Editing Appendix/Ramblings/OriginOfPlanetaryNebulae (section)

==Toy Model==

Let's play with a bipolytropic configuration having <math>(n_c, n_e) = (5, 1)</math> and <math>\mu_e/\mu_c = \tfrac{1}{2}</math>.  Evolution will be dictated by the steady increase of the radial location of the core/envelope interface, <math>\xi_i</math> from zero (initially) to <math>\xi_i|_\mathrm{final} = 2.279258</math> which &#8212; [[SSC/Stability/BiPolytropes#Fundamental_Modes|according to our solution of the bipolytropic LAWE]] &#8212; marks the onset of dynamical instability.

As evolution proceeds along the <math>\mu_e/\mu_c = \tfrac{1}{2}</math> sequence, we want to hold <math>M_\mathrm{tot}</math> constant.  We must choose one additional fixed parameter; it isn't immediately obvious what is the best choice, but we will try setting <math>K_c</math> constant.  ([[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_=_5_and_ne_=_1|Originally]], we held the central density, <math>\rho_0</math>, and <math>K_c</math> constant.)  The relevant renormalization is [[SSC/Structure/BiPolytropes/Analytic51Renormalize|detailed in our chapter tagged SSC/Structure/BiPolytropes/Analytic51Renormalize]].

===Zero-Age Main Sequence (ZAMS) Configuration===
As a check, we note that in the initial (ZAMS) equilibrium configuration <math>(\xi_i=0)</math> there is no "core" &#8212; that is, <math>M_\mathrm{core} = 0</math> &#8212; so its structure should match that of an [[SSC/Structure/Polytropes#n_=_1_Polytrope|isolated <math>n = 1</math> polytrope]].  Specifically, given that, <math>M_\odot = 1.99\times 10^{33}~\mathrm{g}</math>, <math>R_\odot = 6.96\times 10^{10}~\mathrm{cm}</math>, and <math>G = 6.674\times 10^{-8}~\mathrm{cm^3~g^{-1}~s^{-2}}</math> &hellip; 
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>M_\mathrm{tot}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\frac{4}{\pi} \rho_c R^3</math></td>
<td align="center">&nbsp; &nbsp; &nbsp; <math>\Rightarrow</math> &nbsp; &nbsp; &nbsp; </td>
  <td align="right"><math>\rho_c\biggr|_\mathrm{initial}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\frac{\pi}{4} \cdot \frac{M_\odot}{R_\odot^3} = 4.64~\mathrm{g~cm^{-3}} \, .</math></td>
</tr>
</table>
Also,
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>\frac{\rho_c}{\bar\rho}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\frac{\pi^2}{3}</math></td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right"><math>P_c\biggr|_\mathrm{initial}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\frac{\pi G}{8} \biggl( \frac{M_\odot^2}{R_\odot^4} \biggr) = 4.42 \times 10^{15}~\mathrm{g~cm^{-1}~s^{-2}} \, .</math></td>
</tr>
</table>

<table border="1" align="center" cellpadding="5" width="80%"><tr><td align="left">
The sun's luminosity is, <math>L_\odot = 4\pi R_\odot^2 \sigma_\mathrm{SB} T^4 = 3.846\times 10^{33}~\mathrm{ergs~s^{-1}}</math>, where, in terms of {{ Math/C_RadiationConstant }} and {{ Math/C_SpeedOfLight }}, the Stefan-Boltzmann constant is,
<div align="center">
<math>\sigma_\mathrm{SB} = \frac{a_\mathrm{rad} c}{4} = 5.671\times 10^{-5} ~\mathrm{erg~cm^{-2}~s^{-1}~K^{-4}}\, .</math>
</div>
Hence, in the context of our ZAMS toy model, the configuration's surface temperature is,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>T\biggr|_\mathrm{initial}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\biggl[ \frac{L_\odot}{4\pi R_\odot^2 \sigma_\mathrm{SB} } \biggr]^{1 / 4} = 5777~\mathrm{K} \, .</math>
  </td>
</tr>
</table>

