Editing Appendix/Ramblings/NonlinarOscillation (section)

===Summary===

====Function Plots====
The solid green curve in [[#n5MassRadiusSequence|Figure 1]] shows how the equilibrium mass varies with radius for pressure-truncated, n = 5 polytropic spheres, if the polytropic constant and the externally applied pressure are held fixed. The portion of this mass-radius equilibrium sequence that lies inside the black-dashed rectangular box has been displayed again as a (static) solid green curve in the left-most panel of [[#Fig2|Figure 2]]. As in Figure 1, the solid black circular marker identifies the configuration along the equilibrium sequence that has the maximum mass,
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  <td align="right">
<math>~m_\mathrm{max} \equiv \frac{M}{M_\mathrm{SWS}}\biggr|_\mathrm{max}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{3^4\cdot 5^3}{2^{10} \pi} \biggr]^{1 / 2} \, .</math>
  </td>
</tr>
</table>
</div>
Other parameter values associated with this maximum-mass model are given in the row of [[#Table1|Table 1]] that is labeled, "Pairing A" &#8212; for example, <math>~{\tilde\xi}_\pm = 3</math> and <math>~\tilde{C} = 4 \Rightarrow \Delta \tilde{C} = 0</math>.  The (static) solid black curve in the middle panel of [[#Fig2|Figure 2]] provides a quantitative description of the internal structural profile of this maximum-mass model.  <span id="LimitingMassProfile">It displays how the radius of each mass shell varies with the integrated mass that lies internal to that shell; specifically, the black curve in the middle panel of Figure 2 displays the function,</span>

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

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  <td align="right">
<math>~r(m_\xi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{3^2 \cdot 5}{2^6 \pi} \biggr]^{1 / 2} \biggl[ 4m_\xi^{-2/3} -3 \biggr]^{-1 / 2} \, ,</math>
  </td>
</tr>
</table>
</div>
where the fractional mass, <math>~0 \le m_\xi \le 1</math>, is being employed as the Lagrangian radial coordinate. Notice that, when <math>~m_\xi = 1</math>, this expression gives the configuration's equilibrium radius as tabulated in the third column of the row of [[#Table1|Table 1]] that is labeled "Pairing A".  As has been [[#Analytic.2C_Marginally_Unstable_Eigenfunction|explained, above]], the eigenfunction associated with the fundamental mode of radial oscillation for this maximum-mass model is given precisely by the expression,
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<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>~x_P(m_\xi) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{5} \biggl[ \frac{10 -9~m_\xi^{2/3}}{4 -3~m_\xi^{2/3}} \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
This function has been plotted as a (static) black solid curve in the right-most panel of [[#Fig2|Figure 2]]; note that, although its overall normalization is arbitrary, here the eigenfunction has been normalized such that the fractional radial displacement, <math>~x = \delta r/r</math>, is two-fifths at the surface <math>~(m_\xi = 1)</math> of the configuration and unity at the center <math>~(m_\xi = 0)</math>.

As an animation sequence, Figure 2 loops repeatedly through nine separate frames.  In the left-most panel of each animation frame, the solid-blue horizontal line identifies a specific value of the equilibrium configuration's mass <math>~(M/M_\mathrm{SWS} < m_\mathrm{max})</math>, and the two solid-blue circular markers identify where &#8212; that is, at what two values of equilibrium radii &#8212; the horizontal line intersects the equilibrium sequence.  The nine separate values of the configuration mass that are highlighted by the animation (and the values of the pair of radii that are associated with each) are given in column 2 (and, respectively, in columns 3 &amp; 4) of [[#Table1|Table 1]]; they are labeled as "Pairing B" through "Pairing J".  Note that, for each value of the mass, the pair of values of the equilibrium radii were obtained by analytically identifying the pair of real roots of the quartic equation that defines the mass-radius relationship.  For each specified mass, [[#Table1|Table 1]] also lists (columns 5 &amp; 6) the corresponding values of the pair of dimensionless truncation radii, <math>~{\tilde\xi}_\pm</math>, and  (columns 7 &amp; 8) the corresponding values of the pair ''shifted'' parameters, <math>~\Delta C_2</math> and <math>~\Delta C_1</math>.

