Editing SSC/Stability/BiPolytrope00 (section)

===Piecing Together===
Here we illustrate how the two segments of the eigenfunction can be successfully pieced together for the specific case of <math>(\ell,j) = (2,1)</math>.

<span id="STEP1"><font color="red"><b>STEP 1:</b></font></span>  Choose a value of the adiabatic exponent for the envelope, <math>\gamma_e</math>.  Then, the values of both <math>\alpha_e</math> and <math>c_0</math> are known as well; actually, because it is the root of a quadratic equation, <math>c_0</math> can, in general, take on one of a ''pair'' of values.  We will elaborate on this further, below.

<span id="STEP2"><font color="red"><b>STEP 2:</b></font></span>  Acknowledging that the value of <math>q</math> has yet to be determined, fix the value of the leading, overall scaling coefficient, <math>b_0</math>, such that<sup>&dagger;</sup> <math>x_\mathrm{env} = 1</math> at the interface, that is, at <math>\xi = 1</math>.  For the case of <math>\ell=2</math>, this means that, throughout the envelope, the eigenfunction is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>x_{\ell=2} |_\mathrm{env}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\xi^{c_0}\biggl[ \frac{ 1 +  q^3 A_{2} \xi^{3} +  q^6 A_{2}B_{2}\xi^{6} }{ 1 +  q^3 A_{2}  +  q^6 A_{2}B_{2}}\biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
where, the values of the newly introduced coefficients,
<div align="center">
<table border="0" cellpadding="3" align="center">
<tr>
  <td align="right">
<math>A_{2}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{c_0(c_0+5) - (c_0 + 6)(c_0 + 11)}{(c_0 + 3)(c_0+5) - \alpha_e}\biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>B_{2}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{(c_0+3)(c_0+8) - (c_0 + 6)(c_0 + 11)}{(c_0 + 6)(c_0+8) - \alpha_e}\biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
are also both known.

<span id="STEP3"><font color="red"><b>STEP 3:</b></font></span>  Recognizing that this segment of the eigenfunction will only satisfy the envelope's LAWE if we restrict our discussion to equilibrium models for which <math>g^2 = \mathcal{B} = [(1+8q^3)/(1+2q^3)^{2}]</math>, we must insert this same restriction on <math>g^2</math> into the core's eigenfunction.  At the same time, we should fix the value of the leading, overall scaling coefficient, <math>a_0</math>, such that<sup>&dagger;</sup> <math>x_\mathrm{core} = 1</math> at the interface <math>(\xi = 1)</math>.  For the case of <math>j=1</math>, this means that, throughout the core, the eigenfunction is,

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

<tr>
  <td align="right">
<math>x_{j=1} |_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{5(1+8q^3) -  7 (1+2q^3)^2 \xi^2}{5(1+8q^3)-7(1+2q^3)^2} \, .</math>
  </td>
</tr>
</table>
</div>


<span id="STEP4"><font color="red"><b>STEP 4:</b></font></span>  Now we need to match the two eigenfunctions at the interface.  Following the discussion in &sect;&sect;57 &amp; 58 of {{ LW58full }}, the proper treatment is to ensure that fractional perturbation in the gas pressure (see their equation 57.31),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\delta P}{P}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \gamma x \biggl( 3 + \frac{d\ln x}{d\ln \xi} \biggr) \, ,</math>
  </td>
</tr>
</table>
</div>