----

In the context of our examination of a <math>1 M_\odot</math> evolutionary track published by {{ Iben67full }} &#8212; [[#IbenFigure|see below]] &#8212; we can write quite generally that,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>L</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>4\pi R^2 \sigma_\mathrm{SB}T_e^4</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \biggl(\frac{R}{R_\odot}\biggr)^2 </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\frac{L}{L_\odot}\biggl( \frac{T_e}{5777 K}\biggr)^{-4}</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \log\biggl(\frac{R}{R_\odot}\biggr) </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\tfrac{1}{2}\log\biggl(\frac{L}{L_\odot} \biggr)
- 2 \log\biggl( \frac{T_e}{5777 K}\biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\tfrac{1}{2}\log\biggl(\frac{L}{L_\odot} \biggr)
- 2 \log T_e + 7.523 \, .</math>
  </td>
</tr>
</table>

----

Note as well that if an evolving star radiates at an average luminosity, <math>L_\mathrm{avg}/L_\odot</math>, over a specified time interval in units of <math>10^9~\mathrm{yrs}</math>, <math>(\Delta t)_9</math>, during the specified interval of time it will have burned through an amount of mass, <math>\Delta M/M_\odot</math>, given by the expression,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>0.007(\Delta M)c^2</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>(\Delta t)L_\mathrm{avg}</math></td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{(\Delta M)}{M_\odot}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(\Delta t)_9 \cdot \frac{L_\mathrm{avg}}{L_\odot} \biggl[
\frac{L_\odot}{0.007 M_\odot c^2} 
\cdot 3.156\times 10^{16}~\mathrm{s}
\biggr]
=
0.01 \biggl[ (\Delta t)_9 \cdot \frac{L_\mathrm{avg}}{L_\odot}  \biggr] \, ,
</math>
  </td>
</tr>
</table>
where we have assumed that the efficiency of burning hydrogen to helium is 0.7%.
</td></tr></table>

Most notably, the dimensionless radius that appears in the classic Lane-Emden equation is given by the expression,

<div align="center">
<math>
\xi \equiv \frac{r}{a_\mathrm{n = 1}} ,
</math>
</div>
where,
<div align="center">
<math>
a_\mathrm{n=1} \equiv \biggl[\frac{(n+1)K_n}{4\pi G} \cdot \rho_c^{(1-n)/n} \biggr]^{1/2} 
=
\biggl[\frac{K_\mathrm{env}}{2\pi G} \biggr]^{1/2}
\, .
</math>
</div>
And, given that the surface (<math>r = R_\odot</math>) of the isolated <math>n=1</math> polytrope occurs at <math>\xi = \xi_1 = \pi</math>, we find that,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>a_\mathrm{n=1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\frac{R_\odot}{\pi}</math></td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ K_\mathrm{env}\biggr|_\mathrm{initial}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl( \frac{2}{\pi} \biggr) G R_\odot^2 = 2.06 \times 10^{14}~\mathrm{cm^5~g^{-1}~s^{-2}} \, .</math></td>
</tr>
</table>
Double-check:
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>P_c\biggr|_\mathrm{initial}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
K_\mathrm{env}\biggr|_\mathrm{initial}\biggl[ \rho_c\biggr|_\mathrm{initial}\biggr]^2
=
4.42\times 10^{15} ~\mathrm{g~cm^{-1}~s^{-2}} \, .
</math>
  </td>
</tr>
</table>
Yes!


Now, the instant the "core" shows up, we demand that its central density be a factor of <math>(\mu_e/\mu_c)^{-1}</math> larger than the density at the inner edge of the envelope; at the same time, we demand that the pressures be the same.  Hence the relevant polytropic equation of state for the core must be,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>P</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
K_\mathrm{core}\biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \rho \biggr]^{6 / 5} \, .
</math>
  </td>
</tr>
</table>
In our toy model, this means that,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>K_\mathrm{core}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
P \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \rho \biggr]^{-6 / 5} 
=
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{6 / 5} 4.42\times 10^{15} ~\mathrm{g~cm^{-1}~s^{-2}} 
\biggl[ 4.64~\mathrm{g~cm^{-3}} \biggr]^{-6 / 5}
=
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{6 / 5} 7.008\times 10^{14} \mathrm{g^{-1 / 5}~cm^{13 / 5}~s^{-2}}
\, .
</math>
  </td>
</tr>
</table>
Hence, the ratio,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\frac{K_\mathrm{env}}{K_\mathrm{core}}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2.06 \times 10^{14}~\mathrm{cm^5~g^{-1}~s^{-2}}
\biggl\{\biggl(\frac{\mu_e}{\mu_c}\biggr)^{6 / 5} 7.008\times 10^{14} \mathrm{g^{-1 / 5}~cm^{13 / 5}~s^{-2}}\biggr\}^{-1}
=
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-6 / 5}0.293 ~\mathrm{g^{-4 / 5}~cm^{12 / 5}}
\, ,
</math>
  </td>
</tr>
</table>
which may be rewritten as,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\frac{K_\mathrm{env}}{K_\mathrm{core}}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{4/ 5}0.293 \biggl[\mathrm{g~cm^{-3}} \biggr]^{- 4 / 5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
=\biggl[ \underbrace{ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\pi}{4}\cdot \frac{M_\odot}{R_\odot^3} }_\mathrm{central~ density~of~n = 5 ~ core} \biggr]^{- 4 / 5}
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \, .
</math>
  </td>
</tr>
</table>

