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Created page with "__FORCETOC__  =Stability of a BiPolytrope with an Isothermal Core= This analysis pulls largely from {{ Yabushita75full }}; the focus is on bipolytropes having <math>(n_c, n_e) = (\infty, \tfrac{3}{2})</math>. In an accompanying discussion, we summarize the steps that Yabushita took — from 1968 and 1974, to 1975 — that led up to his discovery of an analytic description of the..."

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__FORCETOC__


=Stability of a BiPolytrope with an Isothermal Core=
This analysis pulls largely from {{ Yabushita75full }}; the focus is on bipolytropes having <math>(n_c, n_e) = (\infty, \tfrac{3}{2})</math>. In an [[SSC/Stability/Isothermal#Yabushita_(1975)|accompanying discussion]], we summarize the steps that Yabushita took — from 1968 and 1974, to 1975 — that led up to his discovery of an analytic description of the isothermal displacement function.

==Equilibrium Structure==
We will follow [[SSC/Structure/BiPolytropes#Setup|the accompanying formal recipe]] for building a bipolytropic model, using the [[SSC/Structure/BiPolytropes/Analytic1.53#Our_Derivation|step-by-step construction of Milne's (1930)]] configurations as a guide.

===Step 1===

The {{ Yabushita75 }} bipolytrope has an isothermal core <math>(n_c = \infty)</math> and an <math>n_e = \tfrac{3}{2}</math> polytropic envelope.

===Steps 2 & 3===

Throughout the core, the properties of this bipolytrope can be expressed in terms of the Lane-Emden function, <math>\psi(\chi)</math>, which derives from a solution of the 2<sup>nd</sup>-order ODE,
<div align="center" id="Chandrasekhar">
<font color="maroon"><b>Isothermal Lane-Emden Equation</b></font><br />
{{ Math/EQ_SSLaneEmden02 }}
</div>
subject to the boundary conditions,
<div align="center">
<math>\psi = 1</math>       and       <math>\frac{d\psi}{d\xi} = 0</math>
      at       <math>\xi = 0</math>.
</div>

As has been demonstrated in [[Appendix/Ramblings/PowerSeriesExpressions#IsothermalLaneEmden|our accompanying ''mathematics'' appendix]], a series expansion about the center of the isothermal configuration gives,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\psi(\xi)
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>~\frac{\xi^2}{6} - \frac{\xi^4}{120} + \frac{\xi^6}{1890} - \frac{61 \xi^8}{1,632,960} + \cdots \, .</math>
</td>
</tr>
</table>

The solution, <math>\psi(\xi)</math>, extends to infinity, so the interface between the core and the envelope can be positioned anywhere within the range, <math>0 < \xi_i < \infty</math>.

===Step4: Throughout the core===

<div align="center">
<table border="0" cellpadding="3" width="50%">
<tr>
<td align="center" colspan="4">
Specify: <math>c_s^2</math> and <math>\rho_0 ~\Rightarrow</math>
</td>
</tr>

<tr>
<td align="right">
<math>r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{c_s^2}{4\pi G\rho_0} \biggr]^{1/2} \xi</math>
</td>
<td align="right">(2.2)</td>
</tr>

<tr>
<td align="right">
<math>\rho</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0 e^{-\psi}</math>
</td>
<td align="right">(2.2)</td>
</tr>

<tr>
<td align="right">
<math>P</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>c_s^2 \rho_0 e^{-\psi}</math>
</td>
<td align="right">(2.1)</td>
</tr>

<tr>
<td align="right">
<math>M_r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{c_s^6}{4\pi G^3\rho_0} \biggr]^{1/2} \biggl( \xi^2 \frac{d\psi}{d\xi} \biggr)</math>
</td>
<td align="right">(2.3)</td>
</tr>
<tr><td colspan="4" align="center">{{ Yabushita75 }}, § 2, pp. 442-443</td></tr>
</table>
</div>
After adopting the substitute notation, <math>c_s^2 \rightarrow K_1</math> and <math>\rho_0 \rightarrow \lambda_c</math>, it is clear that these last four parameter-profile expressions are identical to the ones that appear, respectively, as equations (2.2), (2.1) and (2.3) of {{ Yabushita75 }}.

