Editing SSC/Stability/BiPolytrope00Details (section)

===Illustration22===

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Analytically Definable Eigenvectors in <math>~(n_c, n_e) = (0,0)</math> Bipolytropes<br />

<font color="red">Quantum Numbers:</font>&nbsp; <math>~(\ell,j) = (2,2)</math>
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  <th align="center" width="50%"> Analyzable Model Sequence</th>
  <th align="center" width="50%">One Example Eigenfunction</th>
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[[File:Model22MontageCorrected.png|800px|Montage of Stability Results for (ell,j) = (2,1) quantum numbers]]
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<font color="darkblue"><b>''Top-Left Panel:''</b></font>&nbsp; Plotted points show how the location of the core/envelope interface, <math>~q \equiv r_i/R</math>, varies with <math>~\alpha_e \equiv (3-4/\gamma_e)</math> &#8212; where <math>~\gamma_e</math> is the adiabatic exponent of the envelope &#8212; in equilibrium models that are amenable to analytic modal analysis for quantum numbers, <math>~(\ell,j) = (2,2)</math>.  Red (alternatively, blue) markers identify models for which the corresponding value of the adiabatic exponent of the ''core'' [see bottom-left panel] falls inside (alternatively, outside) the physically viable range, namely, <math>~1 \le \gamma_c \le \infty</math>.  The red marker that is farthest to the left identifies the model whose analytically determined eigenfunction is displayed, as an example, in the right-hand panels.

<font color="darkblue"><b>''Bottom-Left Panel:''</b></font> &nbsp; Plotted points (only 2, here!) show how <math>~\alpha_c \equiv (3-4/\gamma_c)</math> varies with <math>~\alpha_e</math> over the physically viable parameter range, <math>~-1 \le \alpha \le 3</math>.  Both axes have been flipped so that incompressible models <math>~(\gamma = \infty)</math> lie on the left/bottom while isothermal models <math>~(\gamma =1)</math> lie on the right/top.  The core is ''more'' compressible than the envelope in models that lie above and to the left of the black-dashed, diagonal line.  

<font color="darkblue"><b>''Top-Right Panel:''</b></font>&nbsp; Displays &#8212; as a function of the fractional radius, <math>~r_0/R = q\xi</math> &#8212;  the analytically determined eigenfunction for the <math>~(\ell,j) = (2,2)</math> mode in the model identified by the red circular marker that is farthest to the left in both left-hand panels, for which,
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<math>~q = \biggl[0.005 + 179\biggl( \frac{0.985}{200} \biggr) \biggr] = 0.886575</math> 
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and, correspondingly, <math>~(c_0, \alpha_e,\alpha_c) = (-2.332785, +0.7763158, -0.9146699)</math>.   Specifically, over the radial interval, <math>~0 \le \xi \le 1</math>, the green markers trace the core's contribution to the combined eigenfunction, namely,

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<math>~x_{j=2} |_\mathrm{core}</math>
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<math>~=</math>
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<math>~
\frac{35(1+8q^3)^2 -  126(1+8q^3) (1+2q^3)^2 \xi^2 + 99 (1+2q^3)^4 \xi^4 }{35(1+8q^3)^2 -  126(1+8q^3) (1+2q^3)^2 + 99 (1+2q^3)^4 } \, ;</math>
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and, over the radial interval, <math>~1 \le \xi \le 1/q</math>, the purple markers trace the envelope's contribution to the combined eigenfunction, namely,
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<math>~x_{\ell=2} |_\mathrm{env}</math>
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<math>~=</math>
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<math>~
\xi^{c_0}\biggl[ \frac{ 1 +  q^3 A_{22} \xi^{3} +  q^6 A_{22}B_{22}\xi^{6} }{ 1 +  q^3 A_{22}  +  q^6 A_{22}B_{22}  }\biggr] \, ,
</math>
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where the coefficients, <math>~A_{22}, B_{22}</math>, are as [[#Setup22|defined above]] in terms of the parameter, <math>~c_0</math>.  The corresponding eigenfrequency is, from the perspective of the core,
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<math>~\frac{3\omega^2}{2\pi G \rho_c}</math>
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<math>~=</math>
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<math>~42\gamma_c - 8 = \frac{8(18 + \alpha_c)}{3-\alpha_c} \approx 34.915496 \, ;</math>
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and, from the perspective of the envelope,
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<math>~\frac{3\omega^2}{2\pi G \rho_c}</math>
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<math>~=</math>
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<math>~3\gamma_e\biggl[\alpha_e + 5c_0 + 22\biggr] \frac{\rho_e}{\rho_c} \approx 34.915496 \, ,</math>
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where the relevant density ratio is, <math>~\rho_e/\rho_c = 2q^3/(1+2q^3) \approx 0.5822407</math>.