In the middle panel of each frame of the animation sequence, we have displayed in a quantitatively precise manner how the internal structural profiles of the relevant pair of equilibrium models compares with the structural profile of the limiting-mass model.  Specifically, the solid blue (dashed blue) curve shows how the radius of each mass shell varies with the integrated mass that lies internal to that shell in the case of the model that has the larger (smaller) equilibrium radius, as defined by the function,

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  <td align="right">
<math>~r_i(m_\xi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{3^2 \cdot 5}{2^2 \pi} \biggr]^{1 / 2} \biggl[ \frac{\tilde{C}_i - 3}{\tilde{C}_i} \biggr] \biggl[ \tilde{C}_i~m_\xi^{-2/3} -3 \biggr]^{-1 / 2} \, ,</math>
  </td>
</tr>
</table>
</div>
where,
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  <td align="right">
<math>~\tilde{C}_i \equiv \frac{3^2}{\tilde\xi_i^2}\biggl( 1 + \frac{\tilde\xi_i^2}{3} \biggr)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\Rightarrow</math> &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~
{\tilde\xi}_i^2 = \frac{9}{\tilde{C}_i - 3} \, .
</math>
  </td>
</tr>

</table>
</div>
Notice that, as it should, this generalized expression for <math>~r_i(m_\xi)</math> reduces to the expression ([[#LimitingMassProfile|shown above]]) that applies specifically to the maximum-mass model when <math>~{\tilde\xi}_i = 3 \Rightarrow {\tilde{C}}_i = 4</math>.  The ''percentage'' that is stamped on the center panel of each animation frame &#8212; varying from 2.9% to 12% &#8212; provides a measure of the degree to which this pair of (blue) profiles departs from the profile of the maximum-mass model; as documented in column 9 of [[#Table1|Table 1]], each percentage ''value'' is determined by dividing one-half the difference in the surface radii of the (blue) model pair by the radius of the (black) limiting-mass model.

References:
* [http://adsabs.harvard.edu/abs/1974A%26A....31..391B G. S. Bisnovatyi-Kogan &amp; S. I. Blinnikov (1974)] [see also [[H_Book#BKB74pt1|here]]]: &nbsp;<font color="darkgreen">If one wants to know from a stability analysis the answer to only one question &#8212; whether the model is stable or not &#8212; then the most straightforward procedure is to use the third, static method (Zeldovich 1963; Dmitrie &amp; Kholin 1963).  For the application of this method, one needs to construct only equilibrium, stationary models, with no further calculation.  Generally the static analysis gives no information about the shape of the modes of oscillation, but, in the vicinity of critical points, where instability sets in, this method makes it possible to find the eigenfunction of the mode which becomes unstable at the critical point.</font>
* &sect;6.8 of [<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>] [see also [[Appendix/Ramblings/NonlinarOscillation#Radial_Oscillations_in_Pressure-Truncated_n_.3D_5_Polytropes|here]]]:&nbsp; Upon further thought, it occurred to me that a careful examination of the internal structure of both models &#8212; especially ''relative'' to one another &#8212; might reveal what the eigenvector of that (nonlinear) oscillation might be.  In support of this idea, I point to the discussion of "Turning-Points and the Onset of Instability" found in &sect;6.8 of [<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>] &#8212; specifically, on p. 149 in the paragraph that follows eq. (6.8.11) &#8212; where we find the following statement:  <b><font color="green">"&hellip; the eigenfunction at a critical point is simply the Lagrangian displacement <math>~\xi</math> that carries an equilibrium configuration on the low-density side of the critical point into an equilibrium configuration on the high-density side."</font></b>
* [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)] [see also [[SSC/Stability/InstabilityOnsetOverview#Stahler83|here]]]:&nbsp; Referencing [http://adsabs.harvard.edu/abs/1974A%26A....31..391B Bisnovatyi-Kogan &amp; Blinnikov (1974)], Stahler points out that, in principle, the marginally unstable, radial-oscillation eigenvector that is associated with each an equilibrium-sequence turning point <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.


<font color="red"><b>Conjecture</b></font> posed by [http://adsabs.harvard.edu/abs/1974A%26A....31..391B G. S. Bisnovatyi-Kogan &amp; S. I. Blinnikov (1974)] &#8212; see the opening paragraph of their &sect; 6:
<table border="1" cellpadding="8" align="center" width="85%">
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<font color="darkgreen">&hellip; a static configuration close to an extremum of the</font> mass-radius equilibrium <font color="darkgreen">curve may be considered as a perturbed state of a model of the same mass situated on the other side of the extremum.  The difference of the two models approximately represents the eigenfunction of the neutral mode.  Let there exist two Models 1 and 2 for a mass <math>~M</math> in the vicinity of an extremum; then the eigenfunction <math>~\mathfrak{x}</math> is</font>
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<math>~\mathfrak{x} = r_2(m_\xi) - r_1(m_\xi) \, ,</math>
</div>
<font color="darkgreen">where <math>~m_\xi</math> is the Lagrangian coordinate and <math>~r_1, r_2</math> Eulerian coordinates of the Models 1 and 2.</font>
</td></tr>
</table>