===Step 5: Interface Conditions===

<div align="center">

<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>\biggl( \frac{\rho_0}{\mu_c} \biggr) e^{-\psi_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_e}{\mu_e} \biggr) \theta^{n_e}_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_e}{\mu_e} \biggr) \theta^{3 / 2}_i</math>
</td>
</tr>

<tr>
<td align="right">
<math>c_s^2 \rho_0 e^{-\psi_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>K_e \rho_e^{1+1/n_e} \theta^{n_e + 1}_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>K_e \rho_e^{5 / 3} \theta^{5 / 2}_i</math>
</td>
</tr>

<tr>
<td align="right">
<math>\biggl[ \frac{c_s^2}{4\pi G\rho_0} \biggr]^{1/2} \xi_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{(n_e + 1)K_e}{4\pi G} \biggr]^{1/2} \rho_e^{(1-n_e)/(2n_e)} \eta_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{(5 / 2)K_e}{4\pi G} \biggr]^{1/2} \rho_e^{(-1 / 6)} \eta_i</math>
</td>
</tr>

<tr>
<td align="right">
<math>\biggl[ \frac{c_s^6}{4\pi G^3\rho_0} \biggr]^{1/2} \biggl( \xi^2 \frac{d\psi}{d\xi} \biggr)_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>4\pi \biggl[ \frac{(n_e + 1)K_e}{4\pi G} \biggr]^{3/2} \rho_e^{(3-n_e)/(2n_e)} \biggl(-\eta^2 \frac{d\theta}{d\eta} \biggr)_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>4\pi \biggl[ \frac{(5 / 2)K_e}{4\pi G} \biggr]^{3/2} \rho_e^{1 / 2} \biggl(-\eta^2 \frac{d\theta}{d\eta} \biggr)_i</math>
</td>
</tr>
</table>


</div>

This means that,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\frac{\rho_e}{\rho_0}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)e^{-\psi_i} \theta_i^{-3 / 2}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right"><math>K_e
\biggl[\rho_0 \biggl(\frac{\mu_e}{\mu_c}\biggr)e^{-\psi_i} \theta_i^{-3 / 2} \biggr]^{5 / 3} \theta^{5 / 2}_i</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
c_s^2 \rho_0 e^{-\psi_i}
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Rightarrow ~~~ \frac{K_e \rho_0^{2 / 3}}{c_s^2}
</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-5 / 3}e^{+2\psi_i/3}
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Rightarrow ~~~ \frac{c_s^2}{K_e}
</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{5 / 2} \rho_0e^{-\psi_i}
\biggr]^{2 / 3}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right"><math>\frac{\eta_i }{\xi_i}
</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[ \frac{c_s^2}{4\pi G\rho_0} \biggr]^{1/2} \biggl[ \frac{4\pi G}{(5 / 2)K_e} \biggr]^{1/2}
\biggl[ \rho_e \biggr]^{1 / 6}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl(\frac{2}{5}\biggr)^{1 / 2}
\biggl[ \frac{c_s^2}{K_e} \biggr]^{1/2}
\biggl[ \rho_0 \biggl(\frac{\mu_e}{\mu_c}\biggr)e^{-\psi_i} \theta_i^{-3 / 2} \biggr]^{1 / 6}\rho_0^{- 1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl(\frac{2}{5}\biggr)^{1 / 2}
\biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{5 / 2} \rho_0e^{-\psi_i} \biggr]^{1/3}
\biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{1 / 6}e^{-\psi_i / 6} \theta_i^{-1 / 4} \biggr]\rho_0^{- 1 / 3}
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl(\frac{2}{5}\biggr)^{1 / 2}
\biggl(\frac{\mu_e}{\mu_c}\biggr) e^{-\psi_i/ 2}
\theta_i^{-1 / 4}
\, .
</math>
</td>
</tr>
</table>
And, finally,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right">
<math>
\biggl(-\eta^2 \frac{d\theta}{d\eta} \biggr)_i
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\frac{1}{4\pi}\biggl[ \frac{c_s^6}{4\pi G^3\rho_0} \biggr]^{1/2}
\biggl[ \frac{4\pi G}{(5 / 2)K_e} \biggr]^{3/2}
\biggl[\rho_e\biggr]^{- 1 / 2}
\biggl( \xi^2 \frac{d\psi}{d\xi} \biggr)_i
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>\biggl(\frac{2}{5}\biggr)^{3 / 2}\rho_0^{- 1 / 2}
\biggl[ \frac{c_s^2}{K_e} \biggr]^{3/2}
\biggl[\rho_0 \biggl(\frac{\mu_e}{\mu_c}\biggr)e^{-\psi_i} \theta_i^{-3 / 2}\biggr]^{- 1 / 2}
\biggl( \xi^2 \frac{d\psi}{d\xi} \biggr)_i
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>\biggl(\frac{2}{5}\biggr)^{3 / 2}\rho_0^{- 1 }
\biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{5 / 2} \rho_0e^{-\psi_i} \biggr]
\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)e^{-\psi_i} \theta_i^{-3 / 2}\biggr]^{- 1 / 2}
\biggl( \xi^2 \frac{d\psi}{d\xi} \biggr)_i
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>\biggl(\frac{2}{5}\biggr)^{3 / 2}
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{2} e^{-\psi_i / 2} \theta_i^{3 / 4}
\biggl( \xi^2 \frac{d\psi}{d\xi} \biggr)_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~
\biggl(-\frac{d\theta}{d\eta} \biggr)_i
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>\biggl(\frac{2}{5}\biggr)^{3 / 2}
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{2} e^{-\psi_i / 2} \theta_i^{3 / 4}
\biggl( \frac{d\psi}{d\xi} \biggr)_i
\biggl[ \biggl(\frac{2}{5}\biggr)^{1 / 2}
\biggl(\frac{\mu_e}{\mu_c}\biggr) e^{-\psi_i/ 2}
\theta_i^{-1 / 4}
\biggr]^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>\biggl(\frac{2}{5}\biggr)^{1 / 2}
e^{+ \psi_i / 2} \theta_i^{5 / 4}
\biggl( \frac{d\psi}{d\xi} \biggr)_i
\, .
</math>
</td>
</tr>
</table>

