User:Tohline/SSC/Stability/BiPolytrope0 0 - Revision history

Tohline: /* Example Solutions */

2017-02-10T21:19:27Z

Example Solutions

← Older revision		Revision as of 21:19, 10 February 2017
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	</div>		</div>

	Each of the six plots displayed in [[#Table4\|Table 4]] (click on a panel in order to view a larger image) was [[User:Tohline/Appendix/Ramblings/NumericallyDeterminedEigenvectors#Numerically_Determined_Eigenvectors_of_a_Zero-Zero_Bipolytrope\|generated numerically]] by integrating the LAWE for the core, outward from the center of the configuration to the core/envelope interface, then integrating the LAWE for the envelope, from the interface outward to the surface. At the interface:   the ''value'' of the envelope eigenfunction is set to the ''value'' of the eigenfunction of the core; and the ''slope'' of the envelope's eigenfunction (highlighted graphically in each plot by the green, dashed line segment) was based on the slope of the core's eigenfunction (highlighted graphically by the orange, dashed line segment) but shifted in a discontinuous fashion [[#Step4\|according to the above "Step 4" discussion]]. Each of the graphically illustrated Table 4 eigenfunctions has been scaled in such a way that the central value is unity; note that the panel labeled <math>~(\ell=2; \sigma_c^2 = +16)</math> displays an eigenfunction that is identical to the ~~one~~ displayed as Model C21 in [[#Table3\|Table 3]], but it has been rescaled — and ~~necessarily~~ inverted — to provide a central value of unity.		Each of the six plots displayed in [[#Table4\|Table 4]] (click on a panel in order to view a larger image) was [[User:Tohline/Appendix/Ramblings/NumericallyDeterminedEigenvectors#Numerically_Determined_Eigenvectors_of_a_Zero-Zero_Bipolytrope\|generated numerically]] by integrating the LAWE for the core, outward from the center of the configuration to the core/envelope interface, then integrating the LAWE for the envelope, from the interface outward to the surface. At the interface:   the ''value'' of the envelope eigenfunction is set to the ''value'' of the eigenfunction of the core; and the ''slope'' of the envelope's eigenfunction (highlighted graphically in each plot by the green, dashed line segment) was based on the slope of the core's eigenfunction (highlighted graphically by the orange, dashed line segment) but shifted in a discontinuous fashion [[#Step4\|according to the above "Step 4" discussion]]. Each of the graphically illustrated Table 4 eigenfunctions has been scaled in such a way that the central value is unity; note that the panel labeled <math>~(\ell=2; \sigma_c^2 = +16)</math> displays an eigenfunction that is identical to the ''analytically defined'' eigenfunction displayed as Model C21 in [[#Table3\|Table 3]], but it has been rescaled — and by necessity inverted — to provide a central value of unity.

Tohline: /* Example Solutions */

2017-02-09T22:46:58Z

Example Solutions

← Older revision		Revision as of 22:46, 9 February 2017
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	</div>		</div>

	Each of the six plots displayed in [[#Table4\|Table 4]] (click on a panel in order to view a larger image) was [[User:Tohline/Appendix/Ramblings/NumericallyDeterminedEigenvectors#Numerically_Determined_Eigenvectors_of_a_Zero-Zero_Bipolytrope\|generated numerically]] by integrating the LAWE for the core, outward from the center of the configuration to the core/envelope interface, then integrating the LAWE for the envelope, from the interface outward to the surface. At the interface:   the ''value'' of the envelope eigenfunction is set to the ''value'' of the eigenfunction of the core; and the ''slope'' of the envelope's eigenfunction (highlighted graphically in each plot by the green, dashed line segment) was based on the slope of the core's eigenfunction (highlighted graphically by the orange, dashed line segment) but shifted in a discontinuous fashion [[#Step4\|according to the above "Step 4" discussion]].		Each of the six plots displayed in [[#Table4\|Table 4]] (click on a panel in order to view a larger image) was [[User:Tohline/Appendix/Ramblings/NumericallyDeterminedEigenvectors#Numerically_Determined_Eigenvectors_of_a_Zero-Zero_Bipolytrope\|generated numerically]] by integrating the LAWE for the core, outward from the center of the configuration to the core/envelope interface, then integrating the LAWE for the envelope, from the interface outward to the surface. At the interface:   the ''value'' of the envelope eigenfunction is set to the ''value'' of the eigenfunction of the core; and the ''slope'' of the envelope's eigenfunction (highlighted graphically in each plot by the green, dashed line segment) was based on the slope of the core's eigenfunction (highlighted graphically by the orange, dashed line segment) but shifted in a discontinuous fashion [[#Step4\|according to the above "Step 4" discussion]]. Each of the graphically illustrated Table 4 eigenfunctions has been scaled in such a way that the central value is unity; note that the panel labeled <math>~(\ell=2; \sigma_c^2 = +16)</math> displays an eigenfunction that is identical to the one displayed as Model C21 in [[#Table3\|Table 3]], but it has been rescaled — and necessarily inverted — to provide a central value of unity.