This matches the [[SSC/Structure/BiPolytropes/Analytic51#Step_5:_Interface_Conditions|interface condition expression]] that relates <math>K_\mathrm{core}</math> to <math>K_\mathrm{env}</math>,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\frac{K_e}{K_c}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\rho_c^{-4 / 5} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \theta_i^{-4} \, ,
</math>
  </td>
</tr>
</table>
where we appreciate that, in the initial bipolytropic configuration (i.e., when <math>\xi_i = 0</math>), <math>\theta_i = 1</math>.

===Adopted Normalizations===
We adopt the normalizations in which we hold <math>M_\mathrm{tot}</math> and <math>K_c</math> constant, as [[SSC/Structure/BiPolytropes/Analytic51Renormalize#NewNormalization|derived in an accompanying discussion]].  That is,

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

<tr>
  <td align="right"><math>r_\mathrm{norm} \equiv \biggl[\biggl( \frac{G}{K_c} \biggr)^{5 / 2} M_\mathrm{tot}^{2}  \biggr]</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>
\biggl[ 6.674\times 10^{-8}~\mathrm{cm^3~g^{-1}~s^{-2}}\biggr]^{5 / 2}  \times \biggl[
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{6 / 5} 7.008\times 10^{14} \mathrm{g^{-1 / 5}~cm^{13 / 5}~s^{-2}} \biggr]^{-5 / 2} 
\biggl( 1.99\times 10^{33}~\mathrm{g} \biggr)^{2} 
</math></td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3} 3.505\times 10^{11}~\mathrm{cm} 
\, ;
</math></td>
</tr>

<tr>
  <td align="right"><math>\rho_\mathrm{norm} \equiv \biggl[\biggl( \frac{K_c}{G} \biggr)^{3 / 2} \frac{1}{M_\mathrm{tot}} \biggr]^{5}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl\{
\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{6 / 5} 7.008\times 10^{14} \mathrm{g^{-1 / 5}~cm^{13 / 5}~s^{-2}} \biggr]^{3 / 2} 
\biggl[ 6.674\times 10^{-8}~\mathrm{cm^3~g^{-1}~s^{-2}} \biggr]^{-3 / 2}
\biggl[1.99\times 10^{33}~\mathrm{g}\biggr]^{-1}
\biggr\}^5 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{9} 0.0462 ~\mathrm{g}~\mathrm{cm^{-3}} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>P_\mathrm{norm} \equiv \biggl[K_c^{10} G^{-9} M_\mathrm{tot}^{-6}  \biggr]</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>
\biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{6 / 5} 7.008\times 10^{14} \mathrm{g^{-1 / 5}~cm^{13 / 5}~s^{-2}} \biggr]^{10} 
\biggl[ 6.674\times 10^{-8}~\mathrm{cm^3~g^{-1}~s^{-2}} \biggr]^{-9} 
\biggl[ 1.99\times 10^{33}~\mathrm{g} \biggr]^{-6} 
  </math></td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{12} \biggl\{
\biggl[ 7.008\times 10^{14}  \biggr] 
\biggl[ 6.674\times 10^{-8} \biggr]^{-0.9} 
\biggl[ 1.99\times 10^{33}~\biggr]^{-0.6} 
\biggr\}^{10}~\mathrm{g ~cm^{-1}~s^{-2} }
  </math></td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{12} 1.751\times 10^{13} ~\mathrm{g ~cm^{-1}~s^{-2} }\, .
  </math></td>
</tr>
</table>

<table border="1" align="center" cellpadding="5">

<tr>
  <td align="center">Parameter</td>
  <td align="center">[[SSC/Structure/BiPolytropes/Analytic51Renormalize#CoreParameters|Core]]</td>
  <td align="center">[[SSC/Structure/BiPolytropes/Analytic51Renormalize#EnvelopeParameters|Envelope]]</td>
</tr>

