Editing SSC/Stability/BiPolytrope00Details (section)

==Try Mode31 (&ell;, j) = (3, 1)==
===Setup31===

In this case we need to replace the envelope eigenfunction segment that was specified in <font color="red"><b>STEP 2</b></font> in our [[SSC/Stability/BiPolytrope00#STEP2|separate discussion]] of mode  <math>(\ell,j) = (2,1)</math> with the following:
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=3} |_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\xi^{c_0}\biggl[ \frac{ 1 +  q^3 A_{31} \xi^{3} +  q^6 A_{31}B_{31}\xi^{6} +  q^9 A_{31}B_{31}C_{31}\xi^{9} }{ 1 +  q^3 A_{31}  +  q^6 A_{31}B_{31} +  q^9 A_{31}B_{31}C_{31} }\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_{31}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{c_0(c_0+5) - (c_0 + 9)(c_0 + 14)}{(c_0 + 3)(c_0+5) - \alpha_e}\biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{c_0^2 + 5c_0 - (c_0^2 + 23c_0 + 126)}{(c_0^2 + 8c_0 + 15) - (c_0^2 + 2c_0)}\biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{- 6(c_0 + 7)}{2c_0 + 5 } \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~B_{31}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{(c_0+3)(c_0+8) - (c_0 + 9)(c_0 + 14)}{(c_0 + 6)(c_0+8) - \alpha_e}\biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{(c_0^2 + 11c_0 + 24) - (c_0^2 + 23c_0 + 126)}{(c_0^2 + 14c_0 +48) - (c_0^2 + 2c_0) }\biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{ - (6c_0 + 51)}{ 6(c_0 +4) } \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~C_{31}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{(c_0+6)(c_0+11) - (c_0 + 9)(c_0 + 14)}{(c_0 + 9)(c_0+11) - \alpha_e}\biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{(c_0^2 +17c_0 + 66) - (c_0^2 + 23 c_0 + 126)}{(c_0^2 + 20 c_0 + 99) -  (c_0^2 + 2c_0) }\biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{- 2( c_0 + 10)}{3(2 c_0 + 11)  } \, .</math>
  </td>
</tr>
</table>
</div>

Then, after defining,
<div align="center">
<table border="0" cellpadding="3" align="center">
<tr>
  <td align="right">
<math>~Q_{31}</math>
  </td>
  <td align="center">
<math>~\equiv</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>