Tohline: /* Eigenvector */

2017-02-09T22:39:54Z

Eigenvector

← Older revision		Revision as of 22:39, 9 February 2017
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	===Core Segment===		===Core Segment===

	<div align="center">		<div align="center" id="Table1">
	<table border="1" cellpadding="8">		<table border="1" cellpadding="8">
			<tr>
			<th align="center" colspan="3"><font size="+1">Table 1:</font>  Analytically Specifiable Core Eigenvectors</th>
			</tr>
	<tr>		<tr>
	<td align="center" colspan="1">		<td align="center" colspan="1">
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	===Envelope Segment===		===Envelope Segment===

	<div align="center">		<div align="center" id="Table2">
	<table border="1" cellpadding="8">		<table border="1" cellpadding="8">
			<tr>
			<th align="center" colspan="3"><font size="+1">Table 2:</font>  Analytically Specifiable Envelope Eigenvectors</th>
			</tr>
	<tr>		<tr>
	<td align="center" colspan="1">		<td align="center" colspan="1">
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	===Example Solutions===		===Example Solutions===

			<div align="center" id="Table3">
	<table border="1" cellpadding="5" align="center">		<table border="1" cellpadding="5" align="center">
	<tr><th align="center" colspan="6">Example Analytic Model Parameters for <math>~(\ell,j) = (2,1)</math><br />NOTE:  <math>\mathfrak{F}_\mathrm{core} = 14</math></th></tr>		<tr><th align="center" colspan="6"><font size="+1">Table 3:</font>  Example Analytic Model Parameters for <math>~(\ell,j) = (2,1)</math><br />NOTE:  <math>\mathfrak{F}_\mathrm{core} = 14</math></th></tr>
	<tr>		<tr>
	<td align="center">Eigenfunction</td>		<td align="center">Eigenfunction</td>
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	</tr>		</tr>
	</table>		</table>
			</div>


	It appears as though the eigenvectors (eigenfunction and eigenfrequency) of other radial oscillation modes can be identified by holding all other parameters fixed but changing the value of the quantum number, <math>~\ell</math>, in the [[#OtherSigmas\|expression provided below]]. Picking the configuration identified as model '''C21''' in ~~the above table~~, for example, in addition to the parameter values provided in the table we have,		It appears as though the eigenvectors (eigenfunction and eigenfrequency) of other radial oscillation modes can be identified by holding all other parameters fixed but changing the value of the quantum number, <math>~\ell</math>, in the [[#OtherSigmas\|expression provided below]]. Picking the configuration identified as model '''C21''' in [[#Table3\|Table 3]], for example, in addition to the parameter values provided in the table we have,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</table>		</table>

	<div align="center">		<div align="center" id="Table4">
	<table border="1" cellpadding="5" align="center">		<table border="1" cellpadding="5" align="center">
	<tr>		<tr>
	<th align="center" colspan="6">Additional Hypothesized Oscillation Modes for Model C21</th>		<th align="center" colspan="6"><font size="+1">Table 4:</font>  Additional Hypothesized Oscillation Modes for Model C21</th>
	</tr>		</tr>
	<tr>		<tr>
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	</table>		</table>
	</div>		</div>

			Each of the six plots displayed in [[#Table4\|Table 4]] (click on a panel in order to view a larger image) was [[User:Tohline/Appendix/Ramblings/NumericallyDeterminedEigenvectors#Numerically_Determined_Eigenvectors_of_a_Zero-Zero_Bipolytrope\|generated numerically]] by integrating the LAWE for the core, outward from the center of the configuration to the core/envelope interface, then integrating the LAWE for the envelope, from the interface outward to the surface. At the interface:   the ''value'' of the envelope eigenfunction is set to the ''value'' of the eigenfunction of the core; and the ''slope'' of the envelope's eigenfunction (highlighted graphically in each plot by the green, dashed line segment) was based on the slope of the core's eigenfunction (highlighted graphically by the orange, dashed line segment) but shifted in a discontinuous fashion [[#Step4\|according to the above "Step 4" discussion]].


