Editing SSC/Stability/BiPolytrope00Details (section)

===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>