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

==Limiting Mass==
===Background===
As early as 1941, Chandraskhar and his collaborators realized that the shape of the model sequence in a <math>\nu</math> versus <math>q</math> diagram, as displayed in Figure 2 above, implies that equilibrium structures can exist only if the fractional core mass lies below some limiting value.  This realization is documented, for example, by the following excerpt from &sect;5 of {{ HC41full }}.

<div align="center">
<table border="1" cellpadding="5" width="60%">
<tr>
  <td align="center" colspan="1">
Text excerpt from &sect;5 (pp. 532 - 533) of<br />{{ HC41figure }}
  </td>
</tr>

<tr>
  <td align="left" colspan="1">
<!-- [[Image:HenrichChandra41a.jpg|600px|center|HenrichChandra1941]] -->
<!-- [[Image:AAAwaiting01.png|600px|center]] -->
<font color="darkgreen">"&hellip; at a fixed central temperature, the fraction of the total mass, <math>\nu</math>, contained in the core increases slowly at first and soon very rapidly as <math>q</math> approaches <math>q_\mathrm{max}</math>.  However, this increase of <math>\nu</math> does not continue indefinitely; <math>\nu</math> soon attains a maximum value <math>\nu_\mathrm{max}</math>.  There exists, therefore, an upper limit to the mass which can be contained in the isothermal core."</font>
  </td>
</tr>

</table>
</div>

Given that our bipolytropic sequence has been defined analytically, it may be possible to analytically determine the limiting core mass of our model.  In order to accomplish this, we need to identify the point along the sequence &#8212; in particular, the value of the dimensionless interface location &#8212; at which <math>d\nu/dq = 0</math> or, equivalently, <math>d\nu/d\xi_i = 0</math>.

Before carrying out the desired differentiations, we will find it useful to rewrite the relevant expressions in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}}  \, ;
</math>
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .
</math>
</div>

We obtain,

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

<tr>
  <td align="right">
<math>\eta_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>m_3 \biggl( \frac{\ell_i}{1 + \ell_i^2} \biggr) \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Lambda_i </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{m_3\ell_i} [ 1 + (1-m_3)\ell_i^2]  ~~~ \Rightarrow ~~~ 
</math>
&nbsp;&nbsp;'''<font color="red">Believe it or not &hellip; &nbsp;</font>'''
<math>
(1 + \Lambda^2) = \frac{(1+\ell_i^2)}{m_3^2 \ell_i^2} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr] \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>A</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{1 + (1-m_3)^2 \ell_i^2}{1 + \ell_i^2} \biggr]^{1/2} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\biggl( \frac{\pi}{2} \biggr)^{1/2} \frac{M_\mathrm{core}}{9}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\ell_i^3}{(1 + \ell_i^2)^{3/2}} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\biggl( \frac{\pi}{2} \biggr)^{1/2} \frac{M_\mathrm{tot}}{9}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{m_3^2} \biggl[ 1 + (1-m_3)^2 \ell_i^2 \biggr]^{1/2}
\biggl\{  \biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr) + m_3 \ell_i (1 + \ell_i^2)^{-1} \biggr\} \, .
</math>
  </td>
</tr>

</table>
</div>

Hence,
<div align="center">
<math>
\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} 
= (m_3^2 \ell_i^3) (1 + \ell_i^2)^{-1/2} [1 + (1-m_3)^2 \ell_i^2]^{-1/2}  \biggl[ m_3\ell_i + (1+\ell_i^2) \biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr) \biggr]^{-1}
</math>
</div>

<table border="1" align="center" width="80%" cellpadding="5"><tr><td align="left">
An interesting limiting case is <math>m_3 = 1</math>, in which case,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>
\nu
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
(\ell_i^3) (1 + \ell_i^2)^{-1/2}  \biggl[ \ell_i + (1+\ell_i^2) \biggl(\frac{\pi}{2} + \tan^{-1} \biggl(\frac{1}{\ell_i}\biggr) \biggr) \biggr]^{-1} \, ,</math>
  </td>
</tr>
</table>
and the maximum value of <math>\nu</math> along this sequence arises when <math>\ell_i \rightarrow \infty</math>, in which case,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>
\nu
</math>
  </td>
  <td align="center">
<math>
\rightarrow
</math>
  </td>
  <td align="left">
<math>
\ell_i^2   \biggl[ \ell_i + (1+\ell_i^2) \biggl(\frac{\pi}{2}  \biggr) \biggr]^{-1} 
\rightarrow
\frac{2}{\pi}
\, .</math>
  </td>
</tr>
</table>
</td></tr></table>


The condition, <math>d\nu/d\xi_i = 0</math>, also will be satisfied if the condition,
<div align="center">
<math>
\frac{d\ln\nu}{d\ln\ell_i} = 0 \, ,
</math>
</div>
is met.