the matching relation becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~Q_{31}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{c_0 + (c_0 + 3)A_{31}\Chi + (c_0 + 6)A_{31}B_{31} \Chi^2 + (c_0 + 9)A_{31}B_{31} C_{31}\Chi^3 }{1 + A_{31}\Chi + A_{31}B_{31}\Chi^2 + A_{31}B_{31}C_{31}\Chi^3 } 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ 0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
[c_0 + (c_0 + 3)A_{31}\Chi + (c_0 + 6)A_{31}B_{31} \Chi^2 + (c_0 + 9)A_{31}B_{31} C_{31}\Chi^3 ] - Q_{31}[1 + A_{31}\Chi + A_{31}B_{31}\Chi^2 + A_{31}B_{31}C_{31}\Chi^3 ]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[c_0 - (c_0 + 3)\cdot \frac{6(c_0 + 7)}{2c_0 + 5 }  \cdot \Chi 
+ (c_0 + 6) \cdot \frac{ 6(c_0 + 7)}{2c_0 + 5 }  \cdot \frac{  (6c_0 + 51)}{ 6(c_0 +4) } \cdot \Chi^2 
- (c_0 + 9)\cdot \frac{ 6(c_0 + 7)}{2c_0 + 5 }  \cdot \frac{  (6c_0 + 51)}{ 6(c_0 +4) } \cdot \frac{ 2( c_0 + 10)}{3( 2c_0 + 11)  } \cdot\Chi^3 \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- Q_{31} \biggl[1 - \frac{ 6(c_0 + 7)}{2c_0 + 5 }  \cdot \Chi + \frac{ 6(c_0 + 7)}{2c_0 + 5 }  \cdot \frac{  (6c_0 + 51)}{ 6(c_0 +4) } \cdot \Chi^2 
- \frac{ 6(c_0 + 7)}{2c_0 + 5 } \cdot \frac{  (6c_0 + 51)}{ 6(c_0 +4) } \cdot \frac{ 2( c_0 + 10)}{3( 2c_0 + 11)  } \cdot \Chi^3 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[3c_0(2c_0 + 5)(c_0 +4)( 2c_0 + 11) - (c_0 + 3)(c_0 +4)3( 2c_0 + 11)\cdot 6(c_0 + 7) \cdot \Chi 
+ (c_0 + 6) 3( 2c_0 + 11)\cdot  (c_0 + 7) \cdot  (6c_0 + 51) \cdot \Chi^2 
- (c_0 + 9)\cdot  (c_0 + 7) \cdot  (6c_0 + 51)\cdot 2( c_0 + 10) \cdot\Chi^3 \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- Q_{31} \biggl[ 3(2c_0 + 5)(c_0 +4)( 2c_0 + 11) 
- 18(c_0 + 7)(c_0 +4)( 2c_0 + 11)   \cdot \Chi 
+ 3( 2c_0 + 11)(c_0 + 7)  \cdot (6c_0 + 51) \cdot \Chi^2 
- 2(c_0 + 7) \cdot (6c_0 + 51) ( c_0 + 10) \cdot \Chi^3 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[(6c_0^2 + 15c_0) (2c_0^2 + 19c_0 + 44) - 18(c_0^2 + 10c_0 + 21) (2c_0^2 + 19c_0 + 44) \cdot \Chi 
+ 3 (2c_0^2 + 23c_0 +66) (6c_0^2 + 93c_0 + 357) \cdot \Chi^2 
- 2(c_0^2 + 19c_0 + 90)  (6c_0^2 + 93c_0 + 357) \cdot\Chi^3 \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- Q_{31} \biggl[ 3(2c_0 + 5) (2c_0^2 + 19c_0 + 44) 
- 18(c_0 + 7) (2c_0^2 + 19c_0 + 44)   \cdot \Chi 
+ 3( 2c_0 + 11) (6c_0^2 + 93c_0 + 357)  \cdot \Chi^2 
- 2 (6c_0^2 + 93c_0 + 357)  ( c_0 + 10) \cdot \Chi^3 \biggr]
</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~\Rightarrow~~~ 0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[(12c_0^4 + 144c_0^3 + 549c_0^2 + 660c_0) - 18(2c_0^4 + 39c_0^3 + 276c_0^2 + 839c_0 + 924) \cdot \Chi 
+ 3 (12c_0^4 + 324c_0^3 + 3249c_0^2 + 14349c_0 + 23562) \cdot \Chi^2 
- 2(6c_0^4 + 207c_0^3 + 2664c_0^2 + 15153c_0 + 32130) \cdot\Chi^3 \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- Q_{31} \biggl[ 3(4c_0^3 + 48c_0^2 + 183c_0 +220) 
- 18(2c_0^3 + 33c_0^2 + 177c_0 + 308)   \cdot \Chi 
+ 3( 12c_0^3 + 252c_0^2 + 1737c_0 + 3927)  \cdot \Chi^2 
- 2 (6c_0^3 + 153c_0^2 + 1287c_0 + 3570) \cdot \Chi^3 \biggr]
</math>
  </td>
</tr>

</table>
</div>

===Root of Quartic Equation===

To solve this equation analytically, we follow the [http://en.wikipedia.org/wiki/Quartic_function#Summary_of_Ferrari.27s_method Summary of Ferrari's method] that is presented in Wikipedia's discussion of the Quartic Function to identify the roots of an arbitrary quartic equation.  