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	As is shown by the plot displayed in the right-hand panel of Figure 1, we have found different values of <math>~q</math> for each choice (STEP 1) of <math>~\gamma_e</math> (or, equivalently, choice of <math>~\alpha_e</math>). In this plot we have purposely flipped the horizontal axis so that the extreme left <math>~(\alpha_e = +3)</math> represents an incompressible <math>~(n = 0)</math> envelope, while the extreme right represents an isothermal <math>~(\gamma_e = 1)</math> envelope.		As is shown by the plot displayed in the right-hand panel of Figure 1, we have found different values of <math>~q</math> for each choice (STEP 1) of <math>~\gamma_e</math> (or, equivalently, choice of <math>~\alpha_e</math>). In this plot we have purposely flipped the horizontal axis so that the extreme left <math>~(\alpha_e = +3)</math> represents an incompressible <math>~(n = 0)</math> envelope, while the extreme right represents an isothermal <math>~(\gamma_e = 1)</math> envelope.
	<!--		<!--
	The green and orange curves, respectively, show as well how the corresponding model parameters, <math>~\nu</math> and <math>~\rho_e/\rho_c</math>, vary with <math>~\alpha_e</math>. The right-hand panel displays one example of an <math>~(\ell, j) = (2,1)</math> eigenfunction that simultaneously satisfies the LAWE of the core and the LAWE of the envelope, and matches smoothly at the interface. This ''particular'' plotted solution corresponds to the case of <math>~\alpha_e = -0.35</math>, for which: <math>~\gamma_e =1.1940299</math>; <math>~n_e = 5.1538462</math>; <math>~q = 0.7943853</math>; <math>~\nu =0.6675302</math>; and <math>~\rho_e/\rho_c =0.5006468</math>.		The green and orange curves, respectively, show as well how the corresponding model parameters, <math>~\nu</math> and <math>~\rho_e/\rho_c</math>, vary with <math>~\alpha_e</math>. The right-hand panel displays one example of an <math>~(\ell, j) = (2,1)</math> eigenfunction that simultaneously satisfies the LAWE of the core and the LAWE of the envelope, and matches smoothly at the interface. This ''particular'' plotted solution corresponds to the case of <math>~\alpha_e = -0.35</math>, for which: <math>~\gamma_e =1.1940299</math>; <math>~n_e = 5.1538462</math>; <math>~q = 0.7943853</math>; <math>~\nu =0.6675302</math>; and <math>~\rho_e/\rho_c =0.5006468</math>.
	-->		-->

Tohline: /* Example Solutions */ Insert images of six hypothetical oscillation modes

2017-02-09T18:01:51Z

Example Solutions: Insert images of six hypothetical oscillation modes

← Older revision		Revision as of 18:01, 9 February 2017
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	<td align="center"><math>~\ell = 4</math><p></p><math>~\sigma_c^2 = +57.033</math></td>		<td align="center"><math>~\ell = 4</math><p></p><math>~\sigma_c^2 = +57.033</math></td>
	<td align="center"><math>~\ell = 5</math><p></p><math>~\sigma_c^2 = +86.255</math></td>		<td align="center"><math>~\ell = 5</math><p></p><math>~\sigma_c^2 = +86.255</math></td>
			</tr>
			<tr>
			<td align="center">[[File:C01hypothetical.png\|150 px\|center]]</td>
			<td align="center">[[File:C11hypothetical.png\|150 px\|center]]</td>
			<td align="center">[[File:C21hypothetical.png\|150 px\|center]]</td>
			<td align="center">[[File:C31hypothetical.png\|150 px\|center]]</td>
			<td align="center">[[File:C41hypothetical.png\|150 px\|center]]</td>
			<td align="center">[[File:C51hypothetical.png\|150 px\|center]]</td>
	</tr>		</tr>
	</table>		</table>

Tohline: /* Example Solutions */ Begin constructing table of additional hypothesized oscillation modes for Model C21

2017-02-09T17:18:05Z

Example Solutions: Begin constructing table of additional hypothesized oscillation modes for Model C21

← Older revision		Revision as of 17:18, 9 February 2017
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	</table>		</table>


