Editing SSC/Dynamics/IsothermalSimilaritySolution (section)

=Examine Connection With &hellip;=
Let's examine whether or not there is overlap between the properties of the above-discussed similarity solutions that give insight into the nonlinear dynamical behavior of collapse and the (a) known structure of the unperturbed, but marginally unstable Bonnor-Ebert sphere, and (b) eigenfunction that describes the radial profile of the marginally unstable radial pulsation mode.  Keep in mind that, as we have [[SSC/Stability/Isothermal#Overview|presented separately]], the truncation radius of the marginally unstable, Bonnor-Ebert sphere has, <math>~\xi_e \approx 6.4510534</math>.

==Pressure-Truncated Equilibrium Structure==
From our [[SSC/Stability/Isothermal#Groundwork|separate discussion of pressure-truncated isothermal spheres]], we can identify the following structural properties of the marginally unstable Bonnor-Ebert sphere.  The function, <math>~\psi(\xi)</math> satisfies the,
<div align="center">
<table border="0" cellpadding="8" align="center">
<tr><td align="center">
<font color="maroon">Isothermal Lane-Emden Equation</font> <p></p>
{{ Math/EQ_SSLaneEmden02 }}
</td></tr>
</table>
</div>

Given the system's sound speed, <math>~c_s</math>, and total mass, <math>~M_{\xi_e}</math>, the expression from [[SSC/Structure/BonnorEbert#Pressure|our presentation]] that shows how the bounding external pressure, <math>~P_e</math>, depends on the dimensionless Lane-Emden function, <math>~\psi</math>, is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P_e</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{c_s^8}{4\pi G^3 M_{\xi_e}^2} \biggr) ~\xi_e^4 \biggl(\frac{d\psi}{d\xi}\biggr)^2_e e^{-\psi_e} \, ;</math>
  </td>
</tr>
</table>
</div>

and, [[SSC/Structure/BonnorEbert#Radius|our expression]] for the truncated configuration's equilibrium radius is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~R</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{GM_{\xi_e}}{c_s^2} \biggl[ \xi \biggl(\frac{d\psi}{d\xi}\biggr) \biggr]_e^{-1} \, .</math>
  </td>
</tr>
</table>
</div>

Also, as has been summarized in our [[SSC/Structure/BonnorEbert#P-V_Diagram|accompanying discussion]], expressions that describe the general run of radius, pressure, and mass are, respectively,

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

<tr>
  <td align="right">
<math>~r_0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{c_s^2}{4\pi G \rho_c} \biggr)^{1/2} \xi \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~P_0 = c_s^2 \rho_0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(c_s^2 \rho_c) e^{-\psi} \, ;</math>
  </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_c} \biggr)^{1/2} \biggl[ \xi^2 \frac{d\psi}{d\xi} \biggr]  \, .</math>
  </td>
</tr>
</table>
</div>

Hence, for isothermal configurations,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~g_0 \equiv \frac{GM_r}{r_0^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~G\biggl( \frac{c_s^6}{4\pi G^3 \rho_c} \biggr)^{1/2} \biggl[ \xi^2 \frac{d\psi}{d\xi} \biggr]
\biggl[ \biggl( \frac{c_s^2}{4\pi G \rho_c} \biggr)^{1/2} \xi\biggr]^{-2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~c_s^2  
\biggl( \frac{4\pi G \rho_c}{c_s^2} \biggr)^{1 / 2} 
\biggl( \frac{d\psi}{d\xi} \biggr) \, .
</math>
  </td>
</tr>
</table>
</div>

From the [[#Summary|above summary of the Hunter (1977) similarity variables]], we also have,
A similarity solution becomes possible for these equations when the single independent variable,
<div align="center">
<math>~\zeta = \frac{c_s t}{r} \, ,</math>
</div>
is used to replace both <math>~r</math> and <math>~t</math>.  Then, if <math>~M_r</math>, <math>~\rho</math>, and <math>~v_r</math> assume the following forms,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~M_r(r,t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{c_s^3 t}{G}\biggr) m(\zeta) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\rho(r,t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{c_s^2 }{4\pi G r^2}\biggr) \Rho (\zeta) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~v_r(r,t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- c_s U(\zeta) \, .</math>
  </td>
</tr>
</table>
</div>