<div align="center" id="Quartic">
<table border="1" cellpadding="8" width="85%">
<tr><td align="left">

First, we adopt the shorthand notation:
<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>~ac_0^4 + bc_0^3 + c c_0^2 +d c_0 +e \, ,</math>
  </td>
</tr>
</table>
where, in our particular case,

<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>~ - 18\cdot 924\Chi +3\cdot 23562\Chi^2 -2\cdot 32130\Chi^3    -Q_{31}[ 3\cdot 220 - 18\cdot 308\Chi +3\cdot 3927\Chi^2 -2\cdot 3570\Chi^3  ]   \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~d</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~660 -18\cdot 839\Chi +3\cdot 14349\Chi^2 -2\cdot 15153\Chi^3   -Q_{31}[3\cdot 183 -18\cdot 177\Chi + 3\cdot 1737\Chi^2 -2\cdot 1287\Chi^3   ]   \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~c</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~549 - 18\cdot 276\Chi +3\cdot 3249\Chi^2 -2\cdot 2664\Chi^3     -Q_{31}[ 3\cdot 48 - 18\cdot 33\Chi + 3\cdot 252\Chi^2 -2\cdot 153 \Chi^3     ]  \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~b</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~144- 18\cdot 39\Chi  +3\cdot 324\Chi^2 -2\cdot 207\Chi^3    - 12 Q_{31}[ 1 - 3\Chi +3\Chi^2 - \Chi^3    ]\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~a</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~12( 1 - 3 \Chi +3\Chi^2 -\Chi^3)    \, .</math>
  </td>
</tr>
</table>

</td></tr>
</table>
</div>
<!-- OMITTED GAMMA 4/3 PARAGRAPH ...

It is prudent to check that this set of coefficients satisfies the quartic expression ''at least'' in the case of the [[#If_the_Envelope_Follows_a_4.2F3_Adiabat|two examples given above]] when <math>~\gamma_e = \tfrac{4}{3}</math>.  When [[#Case_of_c0_.28plus.29|c<sub>0</sub> (plus)]] is examined &#8212; that is, for <math>~c_0 = 0</math> &#8212; we should find that the coefficient, <math>~e</math>, by itself should be zero.  Indeed, it is zero to many significant digits when the empirically derived value of <math>~\Chi = 0.276837296 </math> is plugged into the expression for <math>~e</math>.  Similarly, the quartic expression is satisfied with <math>~c_0 = -2</math> if, as [[#Case_of_c0_.28minus.29|was derived empirically above when c<sub>0</sub> (minus) was examined]], if the value of <math>~\Chi = 0.063819021 </math> is used to determine the five separate coefficient values: 
<table border="0" align="center"><tr><td align="center"><math>~(a,b,c,d,e) = (+11.48754, +73.56560, - 262.9735, -2253.197, - 3049.777)\, .</math></td></tr></table>
END OMITTED PARAGRAPH -->

<div align="center" id="Quartic2">
<table border="1" cellpadding="8" width="85%">
<tr><td align="left">

Now, define,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Delta_0</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ c^2 - 3bd + 12ae  \, ,</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\Delta_1</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ 2c^3 - 9bcd + 27b^2e + 27ad^2 - 72ace  \, ,</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~p</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \frac{8ac - 3b^2}{8a^2}  \, ,</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\kappa</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \frac{b^3 - 4abc + 8a^2 d}{8a^3}  \, ,</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\Kappa</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2^{1 / 3}} \biggl[\Delta_1 + \sqrt{\Delta_1^2 - 4\Delta_0^3}  \biggr]^{1 / 3} \, ,</math>
  </td>
</tr>
<tr>
  <td align="right">
(see [[#Complex|below]]) &nbsp;&nbsp;<math>~\biggl(\Kappa + \frac{\Delta_0}{\Kappa}\biggr)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~2\Delta_0^{1/2}\cos\biggl[ \frac{1}{3} \cos^{-1}\biggl( \frac{\Delta_1^2}{4\Delta_0^3}\biggr)^{1/2}\biggr] \, ,</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~S</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2} \biggl[ - \frac{2p}{3} + \frac{1}{3a}\biggl(\Kappa + \frac{\Delta_0}{\Kappa} \biggr)  \biggr]^{1 / 2} \, .</math>
  </td>
</tr>
</table>
Then the four roots of the quartic equation are,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(c_0)_{1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\frac{b}{4a} - S + \frac{1}{2}\biggl[ -4S^2 - 2p + \frac{\kappa}{S} \biggr]^{1 / 2}  \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~(c_0)_{2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\frac{b}{4a} - S - \frac{1}{2}\biggl[ -4S^2 - 2p + \frac{\kappa}{S} \biggr]^{1 / 2}  \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~(c_0)_{3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\frac{b}{4a} + S + \frac{1}{2}\biggl[ -4S^2 - 2p - \frac{\kappa}{S} \biggr]^{1 / 2}  \, .</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~(c_0)_{4}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\frac{b}{4a} + S - \frac{1}{2}\biggl[ -4S^2 - 2p - \frac{\kappa}{S} \biggr]^{1 / 2}  \, .</math>
  </td>
</tr>
</table>
It is this ''fourth'' root that interests us, here.

