Editing SSC/Stability/BiPolytrope00Details (section)

==Seek Alternate Solution to Mode21 (&ell;, j) = (2, 1)==

===Setup21===
According to [[SSC/Stability/BiPolytrope00#STEP4|STEP 4 in our accompanying summary discussion]], we need to solve the following "derivative matching" expression:
<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">
<math>~\Rightarrow~~~ \frac{\gamma_c}{\gamma_e} \biggl[\frac{14(1+2\Chi)^2}{7(1+2\Chi)^2 - 5(1+8\Chi)}\biggr] + 3\biggl(\frac{\gamma_c}{\gamma_e} -1\biggr) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{c_0 + (c_0 + 3)A_{2}\Chi + (c_0 + 6)A_{2}B_{2} \Chi^2}{1 + A_{2}\Chi + A_{2}B_{2}\Chi^2} 
</math>
  </td>
</tr>
</table>
</div>
where, recognizing that, <math>~\alpha_e = c_0(c_0+2) \, ,</math>
<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">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{c_0^2 + 5c_0 - (c_0^2 + 17c_0 + 66)}{(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>~-\biggl( \frac{ 4c_0 + 22}{2c_0 + 5}\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>

<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 + 17c_0 + 66)}{(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>~-\biggl( \frac{c_0 + 7 }{2c_0+8}\biggr) \, . </math>
  </td>
</tr>
</table>
</div>

Here, we assume that <math>~\Chi \equiv q^3</math> is specified and seek the corresponding value of <math>~c_0</math>.  Given that the LHS of this matching relation is known once <math>~\Chi</math> has been specified, in order to simplify notation we will also define,
<div align="center">
<table border="0" cellpadding="3" align="center">
<tr>
  <td align="right">
<math>~Q</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\gamma_c}{\gamma_e} \biggl[\frac{14(1+2\Chi)^2}{7(1+2\Chi)^2 - 5(1+8\Chi)}\biggr] + 3\biggl(\frac{\gamma_c}{\gamma_e} -1\biggr)  \, .</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>~Q</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{c_0 + (c_0 + 3)A_{21}\Chi + (c_0 + 6)A_{21}B_{21} \Chi^2}{1 + A_{21}\Chi + A_{21}B_{21}\Chi^2} 
</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_{21}\Chi + (c_0 + 6)A_{21}B_{21} \Chi^2 ] - Q[1 + A_{21}\Chi + A_{21}B_{21}\Chi^2 ]
</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)\biggl( \frac{ 4c_0 + 22}{2c_0 + 5}\biggr)\Chi + (c_0 + 6)\biggl( \frac{ 4c_0 + 22}{2c_0 + 5}\biggr)\biggl( \frac{c_0 + 7 }{2c_0+8}\biggr) \Chi^2 \biggr] 
- Q\biggl[1 - \biggl( \frac{ 4c_0 + 22}{2c_0 + 5}\biggr)\Chi + \biggl( \frac{ 4c_0 + 22}{2c_0 + 5}\biggr)\biggl( \frac{c_0 + 7 }{2c_0+8}\biggr)\Chi^2 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[c_0(2c_0+5)(2c_0+8) - (c_0 + 3)(2c_0+8)( 4c_0 + 22)\Chi + (c_0 + 6)( 4c_0 + 22)(c_0 + 7 ) \Chi^2 \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- Q\biggl[(2c_0+5)(2c_0+8) - (2c_0+8)( 4c_0 + 22)\Chi + ( 4c_0 + 22)( c_0 + 7 )\Chi^2 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{c_0(4c_0^2 + 26c_0 + 40) + ( 4c_0 + 22)\Chi [(c_0^2 + 13c_0 + 42 ) \Chi- (2c_0^2 +14c_0 +24)] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- Q\biggl\{ (4c_0^2 + 26c_0 + 40) + ( 4c_0 + 22)\Chi [c_0(\Chi-2) + (7\Chi-8) ] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4c_0^3 +[ 26 - 4Q]c_0^2 + [40  - 26Q]c_0 - 40Q  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ ( 4c_0 + 22)\Chi \biggl\{ [\Chi  - 2]c_0^2 + [13\Chi  -14  -Q  (\Chi-2) ]c_0 + [42\Chi  -Q (7\Chi-8) -24]  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4c_0^3 +[ 26 - 4Q]c_0^2 + [40  - 26Q]c_0 - 40Q  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 4\Chi[\Chi  - 2]c_0^3 + 4\Chi[13\Chi  -14  -Q  (\Chi-2) ]c_0^2 + 4\Chi[42\Chi  -Q (7\Chi-8) -24]c_0  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  22\Chi[\Chi  - 2]c_0^2 + 22\Chi[13\Chi  -14  -Q  (\Chi-2) ]c_0 + 22\Chi[42\Chi  -Q (7\Chi-8) -24]  \, .
</math>
  </td>
</tr>
</table>
</div>

===Solve Cubic Equation===

Here we draw from a [[Appendix/Ramblings/PPToriPt1A#Cubic_Equation_Solution|separate discussion of solutions to a cubic equation]].

