Editing SSC/Stability/BiPolytropes/RedGiantToPN (section)

==Preface==

Go [[SSC/Structure/Polytropes/VirialSummary#StahlerSchematic|here]] for Stahler schematic.

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<table border="1" align="left" cellpadding="2">
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[[File:Stahler1983TitlePage0.png|center|100px|ApJ reference]]
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[[File:Stahler MRdiagram1.png|left|100px|Stahler Schematic]]
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</table>
</td></tr></table>

<table border="0" cellpadding="8" align="right">
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  <th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined2]]Figure 1: &nbsp;Equilibrium Sequences<br />of Pressure-Truncated Polytropes
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[[File:DFBsequenceB.png|300px|Equilibrium sequences of Pressure-Truncated Polytropes]]
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As has been detailed in an [[SSC/Stability/BiPolytropes#Overview|accompanying chapter]], we have [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|successfully analyzed the relative stability of pressure-truncated polytopes]].  The curves shown here on the right in Figure 1 graphically present the mass-radius relationship for pressure-truncated model sequences having a variety of polytropic indexes, as labeled, over the range <math>1 \le n \le 6</math>.  ([[SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Another version of this figure]] includes the isothermal sequence, for which <math>n = \infty</math>.)  

&nbsp;<br />
Along each sequence for which <math>n \ge 3</math>, the green filled circle identifies the model with the largest mass.  This maximum-mass model is the polytropic analogue of the Bonnor-Ebert mass, which was identified independently by {{ Ebert55 }} and {{ Bonnor56 }} in the context of studies of pressure-truncated ''isothermal'' equilibrium configurations.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial M_\mathrm{tot}/\partial \xi \biggr|_\tilde{\xi} = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.  
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>

<font color="red">'''Key Realization:'''</font>  ''Along sequences of pressure-truncated polytropes, the maximum-mass models identify precisely where the onset of dynamical instability occurs.''

----

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  <th align="center"><font size="-1">'''Figure 2:  Equilibrium Sequences of Bipolytropes'''</font> <br /><p>
<font size="-1">'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''</font>
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<tr>
  <td align="center" colspan="1">
[[File:TurningPoints51Bipolytropes.png|300px|Extrema along Various Equilibrium Sequences]]
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Using similar techniques, we have successfully analyzed the relative stability of bipolytropic configurations that have <math>(n_c, n_e) = (5, 1)</math>. Our analytically constructed equilibrium model sequences replicate the ones originally presented by {{ EFC98 }} for this same pair of bipolytropic indexes; and they serve as analogues of the sequences that were constructed numerically by {{ HC41 }} and {{ SC42 }} for bipolytropic configurations that have <math>(n_c, n_e) = (\infty, \tfrac{3}{2})</math>. 

Following {{ HC41hereafter }} and {{ SC42hereafter }}, we have found it particularly useful to label each equilibrium model according to the key structural parameters, <math>q \equiv r_\mathrm{core}/R_\mathrm{surf}</math> and <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>. The curves shown here on the right in Figure 2 graphically present the <math>q - \nu</math> relationship for bipolytropic model sequences that have a variety of molecular-weight jumps, <math>\tfrac{1}{4} \le \mu_e/\mu_c \le 1</math>, at the core-envelope interface, as labeled.  Along each Fig. 2 sequence for which <math>\mu_e/\mu_c \le \tfrac{1}{3}</math>, the green filled circle identifies the model with the largest mass ratio, <math>\nu</math>.  This maximum-mass model is a polytropic analogue of the Sch&ouml;nberg-Chandrasekhar mass limit, which was identified by {{ HC41hereafter }} and {{ SC42hereafter }} in the context of their studies of stars with isothermal cores.
<ol type="1">
<li>The maximum-mass model's position along each sequence has been determined analytically by setting <math>\partial \nu/\partial q  = 0.</math></li>
<li>
By solving the LAWE associated with various models along each equilibrium sequence, we have shown that the eigenfrequency of the fundamental-mode of radial oscillation is zero for each one of these maximum-mass models.  
<ol type="a">
<li>Solutions to the LAWE were initially obtained via numerical integration techniques, in which case we were only able to make this association approximately;</li>
<li>To our surprise, we also have been able to determine ''analytically'' an expression that defines the eigenfunction of the marginally unstable equilibrium model along each sequence <math>(n \ge 3)</math>; as a result we have been able to show that the eigenfrequency associated with the maximum-mass model is ''precisely'' zero.</li>
</ol>
As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.</li>
</ol>



indexes, as labeled, over the range <math>1 \le n \le 6</math>. 

