Editing Appendix/Ramblings/NonlinarOscillation (section)

====Analytic, Marginally Unstable Eigenfunction====
In terms of <math>~\xi</math>, we know that the [[SSC/Stability/n5PolytropeLAWE#Eureka_Moment|eigenfunction of the marginally unstable model]] &#8212; see also [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|a more general discussion]] &#8212; is,
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<math>~x_P = \frac{\delta r}{r_0}</math>
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<math>~=</math>
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<math>~1 - \frac{\xi^2}{15} \, .</math>
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We can now rewrite this eigenfunction in terms of the fractional mass, <math>~m_\xi</math>.  Specifically, given that <math>~\tilde{C} = 4</math> in the marginally unstable configuration, we find that,
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<math>~x_P </math>
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<math>~=</math>
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<math>~1 - \frac{3}{5} \biggl[ 4~m_\xi^{-2/3} -3 \biggr]^{-1} </math>
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&nbsp;
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<math>~=</math>
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<math>~\frac{2}{5} \biggl[ \frac{10~m_\xi^{-2/3} -9}{4~m_\xi^{-2/3} -3} \biggr] </math>
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&nbsp;
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<math>~=</math>
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<math>~\frac{2}{5} \biggl[ \frac{10 -9~m_\xi^{2/3}}{4 -3~m_\xi^{2/3}} \biggr] \, .</math>
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It is important to remember that, although the leading factor of this expression is <math>~\tfrac{2}{5}</math>, in general the overall amplitude of this eigenfunction can be set arbitrarily.  In order to allow for this, we will introduce an overall scaling factor, <math>~A_0</math>, and write,

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<math>~x_P </math>
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<math>~=</math>
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<math>~\frac{2A_0}{5} \biggl[ \frac{10 -9~m_\xi^{2/3}}{4 -3~m_\xi^{2/3}} \biggr] \, .</math>
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</table>
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Then our originating expression for <math>~x_P</math> is retrieved by setting <math>~A_0 = 1</math>, in which case the amplitude of the eigenfunction is unity at the center <math>~(m_\xi = 0)</math> and it is <math>~\tfrac{2}{5}</math> at the surface  <math>~(m_\xi = 1)</math>.