<font color="darkblue"><b>''Bottom-Right Panel:''</b></font>&nbsp; The green and purple markers present the same eigenfunction-amplitude information, <math>~x(r/R)</math>, as in the Top-Right panel, but on a logarithmic scale.  Specifically, in this plot, the vertical displacement of the green and purple markers is given by the expression,
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<math>~y</math>
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<math>~=</math>
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<math>~\tfrac{1}{8} \log_{10}[x^2 + \epsilon^2] + y_\mathrm{shift} \, ,</math>
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where, for plotting purposes, we have used, <math>~\epsilon = 10^{-5}</math>, and have set <math>~y_\mathrm{shift}</math> to a value that ensures that <math>~y \approx 1</math> at the center of the configuration.  In this type of log-amplitude plot, the eigenfunction's various ''nodes'' &#8212; that is, radial locations where <math>~x</math> passes through zero &#8212; are highlighted; here, specifically, there are two nodes inside the core and none in the envelope, although one of the nodes in the core lies just inside the core/envelope interface.  Using the vertical coordinate to represent, instead, the configuration's mass-density normalized to its central value, <math>~\rho/\rho_c</math>, the solid black line segments trace the unperturbed density distribution throughout this specific <math>~(n_c, n_e) = (0,0)</math> bipolytrope.  Throughout the core, <math>~\rho/\rho_c = 1</math>; then, at the location of the interface <math>~(r_i/R = q \approx 0.89)</math>, the density abruptly drops to its envelope value <math>~(\rho/\rho_c = \rho_e/\rho_c \approx 0.58)</math>.

NOTE:  As may be ascertained from our [[SSC/Structure/BiPolytropes/Analytic00#BiPolytrope_with_nc_.3D_0_and_ne_.3D_0|general discussion of the structural properties of <math>~(n_c, n_e) = (0,0)</math> bipolytropes]], equilibrium "zero-zero" bipolytropes can be constructed with the envelope/core interface parameter set to any value across the range, <math>~0 \le q \le 1</math>; and for any chosen value of <math>~q</math>, the envelope/core density ratio can, in principle, be set to ''any'' value, <math>~0 \le \rho_e/\rho_c \le 1</math>.  We have not, however, been able to analytically solve the relevant pair of linear-adiabatic wave equations (LAWEs) for this entire set of equilibrium models.  Instead, our ability to derive analytically prescribed eigenvectors is [[SSC/Stability/BiPolytrope00#KeyConstraint|limited by the constraint]],
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<math>~\frac{\rho_e}{\rho_c}</math>
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<math>~=</math>
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<math>~\frac{2q^3}{1+2q^3} = \frac{2(r_i/R)^3}{1+2(r_i/R)^3}\, .</math>
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The black-dotted curve in the ''Bottom-Right Panel'' displays the behavior of this constraint; accordingly, the step function depicted by the solid black line segments must necessarily drop from unity to a point on this black-dotted curve for any equilibrium model &#8212; such as the example illustrated here &#8212; that has an analytically prescribable radial-oscillation eigenvector.