Editing SSC/Stability/BiPolytropes/RedGiantToPN (section)

=Relevant Instabilities=

==Abstract==

The analysis presented by {{ EFC98 }} is essentially an analysis of the <math>q - v</math> diagram.  We can determine analytically at what value of <math>\xi_i</math> the core-to-total mass ratio reaches a maximum (<math>\nu_\mathrm{max}</math>) for various values of <math>\mu_e/\mu_c \le 1/3</math>.  For example, <math>(\mu_e/\mu_c, \xi_i, \nu_\mathrm{max}) = (\tfrac{1}{4}, 4.9379256, 0.139270157)</math>.  Our LAWE analysis shows that '''none''' of these turning points is associated with the onset of a dynamical instability.

On the other hand, our LAWE analysis '''does''' identify a marginally unstable equilibrium configuration along every sequence; even sequences with <math>\tfrac{1}{3} \le \mu_e/\mu_c \le 1</math>.  [[SSC/Stability/BiPolytropes/Pt3#Eigenvectors_for_Marginally_Unstable_Models_with_(γc,_γe)_=_(6/5,_2)|For example]], <math>(\mu_e/\mu_c, \xi_i, \nu) = (1, 1.6686460157, 0.497747626)</math>.

==Truncated n = 5 Polytrope==

In [[SSC/Structure/PolytropesEmbedded/n5#Fig3|Figure 3 of an accompanying discussion]], we show where various turning points lie along the equilibrium sequence of truncated <math>n=5</math> polytropes.



<div align="center" id="Fig3">
<table border="1" align="center" cellpadding="8" width="1050px">
<tr>
  <td align="center" colspan="6">
<b>Figure 3:</b> &nbsp; Equilibrium Sequences of Pressure-Truncated, n = 5 Polytropic Spheres<br />(viewed from several different astrophysical perspectives)
  </td>
</tr>
<tr>
  <td align="center"><font color="black" size="+2">&#x25CF;</font></td><td align="center"><math>~\xi_e</math></td>
  <td align="center" width="300px"><sup>&dagger;</sup>External Pressure vs. Volume<br /><font size="-1">(Fixed Mass)</font></td>
  <td align="center" width="300px">Mass vs. Radius<br /><font size="-1">(Fixed External Pressure)</font></td>
  <td align="center" width="300px"><sup>&Dagger;</sup>Mass vs. Central Density<br /><font size="-1">(Fixed External Pressure)</font></td>
  <td align="center" width="300px">Mass vs. Central Density<br /><font size="-1">(Fixed Radius)</font></td>
</tr>
<tr>
  <td align="center" colspan="1"><font color="yellow" size="+2">&#x25CF;</font></td>  <td align="center" colspan="1">&radic;3</td>
  <td align="center" colspan="1" rowspan="4">(a)<br />
[[File:N5Sequence01B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
  </td>
  <td align="center" colspan="1" rowspan="4">(b)<br />
[[File:N5Sequence02B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
  </td>
  <td align="center" colspan="1" rowspan="4">(c)<br />
[[File:N5Sequence03B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
  </td>
  <td align="center" colspan="1" rowspan="4">(d)<br />
[[File:N5Sequence04B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
  </td>
</tr>
<tr>
  <td align="center" colspan="1"><font color="darkgreen" size="+2">&#x25CF;</font></td>  <td align="center" colspan="1">3</td>
</tr>
<tr>
  <td align="center" colspan="1"><font color="purple" size="+2">&#x25CF;</font></td>  <td align="center" colspan="1">&radic;15</td>
</tr>
<tr>
  <td align="center" colspan="1"><font color="red" size="+2">&#x25CF;</font></td>  <td align="center" colspan="1">9.01</td>
</tr>
<tr>
  <td align="center" colspan="2">&nbsp;</td>
  <td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^3 \biggl[ \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12} \biggr]_\tilde\xi</math><br /> vs. <br />
<math>\biggl(\frac{\pi}{2\cdot 3}\biggr)^{5/2} \biggl[ \xi^{-15} \biggl(1 + \frac{\xi^2}{3} \biggr)^{9}\biggr]_\tilde\xi</math>
</td>
  <td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2}\biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. 
<br /> <math>\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \biggl[ \xi \biggl(1 + \frac{\xi^2}{3} \biggr)^{-1} \biggr]_\tilde\xi</math></td>
  <td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \biggl(1 + \frac{\xi^2}{3} \biggr)^{5/2}\biggr]_\tilde\xi</math>
  </td>
  <td align="center" colspan="1"><math>\biggl[ \frac{2^3\cdot 3}{\pi} \biggr]^{1 / 4} \biggl[  \xi^{5/2}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \frac{3}{2\pi} \biggr]^{5 / 4} \tilde\xi^{5 / 2}</math>
  </td>
</tr>
</table>
</div>


<ul>
<li>
<font color="red">KEY RESULT:</font>
<ul>
  <li>The maximum "Bonnor-Ebert type" mass and external pressure occurs along the sequence precisely at <math>\tilde\xi = 3</math>.</li>
  <li>It is precisely at this turning point that the equilibrium model is marginally (dynamically) unstable; the eigenfunction is parabolic.</li>
  <li>[[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|For all <math>3 < n < \infty</math>]], the location along the relevant sequence presents an analogous turning point whose location and whose eigenfunction is known analytically.</li>
</ul>

==Bipolytropes with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)==

===q - &nu; Sequence Plots===

In [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Model_Sequences|Figure 1 of an accompanying discussion]], we show &#8212; via a plot in the <math>(q, \nu)</math> diagram &#8212; how the <math>(n_c, n_e) = (5, 1)</math> bipolytrope sequence behaves for various values of the molecular-weight ratio over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le 1</math>.