<tr>
  <td align="right">
<math>r = r_\mathrm{norm} \tilde{r} =</math>
  </td>
  <td align="center">
<math>r_\mathrm{norm} 
\biggl\{
\mathcal{m}_\mathrm{surf}^{-2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} 
\biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi 
\biggr\}
</math>
  </td>
  <td align="center">
<math> r_\mathrm{norm} 
\biggl\{
\mathcal{m}_\mathrm{surf}^{-2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3} \theta^{-2}_i (2\pi)^{-1/2}\eta 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\rho = \rho_\mathrm{norm} \tilde{\rho} =</math>
  </td>
  <td align="center">
<math>\rho_\mathrm{norm}
\biggl\{
\mathcal{m}_\mathrm{surf}^5 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-10} 
\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} 
\biggr\}
</math>
  </td>
  <td align="center">
<math>\rho_\mathrm{norm} 
\biggl\{
\mathcal{m}_\mathrm{surf}^5 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-9} \theta^{5}_i \phi 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P = P_\mathrm{norm} \tilde{P} =</math>
  </td>
  <td align="center">
<math>P_\mathrm{norm} 
\biggl\{
\mathcal{m}_\mathrm{surf}^6 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} 
\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3} 
\biggr\}
</math>
  </td>
  <td align="center">
<math>P_\mathrm{norm} 
\biggl\{
\mathcal{m}_\mathrm{surf}^6 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \theta^{6}_i \phi^{2} 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>M_r = M_\mathrm{tot} \tilde{M} =</math>
  </td>
  <td align="center">
<math>M_\mathrm{tot} 
\biggl\{
\mathcal{m}_\mathrm{surf}^{-1} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{2} 
\biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr]
\biggr\}
</math>
  </td>
  <td align="center">
<math>M_\mathrm{tot} 
\biggl\{
\mathcal{m}_\mathrm{surf}^{-1}~ \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr) 
\biggr\}
</math>
  </td>
</tr>
</table>

'''Polytropic Functions and Their Derivatives &hellip;'''
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">Core:</td>
  <td align="center">
<math>
\theta(\xi) = \biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-1/2}
</math>
  </td>
<td align="center" width="10%">&nbsp;</td>
  <td align="center">
<math>
\frac{d\theta}{d\xi} = - \frac{\xi}{3}\biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-3/2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">Envelope:</td>
  <td align="center">
<math>
\phi = A \biggl[ \frac{\sin(\eta - B)}{\eta} \biggr]
</math>
  </td>
<td align="center">&nbsp;</td>
  <td align="center">
<math>
\frac{d\phi}{d\eta} = \frac{A}{\eta^2} \biggl[ \eta\cos(\eta-B) - \sin(\eta-B) \biggr] 
</math>
  </td>
</tr>
</table>

'''Key Relations'''; numerical values in parentheses assume <math>(\mu_e/\mu_c) = \tfrac{1}{2}</math> and <math>\xi_i = 2.279258</math> &hellip; [[SSC/Stability/BiPolytropes#Equilibrium_Properties_of_Marginally_Unstable_Models|cross-check here]]:
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>\eta 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
3^{1 / 2}\biggl(\frac{\mu_e}{\mu_c}\biggr) \theta_i^2 \xi \, ,
</math>
  </td>
  <td align="center" width="10%">&nbsp;</td>
  <td align="left">(0.722596)</td>
</tr>

<tr>
  <td align="right"><math>\Lambda_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\frac{\xi_i}{\sqrt{3}} 
\biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}\frac{1}{\theta_i^2 \xi_i^2} - 1\biggr] \, ,</math></td>
  <td align="center" width="10%">&nbsp;</td>
  <td align="left">(0.067969)</td>
</tr>

<tr>
  <td align="right"><math>A</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math> 
\eta_i\biggl(1 + \Lambda_i^2\biggr)^{1 / 2} \, ,</math></td>
  <td align="center" width="10%">&nbsp;</td>
  <td align="left">(0.724263)</td>
</tr>

<tr>
  <td align="right"><math>B</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\eta_i + \tan^{-1}(\Lambda_i) - \frac{\pi}{2} \, ,</math></td>
  <td align="center" width="10%">&nbsp;</td>
  <td align="left">(-0.780336)</td>
</tr>

