Editing SSC/Stability/InstabilityOnsetOverview (section)

=====Radial Displacement Function=====
The right panel of Figure 4 shows how the displacement function, <math>~x_P(\xi)</math>, varies with the radial coordinate, <math>~\xi</math>, for five different values of the polytropic index; specifically, as labeled, for n = 3, 3.05, 3.5, 5, and 6.  These are the same index values for which mass-radius equilibrium sequences have been displayed, above, in [[#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Figure 3]].  The Yabushita displacement function, which is relevant to isothermal <math>~(n=\infty)</math> configurations, is also shown, for comparison.  The same set of curves (unlabeled, but having the same colors) have been redrawn on a semi-log plot in the left panel of Figure 4; in this panel the "isothermal" curve is identical to the curve that appears in the top panel of [[#Elaboration|Figure 2]].

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  <th align="center"><br />Figure 4: &nbsp; Fundamental-Mode Eigenfunctions[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheets = YabushitaCombined & YabushitaLinearPlot]]</th>
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[[File:YabushitaMontageB.png|750px|Yabushita Analytic Eigenfunction]]
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In both panels of Figure 4, the curve associated with each polytropic index, <math>~n > 3</math>, has been displayed in two segments:  &nbsp; (a) A solid, colored segment that extends from the center of the configuration out to the radial location where the logarithmic derivative of the relevant displacement function first presents the value, <math>~d\ln x/d\ln\xi = -3</math>; and (b) a dashed, black segment that, for <math>~n < 5</math>, continues on to the natural edge of the isolated polytrope having the same index or, for <math>~n \ge 5</math>, extends to infinity.  (In all cases, the outer coordinate edge of this dashed, black curve segment lies beyond the edge of the plot and is, therefore, not actually shown.)  As we have [[#Analyses_of_Radial_Oscillations|just discussed]], for the special case of n = 3, it is the ''isolated'' polytrope &#8212; not a pressure-truncated configuration &#8212; and the more conventional surface boundary condition that are relevant, so in Figure 4 this particular displacement function has been drawn as a solid (black) curve that extends all the way out to the natural edge of an isolated n = 3 polytrope, which &#8212; see, for example, [[SSC/Structure/Polytropes#Horedt2004|p. 77 of Horedt (2004)]] &#8212; is located at <math>~\xi_\mathrm{surf} \approx 6.89684862</math>.

Drawing direct parallels with our detailed discussion, [[#Elaboration|above]], of Yabushita's displacement function in the context of isothermal spheres, we recognize that the solid curve segments displayed in Figure 4 each:
* Represents a true eigenfunction because its associated (truncated) displacement function satisfies, both, the polytropic LAWE and the appropriate surface boundary condition;
* Identifies the radial profile of the underlying equilibrium configuration's  ''fundamental mode'' of radial oscillation because it exhibits no radial nodes;
* Is associated with a configuration that is marginally [dynamically] unstable because the value of the associated (square of the) oscillation frequency is, <math>~\sigma_c^2 = 0</math>.

On each Figure 4 curve, a small, green circular marker has been placed along the displacement function at the radial coordinate location that is associated with the maximum-mass turning point of the corresponding equilibrium sequence shown in Figure 3.  In every case, this green marker sits at the same location where the transition is made from the solid segment to the dashed segment of the curve.  In each case, this is a graphical illustration of the key point made earlier:  A precise association can be made between the configuration at the maximum-mass (and <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.