<div align="center">
<table border="0" cellpadding="5" width="85%">

<tr>
  <td align="center" colspan="2" bgcolor="white">
[[Image:PlotSequencesBest02.png|500px|center]]
  </td>
</tr>

<tr>
  <td align="left" colspan="2">
'''Figure 1:'''  Analytically determined plot of fractional core mass (<math>\nu</math>) versus fractional core radius (<math>q</math>) for <math>(n_c, n_e) = (5, 1)</math> bipolytrope model sequences having six different values of <math>\mu_e/\mu_c</math>: 1 (blue diamonds), &frac12; (red squares), 0.345 (dark purple crosses), &#8531; (pink triangles), 0.309 (light green dashes),  and &frac14; (purple asterisks).  Along each of the model sequences, points marked by solid-colored circles correspond to models whose interface parameter, <math>\xi_i</math>, has one of three values:  0.5 (green circles), 1 (dark blue circles), or 3 (orange circles); the images linked to Table 2 provide plots of the density, pressure and mass profiles for nine of these identified models.
  </td>
</tr>
</table>
</div>

According to [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Background|our accompanying discussion]], in terms of the parameters,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}}  \, ;
</math>
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, ,
</math>
</div>

the parameter, <math>\nu</math>, varies with <math>\xi</math> as,
<div align="center">
<math>
\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} 
= (m_3^2 \ell_i^3) (1 + \ell_i^2)^{-1/2} [1 + (1-m_3)^2 \ell_i^2]^{-1/2}  \biggl[ m_3\ell_i + (1+\ell_i^2) \biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i \biggr) \biggr]^{-1} \, .
</math>
</div>

<font color="red">KEY RESULT:</font>&nbsp; Over the range, <math>\tfrac{1}{4} \le (\mu_e/\mu_c) \le \tfrac{1}{3}</math>, there is a value of <math>\nu</math> above which no equilibrium configurations exist.  We have determined the location of this "turning point" by setting, <math>d\nu/d\xi = 0</math>; our [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Derivation|derived result]] is,

<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
\underbrace{\biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i\biggr) (1+\ell_i^2) [ 3 + (1-m_3)^2(2-\ell_i^2)\ell_i^2]}_{\mathrm{LHS}} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\underbrace{m_3 \ell_i [(1-m_3)\ell_i^4 - (m_3^2 - m_3 +2)\ell_i^2 - 3]}_{\mathrm{RHS}} \, .
</math> 
  </td>
</tr>
</table>
</div>

<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="12">
<b>Maximum Fractional Core Mass, <math>\nu = M_\mathrm{core}/M_\mathrm{tot}</math> (solid green circular markers)<br />for Equilibrium Sequences having Various Values of <math>\mu_e/\mu_c</math>
  </td>
</tr>
<tr>
  <td align="center">
<math>\frac{\mu_e}{\mu_c}</math>
  </td>
  <td align="center">
<math>\xi_i</math>
  </td>
  <td align="center">
<math>\theta_i</math>
  </td>
  <td align="center">
<math>\eta_i</math>
  </td>
  <td align="center">
<math>\Lambda_i</math>
  </td>
  <td align="center">
<math>A</math>
  </td>
  <td align="center">
<math>\eta_s</math>
  </td>
  <td align="center">
LHS
  </td>
  <td align="center">
RHS
  </td>
  <td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
  </td>
  <td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
  </td>
  <td align="center" rowspan="7">[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]</td>
</tr>

<tr>
  <td align="center">
<math>\frac{1}{3}</math>
  </td>
  <td align="center">
<math>\infty</math> </td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">0.0 </td>
  <td align="center">
<math>\frac{2}{\pi}</math> </td>
</tr>

<tr>
  <td align="center">
0.33
  </td>
  <td align="right">
24.00496  </td>
  <td align="right">
0.0719668  </td>
  <td align="right">
0.0710624  </td>
  <td align="right">
0.2128753  </td>
  <td align="right">
0.0726547  </td>
  <td align="right">
1.8516032  </td>
  <td align="right">
-223.8157  </td>
  <td align="right">
-223.8159  </td>
  <td align="right">
0.038378833  </td>
  <td align="right">
0.52024552  </td>
</tr>

