Editing SSC/Stability/InstabilityOnsetOverview (section)

==Polytropic==
===Isolated Polytropes===
Given a value of the polytropic index, <math>n</math>, the internal structure of an isolated polytrope is provided via the function, <math>\Theta_H(\xi) \equiv (\rho/\rho_c)^{1/n}</math>, which is a solution of the,
<div align="center">
<table border="0" cellpadding="8" align="center">
<tr><td align="center">
<font color="maroon"><b>Polytropic Lane-Emden Equation</b></font> <p></p>
{{ Math/EQ_SSLaneEmden01 }}
</td></tr>
</table>
</div>
subject to the boundary conditions, <math>\Theta_H = 1</math> and <math>d\Theta_H/d\xi = 0</math> at <math>\xi = 0</math>.  In an [[SSC/Structure/Polytropes#Polytropic_Spheres|accompanying chapter]], we have reviewed what the structural properties are of polytropes that have a range of polytropic indexes.  Our emphasis has been on systems (n = 0, 1, and 5) for which the Lane-Emden equation can be solved analytically, but we also have discussed systems of astrophysical interest (n = 2.5, 3.0, 3.5, and 6) whose structural properties can only be described in terms of numerical solutions of this governing equation.  In isolation, systems having <math>n \ge 5</math> extend to infinity &#8212; as does the isothermal sphere discussed above.  But systems with <math>0 \le n < 5</math> have finite radii, that is, the function, <math>\Theta_H(\xi)</math>, naturally drops to zero at a radial-coordinate location, <math>\xi = \xi_\mathrm{surf} < \infty</math>.

===Turning Points along Sequences of Pressure-Truncated Polytropes===
Just as we have discussed above in the context of isothermal spheres, when <math>~n \ge 5</math>, equilibrium configurations of finite extent can be constructed by truncating the function, <math>~\Theta_H</math>, at some radius, <math>~0 < \tilde\xi < \infty</math> &#8212; in which case the surface density is finite and set by the value of <math>~\tilde\theta \equiv \Theta_H(\tilde\xi)</math> &#8212; and embedding them in a hot, tenuous medium that exerts an external pressure, <math>~P_e = K\rho_c^{(n+1)/n}\tilde\theta^{(n+1)/n}</math>, uniformly across the surface of the truncated sphere.  Polytropes having <math>~0 \le n <5</math> may similarly be truncated at any radius, <math>~0 < \tilde\xi < \xi_\mathrm{surf}</math>.  For any value of the index, <math>~n</math>, the internal structure of each such "pressure-truncated" polytrope is completely describable in terms of the same function, <math>~\Theta_H(\xi)</math>, that describes the structure of an isolated polytrope with the same index, except that the function becomes physically irrelevant beyond <math>~\tilde\xi</math>.  