			It appears as though the eigenvectors (eigenfunction and eigenfrequency) of other radial oscillation modes can be identified by holding all other parameters fixed but changing the value of the quantum number, <math>~\ell</math>, in the [[#OtherSigmas\|expression provided below]]. Picking the configuration identified as model '''C21''' in the above table, for example, in addition to the parameter values provided in the table we have,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\alpha_e = 3 - \frac{4}{\gamma_e} = -0.9887044</math>
			</td>
			<td align="center">
			and,
			</td>
			<td align="left">
			<math>~c_0 = \sqrt{1+\alpha_e} - 1 = -0.8937192 \, ,</math>
			</td>
			</tr>
			</table>
			</div>

			so we expect the variation in (the square of) the eigenfrequency, <math>~\sigma_c</math>, with <math>~\ell</math> to be,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\sigma_c^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~12\biggl[\frac{c_0^2 + c_0(2\ell+3) + \ell(3\ell+5) }{(3-\alpha_e) } \biggr]\frac{\rho_e}{\rho_c} </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~0.9671938[c_0^2 + c_0(2\ell+3) + \ell(3\ell+5) ] </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~0.9671938[-1.8824236 +(3.2125616)\ell + 3\ell^2 ] \, .</math>
			</td>
			</tr>
			</table>

			<div align="center">
			<table border="1" cellpadding="5" align="center">
			<tr>
			<th align="center" colspan="6">Additional Hypothesized Oscillation Modes for Model C21</th>
			</tr>
			<tr>
			<td align="center"><math>~\ell = 0</math><p></p><math>~\sigma_c^2 = -1.821</math></td>
			<td align="center"><math>~\ell = 1</math><p></p><math>~\sigma_c^2 = +4.188</math></td>
			<td align="center"><math>~\ell = 2</math><p></p><math>~\sigma_c^2 = +16</math></td>
			<td align="center"><math>~\ell = 3</math><p></p><math>~\sigma_c^2 = +33.615</math></td>
			<td align="center"><math>~\ell = 4</math><p></p><math>~\sigma_c^2 = +57.033</math></td>
			<td align="center"><math>~\ell = 5</math><p></p><math>~\sigma_c^2 = +86.255</math></td>
			</tr>
			</table>
			</div>

Tohline: /* Broad Application */

2017-02-09T16:33:40Z

Broad Application

← Older revision		Revision as of 16:33, 9 February 2017
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	</div>		</div>
	[[File:CommentButton02.png\|right\|100px\|Comment by J. E. Tohline on 4 February 2017: This expression for the density ratio is a necessary but not sufficient relationship. According to each analytic solution, once the pair of quantum numbers <math>~(\ell,j)</math> has been specified, the two adiabatic indexes cannot be specified independently of one another.]]where, as a reminder, <math>~c_0 = [-1 \pm \sqrt{1+\alpha_e}]</math>. Eliminating <math>~\sigma_c^2</math> between these two relations gives,		[[File:CommentButton02.png\|right\|100px\|Comment by J. E. Tohline on 4 February 2017: This expression for the density ratio is a necessary but not sufficient relationship. According to each analytic solution, once the pair of quantum numbers <math>~(\ell,j)</math> has been specified, the two adiabatic indexes cannot be specified independently of one another.]]where, as a reminder, <math>~c_0 = [-1 \pm \sqrt{1+\alpha_e}]</math>. Eliminating <math>~\sigma_c^2</math> between these two relations gives,
	<div align="center">

			<div align="center" id="OtherSigmas">
	<table border="1" cellpadding="8" align="center"><tr><td align="center">		<table border="1" cellpadding="8" align="center"><tr><td align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Example Solutions */ Add plots of eigenfunctions to example table