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


<div align="center" id="Complex">
<table border="1" cellpadding="8" width="85%">
<tr><td align="left">

We have determined empirically that, in our specific case, the quantity, 
<table border="0" align="center"><tr><td align="center"><math>~\Delta_1^2 - 4\Delta_0^3</math></td></tr></table>
is negative over the range of physically interesting values of <math>~\Chi</math>.  Hence, the quantity, <math>~\Kappa^3</math>, is necessarily complex.  Let's work carefully through a determination of <math>~\Kappa</math> and, by consequence, <math>~S</math>, in this situation.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~2\Kappa^3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Delta_1 + \sqrt{\Delta_1^2 - 4\Delta_0^3}   </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Delta_1 + i \sqrt{4\Delta_0^3 - \Delta_1^2}   </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Delta_1 + 2\Delta_0^{3/2} ~ i \biggl[ 1 - \Gamma^2 \biggr]^{1/2}   </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \biggl[ \frac{\Kappa}{\Delta_0^{1/2}} \biggr]^3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Gamma +  i \sqrt{ 1 - \Gamma^2 } \, ,   </math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>\Gamma \equiv \biggl[ \frac{\Delta_1^2}{4\Delta_0^3}\biggr]^{1/2} \, .</math>
</div>
We therefore can state that, in the complex plane, the three roots <math>~(j=0,1,3)</math> of this expression are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \frac{\Kappa}{\Delta_0^{1/2}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~e^{i\theta_\Kappa/3} \cdot e^{i(2\pi j/3)} \, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>\theta_\Kappa \equiv  \cos^{-1}\Gamma \, .</math>
</div>
Focusing on the simplest <math>~(j=0)</math> root, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \Kappa </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Delta_0^{1/2} e^{i\theta_\Kappa/3}  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ \Rightarrow ~~~ \biggl(\Kappa + \frac{\Delta_0}{\Kappa} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Delta_0^{1/2} e^{i\theta_\Kappa/3}  + \Delta_0^{1/2} e^{- i\theta_\Kappa/3}  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\Delta_0^{1/2}\cos\biggl[ \frac{\cos^{-1}\Gamma}{3} \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Because this expression does not contain an imaginary component, we understand that <math>~S</math> is real.

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



Finally, as explained in our [[SSC/Stability/BiPolytrope00Details#Allow_Different_Adiabatic_Exponents|summary discussion]], in order for the two (dimensional) eigenfrequencies to match, the adiabatic exponent for the core must be,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \gamma_c </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{ 6 + 2j(2j+5)] } \biggl\{ 8 +
\gamma_e \biggl[2\alpha_e + (c_0 + 3\ell)(c_0 + 3\ell +5) \biggr]\frac{\rho_e}{\rho_c} \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{ 20} \biggl\{ 8 +
\gamma_e \biggl[2\alpha_e + (c_0 + 9)(c_0 + 14) \biggr]\frac{\rho_e}{\rho_c} \biggr\} \, ,
</math>
  </td>
</tr>
</table>
</div>
where, the last expression follows from plugging in the desired mode's quantum numbers, <math>~(\ell,j) = (3,1)</math>, and, again, using the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Chi = q^3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\rho_e/\rho_c}{2(1-\rho_e/\rho_c)}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{\rho_e}{\rho_c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2\Chi}{1+2\Chi} \, .</math>
  </td>
</tr>
</table>
</div>