<table border="0" cellpadding="8" align="right">
<tr>
  <th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined]]Figure 3: &nbsp;Equilibrium Sequences<br />of Pressure-Truncated Polytropes
  </th>
</tr>
<tr>
  <td align="center" colspan="1">
[[File:MassVsRadiusCombined02.png|350px|Equilibrium sequences of Pressure-Truncated Polytropes]]
  </td>
</tr>
</table>
As is the case for pressure-truncated isothermal spheres, for each index, <math>~n</math>, a ''sequence'' of pressure-truncated polytropes is readily defined by varying the value of <math>~\tilde\xi</math> over the range,  <math>~0 < \tilde\xi < \infty</math> &#8212; or, as the case may be, over the range, <math>~0 < \tilde\xi < \xi_\mathrm{surf}</math>.  In an [[SSC/Structure/PolytropesEmbedded#Additional.2C_Numerically_Constructed_Polytropic_Configurations|accompanying discussion of the properties of such equilibrium configurations]], we have graphically displayed in a single diagram the mass-radius relation of sequences having n = 1, 2.5, 3, 3.05, 3.5, 5, and 6.   In Figure 3, shown here on the right, we have redrawn these mass-radius relations for the subset of sequences that have <math>~n \ge 3</math>, and have inserted as well the mass-radius relation for pressure-truncated isothermal spheres &#8212; copied from the [[#Fig1|middle panel of Figure 1, above]].  More specifically, adopting the mass and radius normalizations introduced by [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|S. W. Stahler (1983)]], each Figure 3 ''polytropic'' sequence has been defined via the pair of parametric relations,
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~\frac{M}{M_\mathrm{SWS} }
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{n^3 }{4\pi} \biggr)^{1 / 2}  \biggl[ \theta^{(n-3)/2} \xi^2 
\biggl| \frac{d\theta}{d\xi} \biggr| ~\biggr]_{\tilde\xi} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\frac{R}{R_\mathrm{SWS}}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~\biggl(  \frac{n }{4\pi} \biggr)^{1 / 2} 
\biggl[ \xi \theta^{(n-1)/2} \biggr]_{\tilde\xi} \, .
</math>
  </td>
</tr>
</table>
</div>
In Figure 3, as in Figure 1, a small, yellow circular marker has been used to identify the configuration along each sequence for which the equilibrium radius reaches its maximum value; also, a small, green circular marker identifies the configuration along each sequence that has the maximum mass.  As we have reviewed in an [[SSC/Structure/PolytropesEmbedded#Other_Limits|accompanying discussion]], [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura (1981)] has shown that, along each such (polytropic) mass-radius sequence, the configuration associated with the "maximum radius" turning point (yellow marker) occurs precisely where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\biggl[~ \frac{\xi}{\theta} \biggl|\frac{d\theta}{d\xi}\biggr|~ \biggr]_{\tilde\xi}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{n-1} \, ,</math>
  </td>
</tr>
</table>
</div>
<span id="MaximumMass">and the configuration associated with the "maximum mass" turning point (green marker) occurs precisely where,</span>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\biggl[ \frac{1}{\theta^{n+1}} \biggl(\frac{d\theta}{d\xi}\biggr)^2\biggr]_{\tilde\xi}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{n-3} \, .</math>
  </td>
</tr>
</table>
</div>
As we have [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Derivation|highlighted elsewhere]], approximately a decade prior to Kimura's work, [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] showed that this last criterion also precisely identifies the configuration that is associated with the <math>~P_e</math>-max turning point along polytropic pressure-volume sequences, analogous to the location of the uppermost green marker in the lefthand panel of our [[#Fig1|Figure 1]].  [[SSC/Structure/PolytropesEmbedded#Table3|Table 3 of an accompanying discussion]] gives the exact, analytically determined coordinate locations of the pair of turning points that arise along the <math>~n=5</math> sequence, as well as approximate, numerically determined coordinate locations of the "maximum radius" (yellow markers) and "maximum mass" (green markers) turning points that have been identified along the other pressure-truncated polytropic sequences displayed in [[#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Figure 3]].  <span id="SpecificN5Reference">Because of its relevance</span> to our [[#n5Analytic|discussion of stability, below]], we note that, along the n = 5 sequence, the configuration that has the maximum radius (yellow marker) is truncated precisely at <math>~\tilde\xi = \sqrt{3}</math>, while the configuration that has the maximum mass (green marker) is truncated precisely at <math>~\tilde\xi = 3</math>.

===Polytropic Stability===
{| class="Our2017Analytic" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|- 
! style="height: 125px; width: 125px; background-color:#ffeeee;" |
<font size="-1">[[H_BookTiledMenu#MoreStabilityAnalyses|<b>Our (2017)<br />Analytic Sol'n for<br /> Marginally Unstable<br />Configurations</b>]]</font><br /><font color="green" size="-1">&hearts;</font>
|}
====Setup====
As we have [[SSC/Stability/Polytropes#Radial_Oscillations_of_Polytropic_Spheres|detailed elsewhere]], in order to identify individual radial oscillation modes in polytropic spheres, we seek solutions to the,
&nbsp;<br />
&nbsp;<br />
&nbsp;<br />
&nbsp;<br />
&nbsp;<br />

<div align="center">
<font color="maroon"><b>Polytropic LAWE (linear adiabatic wave equation)</b></font><br />
{{ Math/EQ_RadialPulsation02 }}
</div>
All physically reasonable solutions are subject to the inner boundary condition,
<div align="center">
<math>\frac{dx}{d\xi} = 0</math> &nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>\xi = 0 \, ,</math>
</div>
but the relevant outer boundary condition depends on whether the underlying equilibrium configuration is isolated (surface pressure is zero), or whether it is a "pressure-truncated" configuration.  As is the case with the pressure-truncated isothermal spheres, discussed above, if the polytropic configuration is truncated by the pressure, <math>P_e</math>, of a hot, tenuous external medium, then the solution to the LAWE is subject to the outer boundary condition,
<div align="center">
<math>-\frac{d\ln x}{d\ln\xi} = 3</math> &nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>\xi = \tilde\xi \, .</math>
</div>
But, for ''isolated'' polytropes &#8212; see the [[#Supporting_Scratch_Work|supporting derivation, below]] &#8212; the sought-after solution is subject to the more conventional boundary condition,
<div align="center">
<math>- \frac{d\ln x}{d\ln \xi} = \biggl(\frac{3-n}{n+1}\biggr) + \frac{n\sigma_c^2}{6(n+1)} \biggl[\frac{\xi}{\theta^'}\biggr] </math> &nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>\xi = \xi_\mathrm{surf} \, .</math><br />
[<font color="red">'''Note added on 8/26/2021:'''</font>  The numerator of the first term on the RHS of this last expression used to be "n-3".  This manuscript error has been fixed.]
</div>

====Analyses of Radial Oscillations====
In one [[SSC/Stability/n3PolytropeLAWE#Radial_Oscillations_of_n_.3D_3_Polytropic_Spheres|accompanying discussion]], we have reviewed and replicated many aspects of the computational analysis presented by [http://adsabs.harvard.edu/abs/1941ApJ....94..245S Schwarzschild (1941)] of radial modes of oscillation in isolated, n = 3 polytropic spheres.  In [[SSC/Stability/n5PolytropeLAWE#Radial_Oscillations_of_n_.3D_5_Polytropic_Spheres|another chapter]] we have presented results from our own analysis of radial oscillations in ''pressure-truncated'', n = 5 polytropic spheres.  