<table border="1" align="center" cellpadding="8">

<tr>
<td align="center" colspan="3">Summary Interface Relations</td>
</tr>

<tr>
<td align="center" colspan="1">Our Derivations</td>
<td align="center" colspan="1">After setting <math>\rho_e = \rho_0</math></td>
<td align="center" colspan="1">Presented by {{ Yabushita75 }}<br />(after setting <math>\mu_e/\mu_c = 1</math>)</td>
</tr>

<tr>



<td align="center" colspan="1">
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\frac{\rho_e}{\rho_0}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)e^{-\psi_i} \theta_i^{-3 / 2}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right"><math>\frac{c_s^2}{K_e}
</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl[
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{5 / 2} \rho_0e^{-\psi_i}
\biggr]^{2 / 3}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right"><math>\frac{\eta_i}{\xi_i}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl(\frac{2}{5}\biggr)^{1 / 2}
\biggl(\frac{\mu_e}{\mu_c}\biggr) e^{-\psi_i/ 2}
\theta_i^{-1 / 4}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\biggl(-\frac{d\theta}{d\eta} \biggr)_i
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>\biggl(\frac{2}{5}\biggr)^{1 / 2}
e^{+ \psi_i / 2} \theta_i^{5 / 4}
\biggl( \frac{d\psi}{d\xi} \biggr)_i
\, .
</math>
</td>
</tr>
</table>
</td>



<td align="center" colspan="1">
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\theta_i^{3 / 2}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)e^{-\psi_i}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right"><math>\biggl[ \frac{c_s^2}{K_e} \biggr]^{3}
</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{5 } \biggl[\rho_0e^{-\psi_i} \biggr]^{2}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right"><math>\frac{\eta_i}{\xi_i}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl(\frac{2}{5}\biggr)^{1 / 2}
\biggl(\frac{\mu_e}{\mu_c}\biggr) e^{-\psi_i/ 2}
\biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)e^{-\psi_i} \biggr]^{-1 / 6}
=
\biggl(\frac{2}{5}\biggr)^{1 / 2}
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{5 / 6} e^{-\psi_i/ 3}
\, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\biggl(-\frac{d\theta}{d\eta} \biggr)_i
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl(\frac{2}{5}\biggr)^{1 / 2}
\biggl( \frac{d\psi}{d\xi} \biggr)_i
e^{+ \psi_i / 2} \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)e^{-\psi_i} \biggr]^{5 / 6}
=
\biggl(\frac{2}{5}\biggr)^{1 / 2}
\biggl( \frac{d\psi}{d\xi} \biggr)_i
e^{- \psi_i / 3} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{5 / 6}
\, .
</math>
</td>
</tr>
</table>
</td>