===Illustration31===

<div align="center">
<table border="1" align="center" cellpadding="8">
<tr>
  <th align="center" colspan="2">
Analytically Definable Eigenvectors in <math>~(n_c, n_e) = (0,0)</math> Bipolytropes<br />

<font color="red">Quantum Numbers:</font>&nbsp; <math>~(\ell,j) = (3,1)</math>
  </th>
</tr>
<tr>
  <th align="center" width="50%"> Analyzable Model Sequence</th>
  <th align="center" width="50%">One Example Eigenfunction</th>
</tr>
<tr>
  <td align="center" colspan="2">
[[File:Model31MontageCorrectedTwice.png|600px|Montage of Stability Results for (ell,j) = (3,1) quantum numbers]]
  </td>
</tr>
</table>
</div>


<font color="darkblue"><b>''Top-Left Panel:''</b></font>&nbsp; Plotted points show how the location of the core/envelope interface, <math>~q \equiv r_i/R</math>, varies with <math>~\alpha_e \equiv (3-4/\gamma_e)</math> &#8212; where <math>~\gamma_e</math> is the adiabatic exponent of the envelope &#8212; in equilibrium models that are amenable to analytic modal analysis for quantum numbers, <math>~(\ell,j) = (3,1)</math>.  Red (alternatively, blue) markers identify models for which the corresponding value of the adiabatic exponent of the ''core'' [see bottom-left panel] falls inside (alternatively, outside) the physically viable range, namely, <math>~1 \le \gamma_c \le \infty</math>.  The yellow circular marker identifies the model whose analytically determined eigenfunction is displayed on the right, as an example.

<font color="darkblue"><b>''Bottom-Left Panel:''</b></font> &nbsp; Plotted points show how <math>~\alpha_c \equiv (3-4/\gamma_c)</math> varies with <math>~\alpha_e</math> over the physically viable parameter range, <math>~-1 \le \alpha \le 3</math>.  Both axes have been flipped so that incompressible models <math>~(\gamma = \infty)</math> lie on the left/bottom while isothermal models <math>~(\gamma =1)</math> lie on the right/top.  The core is ''more'' compressible than the envelope in models that lie above and to the left of the black-dashed, diagonal line.  The yellow circular marker identifies the same example model as it does in the top-left panel.

<font color="darkblue"><b>''Top-Right Panel:''</b></font>&nbsp; Displays &#8212; as a function of the fractional radius, <math>~r_0/R = q\xi</math> &#8212;  the analytically determined eigenfunction for the <math>~(\ell,j) = (3,1)</math> mode in the model identified by the yellow circular marker in both left-hand panels, for which,
<div align="center">
<math>~q = \biggl[0.01 + 40\biggl( \frac{0.98}{99} \biggr) \biggr] \approx 0.4059596</math> 
</div>
and, correspondingly, <math>~(c_0, \alpha_e,\alpha_c) = (-1.7819827, -0.3885031, -0.9647648)</math>.  Specifically, over the radial interval, <math>~0 \le \xi \le 1</math>, the green markers trace the core's contribution to the combined eigenfunction, namely,

<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>
and, over the radial interval, <math>~1 \le \xi \le 1/q</math>, the purple markers trace the envelope's contribution to the combined eigenfunction, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_{\ell=3} |_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\xi^{c_0}\biggl[ \frac{ 1 +  q^3 A_{31} \xi^{3} +  q^6 A_{31}B_{31}\xi^{6} +  q^9 A_{31}B_{31}C_{31}\xi^{9} }{ 1 +  q^3 A_{31}  +  q^6 A_{31}B_{31} +  q^9 A_{31}B_{31}C_{31} }\biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
where the coefficients, <math>~A_{31}, B_{31}, C_{31}</math>, are as [[#Setup31|defined above]] in terms of the parameter, <math>~c_0</math>.  The corresponding eigenfrequency is, from the perspective of the core,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{3\omega^2}{2\pi G \rho_c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~20\gamma_c - 8 = \frac{8(7 + \alpha_c)}{3-\alpha_c} \approx 12.17774\, ;</math>
  </td>
</tr>
</table>
</div>
and, from the perspective of the envelope,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{3\omega^2}{2\pi G \rho_c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_e\biggl[2\alpha_e + (c_0 + 9)(c_0+14)\biggr] \frac{\rho_e}{\rho_c} \approx 12.17774\, ,</math>
  </td>
</tr>
</table>
</div>
where the relevant density ratio is, <math>~\rho_e/\rho_c = 2q^3/(1+2q^3) \approx 0.118016</math>.

