Editing SSC/Structure/BiPolytropes/Analytic51Renormalize (section)

===Trial Displacement Function===

The blue curve in the following figure results from plotting <math>x_\mathrm{core}</math> versus <math>\tilde{r}_\mathrm{core}</math> after setting the leading coefficient, <math>\alpha = - 0.0011</math>.  The red-dotted curve results from plotting <math>(x_\mathrm{env} + x_\mathrm{shift})</math> versus <math>\tilde{r}_\mathrm{env}</math> after setting the leading coefficient, <math>\beta = - 0.000062</math>, and <math>x_\mathrm{shift} = + 0.0105</math>.

<table border="1" align="center" cellpadding="2">
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  <td align="center">[[File:TrialAnalyticEigenfunction01.png|700px|Trial Analytic Eigenfunction]]</td>
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</table>

ASSESSMENT:
<ul>
  <li>Our analytically specified displacement function, <math>x_\mathrm{core}</math>, appears to be an excellent match to the displacement function obtained throughout the core by implementing the [[Appendix/Ramblings/NonlinarOscillation#Radial_Oscillations_in_Pressure-Truncated_n_=_5_Polytropes|B-KB74 conjecture]].</li>
  <li>At first glance, the plot of <math>(x_\mathrm{env} + x_\mathrm{shift})</math> appears to provide a reasonably good fit to the ''approximate'' displacement function that we have obtained throughout the envelope by implementing the B-KB74 conjecture.   But, in reality, there are two fatal flaws:
<ol type="1">
<li>We have presented the behavior of our analytically specified envelope displacement function only up to the radial coordinate, <math>\eta = 2.19707 ~~ (\tilde{r}_\mathrm{env} = 0.19526)</math>. Between this point and the surface, <math>\eta_s = 2.2823226 ~~ (\tilde{r}_\mathrm{env} = 0.2028415)</math> &#8212; where the argument of the cotangent, <math>(\eta_s - B) \rightarrow \pi</math> &#8212; the analytic function dives steeply to negative infinity.  This violently departs from the behavior derived via the [[Appendix/Ramblings/NonlinarOscillation#Radial_Oscillations_in_Pressure-Truncated_n_=_5_Polytropes|B-KB74 conjecture]].</li>
<li>While our analytically specified displacement function, <math>x_\mathrm{env}</math>, satisfies the "n = 1" polytropic LAWE, this satisfaction is destroyed by adding <math>x_\mathrm{shift}</math> to the displacement function.</li>
</ol>
  </li>
</ul>

Let's examine the slope of the displacement function at the interface.  From the perspective of the core, our analytic prescription for the displacement function matches the K-BK74-derived displacement function very well.  An analytic evaluation of the slope at the inferface &#8212; as derived [[#Core|above]] &#8212; gives,
<div align="center">
<math>\frac{dx_\mathrm{core}}{d\tilde{r}}\biggr|_i = -707.53765\alpha = +0.77829</math>.
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The black-dashed line segment that appears in the following figure has this slope and goes through the point of intersection; it appears to be tangent to the analytic displacement function, as expected.  Alternatively, the orange-dashed line segment that appears in this same figure, also goes through the point of intersection, but it has a slope that matches our ''expectation'' for the envelope's displacement function; that is, it has a slope [[#Envelope|as derived]] of,
<div align="center">
<math>\frac{dx_\mathrm{env}}{d\tilde{r}}\biggr|_i = +0.08619</math>.
</div>
This orange-dashed line segment does ''not'' appear to lie tangent to the K-BK74-derived displacement function for the envelope.

<table border="1" align="center" cellpadding="2">
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  <td align="center">[[File:TrialEigenfunctionSlopes01.png|700px|Trial Analytic Eigenfunction with Intersection Slopes]]</td>
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</table>