<td align="center" colspan="1">
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\theta_i^{3 / 2}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
e^{-\psi_i}
\, ;
</math>
</td>
<td align="right" width="5%">(2.9)</td>
</tr>

<tr>
<td align="right"><math>\biggl[ \frac{c_s^2}{K_e} \biggr]^{3 / 2}
</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\lambda_c e^{-\psi_i}
\, ;
</math>
</td>
<td align="right" width="5%">(2.11)</td>
</tr>

<tr>
<td align="right"><math>\frac{\eta_i}{\xi_i}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl(\frac{2}{5}\biggr)^{1 / 2}e^{-\psi_i/ 3}
\, ;
</math>
</td>
<td align="right" width="5%">(2.12)</td>
</tr>

<tr>
<td align="right">
<math>
\biggl(\frac{d\theta}{d\eta} \biggr)_i
</math>
</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
-\biggl(\frac{2}{5}\biggr)^{1 / 2} \biggl( \frac{d\psi}{d\xi} \biggr)_i e^{- \psi_i / 3}
\, .
</math>
</td>
<td align="right" width="5%">(2.13)</td>
</tr>
</table>
</td>
</tr>
</table>

===Step 8: Throughout the Envelope===

Throughout the envelope, we seek the solution, <math>\theta(\eta)</math>, of the following Lane-Emden equation:
<div align="center">
<math>
\frac{1}{\eta^2} \frac{d}{d\eta} \biggl( \eta^2 \frac{d\theta}{d\eta} \biggr) = - \theta^{3 / 2} \, .
</math>
</div>

For the envelope, the [[SSC/Structure/BiPolytropes#Setup|associated key parameter relations]] are:
<table border="0" align="center" cellpadding="3">

<tr>
<td align="right">
<math>\rho</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_e \theta^{3 / 2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>P</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>K_e \rho_e^{5 / 3} \theta^{5 / 2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{(5/2)K_e}{4\pi G} \biggr]^{1/2} \rho_e^{- 1 / 6} \eta</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>4\pi \biggl[ \frac{(5/2)K_e}{4\pi G} \biggr]^{3/2} \rho_e^{1 / 2} \biggl(-\eta^2 \frac{d\theta}{d\eta} \biggr)</math>
</td>
</tr>
</table>
The surface occurs where the polytropic Lane-Emden function, <math>\theta</math>, first goes to zero. We will denote the radius at which this occurs as <math>\eta_s</math> — {{ Yabushita75 }} denotes the same radial location by <math>\eta_1</math> — and the slope of the function at the surface will be denoted as <math>(d\theta/d\eta)_s</math>.

===Normalizations===

Following {{ Yabushita75 }}, along our model sequence we will hold <math>c_s^2</math> and <math>K_e</math> fixed. From the summary interface relations, then, we find that,
<table border="0" align="center" cellpadding="8">

<tr>
<td align="right"><math>\biggl[ \frac{c_s^2}{K_e} \biggr]^{3}
</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{5 } \biggl[\rho_0e^{-\psi_i} \biggr]^{2}
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Rightarrow ~~~\rho_0</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{- 5 / 2} \biggl[ \frac{c_s^2}{K_e} \biggr]^{3 / 2} e^{\psi_i}
\, .
</math>
</td>
</tr>
</table>
This is identical to Eq. (3.1) of {{ Yabushita75 }} if we assume (as did {{ Yabushita75hereafter }}) that <math>\mu_e/\mu_c = 1</math>. Given that, following {{ Yabushita75hereafter }}, we have set <math>\rho_e = \rho_c</math>, the radius of the equilibrium configuration is given by the expression,