is continuous across the interface.  That is to say, at the interface <math>(\xi = 1)</math>, we need to enforce the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \gamma_c x_\mathrm{core} \biggl( 3 + \frac{d\ln x_\mathrm{core}}{d\ln \xi} \biggr) - \gamma_e x_\mathrm{env} \biggl( 3 + \frac{d\ln x_\mathrm{env}}{d\ln \xi} \biggr)\biggr]_{\xi=1}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\gamma_e \biggl[ \frac{\gamma_c}{\gamma_e} \biggl( 3 + \frac{d\ln x_\mathrm{core}}{d\ln \xi} \biggr) - \biggl( 3 + \frac{d\ln x_\mathrm{env}}{d\ln \xi} \biggr)\biggr]_{\xi=1}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~ \frac{d\ln x_\mathrm{env}}{d\ln \xi} \biggr|_{\xi=1}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>3\biggl(\frac{\gamma_c}{\gamma_e}  -1\biggr) + \frac{\gamma_c}{\gamma_e} \biggl( \frac{d\ln x_\mathrm{core}}{d\ln \xi} \biggr)_{\xi=1} \, .</math>
  </td>
</tr>
</table>
</div>

In the context of this interface-matching constraint, {{ LW58 }} state the following (see their equation 62.1): &nbsp; <font color="darkgreen">"In the static</font> (''i.e.,'' unperturbed equilibrium) <font color="darkgreen">model</font> &hellip; <font color="darkgreen">discontinuities in <math>\rho</math> or in <math>\gamma</math> might occur at some [radius]</font>.  <font color="darkgreen">In the first case</font> &#8212; that is, a discontinuity only in density, while <math>\gamma_e = \gamma_c</math> &#8212; the interface conditions <font color="darkgreen">imply the continuity of <math>\tfrac{1}{x} \cdot \tfrac{dx}{d\xi}</math> at that [radius].  In the second case</font> &#8212;  that is, a discontinuity in the adiabatic exponent &#8212; <font color="darkgreen">the dynamical condition may be written</font> as above.  <font color="darkgreen">This implies a discontinuity of the first derivative at any discontinuity of <math>\gamma</math>."</font>

When evaluated at the interface, the logarithmic derivatives of our pair of parameterized eigenfunction expressions are, respectively,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{d\ln x_\mathrm{env}}{d\ln \xi} \biggr|_{\xi=1}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
c_0 + \frac{3A_{2}\Chi +  6A_{2}B_{2} \Chi^2}{1 + A_{2}\Chi + A_{2}B_{2}\Chi^2} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{d\ln x_\mathrm{core}}{d\ln \xi} \biggr|_{\xi=1}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{14(1+2\Chi)^2}{7(1+2\Chi)^2 - 5(1+8\Chi)} \, ,</math>
  </td>
</tr>
</table>
</div>
where we have made the notation substitution, <math>\Chi \equiv q^3</math>.  Allowing for a step function in the adiabatic exponent at the interface, our interface-matching constraint is, therefore,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\frac{\gamma_c}{\gamma_e} \biggl[ \frac{14(1+2\Chi)^2}{7(1+2\Chi)^2 - 5(1+8\Chi)} \biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
c_0 + \frac{3A_{2}\Chi +  6A_{2}B_{2} \Chi^2}{1 + A_{2}\Chi + A_{2}B_{2}\Chi^2}
- 3\biggl(\frac{\gamma_c}{\gamma_e} -1\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\mathfrak{g}_0 + (\mathfrak{g}_0+3)A_{2}\Chi +  (\mathfrak{g}_0+6)A_{2}B_{2} \Chi^2}{1 + A_{2}\Chi + A_{2}B_{2}\Chi^2}
\, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>\mathfrak{g}_0 \equiv c_0 + 3\biggl(1-\frac{\gamma_c}{\gamma_e} \biggr) \, .</math>
</div>

