SSC/Stability/BiPolytrope00Old - Revision history

Joel2: /* Related Discussions */

2024-07-08T21:55:47Z

Related Discussions

← Older revision		Revision as of 17:55, 8 July 2024
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	=Related Discussions=		=Related Discussions=
	* [[~~User:Tohline/~~Appendix/Ramblings/~~Additional_Analytically_Specified_Eigenvectors_for_Zero-Zero_Bipolytropes~~#Searching_for_Additional_Eigenvectors_of_Zero-Zero_Bipolytropes\|Searching for Additional, Analytically Specified Eigenvectors of Zero-Zero Bipolytropes]]		* [[Appendix/Ramblings/AdditionalAnalyticallySpecifiedEigenvectors00Bipolytropes#Searching_for_Additional_Eigenvectors_of_Zero-Zero_Bipolytropes\|Searching for Additional, Analytically Specified Eigenvectors of Zero-Zero Bipolytropes]]
	* [[User:Tohline/Appendix/Ramblings/NumericallyDeterminedEigenvectors#Numerically_Determined_Eigenvectors_of_a_Zero-Zero_Bipolytrope\|Numerically Determined Eigenvectors]]		* [[User:Tohline/Appendix/Ramblings/NumericallyDeterminedEigenvectors#Numerically_Determined_Eigenvectors_of_a_Zero-Zero_Bipolytrope\|Numerically Determined Eigenvectors]]

Joel2: /* Duh! */

2024-07-08T21:54:27Z

Duh!

← Older revision		Revision as of 17:54, 8 July 2024
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	===Duh!===		===Duh!===

	[[File:CommentButton02.png\|right\|100px\|Comment by J. E. Tohline on 4 February 2017: This numerical determination of surface boundary conditions was carried out inside spreadsheet "FDflex22" of Excel file ''analyticeigenvectorcorrected.xlsx''.]]After also checking conformance with the expected boundary condition in ~~[[User:Tohline/~~Appendix/Ramblings/Additional_Analytically_Specified_Eigenvectors_for_Zero-Zero_Bipolytropes#Check_Surface_Boundary_Condition\|the case of analytic eigenfunctions having <math>~\ell = 3</math>]] and, separately (not shown), for numerically generated eigenfunctions having a wide range of oscillation frequencies, it dawned on us that the [[#DesiredBoundaryCondition\|"desired" surface boundary condition]] may actually be a natural outcome of the envelope's LAWE.		[[File:CommentButton02.png\|right\|100px\|Comment by J. E. Tohline on 4 February 2017: This numerical determination of surface boundary conditions was carried out inside spreadsheet "FDflex22" of Excel file ''analyticeigenvectorcorrected.xlsx''.]]After also checking conformance with the expected boundary condition in <!-- Appendix/Ramblings/Additional_Analytically_Specified_Eigenvectors_for_Zero-Zero_Bipolytropes -->[[Appendix/Ramblings/AdditionalAnalyticallySpecifiedEigenvectors00Bipolytropes#Check_Surface_Boundary_Condition\|the case of analytic eigenfunctions having <math>~\ell = 3</math>]] and, separately (not shown), for numerically generated eigenfunctions having a wide range of oscillation frequencies, it dawned on us that the [[#DesiredBoundaryCondition\|"desired" surface boundary condition]] may actually be a natural outcome of the envelope's LAWE.

	By constraining our discussion to models for which <math>~g^2 = \mathcal{B}</math> and <math>~\mathcal{D} = q^3</math>, the [[#The_Envelope.27s_LAWE\|envelope's LAWE]] is,		By constraining our discussion to models for which <math>~g^2 = \mathcal{B}</math> and <math>~\mathcal{D} = q^3</math>, the [[#The_Envelope.27s_LAWE\|envelope's LAWE]] is,
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	</div>		</div>

	And this is ''precisely'' the condition that derives from the astrophysically reasonable boundary condition that we have [[~~User:Tohline/~~SSC/Stability/~~BiPolytrope0_0Details~~#Boundary_Condition\|discussed separately]] and that has been [[#Expectation\|reviewed, above]].		And this is ''precisely'' the condition that derives from the astrophysically reasonable boundary condition that we have [[SSC/Stability/BiPolytrope00Details#Boundary_Condition\|discussed separately]] and that has been [[#Expectation\|reviewed, above]].

	===Broader Analysis===		===Broader Analysis===

Joel2: /* Examining Alignment with Surface Boundary Condition */

2024-07-08T21:21:45Z

Examining Alignment with Surface Boundary Condition

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Joel2: /* Piecing Together */

2024-07-08T21:16:35Z

Piecing Together

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Joel2: /* Piecing Together */

2024-07-08T21:08:34Z

Piecing Together

← Older revision		Revision as of 17:08, 8 July 2024
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	</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 [[SSC/Stability/BiPolytrope00Details#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 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>.
	-->		-->

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	<tr><th align="center" colspan="4">Figure 1</th></tr>		<tr><th align="center" colspan="4">Figure 1</th></tr>
	<tr>		<tr>
	<td align="center" colspan="3"><math>~\alpha_e = -0.35 \, ;~~~c_0 = \sqrt{1+\alpha_e} - 1</math></td>		<td align="center" colspan="3"><math>\alpha_e = -0.35 \, ;~~~c_0 = \sqrt{1+\alpha_e} - 1</math></td>
	<td align="center" rowspan="18">		<td align="center" rowspan="18">

Joel2: /* Two Separate LAWEs */

2024-07-08T21:03:00Z

Two Separate LAWEs

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Joel2: Created page with "FORCETOC =Radial Oscillations of a Zero-Zero Bipolytrope= This is a chapter that summarizes an accompanying, detailed derivation. A newer version of this chapter, that takes into account a ''corrected'' interface matching condition provided by [http://adsabs.harvard.edu/abs/1958HDP....51..353L P. Ledoux & Th. Walraven (1958)] is provided..."

2024-07-08T20:47:07Z

Created page with "__FORCETOC__  =Radial Oscillations of a Zero-Zero Bipolytrope= This is a chapter that summarizes an accompanying, detailed derivation. A newer version of this chapter, that takes into account a ''corrected'' interface matching condition provided by [http://adsabs.harvard.edu/abs/1958HDP....51..353L P. Ledoux & Th. Walraven (1958)] is provided..."

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