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

<tr>
<td align="right">
<math>R^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{5K_e}{8\pi G} \biggr] \rho_e^{- 1 / 3} \eta_s^2</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{5K_e}{8\pi G} \biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{- 5 / 2}
\biggl( \frac{c_s^2}{K_e} \biggr)^{3 / 2} e^{\psi_i} \biggr]^{- 1 / 3} \eta_s^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{5K_e}{8\pi G} \biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{5 / 6}
\biggl( \frac{c_s^2}{K_e} \biggr)^{- 1 / 2} e^{- \psi_i/3} \biggr] \eta_s^2
\, ;
</math>
</td>
</tr>
</table>
and the configuration's total mass is,

<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>4\pi \biggl[ \frac{5K_e}{8\pi G} \biggr]^{3/2} \biggl[ \rho_e \biggr]^{1 / 2} \biggl(-\eta^2 \frac{d\theta}{d\eta} \biggr)_s</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>4\pi \biggl[ \frac{5K_e}{8\pi G} \biggr]^{3/2} \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{- 5 / 2}
\biggl( \frac{c_s^2}{K_e} \biggr)^{3 / 2} e^{\psi_i} \biggr]^{1 / 2} \biggl(-\eta^2 \frac{d\theta}{d\eta} \biggr)_s
\, .
</math>
</td>
</tr>
</table>
If we again set <math>\mu_e/\mu_c = 1</math>, this expression is identical to Eq. (3.2) of {{ Yabushita75 }}. The configuration's mean density is,

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

<tr>
<td align="right">
<math>\bar\rho \equiv \frac{3M_\mathrm{tot}}{4\pi R^3} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>3\biggl[ \frac{5K_e}{8\pi G} \biggr]^{3/2} \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{- 5 / 2}
\biggl( \frac{c_s^2}{K_e} \biggr)^{3 / 2} e^{\psi_i} \biggr]^{1 / 2} \biggl(-\eta^2 \frac{d\theta}{d\eta} \biggr)_s
\biggl\{
\frac{5K_e}{8\pi G} \biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{5 / 6}
\biggl( \frac{c_s^2}{K_e} \biggr)^{- 1 / 2} e^{- \psi_i/3} \biggr] \eta_s^2
\biggr\}^{-3 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>3 \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{- 5 / 4}
\biggl( \frac{c_s^2}{K_e} \biggr)^{3 / 4} e^{+\psi_i / 2} \biggr] \biggl(-\frac{1}{\eta} \cdot \frac{d\theta}{d\eta} \biggr)_s
\biggl[\biggl(\frac{\mu_e}{\mu_c}\biggr)^{- 5 / 4}
\biggl( \frac{c_s^2}{K_e} \biggr)^{3 / 4} e^{+ \psi_i/2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>3 \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{- 5 / 2}
\biggl( \frac{c_s^2}{K_e} \biggr)^{3 / 2} e^{+\psi_i } \biggr] \biggl(-\frac{1}{\eta} \cdot \frac{d\theta}{d\eta} \biggr)_s
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \frac{\bar\rho}{\rho_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>3
\biggl(-\frac{1}{\eta} \cdot \frac{d\theta}{d\eta} \biggr)_s
\, .
</math>
</td>
</tr>
</table>

=See Also=

<ul>
<li>
[https://ui.adsabs.harvard.edu/abs/1974A%26A....32..309G/abstract M. Gabriel & M. L. Roth (1974, A&A, Vol. 32, p. 309)] … ''On the Secular Stability of Models with an Isothermal Core''
</li>
<li>[https://ui.adsabs.harvard.edu/abs/1967AnAp...30..975G/abstract M. Gabriel & P. Ledoux (1967, Annales d'Astrophysique, Vol. 30, p. 975)] … ''Sur la Stabilité Séculaire des Modeéles a Noyaux Isothermes''<br />
<p>
In § 1 (p. 442) of {{ Yabushita75 }} we find the following reference:   <font color="darkgreen">"A somewhat similar problem has been investigated by [https://ui.adsabs.harvard.edu/abs/1967AnAp...30..975G/abstract Gabriel & Ledoux (1967)]. Gaseous configurations with an isothermal core and polytropic envelope of index 3 were studied by {{ HC41 }} and by {{ SC42 }}. As is well known there is an upper limit (Schönberg-Chandrasekhar limit) to the mass of the core for the configurations to be in hydrostatic equilibria. Gabriel & Ledoux have investigated the stability of these configurations and have shown that secular stability is lost at the configuration that corresponds to the Schönberg-Chandrasekhar limit."</font>
</li>
</ul>

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