<tr>
  <td align="center">
0.316943
  </td>
  <td align="right">
10.744571  </td>
  <td align="right">
0.1591479  </td>
  <td align="right">
0.1493938  </td>
  <td align="right">
0.4903393  </td>
  <td align="right">
0.1663869  </td>
  <td align="right">
2.1760793  </td>
  <td align="right">
-31.55254  </td>
  <td align="right">
-31.55254  </td>
  <td align="right">
0.068652714  </td>
  <td align="right">
0.382383875  </td>
</tr>

<tr>
  <td align="center">
0.3090
  </td>
  <td align="right">
8.8301772  </td>
  <td align="right">
0.1924833  </td>
  <td align="right">
0.1750954  </td>
  <td align="right">
0.6130669  </td>
  <td align="right">
0.2053811  </td>
  <td align="right">
2.2958639  </td>
  <td align="right">
-18.47809  </td>
  <td align="right">
-18.47808  </td>
  <td align="right">
0.076265588  </td>
  <td align="right">
0.331475715  </td>
</tr>

<tr>
  <td align="center">
<math>\frac{1}{4}</math>
  </td>
  <td align="right">
4.9379256  </td>
  <td align="right">
0.3309933  </td>
  <td align="right">
0.2342522  </td>
  <td align="right">
1.4179907  </td>
  <td align="right">
0.4064595  </td>
  <td align="right">
2.761622  </td>
  <td align="right">
-2.601255  </td>
  <td align="right">
-2.601257  </td>
  <td align="right">
0.084824137  </td>
  <td align="right">
0.139370157  </td>
</tr>

<tr>
  <td align="left" colspan="11">
Recall that,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}}  \, ;
</math>
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .
</math>
</div>
  </td>
</tr>
</table>

===The EFC98 Sequence Plot===
{{ EFC98 }} also analytically determined the structure of models along various <math>(n_c, n_e) = (5, 1)</math> sequences; their Figure 1 displays the behavior of <math>\nu</math> vs. <math>\log_{10} (\rho_c/\rho_i)</math> for a range of <math>\alpha \equiv (\mu_e/\mu_c)^{-1}</math>.  Note that,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \sqrt{3} \biggl(\frac{\xi_i^3 \theta_i^4}{A\eta_s}\biggr) \, ;</math></td>
</tr>

<tr>
  <td align="right"><math>\log_{10} (\rho_c/\rho_i)</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\log_{10}\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \biggl(1 + \frac{\xi^2}{3}\biggr)^{5/2} \biggr] \, .</math></td>
</tr>
</table>

<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" width="25%"><math>\frac{\mu_e}{\mu_c} = \alpha^{-1}</math></td>
  <td align="center" width="25%><math>\xi_i</math></td>
  <td align="center" width="25%"><math>\nu</math></td>
  <td align="center"><math>\log_{10}\biggl(\frac{\rho_c}{\rho_i}\biggr)</math></td>
</tr>
<tr>
  <td align="center"><math>\frac{1}{4} </math></td>
  <td align="center"><math>4.9379256</math></td>
  <td align="center"><math>0.139370157</math></td>
  <td align="center"><math>3.002964</math></td>
</tr>
</table>
<font color="red">KEY RESULT (to be done):</font>&nbsp; From our original derivation, we have generated a plot intended to replicate Figure 1 from {{ EFC98hereafter }}; then we have marked on each sequence the location of the mass-extremum (i.e., when <math>d\nu/d\xi = 0</math>) as determined by [[#Sequence_Plots|our above analytically derived result]].

===Yabushita75 Guidance===

Alternatively, [[#Fixed_Interface_Pressure_Sequence_Plots|as derived above]], setting <math>dM_\mathrm{tot}/d\ell_i = 0</math> leads to the expression,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math> 
\frac{m_3 \ell_i}{(1+\ell_i^2)(2 - m_3) }\biggl[
( m_3 - 3 ) + (1 - m_3 )\ell_i^2  
\biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \biggr] \, .
</math>
  </td>
</tr>
</table>

For <math>\mu_e/\mu_c = 1.00</math> the <font color="red">solution to this expression is <math>\xi_i = 1.668462981</math></font>. For other parameter choices, see [[SSC/Stability/BiPolytropes/Pt3#Equilibrium_Properties_of_Marginally_Unstable_Models|here, for example]].

<table align="center" border="1" cellpadding="5">
<tr>
  <td align="center" rowspan="2"><math>\frac{\mu_e}{\mu_c}</math></td>
  <td align="center" rowspan="1" colspan="2"><math>\xi_i</math></td>
</tr>

<tr>
  <td align="center" rowspan="1" colspan="1" width="40%">LAWE Sol'n</td>
  <td align="center" rowspan="1" colspan="2" width="40%">Max. M<sub>tot</sub></td>
</tr>

<tr>
  <td align="center" rowspan="1" colspan="1">1</td>
  <td align="center" rowspan="1" colspan="1">1.6686460157</td>
  <td align="center" rowspan="1" colspan="1">1.668462981</td>
</tr>

<tr>
  <td align="center" rowspan="1" colspan="1"><math>\tfrac{1}{2}</math></td>
  <td align="center" rowspan="1" colspan="1">2.2792811317</td>
  <td align="center" rowspan="1" colspan="1">n/a</td>
</tr>

</table>