<!--
<font color="red">'''The principal question is:'''</font>  ''Along bipolytropic sequences, are maximum-mass models (identified by the solid green circular markers in Fig. 2) associated with the onset of dynamical instabilities?''</span> For more details, look [[SSC/Stability/BiPolytropes/51Models#Structure|here]].
-->

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'''Figure 2:  Equilibrium Sequences of Bipolytropes''' <br /><p>
'''with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math>'''
  </td>
  <td align="center" colspan="4">
<b>Analytically Determined Parameters<sup>&dagger;</sup><br />for Models that have the Maximum Fractional Core Mass<br />(solid green circular markers)<br />Along Various Equilibrium Sequences
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[[File:TurningPoints51Bipolytropes.png|450px|Extrema along Various Equilibrium Sequences]]
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<math>\frac{\mu_e}{\mu_c}</math>
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<math>\xi_i</math>
  </td>
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<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
  </td>
  <td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
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<tr>
  <td align="center">
<math>\frac{1}{3}</math>
  </td>
  <td align="center">
<math>\infty</math> </td>
  <td align="center">0.0 </td>
  <td align="center">
<math>\frac{2}{\pi}</math> </td>
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<tr>
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0.33
  </td>
  <td align="right">
24.00496  </td>
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0.038378833  </td>
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0.52024552  </td>
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0.316943
  </td>
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10.744571  </td>
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0.068652714  </td>
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0.382383875  </td>
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0.31
  </td>
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9.014959766  </td>
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0.0755022550  </td>
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0.3372170064  </td>
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0.3090
  </td>
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8.8301772  </td>
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0.076265588  </td>
  <td align="right">
0.331475715  </td>
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<math>\frac{1}{4}</math>
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4.9379256  </td>
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0.084824137  </td>
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0.139370157  </td>
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<sup>&dagger;</sup>Additional model parameters [[SSC/Stability/BiPolytropes/51Models#Structure|can be found here]].
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<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
In terms of mass <math>(m)</math>, length <math>(\ell)</math>, and time <math>(t)</math>, the units of various physical constants and variables are:
<table border="0" align="center" cellpadding="5">

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  <td align="right">
Mass-density
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<math>\sim</math>
  </td>
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<math>
m \ell^{-3} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
Pressure (energy-density)
  </td>
  <td align="center">
<math>\sim</math>
  </td>
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<math>
m \ell^{-1} t^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
Newtonian gravitational constant, <math>G</math>
  </td>
  <td align="center">
<math>\sim</math>
  </td>
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<math>
m^{-1} \ell^{3} t^{-2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
The core's polytropic constant, <math>K_c</math>
  </td>
  <td align="center">
<math>\sim</math>
  </td>
  <td align="left">
<math>
\biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5}
</math>
  </td>
</tr>

<tr>
  <td align="right">
The envelope's polytropic constant, <math>K_e</math>
  </td>
  <td align="center">
<math>\sim</math>
  </td>
  <td align="left">
<math>
m^{-1} \ell^{5} t^{-2} 
</math>
  </td>
</tr>
</table>

----

As a result, for example (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Central_Density|details below]]), if we hold the central-density <math>(\rho_0)</math>  &#8212; as well as <math>G</math> and <math>K_c</math> &#8212; constant along an equilibrium sequence, mass will scale as &hellip;
<table border="0" align="center" cellpadding="5">

<tr>
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Mass
  </td>
  <td align="center">
<math>\sim</math>
  </td>
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<math>
\biggl[K_c^{3/2} G^{-3/2} \rho_0^{-1 / 5} \biggr] 
</math>
  </td>
</tr>

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&nbsp;
  </td>
  <td align="center">
<math>\sim</math>
  </td>
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<math>
\biggl\{ \biggl[ m^{-1} \ell^{13} t^{-10} \biggr]^{1 / 5} \biggr\}^{3/2} 
~\biggl[ m^{-1} \ell^{3} t^{-2}  \biggr]^{-3/2} 
~\biggl[ m \ell^{-3} \biggr]^{-1 / 5} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>\sim</math>
  </td>
  <td align="left">
<math>
\biggl[ m^{-3/10 + 3/2 - 1/5}  \biggr] 
~\biggl[ \ell^{39/10 - 9/2 + 3/5 }   \biggr]
~\biggl[ t^{-3 + 3 }  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>\sim</math>
  </td>
  <td align="left">
<math>
\biggl[ m^{+1}  \biggr] 
~\biggl[ \ell^{0 }   \biggr]
~\biggl[ t^{0}  \biggr] 
\, .
</math>
  </td>
</tr>
</table>

If instead (see [[SSC/Stability/BiPolytropes/RedGiantToPN#Fixed_Interface_Pressure|details below]]) we hold <math>K_e</math>  &#8212; as well as <math>G</math> and <math>K_c</math> &#8212; constant along an equilibrium sequence, mass will scale as &hellip;
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
Mass
  </td>
  <td align="center">
<math>\sim</math>
  </td>
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<math>
\biggl[K_c^{5}K_e G^{-6}\biggr]^{1 / 4}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>\sim</math>
  </td>
  <td align="left">
<math>
\biggl\{~\biggl[ m^{-1} \ell^{13} t^{-10} \biggr] 
\biggl[m^{-1} \ell^{5} t^{-2}\biggr] 
\biggl[m^{-1} \ell^{3} t^{-2}\biggr]^{-6} ~\biggr\}^{1 / 4}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>\sim</math>
  </td>
  <td align="left">
<math>
\biggl\{~\biggl[ m^{-1 -1 + 6} \biggr] \biggl[ \ell^{13 + 5 - 18}\biggr] 
\biggl[  t^{-10 - 2 + 12}\biggr] 
~\biggr\}^{1 / 4}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>\sim</math>
  </td>
  <td align="left">
<math>
\biggl[ m^{+1}  \biggr] 
~\biggl[ \ell^{0 }   \biggr]
~\biggl[ t^{0}  \biggr] 
\, .
</math>
  </td>
</tr>
</table>

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