<div align="center">
<table border="1" cellpadding="8" width="80%">
<tr>
  <td align="left">
Using <math>~y</math> in place of <math>~c_0</math>, this "derivative matching" relation can be written in the form of a standard cubic equation.  Specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~a y^3 + b y^2 + c y + d</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0 \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
4 + 4\Chi(\Chi - 2)\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~b</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
( 26 - 4Q_{21}) + 4\Chi[ 13\Chi  -14  - Q_{21}  (\Chi-2) ] +  22\Chi (\Chi  - 2)\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~c</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
 [40  - 26 Q_{21}]+ 4\Chi[42\Chi  - Q_{21} (7\Chi-8) -24]  + 22\Chi[13\Chi  -14  - Q_{21}  (\Chi-2) ] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~d</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
- 40 Q_{21}+ 22\Chi[42\Chi  -Q_{21} (7\Chi-8) -24] \, .
</math>
  </td>
</tr>
</table>
</div>

As is well known and documented &#8212; see, for example [http://mathworld.wolfram.com/CubicFormula.html Wolfram MathWorld] or [http://en.wikipedia.org/wiki/Cubic_function Wikipedia's discussion] of the topic &#8212; the roots of any cubic equation can be determined analytically.  In order to evaluate the root(s) of our particular cubic equation, we have drawn from the utilitarian [http://www.math.vanderbilt.edu/~schectex/courses/cubic/ online summary provided by Eric Schechter at Vanderbilt University].  For a cubic equation of the general form,
<div align="center">
<math>~ay^3 + by^2 + cy + d = 0 \, ,</math>
</div> 
a real root is given by the expression,
<div align="center">
<math>~
y = p + \{z + [z^2 + (r-p^2)^3]^{1/2}\}^{1/3} + \{z - [z^2 + (r-p^2)^3]^{1/2}\}^{1/3}
\, ,</math>
</div> 
where,
<div align="center">
<math>~p \equiv -\frac{b}{3a} \, ,</math> &nbsp;&nbsp;&nbsp;&nbsp;
<math>~z \equiv \biggl[p^3 + \frac{bc-3ad}{6a^2} \biggr] \, ,</math>
&nbsp;&nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp; 
<math>~r=\frac{c}{3a} \, .</math>
</div>
(There is also a pair of imaginary roots, but they are irrelevant in the context of our overarching astrophysical discussion.)

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


Upon evaluation, we have found that the expression inside of the square root is negative over the region of parameter space that is of most physical interest.  Hence, we need to call upon [[Appendix/Ramblings/PPToriPt1A#Cubic_Equation_Solution|a separate discussion]] in which the cube root of complex numbers was discussed.


<div align="center">
<table border="1" cellpadding="8" width="80%">
<tr>
  <td align="left">
We'll shift to Wolfram's notation; specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a_0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d}{a}\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~a_1 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{c}{a}\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~a_2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{b}{a}\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~R </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{3^2 a_2 a_1 - 3^3a_0 - 2a_2^3}{2\cdot 3^3}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{bc - 3ad }{6 a^2} - \biggl( \frac{ b}{3 a} \biggr)^3 = z \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~Q </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{3 a_1 - a_2^2}{3^2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{ c }{3a} - \biggl(\frac{b}{3a}\biggr)^2 = r-p^2 \, .</math>
  </td>
</tr>
</table>
</div>
Then, after defining,
<div align="center">
<math>~D \equiv Q^3 + R^2 = z^2 + (r-p^2)^3 \, ;</math>
&nbsp; &nbsp; &nbsp; &nbsp;
<math>~S^3 \equiv R+ \sqrt{D} \, ;</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>~T^3 \equiv R- \sqrt{D} \, ;</math>
</div>
Wolfram states that the three roots of the cubic equation are (the first one being identical to the "real" root identified above),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~y_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~p + (S + T) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~y_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~p - \frac{1}{2}(S + T) + \frac{1}{2} i \sqrt{3}(S-T) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~y_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~p - \frac{1}{2}(S + T) - \frac{1}{2} i \sqrt{3}(S-T) \, .</math>
  </td>
</tr>
</table>
</div>
Now, whenever <math>~D</math> is intrinsically negative, we need to treat both <math>~S^3</math> and <math>~T^3</math> as complex numbers.  If we define,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{r} \equiv (R^2 + |D|)^{1 / 2} \, ,</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\theta \equiv \tan^{-1}\biggl[ \frac{\sqrt{|D|}}{R} \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>

then we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~S^3 = \mathfrak{r} e^{+i\theta} \, ,</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~T^3 = \mathfrak{r} e^{-i\theta} \, .</math>
  </td>
</tr>
</table>
</div>

