Editing SSC/Structure/BiPolytropes/Analytic15 (section)

===Comment on Murphy's Scalings===

The {{ Murphy83a }} derivations also include an homology factor, <math>~A</math>, and an overall scaling factor, <math>~B</math>, but they are calculated differently from our <math>~A_0</math> and <math>~B_0</math>.  In the righthand column of the third page of his paper, Murphy states that,
<div align="center">
<math>~A = \frac{\xi_J}{\zeta_J} \, ,</math>
</div>
which, when translated into our notation <math>~(\zeta_J \rightarrow \xi_i </math> and <math>~\xi_J \rightarrow A_0\eta_\mathrm{root})</math> gives,
<div align="center">
<math>~A = \frac{A_0 \eta_\mathrm{root}}{\xi_i} \, .</math>
</div>
Now, in our derivation, <math>~\eta_\mathrm{root}</math> is synonymous with the location of the envelope interface, <math>~\eta_i</math>, as expressed in terms of the dimensionless radial coordinate associated with Srivastava's Lane-Emden function, so we can equally well state that, 
<div align="center">
<math>~A = \frac{A_0 \eta_i}{\xi_i} \, .</math>
</div>
Recalling that <math>~\phi_i = 1</math>, we know from the [[SSC/Structure/BiPolytropes/Analytic15#Step_5:_Interface_Conditions|interface conditions detailed above]] that,
<div align="center">
<math>~\frac{\eta_i}{\xi_i} = \frac{1}{3^{1/2}} \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .</math>
</div>
Hence Murphy's homology factor, <math>~A</math>, is related to our homology factor, <math>~A_0</math> via the expression,
<div align="center">
<math>~A =  \frac{A_0}{3^{1/2}} \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .</math>
</div>
It is usually the value of this quantity, rather than simply our derived value of <math>~A_0</math>, that is tabulated below &#8212; both [[SSC/Structure/BiPolytropes/Analytic15#Murphy.27s_Example_Model_Characteristics|here]] and [[SSC/Structure/BiPolytropes/Analytic15#Murphy_and_Fiedler_.281985.29|here]] &#8212; as we make quantitative comparisons between the characteristics of our derived models and those published by {{ Murphy83a }} and by {{ MF85a }}.


In the lefthand column of the fourth page of his paper, {{ Murphy83a }} defines the coefficient <math>B</math> in such a way that the ''value'' of the envelope function, <math>\phi_{5F}</math>, equals the ''value'' of the core function, <math>\theta_{1E}</math>, at the interface.  Specifically, he sets,

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

<tr>
  <td align="right">
<math>~B</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{\zeta_J}{\sin\zeta_J} \biggr] \biggl[  \frac{A^{1/2} \sin(\ln\sqrt{A\zeta_J}) }{(A\zeta_J)^{1/2} \{2 + \cos[\ln(A\zeta_J)]\}^{1/2}}\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{\zeta_J}{\sin\zeta_J} \biggr] \biggl[  \frac{\sin(\ln\sqrt{A\zeta_J}) }{\zeta_J^{1/2} \{3 - 2\sin^2(\ln\sqrt{A\zeta_J}) \}^{1/2}}\biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Switching to our terminology, that is, setting, 
<div align="center">
<math>\ln\sqrt{A\zeta_J} \rightarrow \Delta_i</math>&nbsp; &nbsp;&nbsp; and, as before, &nbsp; &nbsp;&nbsp; 
<math>~\zeta_J \rightarrow \xi_i \, ,</math>
</div>
gives,

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

<tr>
  <td align="right">
<math>~B</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{\xi_i}{\sin\xi_i} \biggr] \biggl[  \frac{\sin\Delta_i }{\xi_i^{1/2}  (3 - 2\sin^2\Delta_i )^{1/2}}\biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\theta_i^{-1} \biggl[  \xi_i^{-1}  \biggl(\frac{3}{\sin^2\Delta_i} - 2\biggr)^{-1} \biggr]^{1/2} \, .</math>
  </td>
</tr>
</table>
</div>
Hence, in terms of the definition of ''our'' scaling coefficient, <math>~B_0</math>, derived above, we have,

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

<tr>
  <td align="right">
<math>~B</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{B_0}{3^{1/4}} \biggl(  \frac{\mu_e}{\mu_c} \biggr)^{1/2} \theta_i^{-1} \, .</math>
  </td>
</tr>
</table>
</div>

As we make quantitative comparisons between the characteristics of our derived models and those published by [http://adsabs.harvard.edu/abs/1983PASAu...5..175M Murphy (1983)] and by  [http://adsabs.harvard.edu/abs/1985PASAu...6..219M Murphy &amp; Fiedler (1985a)], below, we usually will tabulate the value of this quantity, 
rather than simply our derived value of <math>~B_0</math>.