2017-02-08T20:43:47Z

Example Solutions: Add plots of eigenfunctions to example table

← Older revision		Revision as of 20:43, 8 February 2017
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	<table border="1" cellpadding="5" align="center">		<table border="1" cellpadding="5" align="center">
	<tr><th align="center" colspan="6">Example Model Parameters for <math>~(\ell,j) = (2,1)</math></th></tr>		<tr><th align="center" colspan="6">Example Analytic Model Parameters for <math>~(\ell,j) = (2,1)</math><br />NOTE:  <math>\mathfrak{F}_\mathrm{core} = 14</math></th></tr>
	<tr>		<tr>
	<td align="center">~~Model~~</td>		<td align="center">Eigenfunction</td>
	<td align="center"><math>~\gamma_c ~(n_c)</math></td>		<td align="center"><math>~\gamma_c ~(n_c)</math></td>
	<td align="center"><math>~\gamma_e</math></td>		<td align="center"><math>~\gamma_e</math></td>
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	</tr>		</tr>
	<tr>		<tr>
	<td ~~align="~~center~~">'''~~A21~~'''</td>~~		<td>[[File:A21.png\|center\|thumb\|100px\|Model A21]]
	<td align="center"> <math>~\frac{5}{3} ~~\biggl(\frac{3}{2} \biggr)</math></td>		<td align="center"> <math>~\frac{5}{3} ~~\biggl(\frac{3}{2} \biggr)</math></td>
	<td align="center">1.1340607</td>		<td align="center">1.1340607</td>
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	</tr>		</tr>
	<tr>		<tr>
	<td ~~align="~~center~~">'''~~B21~~'''</td>~~		<td>[[File:B21.png\|center\|thumb\|100px\|Model B21]]
	<td align="center"> <math>~\frac{4}{3} ~~\biggl( 3 \biggr)</math></td>		<td align="center"> <math>~\frac{4}{3} ~~\biggl( 3 \biggr)</math></td>
	<td align="center">1.0263212</td>		<td align="center">1.0263212</td>
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	<td align="center">0.3424445</td>		<td align="center">0.3424445</td>
	<td align="center">18.666666</td>		<td align="center">18.666666</td>
			</tr>
			<tr>
			<td>[[File:C21.png\|center\|thumb\|100px\|Model C21]]
			<td align="center"> <math>~\frac{6}{5} ~~\biggl( 5 \biggr)</math></td>
			<td align="center">1.0028319</td>
			<td align="center">0.6187646</td>
			<td align="center">0.3214875</td>
			<td align="center">16.000000</td>
	</tr>		</tr>
	</table>		</table>



	{{LSU_WorkInProgress}}		{{LSU_WorkInProgress}}

	===Old, Incorrect Solutions===		===Old, Incorrect Solutions===
	As is shown by the plot displayed in the right-hand panel of Figure 1, we have found different values of <math>~q</math> for each choice (STEP 1) of <math>~\gamma_e</math> (or, equivalently, choice of <math>~\alpha_e</math>). In this plot we have purposely flipped the horizontal axis so that the extreme left <math>~(\alpha_e = +3)</math> represents an incompressible <math>~(n = 0)</math> envelope, while the extreme right represents an isothermal <math>~(\gamma_e = 1)</math> envelope.		As is shown by the plot displayed in the right-hand panel of Figure 1, we have found different values of <math>~q</math> for each choice (STEP 1) of <math>~\gamma_e</math> (or, equivalently, choice of <math>~\alpha_e</math>). In this plot we have purposely flipped the horizontal axis so that the extreme left <math>~(\alpha_e = +3)</math> represents an incompressible <math>~(n = 0)</math> envelope, while the extreme right represents an isothermal <math>~(\gamma_e = 1)</math> envelope.

Tohline: /* Piecing Together */

2017-02-08T20:06:45Z

Piecing Together

Show changes

Tohline: /* Eigenvector */ Begin tabulating model solutions with discontinuity in adiabatic exponent

2017-02-08T20:01:46Z

Eigenvector: Begin tabulating model solutions with discontinuity in adiabatic exponent

← Older revision		Revision as of 20:01, 8 February 2017
Line 1,023:		Line 1,023:
	The physically relevant (real) root of this quartic equation in <math>~\Chi</math> — see our [[User:Tohline/SSC/Stability/BiPolytrope0_0Details#Quartic\|accompanying detailed presentation]] — gives us the ''specific'' value of the dimensionless interface location, <math>~q</math>, for which both the values and the first derivatives of the two eigenfunction segments match at the interface.		The physically relevant (real) root of this quartic equation in <math>~\Chi</math> — see our [[User:Tohline/SSC/Stability/BiPolytrope0_0Details#Quartic\|accompanying detailed presentation]] — gives us the ''specific'' value of the dimensionless interface location, <math>~q</math>, for which both the values and the first derivatives of the two eigenfunction segments match at the interface.