As is explained in [http://math.stackexchange.com/questions/8760/what-are-the-three-cube-roots-of-1 this online resource], both <math>~S</math> and <math>~T</math> must formally submit to three separate roots tagged by the integer index, <math>~(j=0,1,2)</math>.  Working only with the <math>~j=0</math> root for both, we find that the above expressions for the three roots of our cubic equation become,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~y_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~p + 2 \mathfrak{r}^{1 / 3} \cos\biggl(\frac{\theta}{3}\biggr) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~y_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~p - \mathfrak{r}^{1 / 3} \biggl[ \cos\biggl(\frac{\theta}{3}\biggr) + \sqrt{3} \sin\biggl(\frac{\theta}{3}\biggr) \biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~y_3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~p - \mathfrak{r}^{1 / 3} \biggl[ \cos\biggl(\frac{\theta}{3}\biggr) - \sqrt{3} \sin\biggl(\frac{\theta}{3}\biggr) \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
  </td>
</tr>
</table>
</div>

We have deduced empirically that <math>~y_3</math> is the root that is physically relevant in our case.  That is to say, for a given <math>~0 < q < 1</math>,
<div align="center">
<math>~c_0 = p - \mathfrak{r}^{1 / 3} \biggl[ \cos\biggl(\frac{\theta}{3}\biggr) - \sqrt{3} \sin\biggl(\frac{\theta}{3}\biggr) \biggr] \, .</math>
</div>
In turn, for a given value of <math>~q</math>, the corresponding value of <math>~\alpha_e</math> is obtained via the relation, <math>~\alpha_e = c_0(c_0 + 2)</math>.

The right-hand panel of Figure 1 presents a plot of this <math>~c_0(q)</math> function; actually what has been plotted is the inverted relation, <math>~q(\alpha_e)</math>.  The open, red circular markers trace the portion of the function that provides physically viable solutions, in the sense that the corresponding value of <math>~\alpha_c</math> lies between the values, negative one and three; the filled, light=blue circular markers identify roots of the cubic equation that are not physically viable.

In the left-hand panel of Figure 1, we re-display a plot that has been discussed in an [[SSC/Stability/BiPolytrope00#Figure1|accompanying chapter]].  It contains a plot (blue markers) of the same same <math>~q(\alpha_e)</math> function but, this time, as determined from the root of a quartic equation.  In order to illustrate more clearly that the two curves are the same, we have plotted the quartic solution (small, purple circular markers) ''on top of'' the cubic solution in the right-hand panel.  

<div align="center">
<table border="1" cellpadding="5"><tr><th align="center">
Figure 1:&nbsp; Comparing Roots to Quartic and Cubic Equations
</th></tr>
<tr><td align="center">
[[File:Quartic21Solution02Corrected.png|300px|quartic solution]]
[[File:Cubic21Solution01Corrected.png|300px|cubic solution]]
</td></tr>
</table>
</div>

<!-- DELETE IMAGE (it appears immediately below)
<div align="center">
[[File:Model21Montage2.png|750px|Montage of Stability Results for (ell,j) = (2,1) quantum numbers]]
</div>
-->

===Illustration21===

<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) = (2,1)</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:Model21MontageCorrected.png|800px|Montage of Stability Results for (ell,j) = (2,1) 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) = (2,1)</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) = (2,1)</math> mode in the model identified by the yellow circular marker in both left-hand panels, for which,
<div align="center">
<math>~\alpha_e =-0.35</math> &nbsp; &nbsp; &nbsp; <math>~\Rightarrow</math> &nbsp; &nbsp; &nbsp; <math>~c_0~\mathrm{(plus)} = -0.1937742</math>
</div>
and, correspondingly (via our [[SSC/Stability/BiPolytrope00Details#Quartic|separate solution of the governing ''quartic'' equation]]), <math>~(q,\alpha_c) = (0.6840119, +0.8326585)</math>.  (This same plot appears in [[SSC/Stability/BiPolytrope00#Figure1|Figure 1 of our accompanying summary]].)  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=2} |_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\xi^{c_0}\biggl[ \frac{ 1 +  q^3 A_{21} \xi^{3} +  q^6 A_{21}B_{21}\xi^{6} }{ 1 +  q^3 A_{21}  +  q^6 A_{21}B_{21}  }\biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
where the coefficients, <math>~A_{21}, B_{21}</math>, are as [[#Setup21|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 28.9115809\, ;</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>~3\gamma_e\biggl[\alpha_e + 5c_0 + 22\biggr] \frac{\rho_e}{\rho_c} \approx 28.9115807\, ,</math>
  </td>
</tr>
</table>
</div>
where the relevant density ratio is, <math>~\rho_e/\rho_c = 2q^3/(1+2q^3) \approx 0.3902664</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 one node resides 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 0.684)</math>, the density abruptly drops to its envelope value <math>~(\rho/\rho_c = \rho_e/\rho_c \approx 0.39)</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.