Editing SSC/Structure/PolytropesEmbedded/Other (section)

===Other Limits===

In a similar fashion, [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura (1981b)] derived mathematical expressions that identify the location of other turning points along equilibrium sequences of bounded polytropic configurations.  An M<sub>1</sub> sequence &#8212; as displayed, for example, in the set of P-R diagrams shown in [[SSC/Structure/PolytropesEmbedded#WhitworthFig1b|Figure 1, above]] &#8212; exhibits not only an "extremal of p<sub>1</sub>" but also an "extremal of r<sub>1</sub>."  As we have [[SSC/Structure/PolytropesEmbedded#Location_of_Pressure_Limit|just reviewed]], the first of these is identified by setting <math>~(d\ln p_1/d\ln r_1)_{M} = 0</math> or, using Kimura's more compact terminology, the first occurs at a location that satisfies the condition, 
<div align="center">
<math>h_G = 0 \, ,</math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
that is, where &hellip; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>~\tilde\theta^{n+1} (\tilde\theta^')^{-2} = (n-3)/2 \, .</math>
</div>
Similarly, Kimura points out that an "extremal in r<sub>1</sub>" along an M<sub>1</sub> sequence occurs at a location that satisfies the condition,
<div align="center">
<math>k_G = 0 \, ,</math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
that is, where &hellip; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>~\tilde\xi \tilde\theta^{n} (-\tilde\theta^')^{-1} = (n-3)/(n-1) \, .</math>
</div>

As is illustrated by the plots presented in [[SSC/Structure/PolytropesEmbedded#Stahler1983Fig17|Figure 2, above]], turning points also arise in the mass-radius relationship of bounded polytropic configurations having <math>~n > 3</math>.  These are identified by Kimura as "p<sub>1</sub> sequences" because the external pressure is held fixed while the system's mass and corresponding equilibrium radius is varied.  In &sect;5 of his [http://adsabs.harvard.edu/abs/1981PASJ...33..299K "Paper II,"] Kimura points out that the same two conditions &#8212; namely, <math>~h_G = 0</math> and <math>~k_G = 0</math> &#8212; also identify the location of extrema in M<sub>1</sub> along, respectively, p<sub>1</sub> sequences and r<sub>1</sub> sequences.


We can also identify extrema in r<sub>1</sub> along p<sub>1</sub> sequences by setting <math>~(\dot{p}_1/p_1) = 0</math> in [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura's] equation (17), then substituting the resulting expression for the function <math>~Z</math>, namely,
<div align="center">
<math>~Z = v_1 \, ,</math>
</div>
into his equations (15) and (16).  The ratio of these two resulting expressions gives,
<div align="center">
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>~\frac{d\ln M_1}{d \ln r_1}\biggr|_{p_1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{u_1 -(u_G/v_G)v_1}{1 - v_1/v_G}
=
[u_1 v_G - u_G v_1][v_G - v_1]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{2(n+1)}{(n-1)} \cdot \frac{\xi \theta^n}{(-\theta^')}  - \frac{(n-3)}{(n-1)} \cdot \frac{(n+1)\xi (-\theta^')}{\theta} \biggr] 
\biggl[\frac{2(n+1)}{(n-1)} - \frac{(n+1)\xi (-\theta^')}{\theta}  \biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\xi }{(-\theta^')} \biggl[ \frac{2 \theta^{n+1}  - (n-3) (-\theta^')^2 }{2\theta - (n-1)\xi (-\theta^')  } \biggr]
</math>
  </td>
</tr>
</table>
</div>

<span id="TurningPointXmax">As has just been reviewed, the condition <math>~h_G=0</math> results from setting the numerator of this expression equal to zero and identifies extrema in M<sub>1</sub> along p<sub>1</sub> sequences.  In addition, now, we can identify the condition for extrema in r<sub>1</sub> along p<sub>1</sub> sequences by setting the denominator to zero.</span>  The condition is,
<div align="center">
<math>~\frac{\xi (-\theta^')}{\theta} = \frac{2}{(n-1)} \, .</math>
</div>