Editing SSCpt1/Virial/FormFactors/Pt3 (section)

===Mass2 (n = 1)===

Alternatively, as has been laid out in our [[SSCpt1/Virial#Summary_of_Normalized_Expressions|accompanying summary of normalized expressions that are relevant to free-energy calculations]],
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<math>\frac{M_r(x)}{M_\mathrm{tot}}  </math>
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<math>=</math>
  </td>
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<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{x}  3x^2 \biggl[ \frac{\rho(x)}{\rho_0} \biggr]  dx \, ,</math>
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</tr>
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where, <math>M_\mathrm{limit}</math> is the "total" mass of the polytropic configuration that is truncated at <math>R_\mathrm{limit}</math>; keep in mind that, here,
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<math>M_\mathrm{tot} = \biggr[ \frac{2\pi K^3}{G^3} \biggr]^{1/2}  \rho_0  \, ,</math>
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is the total mass of the ''isolated'' <math>n=1</math> polytrope, that is, a polytrope whose ''Lane-Emden'' radius extends all the way to <math>\xi_1 = \pi</math>.  In our discussions of truncated polytropes, we often will use <math>\tilde\xi \le \xi_1</math> to specify the truncated radius in terms of the familiar, dimensionless Lane-Emden radial coordinate, so here we will set,
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<math>R_\mathrm{limit} = a_1 \tilde\xi ~~~~\Rightarrow ~~~~ x = \frac{r}{R_\mathrm{limit}} = \frac{a_1 \xi}{a_1 \tilde\xi} = \frac{\xi}{\tilde\xi} \, .</math>
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Hence, in terms of the desired integration coordinate, <math>x</math>, the density profile provided above becomes,

<div align="center" id="rhoofx1">
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<math>\frac{\rho(x)}{\rho_0}</math>
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<math>=</math>
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<math>\frac{\sin(\tilde\xi x)}{\tilde\xi x} \, ,</math>
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</tr>
</table>

</td></tr>
</table>
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and the integral defining <math>M_r(x)</math> becomes,
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  <td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}}  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
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<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\frac{3}{\tilde\xi} \int_0^{x}  x \sin(\tilde\xi x)  dx </math>
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</tr>

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&nbsp;
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<math>=</math>
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<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{3}{\tilde\xi^3} 
[\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] \, . </math>
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</table>
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In this case, integrating "all the way out to the surface" means setting <math>r = R_\mathrm{limit}</math> and, hence, <math>x = 1</math>; by definition, it also means <math>M_r(x) = M_\mathrm{limit}</math>.  Therefore we have,
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  <td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot}}  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\frac{3}{\tilde\xi^3} [\sin \tilde\xi  - \tilde\xi \cos \tilde\xi  ] </math>
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<math>\Rightarrow ~~~ \biggl( \frac{\bar\rho}{\rho_c} \biggr)_\mathrm{eq} </math>
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<math>=</math>
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<math>\frac{3}{\tilde\xi^3} [\sin \tilde\xi  - \tilde\xi \cos \tilde\xi  ]   \, .</math>
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</table>
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Using this expression for the mean-to-central density ratio along with the expression for the ratio, <math>M_\mathrm{limit}/M_\mathrm{tot}</math>, derived in the preceding subsection, we also can state that for truncated <math>n=1</math> polytropes,
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<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}}  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
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<math>\frac{1}{\pi} (\sin\tilde\xi - \tilde\xi \cos\tilde\xi) 
\biggl\{ \frac{[\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ]}{( \sin \tilde\xi  - \tilde\xi \cos \tilde\xi  )} \biggr\} </math>
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&nbsp;
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<math>=</math>
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<math>\frac{1}{\pi} [\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] </math>
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By making the substitution, <math>x \rightarrow \xi/\tilde\xi</math>, this expression becomes identical to the <math>M_r/M_\mathrm{tot}</math> [[#NormalizedProfiles1|profile presented just before the "Mass1" subsection]], above.  In summary, then, we have the following two equally valid expressions for the <math>M_r</math> profile &#8212; one expressed as a function of <math>\xi</math> and the other expressed as a function of <math>x</math>:

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<math>\frac{M_r(\xi)}{M_\mathrm{tot}}</math> 
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<math>=</math>
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<math>\frac{1}{\pi} (\sin\xi - \xi \cos\xi  ) \, ;</math>
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</tr>

<tr>
  <td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}}</math> 
  </td>
  <td align="center">
<math>=</math>
  </td>
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<math>\frac{1}{\pi} [\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] \, .</math>
  </td>
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</td></tr>
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