<tr>
  <td align="right"><math>\eta_s</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math> 
\frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \, ,</math></td>
  <td align="center" width="10%">&nbsp;</td>
  <td align="left">(2.361257)</td>
</tr>

<tr>
  <td align="right"><math>\mathcal{m}_\mathrm{surf}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i} \, ,</math></td>
  <td align="center" width="10%">&nbsp;</td>
  <td align="left">(2.255246)</td>
</tr>

<tr>
  <td align="right"><math>\frac{\rho_c}{\bar\rho}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s^2}{3A\theta_i^5} \, ,</math></td>
  <td align="center" width="10%">&nbsp;</td>
  <td align="left">(63.29514)</td>
</tr>

<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>
  <td align="center" width="10%">&nbsp;</td>
  <td align="left">(0.401776)</td>
</tr>

<tr>
  <td align="right"><math>q \equiv \frac{r_\mathrm{core}}{R}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr) \sqrt{3} ~ \biggl[ \frac{\xi_i \theta_i^2}{\eta_s}\biggr] \, .</math></td>
  <td align="center" width="10%">&nbsp;</td>
  <td align="left">(0.306022)</td>
</tr>

<tr>
  <td align="right"><math>R_\mathrm{surf}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>r_\mathrm{norm} 
\biggl\{
\mathcal{m}_\mathrm{surf}^{-2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3} \theta^{-2}_i (2\pi)^{-1/2}\eta_s 
\biggr\}</math></td>
  <td align="center" width="10%">&nbsp;</td>
  <td align="left"><math>(0.063242 ~r_\mathrm{norm} = 1.773\times 10^{11} ~\mathrm{cm} = 2.548~R_\odot)</math></td>
</tr>
</table>

===Example Evolution===

Let's attempt to construct the post-main-sequence evolutionary trajectory for <math>1 M_\odot</math> stars having a couple of different values of the mean-molecular weight jump, <math>\mu_e/\mu_c</math>.  This will be for comparison with the <math>1 M_\odot</math> evolutionary trajectory published by {{ Iben67 }}; hereafter, {{ Iben67hereafter }}.  A remake of this published trajectory is [[#IbenFigure|displayed below]].  

Our toy model  does not handle radiation, so we cannot plot luminosity versus effective (surface) temperature, that is, we cannot generate a traditional HR diagram.  However, our toy model predicts how the radius of the star should vary with the mass of its core.  In the following figure, the solid-green circular markers show how the fractional mass, <math>\nu</math>, varies with <math>R_\mathrm{surf}/R_\odot</math> for the model sequence in which <math>\mu_e/\mu_c = 1 / 2</math> &#8212; note, for example, that among our above numerically evaluated "key relations", <math>(\nu, R_\mathrm{surf}/R_\odot) = (0.40178, 2.548)</math>; this point along the curve is marked by the solid-red circular marker.  The solid-yellow circular markers show the same information for our toy model sequence in which <math>\mu_e/\mu_c = 1 / 4</math>.

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="center">
[[File:EvolveComparison3Labeled.png|center|500px|One Solar Mass Evolution]]
  </td>
  <td align="center">
<table border="1" align="center" cellpadding="8">

<tr>
  <td align="center" colspan="6">Parameters values deduced from Figure 1 and Table 1 of {{ Iben67figure }}</td>
</tr>

<tr>
  <td align="center">Interval</td>
  <td align="center"><math>(\Delta t)_9</math></td>
  <td align="center"><math>\biggl[\log \biggl(\frac{L}{L_\odot}\biggr)\biggr]_\mathrm{avg}</math></td>
  <td align="center"><math>\biggl[\log \biggl(\frac{R}{R_\odot}\biggr)\biggr]_\mathrm{avg}</math></td>
  <td align="center"><math>\Delta (M/M_\odot)</math></td>
  <td align="center"><math>\nu</math></td>
</tr>

<tr>
  <td align="center">1 - 2</td>
  <td align="center">3.770</td>
  <td align="center">-0.070</td>
  <td align="center">-0.05</td>
  <td align="center">0.0311</td>
  <td align="center">0.0311</td>
</tr>

<tr>
  <td align="center">2 - 3</td>
  <td align="center">2.889</td>
  <td align="center">0.06</td>
  <td align="center">-0.003</td>
  <td align="center">0.0322</td>
  <td align="center">0.0633</td>
</tr>

