Editing SSC/Stability/InstabilityOnsetOverview (section)

===Elaboration===

<table border="0" cellpadding="8" align="right">
<tr><th align="center">Figure 2: &nbsp;Yabushita Displacement Function</th></tr>
<tr><td align="center">
[[File:Yabushita1975Combined.png|400px|Yabushita Analytic Eigenfunction]]
</td></tr></table>
The curves drawn in all three panels of Figure 2, shown here on the right, are identical to one another.  They each trace on a semi-log plot the behavior of, what we will henceforth refer to as, the Yabushita displacement function,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~x_Y</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~1 - \biggl( \frac{1}{\xi e^{-\psi}}\biggr) \frac{d\psi}{d\xi} \, .</math>
  </td>
</tr>
</table>
While this function is formally valid over the entire coordinate range, <math>~0 \le \xi \le \infty</math>, for practical purposes, Figure 1 only displays the function segment that extends over the radial coordinate range, <math>~\xi = 0.1</math> to <math>~\xi = 1830</math>.  It is worth noting, as has been [[SSC/Stability/Isothermal#CentralValue|demonstrated separately]], that, <math>~x_Y \rightarrow \tfrac{2}{3}</math> in the limit of <math>~\xi \rightarrow 0</math>.

The solid, blue portion of the curve that appears in the top panel of Figure 2 extends from (near) the center of the isothermal sphere, out to the radial-coordinate location where the logarithmic derivative of the Yabushita displacement function first presents the value,  
<div align="center">
<math>~\frac{d\ln x_Y}{d\ln\xi} = -3 \, .</math>
</div>
If we truncate the isothermal sphere at this radius and, accordingly, truncate <math>~x_Y</math> at this radius as well (a short, blue vertical line segment has been drawn at this radial location to emphasize the truncation), then the (truncated) Yabushita displacement function not only satisfies the isothermal LAWE, but it also satisfies the desired surface boundary condition.  We can conclude, therefore, that in tandem with the value of the eigenfrequency that accompanies Yabushita's solution &#8212; namely, <math>~\sigma_c^2 = 0</math> &#8212; the blue curve segment in the top panel of Figure 2 represents a true ''eigenfunction.''  Because it exhibits no radial nodes, we conclude that it is the eigenfunction associated with a ''fundamental mode'' oscillation.

A small, green circular marker also appears in the top panel of Figure 2.  It has been placed on the Yabushita displacement function curve at the radial-coordinate location, <math>~\tilde\xi = 6.45</math>, corresponding to the truncation radius of the equilibrium configuration that sits at the first pressure-extremum in the left panel of Figure 1.  The fact that the location of this green marker coincides with the truncation radius that naturally attends the Yabushita analysis provides a graphical illustration of the key point made above: &nbsp; a precise association can be made between the configuration at the <math>~P_e</math>-max turning point and the configuration along the equilibrium sequence whose fundamental, radial mode of oscillation has an oscillation frequency of zero and, therefore, is marginally [dynamically] unstable.

<span id="Harmonics">Because the Yabushita displacement function undulates</span> back and forth between positive and negative values, we were not surprised to find that there are additional radial-coordinate locations (besides the first one, <math>~\tilde\xi \approx 6.45</math>) at which the function's logarithmic derivative equals negative three.  In the middle panel of Figure 2, the blue curve segment has been extended to, and truncated at, the point along the Yabushita displacement function where the function presents, for the second time, the value,
<div align="center">
<math>~\frac{d\ln x_Y}{d\ln\xi} = -3 \, .</math>
</div>
If we truncate the isothermal sphere at this radius then, as before, the (truncated) Yabushita displacement function not only satisfies the isothermal LAWE, but it also satisfies the desired "surface" boundary condition.  We can conclude, therefore, that the blue curve segment in the middle panel of Figure 2 represents another true ''eigenfunction.''  Because this eigenfunction exhibits one radial node, we conclude that it is the eigenfunction associated with a ''first harmonic'' mode of oscillation; and because the (square of the) eigenfrequency that accompanies the Yabushita displacement function is, <math>~\sigma_c^2 = 0</math>, we conclude that the blue curve segment in the middle panel of Figure 2 is an eigenfunction of the configuration in which the first harmonic, rather than the fundamental, mode is marginally [dynamically] unstable.  