For each specified value of the truncated configuration's total mass, <math>~m</math>, measured relative to the maximum-allowed mass,
<div align="center">
<math>~\mu \equiv \biggl[ 1 - \biggl( \frac{m}{m_\mathrm{max}} \biggr)^2 \biggr]^{1 / 2}</math> &nbsp; &nbsp; &nbsp; <math>~\Rightarrow</math> &nbsp; &nbsp; &nbsp; <math>~m^2 =
 m_\mathrm{max}^2(1-\mu^2) = \frac{3^4\cdot 5^3}{2^{10} \pi}(1-\mu^2) \, ,</math>
</div>
there will be a pair of equilibrium states, whose identifying truncation radii, <math>~(\tilde{C}_1, \tilde{C}_2)</math>, are determined analytically as [[#Roots_of_Quartic_Equation|roots of a quartic equation]].  Ten such pairings are identified in [[#Table1|Table  1]].

As [[#Layout|shown above]], we can define a (nonlinear) displacement function as the fractional difference between the radial profiles of this pair of equal-mass states,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{x}(m_\xi) </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{\tfrac{1}{2}\Delta r_\mathrm{SWS}}{<r_\mathrm{SWS}>} =
\frac{ \tfrac{1}{2} [r_2(m_\xi) - r_1(m_\xi) ]}{\tfrac{1}{2} [ r_2(m_\xi) + r_1(m_\xi)]}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \tilde{C}_1 ( \tilde{C}_2 - 3 ) \biggl[ \tilde{C}_1 -3~m_\xi^{2/3} \biggr]^{1 / 2}  
- \tilde{C}_2 ( \tilde{C}_1 - 3 ) \biggl[ \tilde{C}_2 -3 ~m_\xi^{2/3}\biggr]^{1 / 2} \biggr\}
\biggl\{ \tilde{C}_1 ( \tilde{C}_2 - 3 ) \biggl[ \tilde{C}_1 -3~m_\xi^{2/3} \biggr]^{1 / 2}  
+ \tilde{C}_2 ( \tilde{C}_1 - 3 ) \biggl[ \tilde{C}_2 -3~m_\xi^{2/3} \biggr]^{1 / 2} \biggr\}^{-1} \, .
</math>
  </td>
</tr>
</table>
</div>
Or, in terms of the ''shifted'' parameters,
<div align="center">
<math>~\Delta C_i \equiv {\tilde{C}}_i - 4 </math> &nbsp; &nbsp; &nbsp;<math>~\Rightarrow</math>  &nbsp; &nbsp; &nbsp;<math>~\tilde{C}_i = \Delta C_i + 4 \, ,</math>
</div>

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

<tr>
  <td align="right">
<math>~\mathfrak{x}[\beta(m_\xi)] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \biggl[ 1 + \frac{\Delta C_1}{4} + \Delta C_2 + \frac{\Delta C_1 \cdot \Delta C_2}{4} \biggr](1 + \beta^2 \Delta C_1  )^{1 / 2}  
- \biggl[ 1 + \frac{\Delta C_2}{4} + \Delta C_1 + \frac{\Delta C_1 \cdot \Delta C_2}{4} \biggr] ( 1+\beta^2 \Delta C_2 )^{1 / 2} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~\times
\biggl\{ \biggl[ 1 + \frac{\Delta C_1}{4} + \Delta C_2 + \frac{\Delta C_1 \cdot \Delta C_2}{4} \biggr] ( 1 + \beta^2\Delta C_1 )^{1 / 2}  
+ \biggl[ 1 + \frac{\Delta C_2}{4} + \Delta C_1 + \frac{\Delta C_1 \cdot \Delta C_2}{4} \biggr] ( 1 + \beta^2\Delta C_2 )^{1 / 2} \biggr\}^{-1} \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~\beta \equiv (4 -3~m_\xi^{2/3})^{-1 / 2} \, .</math>
</div>

The conjecture is that this ''nonlinear'' displacement function, <math>~\mathfrak{x}</math>, should match the analytically specified radial oscillation eigenfunction, <math>~x_P</math>, in the limit of <math>~m/m_\mathrm{max} \rightarrow 1</math>.

This is an excerpt from a [[SSC/Stability/InstabilityOnsetOverview#Elaboration|related discussion]]:  &nbsp;Referencing [http://adsabs.harvard.edu/abs/1974A%26A....31..391B Bisnovatyi-Kogan &amp; Blinnikov (1974)], Stahler 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>. As far as we are aware, such an analysis has never been completed.

====Animation====

<div align="center" id="Fig2">
<table border="1" cellpadding="5" align="center">
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  <th align="center">
Figure 2:  &nbsp; Structural properties of Model Pairs having the same Equilibrium Masses
  </th>
</tr>
<tr><td align="center">
[[File:Displacement3.gif|1000px|movie]]
</td></tr>
</table>
</div>


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