<tr>
  <td align="center">3 - 4</td>
  <td align="center">1.462</td>
  <td align="center">0.170</td>
  <td align="center">0.044</td>
  <td align="center">0.0210</td>
  <td align="center">0.0842</td>
</tr>

<tr>
  <td align="center">4 - 5</td>
  <td align="center">1.029</td>
  <td align="center">0.275</td>
  <td align="center">0.0975</td>
  <td align="center">0.0188</td>
  <td align="center">0.1030</td>
</tr>

<tr>
  <td align="center">5 - 6</td>
  <td align="center">0.7018</td>
  <td align="center">0.38</td>
  <td align="center">0.170</td>
  <td align="center">0.0163</td>
  <td align="center">0.1193</td>
</tr>

<tr>
  <td align="center">6 - 7</td>
  <td align="center">0.292</td>
  <td align="center">0.445</td>
  <td align="center">0.253</td>
  <td align="center">0.0079</td>
  <td align="center">0.1272</td>
</tr>

<tr>
  <td align="center">7 - 10</td>
  <td align="center">0.157</td>
  <td align="center">0.455</td>
  <td align="center">0.333</td>
  <td align="center">0.0043</td>
  <td align="center">0.1316</td>
</tr>

<tr>
  <td align="center">10 - 11</td>
  <td align="center">0.213</td>
  <td align="center">0.51</td>
  <td align="center">0.432</td>
  <td align="center">0.0067</td>
  <td align="center">0.1383</td>
</tr>

<tr>
  <td align="center">11 - 12</td>
  <td align="center">0.185</td>
  <td align="center">0.68</td>
  <td align="center">0.560</td>
  <td align="center">0.0086</td>
  <td align="center">0.1468</td>
</tr>

<tr>
  <td align="center">12 - 13</td>
  <td align="center">0.125</td>
  <td align="center">0.925</td>
  <td align="center">0.708</td>
  <td align="center">0.0102</td>
  <td align="center">0.1570</td>
</tr>
</table>
  </td>
</tr>
</table>

The parameter values displayed in the table shown here &#8212; immediately on the right of our figure &#8212; were determined as follows, referencing the data from [[#IbenFigure|Iben67 presented below]]:
<ul>
  <li>Interval: &#8212;The points displayed along the <math>1 M_\odot</math> evolutionary trajectory in Figure 1 of {{ Iben67hereafter }} are numbered in chronological order, from 1 to 13 (except points 8 and 9 are not included).  Here, our parameter values reference the ''time intervals'' between adjacent points; for example, our table refers to interval "1 - 2," interval "2 - 3," etc.</li>
  <li><math>(\Delta t)_9</math>: &nbsp; This refers to the amount of evolutionary time that separates the two points of an identified interval.  For example, Table 1 of {{ Iben67hereafter }} states that the star's age at "point 1" along its evolutionary trajectory is <math>0.0506\times 10^{9}~\mathrm{yrs}</math> and at "point 2" its age is <math>3.8209\times 10^{9}~\mathrm{yrs}</math>; hence the interval of time between points 1 and 2 is <math>(\Delta t)_9 = (3.8209 - 0.0506)\times 10^{9}~\mathrm{yrs}= 3.770 \times 10^{9}~\mathrm{yrs}</math>.</li>
  <li><math>[\log_{10}(L/L_\odot)]_\mathrm{avg}</math>:&nbsp; We have performed a linear interpolation ''in log<sub>10</sub>'' of the luminosity listed for each point along the evolutionary trajectory in order to obtain an ''average'' value of the luminosity across each interval.  For example, the star's luminosity at "point 1" is approximately <math>\log(L/L_\odot) \approx -0.140</math> while at "point 2" it is <math>\log(L/L_\odot) \approx 0.0</math>.  Hence, while evolving across the 1 - 2 interval, we assume that its luminosity is constant and has the value, <math>[\log_{10}(L/L_\odot)]_\mathrm{avg} = \tfrac{1}{2}[0.0 + (- 0.140)] = -0.07</math>.</li>
  <li><math>[\log_{10}(R/R_\odot)]_\mathrm{avg}</math>:&nbsp; The average stellar radius over an interval is obtained in the same way.  For example, while evolving across the 1 - 2 interval, we assume that its radius is constant and has the value, <math>[\log_{10}(R/R_\odot)]_\mathrm{avg} = \tfrac{1}{2}[-0.073 + (- 0.027)] = -0.05</math>.</li>
  <li><math>\Delta (M/M_\odot)</math>: &nbsp; For a given interval, if we multiply the star's average luminosity times the time spent crossing that interval, we obtain a reasonable estimate of the total amount of energy, <math>E_\mathrm{\Delta t}</math>, that the star has released over that time interval.  Presumably, the same amount of energy was generated via nuclear reactions over that interval.  Assuming that the primary reaction is the conversion of hydrogen into helium, the conversion of mass into energy occurs at an efficiency of 0.7%, that is, a reasonable estimate of the amount of mass that gets burned during that interval is,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>0.007 (\Delta M)c^2</math></td>
  <td align="right"><math>\approx</math></td>
  <td align="left"><math>\Delta t \cdot L_\mathrm{avg}</math></td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{\Delta M}{M_\odot}</math></td>
  <td align="right"><math>\approx</math></td>
  <td align="left"><math>\biggl[\frac{(\Delta t)_9 \times 3.156\times 10^{16}~\mathrm{s}}{0.007 c^2}\biggr] \cdot 10^{[\log(L/L_\odot)]_\mathrm{avg}} \biggl\{ \frac{L_\odot}{M_\odot} \biggr\}
= 0.00970 \times (\Delta t)_9\cdot 10^{[\log(L/L_\odot)]_\mathrm{avg}} \, .
</math></td>
</tr>
</table>
For example, for interval 1 - 2, <math>\Delta M/M_\odot \approx (0.0097 \times 3.770 \times 10^{-0.070}) = 0.0311</math>.
</li>
  <li><math>\nu</math>: &nbsp; This is the fraction of the star's total mass that has accumulated in the core by the end of the specified time interval; in our toy model, the total mass is <math>1 M_\odot</math> so <math>\nu</math> is also the core mass expressed in solar masses.  Here the value of <math>\nu</math> is obtained by summing over the mass increments, <math>\Delta(M/M_\odot)</math>, that have been burned at all previous evolutionary time intervals, including the present one; for example, at the end of time interval 3 - 4, <math>\nu = 0.0311+0.0322+0.0210 = 0.0842</math>.</li>
</ul>

