Editing SSC/Structure/BiPolytropes/Analytic51/Pt3 (section)

===Limit when m<sub>3</sub> = 0===
It is instructive to examine the root of this equation in the limit where <math>m_3 = 0</math> &#8212; that is, when <math>\mu_e/\mu_c = 0</math>.  First, we note that,
<div align="center">
<math>\Lambda_i\biggr|_{m_3 \rightarrow 0} = \biggl\{ \frac{1}{m_3\ell_i} [ 1 + (1-m_3)\ell_i^2] \biggr\}_{m_3 \rightarrow 0} = \infty \, .</math> 
</div>
Hence,
<div align="center">
<math>\biggl[\tan^{-1}\Lambda_i\biggr]_{m_3 \rightarrow 0} = \frac{\pi}{2} \, ,</math> 
</div>
and the limiting relation becomes,
<div align="center">
<math>
\pi (1+\ell_i^2) [ 3 + (2-\ell_i^2)\ell_i^2] = 0 \, ,
</math>
</div>
or, more simply,
<div align="center">
<math>
\ell_i^4 - 2\ell_i^2 - 3 = 0 \, .
</math>
</div>
The real root is,
<div align="center">
<math>\ell_i^2 = \frac{1}{2} \biggl[ 2 + \sqrt{4 + 12} \biggr] = 3 ~~~~ \Rightarrow ~~~~ \xi_i = 3 \, .</math>
</div>
For <math>\xi_i = 3</math>, the radius of the core, the mass of the core, and the pressure at the edge of the core are, respectively,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>r^*_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{3^3}{2\pi}\biggr)^{1/2} </math>
  </td>

<td align="center">&nbsp; &nbsp; <math>\Rightarrow</math>  &nbsp; &nbsp; </td>

  <td align="right">
<math>r_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{3^3}{2\pi}\biggr)^{1/2} \biggl[ \frac{K_c^{1/2}}{G^{1/2} \rho_0^{2/5}} \biggr] \, ;</math>
  </td>

</tr>

<tr>
  <td align="right">
<math>M^*_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{3^7}{2^5\pi}\biggr)^{1/2} </math>
  </td>

<td align="center">&nbsp; &nbsp; <math>\Rightarrow</math>  &nbsp; &nbsp; </td>

  <td align="right">
<math>M_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{3^7}{2^5\pi}\biggr)^{1/2}  \biggl[ \frac{K_c^{3/2}}{G^{3/2} \rho_0^{1/5}} \biggr] \, ;</math>
  </td>

</tr>

<tr>
  <td align="right">
<math>P^*_i</math>
  </td>
  <td align="right">
<math>=</math>
  </td>
  <td align="left">
<math>2^{-6} </math>
  </td>

<td align="center">&nbsp; &nbsp; <math>\Rightarrow</math>  &nbsp; &nbsp; </td>

  <td align="right">
<math>P_i</math>
  </td>
  <td align="right">
<math>=</math>
  </td>
  <td align="left">
<math>2^{-6} [ K_c \rho_0^{6/5}] \, .</math>
  </td>

</tr>
</table>
</div>

If we invert the middle expression to obtain <math>\rho_0</math> in terms of <math>M_\mathrm{core}</math>, specifically,
<div align="center">
<math>\rho_0^{1/5} = \biggl(\frac{3^7}{2^5\pi}\biggr)^{1/2}  \biggl[ \frac{K_c^{3/2}}{G^{3/2} M_\mathrm{core}} \biggr] \, ,</math> 
</div>
then we can rewrite <math>r_\mathrm{core}</math> and <math>P_i</math> in terms of, respectively, the ''reference'' radius,
<math>R_\mathrm{rf}</math>, and reference pressure, <math>P_\mathrm{rf}</math>, as defined in 
[[SSC/Structure/PolytropesEmbedded#Extension_to_Bounded_Sphere_2|our discussion of isolated <math>n=5</math>
polytropes embedded in an external medium]].  Specifically, we obtain,
<div align="center">
<table border="1" cellpadding="5">
<tr>

  <td align="right">
<math>r_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{3^3}{2\pi}\biggr)^{1/2} \biggl[ \frac{K_c^{1/2}}{G^{1/2} } \biggr] \biggl(\frac{3^7}{2^5\pi}\biggr)^{-1}  \biggl[ \frac{K_c^{3/2}}{G^{3/2} M_\mathrm{core}} \biggr]^{-2}</math>
  </td>

  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{2^9 \pi}{3^{11}}\biggr)^{1/2} \biggl[ \frac{G^{5/2} M^2_\mathrm{core}}{K_c^{5/2}} \biggr]</math>
  </td>

  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{2^9 \pi}{3^{11}}\biggr)^{1/2} \frac{3^3}{2^6} \biggl( \frac{5^5}{\pi} \biggr)^{1/2} R_\mathrm{rf} \biggr|_{n=5}</math>
  </td>

  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{5^5}{2^3 \cdot 3^5} \biggr)^{1/2} R_\mathrm{rf} \biggr|_{n=5}</math>
  </td>

</tr>
<tr>

  <td align="right">
<math>P_i</math>
  </td>
  <td align="right">
<math>=</math>
  </td>
  <td align="left">
<math>2^{-6} [ K_c ] \biggl(\frac{3^7}{2^5\pi}\biggr)^{3}  \biggl[ \frac{K_c^{3/2}}{G^{3/2} M_\mathrm{core}} \biggr]^6 </math>
  </td>

  <td align="right">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{3^7}{2^{7}\pi}\biggr)^{3}  \biggl[ \frac{K_c^{10}}{G^{9} M^6_\mathrm{core}} \biggr] </math>
  </td>

  <td align="right">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{3^7}{2^7\pi}\biggr)^{3}  \biggl( \frac{2^{26} \pi^3}{3^{12} 5^9} \biggr) P_\mathrm{rf} \biggr|_{n=5}</math>
  </td>

  <td align="right">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{2^{5}\cdot 3^9 }{5^9} \biggr) P_\mathrm{rf} \biggr|_{n=5}</math>
  </td>

</tr>
</table>
</div>

['''<font color="red">26 May 2013</font>''' with further elaboration on '''<font color="red">28 May 2013</font>'''] This is the same result that was obtained when we
[[SSC/Structure/PolytropesEmbedded#Extension_to_Bounded_Sphere_2|embedded an isolated <math>n=5</math> polytrope in an external medium]].  Apparently, therefore, the physics that leads to the mass limit for a [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|Bonnor-Ebert sphere]] is the same physics that sets the {{ SC42 }} mass limit.