Defining a new, dimensionless time as,
<div align="center">
<math>~\tau \equiv (4\pi G \rho_c t^2)^{1 / 2} \, ,</math>
</div>
then inserting the equilibrium structures into the expressions for the similarity variables gives, for example,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~m(\zeta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{G}{c_s^3 t}\biggr) 
\biggl( \frac{c_s^6}{4\pi G^3 \rho_c} \biggr)^{1/2} \biggl[ \xi^2 \frac{d\psi}{d\xi} \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{1}{\tau}\biggr) \biggl[ \xi^2 \frac{d\psi}{d\xi} \biggr]  \, ;
</math>
  </td>
</tr>
</table>
</div>

and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Rho (\zeta) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{4\pi G r^2}{c_s^2 }\biggr) \rho_c e^{-\psi} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{\tau^2}{\zeta^2 }\biggr] e^{-\psi} \, ,</math>
  </td>
</tr>
</table>
</div>

while,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{\zeta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{1}{c_s t}\biggr) \biggl( \frac{c_s^2}{4\pi G \rho_c} \biggr)^{1/2} \xi 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\xi }{\tau} \, .
</math>
  </td>
</tr>
</table>
</div>
Putting these last two expressions together also gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Rho (\zeta) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi^2 e^{-\psi} \, .</math>
  </td>
</tr>
</table>
</div>


==Yabushita's Radial Pulsation Eigenvector==
As we have, separately, [[SSC/Perturbations#The_Eigenvalue_Problem|discussed in detail]], the eigenvalue problem is defined in terms of the following ''perturbed'' variables,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P(m,t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~P_0(m) + P_1(m,t) = P_0(m) \biggl[1 + p(m) e^{i\omega t}  \biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\rho(m,t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_0(m) + \rho_1(m,t) = \rho_0(m) \biggl[1 + d(m) e^{i\omega t}  \biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~r(m,t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_0(m) + r_1(m,t) = r_0(m) \biggl[1 + x(m) e^{i\omega t}  \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
And the following three linearized equations describe the physical relationship between the three dimensionless perturbation amplitudes <math>~p(r_0)</math>, <math>~d(r_0)</math> and <math>~x(r_0)</math>, for various characteristic eigenfrequencies, <math>~\omega</math>:

<div align="center">
<table border="1" cellpadding="10">
<tr><td align="center">
<font color="#770000">'''Linearized'''</font><br />
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />
<math>
r_0 \frac{dx}{dr_0} = - 3 x - d ,
</math><br />

<font color="#770000">'''Linearized'''</font><br />
<span id="PGE:Euler"><font color="#770000">'''Euler + Poisson Equations'''</font></span><br />
<math>
\frac{P_0}{\rho_0} \frac{dp}{dr_0}  = (4x + p)g_0 + \omega^2 r_0 x ,
</math><br />

<font color="#770000">'''Linearized'''</font><br />
<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />
<math>
p = \gamma_\mathrm{g} d  \, .
</math>
</td></tr>
</table>
</div>

And, as was first [[SSC/Stability/InstabilityOnsetOverview#Yabushita.27s_Insight_Regarding_Stability|demonstrated by Yabushita (1975)]], 
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="3"><font color="maroon"><b>Exact Solution to the Isothermal LAWE</b></font></td>
</tr>
<tr>
  <td align="right">
<math>~\sigma_c^2 = 0</math>
  </td>
  <td align="center">
&nbsp;and &nbsp;
  </td>
  <td align="left">
<math>~x = 1 - \biggl( \frac{1}{\xi e^{-\psi}}\biggr) \frac{d\psi}{d\xi} \, .</math>
  </td>
</tr>
</table>
</div>