From Schwarzschild's work, we have come to appreciate that, if the adiabatic exponent is assigned the value, <math>~\gamma_g = (n+1)/n = 4/3</math> &#8212; in which case the parameter, <math>~\alpha = 0</math> &#8212; 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 (n = 3) Polytropic 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 </math>
  </td>
</tr>
</table>
</div>
that satisfies both the desired central boundary condition and the conventional surface boundary condition.  This is easy to verify.  Because the displacement function is unity and, therefore, independent of the radial coordinate, <math>~\xi</math>, its first and second derivatives &#8212; and, hence, the first two terms on the RHS of the polytropic LAWE &#8212; are zero.  The third term on the RHS of the polytropic LAWE is also zero because both <math>~\sigma_c^2</math> and <math>~\alpha</math> are zero.  Hence the sum of these three terms is zero, so the LAWE is satisfied.  And, because <math>~n =3</math> and <math>~\sigma_c^2=0</math>, we expect from the conventional surface boundary condition that <math>~d\ln x/d\ln \xi = 0</math> at <math>~\xi = \xi_\mathrm{surf}</math>, which of course it is because the the first derivative of the displacement function is zero at all radial locations, including the surface.

<span id="n6Analytic">In the course of our study</span> of the properties of radial oscillations in pressure-truncated, n = 5 polytropes, [[SSC/Stability/n5PolytropeLAWE#Search_for_Analytic_Solutions_to_the_LAWE|we discovered]] that, if the adiabatic exponent is assigned the value, <math>~\gamma_g = (n+1)/n = 6/5</math> &#8212; in which case the parameter, <math>~\alpha = - \tfrac{1}{3}</math> &#8212; 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 (n = 5) Polytropic 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 - \frac{\xi^2}{15} </math>
  </td>
</tr>
</table>
</div>
<span id="NeutralMode">which satisfies</span> both of the desired boundary conditions if the configuration is truncated at the radial coordinate, <math>~\tilde\xi = 3</math>.  From our [[#SpecificN5Reference|above reference]] to the n = 5 equilibrium sequence, we know that <math>~\tilde\xi = 3</math> identifies the configuration that sits at the maximum-mass turning point.  In this case, then, we recognize that our analytically defined eigenvector describes the properties of the (marginally unstable) fundamental mode of radial oscillation specifically for the configuration at the maximum-mass turning point &#8212; which, recall, is also the configuration at the <math>~P_e</math>-max turning point of the P-V sequence.  Our ability to identify this analytic solution to the polytropic LAWE was significantly aided by the fact that the structural properties of the underlying equilibrium configuration are analytically specifiable.

[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 19 March 2017:  As far as we have been able to determine, it has not previously been recognized that this eigenvector provides a precise solution to the Polytropic LAWE.]]Building on our recognition of these two analytically specifiable eigenvectors for polytropes, and Yabushita's extraordinary discovery of the set of analytically specifiable eigenvectors that are associated with pressure-truncated isothermal spheres, [[SSC/Stability/Isothermal#Try_to_Generalize|we have made the following additional, and more broadly applicable discovery]]:  &nbsp;For any value of the polytropic index in the range, <math>~3 \le n < \infty</math>, if the adiabatic exponent is assigned the value, <math>~\gamma_g = (n+1)/n</math> &#8212; in which case the parameter, <math>~\alpha = (3-n)/(n+1)</math> &#8212; the following eigenvector specification provides an,

<div align="center" id="ExactPolytropicSolution">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="3"><font color="maroon"><b>Exact Solution to the <math>~(3 \le n < \infty)</math> Polytropic LAWE</b></font></td>
</tr>
<tr>
  <td align="right">
<math>~\sigma_c^2 = 0</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~x_P \equiv \frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{1}{\xi \theta^{n}}\biggr) \frac{d\theta}{d\xi}\biggr] \, .</math>
  </td>
</tr>
</table>
</div>

Furthermore, for all values of <math>~n</math> in this specified range, this displacement function will satisfy both of the desired boundary conditions if the associated polytropic configuration is truncated at the radial coordinate, <math>~\tilde\xi</math>, that satisfies the condition,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\biggl[ \frac{1}{\theta^{n+1}} \biggl(\frac{d\theta}{d\xi}\biggr)^2\biggr]_{\tilde\xi}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{n-3} \, .</math>
  </td>
</tr>
</table>
</div>

As has been pointed out, [[#MaximumMass|above, in the context of our discussion of equilibrium sequences]], in all cases <math>~(3 \le n < \infty)</math>, this is a condition that also is associated with the "maximum mass" (and <math>~P_e</math>-max) turning point along the corresponding equilibrium sequence &#8212; see the green markers in Figure 3.  We conclude, therefore, that for all polytropic indexes greater than or equal to three (including isothermal structures), the configuration that is marginally [dynamically] unstable <math>~(\sigma_c^2 = 0)</math> is precisely associated with the maximum-mass and <math>~P_e</math>-max turning point along the relevant equilibrium sequence!

