Editing SSC/Stability/InstabilityOnsetOverview (section)

===Yabushita's Insight Regarding Stability===
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<font size="-1">[[H_BookTiledMenu#MoreStabilityAnalyses|<b>Yabushita's<br />Analytic Sol'n for<br /> Marginally Unstable<br />Configurations</b>]]<br />(1974)</font>
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For any individual model along the equilibrium sequences depicted in Figure 1 &#8212; more specifically, for any choice of the truncation radius, <math>~\tilde\xi</math> &#8212; we can call upon traditional radial pulsation theory to determine the eigenvectors associated with that configuration's natural modes of radial oscillation.  The determination of an eigenvector for a particular mode in a particular equilibrium model requires the simultaneous determination of how the fractional displacement, <math>~x \equiv \delta r/r</math>, varies with radial location throughout the configuration <math>~(0 < \xi \le \tilde\xi)</math>, and the associated (square of the) frequency of oscillation, <math>~\sigma_c^2 \equiv 3\omega^2/(2\pi G\rho_c)</math>, of that mode.  Formally, in the context of pressure-truncated isothermal spheres, this is accomplished by solving the eigenvalue problem that is defined mathematically by the,

<div align="center" id="IsothermalLAWE">
<font color="maroon"><b>Isothermal LAWE</b> (linear adiabatic wave equation)</font><br />
{{ Math/EQ_RadialPulsation03 }}
</div>

subject to the boundary conditions identified by [http://adsabs.harvard.edu/abs/1974MNRAS.168..427T Taff &amp; Van Horn (1974)], namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dx}{d\xi} = 0</math>&nbsp; &nbsp; at &nbsp; &nbsp; <math>~\xi = 0</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~\frac{d\ln x}{d\ln\xi} = - 3</math>&nbsp; &nbsp; at &nbsp; &nbsp; <math>~\xi = \tilde\xi</math>.
  </td>
</tr>
</table>
</div>

When pursuing this type of analysis, it is broadly appreciated that a ''stable'' mode of radial oscillation will exhibit an eigenvector with a ''positive'' value of <math>~\sigma_c^2</math>, while an eigenvector for which <math>~\sigma_c^2</math> is ''negative'' identifies a ''dynamically unstable'' mode of oscillation.  

In an [[SSC/Stability/Isothermal#From_the_Analysis_of_Taff_and_Van_Horn_.281974.29|accompanying discussion]], we have reviewed and replicated many aspects of the computational analysis presented by [http://adsabs.harvard.edu/abs/1974MNRAS.168..427T Taff &amp; Van Horn (1974)] of radial modes of oscillation in pressure-truncated isothermal spheres.  They restricted their analysis to models having <math>~2 \le \tilde\xi \le 10</math>, that is, to models in the vicinity of the <math>~P_e</math>-max turning point; see the left panel of our Figure 1, above.  While focusing specifically on the ''fundamental'' mode of oscillation, Taff and Van Horn showed &#8212; as had [http://adsabs.harvard.edu/abs/1968MNRAS.140..109Y Yabushita (1968)] before them &#8212; that <math>~\sigma_c^2</math> is positive for configurations that lie to the right of the <math>~P_e</math>-max turning point, while it is negative for configurations that lie to the left of this turning point. This result supported the astrophysics community's widely held expectation that the <math>~P_e</math>-max turning point is associated with the onset of a dynamical instability in pressure-truncated isothermal spheres.

In a pair of papers that, to date, have not been widely cited, Yabushita ([http://adsabs.harvard.edu/abs/1974MNRAS.167...95Y 1974], [http://adsabs.harvard.edu/abs/1975MNRAS.172..441Y 1975]) provided solid ''proof'' that the <math>~P_e</math>-max turning point precisely marks the onset of dynamical instability in pressure-truncated isothermal spheres.  Specifically, he showed that, if the adiabatic exponent is assigned the value, <math>~\gamma_g = 1</math>, in which case the parameter, <math>~\alpha = -1</math>, the following analytically defined eigenvector provides an 
<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>
(Note that, because <math>~\sigma_c^2 = 0</math>, this eigenvector must be associated with a configuration that is marginally [dynamically] unstable.) In an [[SSC/Stability/Isothermal#Yabushita_.281975.29|accompanying discussion]], we have demonstrated explicitly that this eigenvector is a solution to the isothermal LAWE.  Along the way, we have also shown that the first logarithmic derivative of Yabushita's eigenfunction is given by the expression,

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

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  <td align="right">
<math>~ - \frac{d\ln x}{d\ln\xi} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{x}\biggl[ \biggl(\frac{d\psi }{d\xi} \biggr)^2 e^\psi  + 3x-2\biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Hence, if at the surface we impose the above-specified boundary condition, namely, <math>~[d\ln x/d\ln\xi]_\tilde\xi = -3</math>, then this also means that, at the surface,

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

<tr>
  <td align="right">
<math>~\biggl[ e^\psi   \biggl(\frac{d\psi }{d\xi} \biggr)^2 \biggr]_{\tilde\xi} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2 \, .</math>
  </td>
</tr>
</table>
</div>
And this is precisely the condition, [[#BonnorCondition|highlighted above]], that Bonnor showed was associated with the critical <math>~P_e</math>-max turning point along the equilibrium sequence.  Hence, as [http://adsabs.harvard.edu/abs/1975MNRAS.172..441Y Yabushita (1975)] recognized, we can precisely associate the configuration at the turning point with the marginally [dynamically] unstable configuration.