===Commentary===

====Evolutionary Tracks====
When we embarked on this investigation, we thought that it would be difficult to quantitatively compare the evolution of our <math>(n_c, n_e) = (5, 1)</math> bipolytrope with more realistic stellar evolutionary tracks &#8212; as published, for example, by {{ Iben67full }} &#8212; because our toy model does not provide a mechanism for assessing the variation in time of the configuration's surface temperature.  In the preceding subsection of this discussion, we have demonstrated that a comparison of evolutions is possible if we focus on an examination of how the star's radius varies as the mass of its core steadily increases.  

Not surprisingly, we find that the ''trend'' is the same in our toy model as it is in {{ Iben67hereafter }}'s <math>1 M_\odot</math> model: &nbsp; as the core's mass monotonically increases, the star's radius monotonically increases as well.  But there appear to be two significant mismatches.  (1) In the earliest stage of its evolution, {{ Iben67hereafter }}'s model exhibits a steeper rise in the mass of the core for a given radius.  (2) If we set <math>\mu_e/\mu_c = 1/2</math> &#8212; which is the ''expected'' value when considering the fusion of hydrogen into helium &#8212; overall, our toy model dumps significantly more mass into the core than does {{ Iben67hereafter }}'s evolutionary model. Dropping the mean-molecular-weight ratio from <math>1 / 2</math> to <math>1 / 4</math> provides a better match, but it is difficult to come up with an astrophysical argument that would justify such a low ratio. 

Given that the ''trend'' matches, we will consider this a win!  Hopefully a more quantitative match will be obtained when we switch to a bipolytrope that has an isothermal <math>(n_c = \infty)</math> core, instead of an <math>n_c = 5</math> core.

====Dynamical Instability====

We have placed one solid-red circular marker on each of the evolutionary tracks that have been drawn from our toy model.  In both cases, this marks the point along the track &#8212; at a radius less than <math>4 R_\odot</math> &#8212; where the <math>(n_c, n_e) = (5, 1)</math> model becomes dynamically unstable.  It is tempting to suggest that this indicates that {{ Iben67hereafter }}'s model should also become dynamically unstable before expanding to a radius greater than <math>4 R_\odot</math>.  Let's pursue building a bipolytrope that has an isothermal core and see if that system strengthens this argument.