====Properties of the Marginally Unstable, Fundamental-Mode Eigenfunctions====

=====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]].

<table border="1" cellpadding="8" align="center">
<tr>
  <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>
</tr>
<tr>
  <td align="center">
[[File:YabushitaMontageB.png|750px|Yabushita Analytic Eigenfunction]]
</td></tr>
</table>

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.

=====Pressure and Density Displacement Functions=====

Referring back to the [[SSC/Perturbations#Summary_Set_of_Linearized_Equations|summary set of linearized governing equations]], we can derive analytic expressions for the pressure, <math>p_P(\xi)</math>, and/or density, <math>d_P(\xi)</math>, displacement functions that correspond to our expression for <math>x_P(\xi)</math>.  More specifically, from the linearized equation of continuity, we appreciate that,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>d_P</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>-3x_P - r_0\frac{dx_P}{dr_0}</math></td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>-3x_P - \xi \cdot \frac{dx_P}{d\xi} \, .</math></td>
</tr>
</table>
Now, from our above-derived,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="1"><font color="maroon"><b>Exact Solution to the <math>~(3 \le n < \infty)</math> Polytropic LAWE</b></font></td>
</tr>
<tr>
  <td align="left">
<math>x_P \equiv \frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{1}{\xi \theta^{n}}\biggr) \frac{d\theta}{d\xi}\biggr] \, ,</math>
  </td>
</tr>
</table>
we see that,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\frac{dx_P}{d\xi}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{3(n-3)}{2n}\biggl[ 
-\xi^{-2}\theta^{-n} \frac{d\theta}{d\xi}
-
n\xi^{-1}\theta^{-n-1} \biggl(\frac{d\theta}{d\xi}\biggr)^2
+
\xi^{-1}\theta^{-n} \frac{d^2\theta}{d\xi^2}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{3(n-3)}{2n}\biggl[ 
-\xi^{-1}\theta^{-n} \frac{d\theta}{d\xi}
-
n\theta^{-n-1} \biggl(\frac{d\theta}{d\xi}\biggr)^2
+
\theta^{-n} \frac{d^2\theta}{d\xi^2}
\biggr]\frac{1}{\xi} \, .
</math>
  </td>
</tr>
</table>
And, from the [[SSC/Structure/Polytropes#Lane-Emden_Equation|Lane-Emden equation]], we appreciate that,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>-\theta^n</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\xi^2} \frac{d}{d\xi}\biggl(\xi^2 \frac{d\theta}{d\xi}\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(\frac{2}{\xi}\biggr) \frac{d\theta}{d\xi} 
+
\frac{d^2\theta}{d\xi^2} 
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{d^2\theta}{d\xi^2}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
-\biggl(\frac{2}{\xi}\biggr) \frac{d\theta}{d\xi} 
-\theta^n
 
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \theta^{-n}\frac{d^2\theta}{d\xi^2}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
-2\xi^{-1}\theta^{-n}\frac{d\theta}{d\xi} -1 
\, .
</math>
  </td>
</tr>
</table>

As a result,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\frac{dx_P}{d\xi}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{3(n-3)}{2n}\biggl[ 
-3\xi^{-1}\theta^{-n} \frac{d\theta}{d\xi}
-
n\theta^{-n-1} \biggl( \frac{d\theta}{d\xi} \biggr)^2
-1 
\biggr]\frac{1}{\xi} \, ;
</math>
  </td>
</tr>
</table>
and,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>d_P</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>-3\biggl\{ \frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{1}{\xi \theta^{n}}\biggr) \frac{d\theta}{d\xi}\biggr] \biggr\} 
- 
\biggl\{ \frac{3(n-3)}{2n}\biggl[ -3\xi^{-1}\theta^{-n} \frac{d\theta}{d\xi} - n\theta^{-n-1} \biggl(\frac{d\theta}{d\xi}\biggr)^2 - 1 
\biggr]\biggr\} </math></td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>-  \biggl[\frac{9(n-1)}{2n} 
+ \frac{9(n-3)}{2n} \biggl( \frac{1}{\xi \theta^{n}}\biggr) \frac{d\theta}{d\xi}\biggr]  
+ 
\biggl\{ \frac{3(n-3)}{2n}\biggl[ 3\xi^{-1}\theta^{-n} \frac{d\theta}{d\xi} + n\theta^{-n-1} \biggl(\frac{d\theta}{d\xi}\biggr)^2 + 1 
\biggr]\biggr\} </math></td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>-  \biggl[\frac{9(n-1)}{2n} 
\biggr]  
+ 
\frac{3(n-3)}{2n}\biggl[ \frac{n}{\theta^{n+1}} \biggl(\frac{d\theta}{d\xi}\biggr)^2 + 1 
\biggr] </math></td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left"><math> 
-3 \biggl\{1   
- 
\frac{(n-3)}{2}\biggl[ \frac{1}{\theta^{n+1}} \biggl( \frac{d\theta}{d\xi}\biggr)^2 \biggr] \biggr\}
\, .
</math></td>
</tr>
</table>
Finally, according to the [[SSC/Perturbations#Summary_Set_of_Linearized_Equations|linearized (adiabatic form of the) first law of thermodynamics]], <math>p = \gamma_g d</math>.  That is, when we set <math>\gamma_g = (n+1)/n</math> along with <math>\sigma_c^2 = 0</math>, we find that the pressure displacement function is given by the expression,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>p_P</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math> 
-\frac{3(n+1)}{n} \biggl\{1   
- 
\frac{(n-3)}{2}\biggl[ \frac{1}{\theta^{n+1}} \biggl( \frac{d\theta}{d\xi}\biggr)^2 \biggr] \biggr\}
\, .
</math></td>
</tr>
</table>