			===Example Solutions===

			<table border="1" cellpadding="5" align="center">
			<tr><th align="center" colspan="6">Example Model Parameters for <math>~(\ell,j) = (2,1)</math></th></tr>
			<tr>
			<td align="center">Model</td>
			<td align="center"><math>~\gamma_c ~(n_c)</math></td>
			<td align="center"><math>~\gamma_e</math></td>
			<td align="center"><math>~q</math></td>
			<td align="center"><math>~\frac{\rho_e}{\rho_c}</math></td>
			<td align="center"><math>~\sigma_c^2</math></td>
			</tr>
			<tr>
			<td align="center">'''A21'''</td>
			<td align="center"> <math>~\frac{5}{3} ~~\biggl(\frac{3}{2} \biggr)</math></td>
			<td align="center">1.1340607</td>
			<td align="center">0.6684554</td>
			<td align="center">0.3739731</td>
			<td align="center">25.333333</td>
			</tr>
			<tr>
			<td align="center">'''B21'''</td>
			<td align="center"> <math>~\frac{4}{3} ~~\biggl( 3 \biggr)</math></td>
			<td align="center">1.0263212</td>
			<td align="center">0.6385711</td>
			<td align="center">0.3424445</td>
			<td align="center">18.666666</td>
			</tr>
			</table>

	{{LSU_WorkInProgress}}		{{LSU_WorkInProgress}}

Tohline: /* Radial Oscillations of a Zero-Zero Bipolytrope */ Insert "WorkInProgress" template to isolate old, incorrect solutions

2017-02-08T19:03:07Z

Radial Oscillations of a Zero-Zero Bipolytrope: Insert "WorkInProgress" template to isolate old, incorrect solutions

← Older revision		Revision as of 19:03, 8 February 2017
Line 1,021:		Line 1,021:
	</td></tr></table>		</td></tr></table>

	The physically relevant (real) root of this quartic equation in <math>~\Chi</math> — see our [[User:Tohline/SSC/Stability/BiPolytrope0_0Details#Quartic\|accompanying detailed presentation]] — gives us the ''specific'' value of the dimensionless interface location, <math>~q</math>, for which both the values and the first derivatives of the two eigenfunction segments match at the interface. As is shown by the plot displayed in the right-hand panel of Figure 1, we have found different values of <math>~q</math> for each choice (STEP 1) of <math>~\gamma_e</math> (or, equivalently, choice of <math>~\alpha_e</math>). In this plot we have purposely flipped the horizontal axis so that the extreme left <math>~(\alpha_e = +3)</math> represents an incompressible <math>~(n = 0)</math> envelope, while the extreme right represents an isothermal <math>~(\gamma_e = 1)</math> envelope.		The physically relevant (real) root of this quartic equation in <math>~\Chi</math> — see our [[User:Tohline/SSC/Stability/BiPolytrope0_0Details#Quartic\|accompanying detailed presentation]] — gives us the ''specific'' value of the dimensionless interface location, <math>~q</math>, for which both the values and the first derivatives of the two eigenfunction segments match at the interface.


			{{LSU_WorkInProgress}}
			===Old, Incorrect Solutions===
			As is shown by the plot displayed in the right-hand panel of Figure 1, we have found different values of <math>~q</math> for each choice (STEP 1) of <math>~\gamma_e</math> (or, equivalently, choice of <math>~\alpha_e</math>). In this plot we have purposely flipped the horizontal axis so that the extreme left <math>~(\alpha_e = +3)</math> represents an incompressible <math>~(n = 0)</math> envelope, while the extreme right represents an isothermal <math>~(\gamma_e = 1)</math> envelope.
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	The green and orange curves, respectively, show as well how the corresponding model parameters, <math>~\nu</math> and <math>~\rho_e/\rho_c</math>, vary with <math>~\alpha_e</math>. The right-hand panel displays one example of an <math>~(\ell, j) = (2,1)</math> eigenfunction that simultaneously satisfies the LAWE of the core and the LAWE of the envelope, and matches smoothly at the interface. This ''particular'' plotted solution corresponds to the case of <math>~\alpha_e = -0.35</math>, for which: <math>~\gamma_e =1.1940299</math>; <math>~n_e = 5.1538462</math>; <math>~q = 0.7943853</math>; <math>~\nu =0.6675302</math>; and <math>~\rho_e/\rho_c =0.5006468</math>.		The green and orange curves, respectively, show as well how the corresponding model parameters, <math>~\nu</math> and <math>~\rho_e/\rho_c</math>, vary with <math>~\alpha_e</math>. The right-hand panel displays one example of an <math>~(\ell, j) = (2,1)</math> eigenfunction that simultaneously satisfies the LAWE of the core and the LAWE of the envelope, and matches smoothly at the interface. This ''particular'' plotted solution corresponds to the case of <math>~\alpha_e = -0.35</math>, for which: <math>~\gamma_e =1.1940299</math>; <math>~n_e = 5.1538462</math>; <math>~q = 0.7943853</math>; <math>~\nu =0.6675302</math>; and <math>~\rho_e/\rho_c =0.5006468</math>.