===Derivation===
My manual derivation gives,

<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
(1+\ell_i^2)  \biggl[ \frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr]
\biggl\{ 3  - \frac{(1-m_3)^2 \ell_i^2 (1+\ell_i^2)}{[ 1 + (1-m_3)^2 \ell_i^2 ]} \biggr\} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> (1+\ell_i^2)
\frac{\partial \tan^{-1}\Lambda_i}{\partial \ln \ell_i} 
-m_3\ell_i \biggl\{ \ell_i^2 + 2  - \frac{(1-m_3)^2 \ell_i^2 (1+\ell_i^2)}{[ 1 + (1-m_3)^2 \ell_i^2 ]} \biggr\}
</math> 
  </td>
</tr>
</table>
</div>

where,

<div align="center">
<math>
\frac{\partial \tan^{-1}\Lambda_i}{\partial \ln \ell_i} =  
\frac{[(1-m_3)\ell_i^2 - 1 ] }{m_3\ell_i (1 + \Lambda_i^2)}
= \frac{m_3 \ell_i [(1-m_3)\ell_i^2 - 1 ]}{(1 + \ell_i^2) [ 1 + (1-m_3)^2 \ell_i^2]}  \, .
</math>
</div>

Upon rearrangement, this gives,

<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
(1+\ell_i^2)  \biggl[ \frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr]
\biggl\{ 3[ 1 + (1-m_3)^2 \ell_i^2 ]  - (1-m_3)^2 \ell_i^2 (1+\ell_i^2) \biggr\} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
 m_3\ell_i \biggl\{[(1-m_3)\ell_i^2 - 1 ] -(\ell_i^2 + 2)[ 1 + (1-m_3)^2 \ell_i^2 ]  + (1-m_3)^2 \ell_i^2 (1+\ell_i^2) \biggr\} \, ,
</math> 
  </td>
</tr>
</table>
</div>


and further simplification <font color="red">[completed on 19 May 2013]</font> gives,

<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
\underbrace{\biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i\biggr) (1+\ell_i^2) [ 3 + (1-m_3)^2(2-\ell_i^2)\ell_i^2]}_{\mathrm{LHS}} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\underbrace{m_3 \ell_i [(1-m_3)\ell_i^4 - (m_3^2 - m_3 +2)\ell_i^2 - 3]}_{\mathrm{RHS}} \, .
</math> 
  </td>
</tr>
</table>
</div>

<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="12">
<b>Maximum Fractional Core Mass, <math>\nu = M_\mathrm{core}/M_\mathrm{tot}</math> (solid green circular markers)<br />for Equilibrium Sequences having Various Values of <math>\mu_e/\mu_c</math>
  </td>
</tr>
<tr>
  <td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
  </td>
  <td align="center">
<math>\xi_i</math>
  </td>
  <td align="center">
<math>\theta_i</math>
  </td>
  <td align="center">
<math>\eta_i</math>
  </td>
  <td align="center">
<math>\Lambda_i</math>
  </td>
  <td align="center">
<math>A</math>
  </td>
  <td align="center">
<math>\eta_s</math>
  </td>
  <td align="center">
LHS
  </td>
  <td align="center">
RHS
  </td>
  <td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
  </td>
  <td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
  </td>
  <td align="center" rowspan="7">[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]</td>
</tr>

<tr>
  <td align="center">
<math>\frac{1}{3}</math>
  </td>
  <td align="center">
<math>\infty</math> </td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">0.0 </td>
  <td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
  <td align="center">
0.33
  </td>
  <td align="right">
24.00496  </td>
  <td align="right">
0.0719668  </td>
  <td align="right">
0.0710624  </td>
  <td align="right">
0.2128753  </td>
  <td align="right">
0.0726547  </td>
  <td align="right">
1.8516032  </td>
  <td align="right">
-223.8157  </td>
  <td align="right">
-223.8159  </td>
  <td align="right">
0.038378833  </td>
  <td align="right">
0.52024552  </td>
</tr>

<tr>
  <td align="center">
0.316943
  </td>
  <td align="right">
10.744571  </td>
  <td align="right">
0.1591479  </td>
  <td align="right">
0.1493938  </td>
  <td align="right">
0.4903393  </td>
  <td align="right">
0.1663869  </td>
  <td align="right">
2.1760793  </td>
  <td align="right">
-31.55254  </td>
  <td align="right">
-31.55254  </td>
  <td align="right">
0.068652714  </td>
  <td align="right">
0.382383875  </td>
</tr>

<tr>
  <td align="center">
0.3090
  </td>
  <td align="right">
8.8301772  </td>
  <td align="right">
0.1924833  </td>
  <td align="right">
0.1750954  </td>
  <td align="right">
0.6130669  </td>
  <td align="right">
0.2053811  </td>
  <td align="right">
2.2958639  </td>
  <td align="right">
-18.47809  </td>
  <td align="right">
-18.47808  </td>
  <td align="right">
0.076265588  </td>
  <td align="right">
0.331475715  </td>
</tr>

<tr>
  <td align="center">
<math>\frac{1}{4}</math>
  </td>
  <td align="right">
4.9379256  </td>
  <td align="right">
0.3309933  </td>
  <td align="right">
0.2342522  </td>
  <td align="right">
1.4179907  </td>
  <td align="right">
0.4064595  </td>
  <td align="right">
2.761622  </td>
  <td align="right">
-2.601255  </td>
  <td align="right">
-2.601257  </td>
  <td align="right">
0.084824137  </td>
  <td align="right">
0.139370157  </td>
</tr>

<tr>
  <td align="left" colspan="11">
Recall that,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}}  \, ;
</math>
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .
</math>
</div>
  </td>
</tr>
</table>

===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.