For later use ([[#Derivative|immediately below]]) we note as well that,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\frac{d(p_P)}{d\xi}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math> 
\frac{3(n+1)(n-3)}{2n}   \frac{d}{d\xi}
\biggl[ \theta^{-(n+1)} \biggl( \frac{d\theta}{d\xi}\biggr)^2 \biggr] 
</math></td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left"><math> 
\frac{3(n+1)(n-3)}{2n}   
\biggl[ -\frac{(n+1)}{\theta^{n+2}}\biggl( \frac{d\theta}{d\xi}\biggr)^2
+ 
2\theta^{-(n+1)} \frac{d^2\theta}{d\xi^2} 
\biggr] \biggl(\frac{d\theta}{d\xi}\biggr) 
</math></td>
</tr>
</table>

Again, from the [[SSC/Structure/Polytropes#Lane-Emden_Equation|Lane-Emden equation]], we appreciate that,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\theta^{-n}\frac{d^2\theta}{d\xi^2}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
-2\xi^{-1}\theta^{-n}\frac{d\theta}{d\xi} -1 
\, .
</math>
  </td>
</tr>
</table>
Hence,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\frac{d(p_P)}{d\xi}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math> 
\frac{3(n+1)(n-3)}{2n}   
\biggl\{ -\frac{(n+1)}{\theta^{n+2}}\biggl( \frac{d\theta}{d\xi}\biggr)^2
+ 
\frac{2}{\theta} \biggl[-2\xi^{-1}\theta^{-n}\frac{d\theta}{d\xi} -1\biggr]
\biggr\} \biggl(\frac{d\theta}{d\xi}\biggr) 
</math></td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left"><math> 
-\frac{3(n+1)(n-3)}{n}   
\biggl\{ \frac{(n+1)}{2\theta^{n+1}}\biggl( \frac{d\theta}{d\xi}\biggr)^2
+ 
2\xi^{-1}\theta^{-n}\biggl(\frac{d\theta}{d\xi}\biggr) + 1
\biggr\} \biggl(\frac{d\theta}{d\xi}\biggr) \frac{1}{\theta} 
</math></td>
</tr>
</table>

=====Cross-Check=====

Can you determine <math>x_P</math>, given this expression for <math>p_P</math>?