<font color="darkblue"><b>''Bottom-Right Panel:''</b></font>&nbsp; The green and purple markers present the same eigenfunction-amplitude information, <math>~x(r/R)</math>, as in the Top-Right panel, but on a logarithmic scale.  Specifically, in this plot, the vertical displacement of the green and purple markers is given by the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tfrac{1}{8} \log_{10}[x^2 + \epsilon^2] + y_\mathrm{shift} \, ,</math>
  </td>
</tr>
</table>
</div>
where, for plotting purposes, we have used, <math>~\epsilon = 10^{-5}</math>, and have set <math>~y_\mathrm{shift}</math> to a value that ensures that <math>~y \approx 1</math> at the outer edge.  In this type of log-amplitude plot, the eigenfunction's various ''nodes'' &#8212; that is, radial locations where <math>~x</math> passes through zero &#8212; are highlighted; here, specifically, there is one node inside the core and two nodes reside in the envelope.  Using the vertical coordinate to represent, instead, the configuration's mass-density normalized to its central value, <math>~\rho/\rho_c</math>, the solid black line segments trace the unperturbed density distribution throughout this specific <math>~(n_c, n_e) = (0,0)</math> bipolytrope.  Throughout the core, <math>~\rho/\rho_c = 1</math>; then, at the location of the interface <math>~(r_i/R = q \approx 0.41)</math>, the density abruptly drops to its envelope value <math>~(\rho/\rho_c = \rho_e/\rho_c \approx 0.12)</math>.

NOTE:  As may be ascertained from our [[SSC/Structure/BiPolytropes/Analytic00#BiPolytrope_with_nc_.3D_0_and_ne_.3D_0|general discussion of the structural properties of <math>~(n_c, n_e) = (0,0)</math> bipolytropes]], equilibrium "zero-zero" bipolytropes can be constructed with the envelope/core interface parameter set to any value across the range, <math>~0 \le q \le 1</math>; and for any chosen value of <math>~q</math>, the envelope/core density ratio can, in principle, be set to ''any'' value, <math>~0 \le \rho_e/\rho_c \le 1</math>.  We have not, however, been able to analytically solve the relevant pair of linear-adiabatic wave equations (LAWEs) for this entire set of equilibrium models.  Instead, our ability to derive analytically prescribed eigenvectors is [[SSC/Stability/BiPolytrope00#KeyConstraint|limited by the constraint]],
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\rho_e}{\rho_c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2q^3}{1+2q^3} = \frac{2(r_i/R)^3}{1+2(r_i/R)^3}\, .</math>
  </td>
</tr>
</table>
</div>

The black-dotted curve in the ''Bottom-Right Panel'' displays the behavior of this constraint; accordingly, the step function depicted by the solid black line segments must necessarily drop from unity to a point on this black-dotted curve for any equilibrium model &#8212; such as the example illustrated here &#8212; that has an analytically prescribable radial-oscillation eigenvector.