But where, along the equilibrium sequence, does this configuration lie?  The small, green circular marker that appears in the middle panel of Figure 2 has been placed on the Yabashita displacement function curve at the radial-coordinate location, <math>~\tilde\xi = 67</math>, that corresponds to the truncation radius of the equilibrium configuration that sits at the ''second'' pressure-extremum in the left panel of Figure 1.  This graphically illustrates that a direct association can be made between the configuration that sits at the second pressure extremum along the equilibrium curve and the configuration along the sequence that is marginally [dynamically] unstable to the ''first harmonic'' mode of oscillation.  This is not simply a graphically approximate association but, rather, a ''precise'' one because the parameters that define ''both'' the configuration at the relevant turning point in Figure 1 and the configuration whose first harmonic mode of oscillation is displayed in the middle panel of Figure 2, satisfy the condition,
<div align="center">
<math>~\biggl[e^{\psi} \biggl( \frac{d\psi}{d\xi}\biggr)^2\biggr]_{\tilde\xi} = 2 \, .</math>
</div>

In the bottom panel of Figure 2, the blue curve segment has been extended to, and truncated at, the point along the Yabushita displacement function where the function presents, for the third time, the value,
<div align="center">
<math>~\frac{d\ln x_Y}{d\ln\xi} = -3 \, .</math>
</div>
Following a line of reasoning that is analogous to the others, just presented, we conclude that this blue curve segment represents a third true ''eigenfunction.''  This time, because the eigenfunction exhibits two radial nodes, we conclude that it is the eigenfunction associated with a ''second harmonic'' mode of oscillation; and because the (square of the) eigenfrequency that accompanies the Yabushita displacement function is, <math>~\sigma_c^2 = 0</math>, we conclude that the blue curve segment in the bottom panel of Figure 2 is an eigenfunction of the configuration in which the second harmonic mode is marginally [dynamically] unstable.  Finally, as the placement of the small, green circular dot in the bottom panel of Figure 2 illustrates, we find that there is a precise association between the configuration that sits at the third pressure extremum along the equilibrium curve and the configuration along the sequence that is marginally [dynamically] unstable to the ''second harmonic'' mode of oscillation.

Presumably many more associations of this type can be made, with the configuration located at successive pressure extrema being identified as a configuration that is marginally unstable to the next higher harmonic, radial mode of oscillation.  We note, also, that although we have focused on drawing an association between configurations that are marginally unstable and configurations that sit at turning points along the <math>~P-V</math> curve (left panel of Figure 1), the same association can be made with configurations that sit at turning points (mass extrema) along the equilibrium curves displayed in the center panel and the right panel of Figure 1.

<span id="Stahler83">It is important to recognize</span> that the physical significance of these turning points, along with the qualitative nature of the eigenfunction associated with each, has been previously deduced and described by [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)] in the context of a free-energy analysis of pressure-truncated, isothermal configurations.  Specifically referring to the behavior of the equilibrium sequence in an <math>~M(\rho_c)</math> diagram &#8212; see, for example, the right panel of our Figure 1 &#8212; Stahler concludes that (see his &sect;IIIb), <font color="darkgreen">&hellip; it also follows that the first maximum in <math>~M(\rho_c)</math> corresponds to the onset of instability in the lowest-frequency, fundamental mode (zero nodes), the first minimum to instability of the first harmonic (one node), etc.</font>  Referencing [http://adsabs.harvard.edu/abs/1974A%26A....31..391B Bisnovatyi-Kogan &amp; Blinnikov (1974)], Stahler also points out that, in principle, the marginally unstable, radial-oscillation eigenvector that is associated with each of these turning points <font color="darkgreen">&hellip; can be computed as the zero-frequency displacement connecting neighboring equilibria.</font>  Such an analysis would, presumably, reveal the same eigenfunction that was derived by [http://adsabs.harvard.edu/abs/1975MNRAS.172..441Y Yabushita (1975)] &#8212; that is, the function defined above as, <math>~x_Y(\xi)</math>. But, as far as we are aware, such an analysis has never been completed.