When the square of the radial-oscillation frequency is set to zero, the [[SSC/Perturbations#Summary_Set_of_Linearized_Equations|linearized Euler + Poisson equations]] states,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\biggl[ \frac{P_0}{\rho_0 r_0} \biggr] \frac{d(p)}{d\ln r_0}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math> 
(4x + p)g_0 + \cancelto{0}{\omega^2 r_0 x} 
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ 4x_P</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math> 
\biggl[ \frac{P_0}{\rho_0 r_0 g_0} \biggr] \frac{d(p_P)}{d\ln \xi}
-
p_P
</math>
  </td>
</tr>
</table>
Now, according to our [[SSC/Stability/Polytropes#Groundwork|accompanying discussion of the equilibrium structure of isolated polytopes]],
<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>~a_n \xi \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\rho_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_c \theta^{n} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~P_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>K\rho_0^{(n+1)/n} = K\rho_c^{(n+1)/n} \theta^{n+1} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~g_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{GM(r_0)}{r_0^2} = \frac{G}{r_0^2} \biggl[ 4\pi a_n^3 \rho_c \biggl(-\xi^2 \frac{d\theta}{d\xi}\biggr) \biggr]
\, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>a_n</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[\frac{(n+1)K}{4\pi G} \cdot \rho_c^{(1-n)/n} \biggr]^{1/2} </math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; <math>\Rightarrow</math>&nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>K</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[\frac{4\pi G a_n^2}{(n+1)} \cdot \rho_c^{(n-1)/n} \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Therefore,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{P_0}{\rho_0 r_0 g_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[K\rho_c^{(n+1)/n} \theta^{n+1}\biggr]
\cdot \biggl[ \frac{1}{\rho_c \theta^n} \biggr]
\cdot \frac{1}{\xi a_n}  
\cdot \frac{a_n^2 \xi^2}{G} \biggl[ 4\pi \rho_c a_n^3 \biggl(-\xi^2 \frac{d\theta}{d\xi}\biggr) \biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{4\pi G a_n^2}{(n+1)} \cdot \rho_c^{(n-1)/n} \biggr]
\biggl[\rho_c^{(n+1)/n} \theta^{n+1}\biggr]
\cdot \biggl[ \frac{1}{\rho_c \theta^n} \biggr]  
\cdot \frac{\xi}{4\pi \rho_c a_n^2 G} \biggl(-\xi^2 \frac{d\theta}{d\xi}\biggr)^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{1}{(n+1)}  \biggr]
\cdot \xi \theta \biggl(-\xi^2 \frac{d\theta}{d\xi}\biggr)^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{1}{(n+1)}
\cdot \frac{\theta}{\xi} \biggl(\frac{d\theta}{d\xi}\biggr)^{-1} 
\, .
</math>
  </td>
</tr>
</table>

<span id="Derivative">As a result</span>, we find that,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\biggl[ \frac{P_0}{\rho_0 r_0 g_0} \biggr] \frac{d(p_P)}{d\ln \xi}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math> 
\biggl\{
\frac{1}{(n+1)}
\cdot \frac{\theta}{\xi} \biggl(\frac{d\theta}{d\xi}\biggr)^{-1} 
\biggr\}
\frac{3(n+1)(n-3)}{n}   
\biggl\{ \frac{(n+1)}{2\theta^{n+1}}\biggl( \frac{d\theta}{d\xi}\biggr)^2
+ 
2\xi^{-1}\theta^{-n}\biggl(\frac{d\theta}{d\xi}\biggr) + 1
\biggr\} \biggl(\frac{d\theta}{d\xi}\biggr) \frac{\xi}{\theta} 
</math></td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left"><math> 
\frac{3(n-3)}{n}   
\biggl\{ \frac{(n+1)}{2\theta^{n+1}}\biggl( \frac{d\theta}{d\xi}\biggr)^2
+ 
2\xi^{-1}\theta^{-n}\biggl(\frac{d\theta}{d\xi}\biggr) + 1
\biggr\} \, ;
</math></td>
</tr>
</table>
and

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>4x_P</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math> 
\biggl[ \frac{P_0}{\rho_0 r_0 g_0} \biggr] \frac{d(p_P)}{d\ln \xi}
-
p_P
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math> 
\frac{3(n-3)}{n}   
\biggl\{ \frac{(n+1)}{2\theta^{n+1}}\biggl( \frac{d\theta}{d\xi}\biggr)^2
+ 
2\xi^{-1}\theta^{-n}\biggl(\frac{d\theta}{d\xi}\biggr) + 1
\biggr\} 
+ \frac{3(n+1)}{n} \biggl\{1   
- 
\frac{(n-3)}{2}\biggl[ \frac{1}{\theta^{n+1}} \biggl( \frac{d\theta}{d\xi}\biggr)^2 \biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math> 
\frac{3(n-3)}{n}\biggl\{   
2\xi^{-1}\theta^{-n}\biggl(\frac{d\theta}{d\xi}\biggr) + 1
\biggr\} 
+ \frac{3(n+1)}{n} 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math> 
\frac{3(n-3)}{n}\biggl[   
2\xi^{-1}\theta^{-n}\biggl(\frac{d\theta}{d\xi}\biggr)
\biggr] 
+ \frac{6(n-1)}{n} 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{6(n-1)}{n} \biggl\{ 
1 + 
\frac{n}{6(n-1)} \cdot \frac{3(n-3)}{n}\biggl[   
2\xi^{-1}\theta^{-n}\biggl(\frac{d\theta}{d\xi}\biggr)
\biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{6(n-1)}{n} \biggl\{ 
1 + 
\frac{(n-3)}{(n-1)}\biggl[   
\xi^{-1}\theta^{-n}\biggl(\frac{d\theta}{d\xi}\biggr)
\biggr] 
\biggr\} \, .
</math>
  </td>
</tr>
</table>

Q. E. D.

=====Displacement Functions Summary=====

<table border="1" align="center" width="60%" cellpadding="8"><tr><td align="left">

<div align="center"><b>Summary &hellip;</b></div>

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>x_P</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math> 
\frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{1}{\xi \theta^{n}}\biggr) 
\frac{d\theta}{d\xi}\biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right"><math>d_P</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math> 
-3 \biggl\{1   
- 
\frac{(n-3)}{2}\biggl[ \frac{1}{\theta^{n+1}} \biggl( \frac{d\theta}{d\xi}\biggr)^2 \biggr] \biggr\}
\, ,
</math></td>
</tr>

