivations, we can describe the properties of a spherical {{Math/MP_PolytropicIndex}} = 1 polytrope as follows:
* <font color="red">Mass</font>:  
: Given the density, <math>\rho_c</math>, and the radius, <math>R</math>, of the configuration, the total mass is,
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<math>M = \frac{4}{\pi} \rho_c R^3 </math> ;
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: and, expressed as a function of <math>M</math>, the mass that lies interior to radius <math>r</math> is,
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<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> .
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* <font color="red">Pressure</font>: 
: 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,
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<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> ; 
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: and, expressed in terms of the central pressure <math>P_c</math>, the variation with radius of the pressure is,
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<math>P(r)= P_c \biggl[\frac{R}{\pi r} \sin\biggl(\frac{\pi r}{R}\biggr)  \biggr]^2</math> .
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* <font color="red">Enthalpy</font>: 
: Throughout the configuration, the enthalpy is given by the relation,
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<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> .
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* <font color="red">Gravitational potential</font>: 
: Throughout the configuration &#8212; that is, for all <math>r \leq R</math> &#8212; the gravitational potential is given by the relation,
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<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> .
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: Outside of this spherical configuration&#8212; that is, for all <math>r \geq R</math> &#8212;  the potential should behave like a point mass potential, that is,
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<math>\Phi(r) = - \frac{GM}{r} </math> .
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: 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/MP_PolytropicIndex}} = 1 polytropic configuration:
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<math>\Phi(r) = - \frac{G M}{R} \biggl\{ 1 + \biggl[\frac{R}{\pi r} \sin\biggl(\frac{\pi r}{R}\biggr)  \biggr] \biggr\} </math> .
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* <font color="red">Mass-Radius relationship</font>:
: We see that, for a given value of <math>\rho_c</math>, the relationship between the configuration's total mass and radius is,
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<math>M \propto R^3  ~~~~~\mathrm{or}~~~~~R \propto M^{1/3} </math> .
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* <font color="red">Central- to Mean-Density Ratio</font>:
: The ratio of the configuration's central density to its mean density is,
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<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> .
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<table border="1" cellpadding="8" align="right">
<tr>
  <th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/n1.xlsx --- worksheet = Sheet1]]Figure 1: &nbsp; Mass vs. Radius <br />for n = 1 polytrope</th>
</tr>
<tr>
  <td align="center">[[File:N100SequenceA.png|300px|n = 1 mass vs. radius diagram]]</td>
</tr>
</table>
For the purposes of comparing the internal structure of configurations having different polytropic indexes &#8212; see, for example [[SSC/Structure/Polytropes/Numerical#Fig4|Figure 4 in an accompanying chapter]] &#8212; we have found it useful in each case to graphically illustrate how the normalized mass, <math>~M/M_\mathrm{SWS}</math>, varies with the normalized radius, <math>~R/R_\mathrm{SWS}</math>, where the definition of these two functions is drawn from an [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|accompanying discussion of pressure-truncated polytropic configurations]]. In the case of an <math>~n=1</math> polytrope, both functions are expressible analytically; specifically, we have, 
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<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~\frac{R}{R_\mathrm{SWS}}\biggr|_{n=1}
</math>
  </td>
  <td align="center">
<math>~\equiv~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{1}{4\pi} \biggr)^{1/2} \xi   \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\frac{M}{M_\mathrm{SWS}}\biggr|_{n=1}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{1}{4\pi} \biggr)^{1/2} \biggl[ \frac{\xi^2}{\theta_n} 
\biggl| \frac{d\theta_n}{d\xi} \biggr|  ~\biggr]_{n=1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{1}{4\pi} \biggr)^{1/2} \frac{\xi^3}{\sin\xi} 
\biggl[\frac{\sin\xi - \xi\cos\xi}{\xi^2} \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{1}{4\pi} \biggr)^{1/2} \xi 
\biggl[1 - \xi\cot\xi \biggr]  \, .
</math>
  </td>
</tr>
</table>
</div>

As Figure 1 illustrates, this normalized mass increases monotonically with radius.  Given that the surface of the configuration is associated with the parameter value, <math>~\xi = \pi</math>, we recognize that, at the surface,
<math>~R/R_\mathrm{SWS} = \sqrt{\pi/4} \approx 0.8862269</math> and <math>~M/M_\mathrm{SWS}</math> formally climbs to infinity.