This can be rewritten as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
0
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
[\mathfrak{g}_0 + (\mathfrak{g}_0 + 3)A_{2}\Chi + (\mathfrak{g}_0 + 6)A_{2}B_{2} \Chi^2] [7(1+2\Chi)^2 - 5(1+8\Chi)]
- 14(\gamma_c/\gamma_e) (1+2\Chi)^2 [1 + A_{2}\Chi + A_{2}B_{2}\Chi^2] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
[\mathfrak{g}_0 + (\mathfrak{g}_0 + 3)A_{2}\Chi + (\mathfrak{g}_0 + 6)A_{2}B_{2} \Chi^2] [2 - 12\Chi + 28\Chi^2 ]
- (14 + 56\Chi + 56 \Chi^2)(\gamma_c/\gamma_e) [1 + A_{2}\Chi + A_{2}B_{2}\Chi^2] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2[\mathfrak{g}_0 + (\mathfrak{g}_0 + 3)A_{2}\Chi + (\mathfrak{g}_0 + 6)A_{2}B_{2} \Chi^2] 
-12\Chi [\mathfrak{g}_0 + (\mathfrak{g}_0 + 3)A_{2}\Chi + (\mathfrak{g}_0 + 6)A_{2}B_{2} \Chi^2] 
+ 28\Chi^2 [\mathfrak{g}_0 + (\mathfrak{g}_0 + 3)A_{2}\Chi + (\mathfrak{g}_0 + 6)A_{2}B_{2} \Chi^2] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
- 14(\gamma_c/\gamma_e) [1 + A_{2}\Chi + A_{2}B_{2}\Chi^2] 
- 56(\gamma_c/\gamma_e)\Chi  [1 + A_{2}\Chi + A_{2}B_{2}\Chi^2] 
- 56 (\gamma_c/\gamma_e)\Chi^2 [1 + A_{2}\Chi + A_{2}B_{2}\Chi^2] \, .
</math>
  </td>
</tr>
</table>
</div>

Or we have, equivalently,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>a\Chi^4 + b\Chi^3 + c\Chi^2 +d\Chi +e </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0 \, ,</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>e</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>2\mathfrak{g}_0 - 14(\gamma_c/\gamma_e)  \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>d</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>2(\mathfrak{g}_0+3)A_{2} - 12\mathfrak{g}_0 - 14(\gamma_c/\gamma_e)A_{2} - 56(\gamma_c/\gamma_e)</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2[\mathfrak{g}_0 + 3 -7(\gamma_c/\gamma_e)]A_{2} - 4[14(\gamma_c/\gamma_e) + 3\mathfrak{g}_0] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>c</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>2(\mathfrak{g}_0+6)A_{2}B_{2} - 12(\mathfrak{g}_0+3)A_{2} + 28\mathfrak{g}_0 - 14(\gamma_c/\gamma_e)A_{2}B_{2} - 56(\gamma_c/\gamma_e)A_{2} - 56(\gamma_c/\gamma_e)  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2[\mathfrak{g}_0 + 6 -7(\gamma_c/\gamma_e)] A_{2}B_{2} - 4[9 + 14(\gamma_c/\gamma_e) + 3\mathfrak{g}_0]A_{2} + 28[\mathfrak{g}_0 - 2(\gamma_c/\gamma_e)]  \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>b</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>-12(\mathfrak{g}_0+6)A_{2}B_{2} + 28(\mathfrak{g}_0+3)A_{2} - 56(\gamma_c/\gamma_e)A_{2}B_{2} - 56(\gamma_c/\gamma_e)A_{2} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- 4[3\mathfrak{g}_0+18+14(\gamma_c/\gamma_e)]A_{2}B_{2} + 28[\mathfrak{g}_0+3-2(\gamma_c/\gamma_e)]A_{2} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>a</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>28[\mathfrak{g}_0+6 - 2(\gamma_c/\gamma_e)]A_{2}B_{2} \, .</math>
  </td>
</tr>
</table>


The physically relevant (real) root of this quartic equation in <math>\Chi</math> &#8212; see our [[SSC/Stability/BiPolytrope00Details#Quartic|accompanying detailed presentation]] &#8212; gives us the ''specific'' value of the dimensionless interface location, <math>q</math>, for which the values of the two eigenfunctions match at the interface, and for which the first derivatives of the two eigenfunctions are discontinuous by the properly prescribed amount at the interface.