<tr>
  <td align="right"><math>p_P</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math> 
\biggl( \frac{n+1}{n} \biggr) d_P
\, .
</math></td>
</tr>
</table>

</td></tr></table>

====Related Thoughts====

=====D&eacute;j&agrave; Vu=====
It should not be a surprise, but is nevertheless comforting to see, that our general expression for the polytropic displacement function, 

<div align="center">
<math>~x_P \equiv \frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{1}{\xi \theta^{n}}\biggr) \frac{d\theta}{d\xi}\biggr] \, ,</math>
</div>
morphs into the analytically defined, polytropic displacement functions that had previously been identified, namely, for n = 3 and n = 5 configurations.  The first of these transformations is trivial to demonstrate because, when n = 3, the leading coefficient of the radially dependent portion of the expression for <math>~x_P</math> is zero.  Hence,
<div align="center">
<math>~x_P\biggr|_{n=3} \rightarrow \biggl[\frac{3(n-1)}{2n}\biggl]_{n=3} =  1 \, .</math>
</div>
In the second case, drawing on the definition of <math>~\theta(\xi)</math> for n = 5 polytropes, as given [[SSC/Structure/Polytropes#Primary_E-Type_Solution_2|in an accompanying chapter]], we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_P\biggr|_{n=5}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{6}{5}\biggl[1 + \frac{1}{2}\biggl( \frac{1}{\xi \theta^{5}}\biggr) \frac{d\theta}{d\xi}\biggr]_{n=5} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{6}{5} - \frac{3}{5\xi} \biggl( 1 + \frac{\xi^2}{3} \biggr)^{5/2} \frac{\xi}{3} \biggl( 1 + \frac{\xi^2}{3} \biggr)^{-3/2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{6}{5} - \frac{1}{5} \biggl( 1 + \frac{\xi^2}{3} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 - \frac{\xi^2}{15} \, .
</math>
  </td>
</tr>
</table>
</div>

=====Multiple Harmonics=====
In addition to the green circular marker, two small open circular markers have been placed along the dashed segment of the isothermal displacement function that is displayed in the left panel of Figure 4 &#8212; see also the top panel of Figure 2.  As has been [[#Harmonics|explained above]], in connection with the center and bottom panels of Figure 2, these additional markers have been placed at the next two locations along the undulating displacement function where <math>~d\ln x_Y/d\ln\xi = -3</math>.  They also correspond to the next two positions along the isothermal equilibrium sequence where an extremum in the mass occurs.  

In the left panel of Figure 4, the undulating displacement function that provides a solution to the LAWE for pressure-truncated, n = 6 polytropic configurations has been similarly marked at the two additional locations where <math>~d\ln x_P/d\ln\xi = -3</math>.  It is clear that these two additional markers correspond to the next two positions along the companion n = 6 equilibrium sequence (Figure 3) where an extremum in the mass occurs.  By analogy with the isothermal case, then, we can immediately identify the segments of the n = 6 displacement function that represent the eigenfunctions of the marginally unstable, first harmonic and second harmonic modes of radial oscillation.  Presumably, similar undulations occur in, and precisely this same type of structural information about multiple harmonic-mode eigenfunctions can be extracted from, our analytically specified polytropic displacement function, <math>~x_P</math>, for configurations having any value of the polytropic index within the range, <math>~5 < n < \infty</math>.

=====Configurations Having an Index Less Than Three=====

Up to this point, we have focused our stability analysis on pressure-truncated equilibrium sequences for which the polytropic index, <math>~n \ge 3</math>, because these sequences exhibit turning points associated with physically interesting mass/pressure limits.  As it turns out, if we assume that <math>~\sigma_c^2</math> is zero, the generalized <math>~x_P(\xi)</math> displacement function defined above also provides a solution to the polytropic LAWE when <math>~n < 3</math>.  This can be demonstrated explicitly when <math>~n=1</math> because the equilibrium structural function, <math>~\Theta_H(\xi)</math>, is expressible analytically; [[SSC/Structure/Polytropes#Primary_E-Type_Solution_2|specifically]], 
<div align="center">
<math>~\Theta_H = \frac{\sin\xi}{\xi} \, .</math>
</div>

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

<tr>
  <td align="right">
<math>~x_P\biggr|_{n=1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-3 \biggl[ \biggl( \frac{1}{\xi \theta}\biggr) \frac{d\theta}{d\xi}\biggr]_{n=1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3}{\xi} \biggl( \frac{\xi}{\sin\xi}\biggr) \biggl[\frac{\sin\xi}{\xi^2} - \frac{\cos\xi}{\xi}  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3}{\xi^2}\biggl[ 1- \xi \cot\xi  \biggr]  = 1 + \frac{\xi^2}{15} + \frac{2\xi^4}{315} + \frac{\xi^6}{1575} + \cdots \, .
</math>
  </td>
</tr>
</table>
</div>

In an [[SSC/Stability/n1PolytropeLAWE#Succinct_Demonstration|accompanying discussion]] <font color="red"><b>[<== Very Useful Link]</b></font> we show that this displacement function precisely satisfies the n = 1, polytropic LAWE when <math>~\sigma_c^2 = 0</math>.
 