===Check Surface Boundary Condition===

Let's determine analytically the logarithmic derivative of the envelope segment of the eigenfunction, at the surface <math>~(\xi = 1/q)</math> of the configuration.  Given that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_{\ell=3} |_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\xi^{c_0}\biggl[ \frac{ 1 +  q^3 A_{3} \xi^{3} +  q^6 A_{3}B_{3}\xi^{6} +  q^9 A_{3}B_{3}C_{3}\xi^{9} }{ 1 +  q^3 A_{3}  +  q^6 A_{3}B_{3} +  q^9 A_{3}B_{3}C_{3} }\biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>

where,
<div align="center">
<table border="0" cellpadding="3" align="center">
<tr>
  <td align="right">
<math>~A_{3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{- 6(c_0 + 7)}{(2c_0 + 5 )} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~B_{3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{ - (6c_0 + 51)}{ 6(c_0 +4) } \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~C_{3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{- 2( c_0 + 10)}{3(2 c_0 + 11)  } \, ,</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~\frac{ d\ln x_{\ell=3} }{d\ln \xi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{x} \biggl\{
c_0\xi^{c_0}\biggl[ \frac{ 1 +  q^3 A_{3} \xi^{3} +  q^6 A_{3}B_{3}\xi^{6} +  q^9 A_{3}B_{3}C_{3}\xi^{9} }{ 1 +  q^3 A_{3}  +  q^6 A_{3}B_{3} +  q^9 A_{3}B_{3}C_{3} }\biggr] 
+ \xi^{c_0}\biggl[ \frac{ 3 q^3 A_{3} \xi^{3} +  6q^6 A_{3}B_{3}\xi^{6} + 9 q^9 A_{3}B_{3}C_{3}\xi^{9} }{ 1 +  q^3 A_{3}  +  q^6 A_{3}B_{3} +  q^9 A_{3}B_{3}C_{3} }\biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
c_0
+ \biggl[ \frac{ 3 q^3 A_{3} \xi^{3} +  6q^6 A_{3}B_{3}\xi^{6} + 9 q^9 A_{3}B_{3}C_{3}\xi^{9} }{ 1 +  q^3 A_{3} \xi^{3} +  q^6 A_{3}B_{3}\xi^{6} +  q^9 A_{3}B_{3}C_{3}\xi^{9} }\biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

This means that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{ d\ln x_{\ell=3} }{d\ln \xi} \biggr|_{\xi=1/q}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
c_0
+ \biggl[ \frac{ 3 A_{3}  +  6A_{3}B_{3} + 9 A_{3}B_{3}C_{3} }{ 1 +  A_{3}  +  A_{3}B_{3}+  A_{3}B_{3}C_{3} }\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
c_0
+ \biggl\{ \frac{ -3 [108(c_0 + 7)(c_0 +4)(2 c_0 + 11)]  
+  6[18(c_0 + 7) (6c_0 + 51)(2 c_0 + 11)] - 9 [ 12(c_0 + 7)(6c_0 + 51)( c_0 + 10) ] }{ 18(2c_0+5)(c_0+4)(2c_0+11) -  108(c_0 + 7)(c_0 +4)(2 c_0 + 11)  
+  18(c_0 + 7) (6c_0 + 51)(2 c_0 + 11) -  12(c_0 + 7)(6c_0 + 51)( c_0 + 10) }\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
c_0
+ 108\biggl\{ \frac{ -  3(2c_0^3 + 33c_0^2 + 177c_0 + 308)  
+  (12c_0^3 + 252c_0^2 + 1737c_0 + 3927) - (6c_0^3 + 153c_0^2 + 1287c_0 + 3570)  }{ 18(4c_0^3 + 48c_0^2 + 183c_0 +220) -  108(2c_0^3 + 33c_0^2 + 177c_0 + 308)  
+  18(12c_0^3 + 252c_0^2 + 1737c_0 + 3927) -  12(6c_0^3 + 153c_0^2 + 1287c_0 + 3570) }\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
c_0
+ 108\biggl\{ \frac{0 c_0^3  + 0 c_0^2 -81 c_0 -567 }{ 0 c_0^3  + 0c_0^2 + 0c_0 -1458  }\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
c_0 + 6(c_0 + 7)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
7(c_0 + 6) \, ,
</math>
  </td>
</tr>
</table>
</div>

which precisely matches the desired surface boundary condition for <math>\ell=3</math> that has been detailed in [[SSC/Stability/BiPolytrope00#DesiredBoundaryCondition|an accompanying summary discussion]].


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