Does this mean that at least one configuration along the equilibrium sequence of pressure-truncated, n = 1 polytropes &#8212; see [[SSC/Structure/PolytropesEmbedded#Additional.2C_Numerically_Constructed_Polytropic_Configurations|the right panel of Figure 3 in an accompanying discussion]] &#8212; is marginally [dynamically] unstable? The answer is, "No," because, when it is evaluated at the surface <math>~(\tilde\xi)</math> of the truncated configuration, the logarithmic derivative of the displacement function ,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\ln x_P}{d\ln\xi} \biggr|_{n=1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( 1- \tilde\xi \cot\tilde\xi  )^{-1}  
( \tilde\xi \cot\tilde\xi  + \tilde\xi^2 + \tilde\xi^2 \cot^2\tilde\xi -2)\, ,
</math>
  </td>
</tr>
</table>
</div>

is positive along the entire equilibrium sequence <math>~(0 < \tilde\xi < \xi_\mathrm{surf} = \pi)</math>.  Hence, the desired surface boundary condition, <math>~d\ln x/d\ln\xi = - 3</math>, is not satisfied at any location along the sequence.  As a consequence, this displacement function cannot serve as a physically satisfactory, radial-mode eigenfunction.  

Presumably the same logic &#8212; and ultimate consequence &#8212; applies to all other equilibrium, pressure-truncated polytropic sequences that have <math>~n < 3</math>.

====Focus on n = 1 Configurations====
[[SSC/Structure/Polytropes#Primary_E-Type_Solution_2|Specifically for n = 1 configurations]], 
<div align="center">
<math>~\Theta_H = \frac{\sin\xi}{\xi} \, .</math>
</div>

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

<tr>
  <td align="right">
<math>~x_P\biggr|_{n=1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-3 \biggl[ \biggl( \frac{1}{\xi \theta}\biggr) \frac{d\theta}{d\xi}\biggr]_{n=1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3}{\xi} \biggl( \frac{\xi}{\sin\xi}\biggr) \biggl[\frac{\sin\xi}{\xi^2} - \frac{\cos\xi}{\xi}  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3}{\xi^2}\biggl[ 1- \xi \cot\xi  \biggr]   \, ;
</math>
  </td>
</tr>
</table>

and the logarithmic derivative of the displacement function,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\ln x_P}{d\ln\xi} \biggr|_{n=1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( 1- \tilde\xi \cot\tilde\xi  )^{-1}  
\biggl[ -2 + \tilde\xi \cot\tilde\xi  + \frac{{\tilde\xi}^2}{\sin^2 \tilde\xi} \biggr]\, .
</math>
  </td>
</tr>
</table>
<!-- 
<table border="1" cellpadding="8" align="center">
<tr><td align="center">
<font color="red">Originally Typed Expression:</font>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\ln x_P}{d\ln\xi} \biggr|_{n=1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( 1- \tilde\xi \cot\tilde\xi  )^{-1}  
( \tilde\xi \cot\tilde\xi  + \tilde\xi^2 + \tilde\xi^2 \cot^2 \tilde\xi -2)\, .
</math>
  </td>
</tr>
</table>

----

<font color="red">Preferable Expression (9/30/2025)</font>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\ln x_P}{d\ln\xi} \biggr|_{n=1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( 1- \tilde\xi \cot\tilde\xi  )^{-1}  
\biggl[ -2 + \tilde\xi \cot\tilde\xi  + \frac{{\tilde\xi}^2}{\sin\tilde\xi} \biggr]\, .
</math>
  </td>
</tr>
</table>

</td></tr>
</table>
-->
If the polytropic configuration is truncated by the pressure, <math>P_e</math>, of a hot, tenuous external medium, then the solution to the LAWE is subject to the outer boundary condition,
<div align="center">
<math>-\frac{d\ln x}{d\ln\xi} = 3</math> &nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>\xi = \tilde\xi \, .</math>
</div>
But, for ''isolated'' polytropes &#8212; see the [[#Supporting_Scratch_Work|supporting derivation]] &#8212; the sought-after solution is subject to the more conventional boundary condition,
<div align="center">
<math>- \frac{d\ln x}{d\ln \xi} = \biggl(\frac{3-n}{n+1}\biggr) + \frac{n\sigma_c^2}{6(n+1)} \biggl[\frac{\xi}{\theta^'}\biggr] </math> &nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>\xi = \xi_\mathrm{surf} \, .</math><br />


</div>