Appendix/Ramblings/NumericallyDeterminedEigenvectors - Revision history

Jet53man: /* Additional Possibilities */

2022-04-12T20:25:33Z

Additional Possibilities

← Older revision		Revision as of 16:25, 12 April 2022
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	* Tie this new (numerical) technique into the [[SSC/Stability/BiPolytrope00CompareApproaches#Comparing_Stability_Analyses_of_Zero-Zero_Bipolytropes\|associated discussion]] of the relationship between stability determinations via LAWEs and global free-energy considerations.		* Tie this new (numerical) technique into the [[SSC/Stability/BiPolytrope00CompareApproaches#Comparing_Stability_Analyses_of_Zero-Zero_Bipolytropes\|associated discussion]] of the relationship between stability determinations via LAWEs and global free-energy considerations.

	* The Figure 2 animation displays numerically determined solutions, <math>~x(r/R)</math>, to the combined/matching LAWEs only for ''non-negative'' values of <math>~\sigma_c^2</math>; these should be relevant only to ''stable'' radial oscillations. We should also see what can be learned from solutions associated with ''negative'' values of <math>~\sigma_c^2</math>, which may be relevant to unstable radial modes of oscillation.		* The Figure 2 animation displays numerically determined solutions, <math>x(r/R)</math>, to the combined/matching LAWEs only for ''non-negative'' values of <math>\sigma_c^2</math>; these should be relevant only to ''stable'' radial oscillations. We should also see what can be learned from solutions associated with ''negative'' values of <math>~\sigma_c^2</math>, which may be relevant to unstable radial modes of oscillation.

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Jet53man: /* Unconstrained LAWEs */

2022-04-12T20:22:41Z

Unconstrained LAWEs

← Older revision		Revision as of 16:22, 12 April 2022
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	where,		where,
	<div align="center">		<div align="center">
	<math>~\eta \equiv \frac{\xi}{g} \, ,</math>		<math>\eta \equiv \frac{\xi}{g} \, ,</math>
	and		and
	<math>~\mathfrak{F}_\mathrm{core} \equiv \frac{3\omega_\mathrm{core}^2}{2\pi G\gamma_c \rho_c} - 2\alpha_c = \frac{(\sigma_c^2 +8)}{\gamma_c} -6 \, ,</math>		<math>\mathfrak{F}_\mathrm{core} \equiv \frac{3\omega_\mathrm{core}^2}{2\pi G\gamma_c \rho_c} - 2\alpha_c = \frac{(\sigma_c^2 +8)}{\gamma_c} -6 \, ,</math>
	</div>		</div>
	and,		and,
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~g^2</math>		<math>g^2</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~\equiv</math>		<math>\equiv</math>
	</td>		</td>
	<td align="left">		<td align="left">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~0</math>		<math>0</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	(g^2 - \xi^2)\frac{d^2x}{d\xi^2} +		(g^2 - \xi^2)\frac{d^2x}{d\xi^2} +
	( 4g^2 - 6\xi^2 ) \frac{1}{\xi} \cdot \frac{dx}{d\xi}		( 4g^2 - 6\xi^2 ) \frac{1}{\xi} \cdot \frac{dx}{d\xi}
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	and we should be able to numerically integrate from the center, outward through the core, exactly as [[#Integrating_Outward_Through_the_Core\|described above]].		and we should be able to numerically integrate from the center, outward through the core, exactly as [[#Integrating_Outward_Through_the_Core\|described above]].

	Separately, the [[~~User:Tohline/~~SSC/Stability/~~BiPolytrope0_0~~#The_Envelope.27s_LAWE\|LAWE for the envelope]] is,		Separately, the [[SSC/Stability/BiPolytrope00#The_Envelope.27s_LAWE\|LAWE for the envelope]] is,

	<div align="center">		<div align="center">

Jet53man: /* Motivated by Analytic31 */

2022-04-12T20:21:27Z

Motivated by Analytic31

← Older revision		Revision as of 16:21, 12 April 2022
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	====Motivated by Analytic31====		====Motivated by Analytic31====

	We have also numerically constructed an eigenfunction that matches our [[~~User:Tohline/~~Appendix/Ramblings/~~Additional_Analytically_Specified_Eigenvectors_for_Zero-Zero_Bipolytropes~~#Illustration31\|accompanying analytic Illustration31]]. In Figure 4, the numerically derived solution has been plotted on top of the analytically derived solution.		We have also numerically constructed an eigenfunction that matches our [[Appendix/Ramblings/AdditionalAnalyticallySpecifiedEigenvectors00Bipolytropes#Illustration31\|accompanying analytic Illustration31]]. In Figure 4, the numerically derived solution has been plotted on top of the analytically derived solution.
	<div align="center" id="Figure4">		<div align="center" id="Figure4">
	<table border="1" align="center" cellpadding="5">		<table border="1" align="center" cellpadding="5">

Jet53man: /* Motivated by Analytic22 */

2022-04-12T20:17:57Z

Motivated by Analytic22

← Older revision		Revision as of 16:17, 12 April 2022
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	====Motivated by Analytic22====		====Motivated by Analytic22====

	We have also numerically constructed an eigenfunction that matches our [[~~User:Tohline/~~Appendix/Ramblings/~~Additional_Analytically_Specified_Eigenvectors_for_Zero-Zero_Bipolytropes~~#Illustration22\|accompanying analytic Illustration22]]. In Figure 3, the numerically derived solution has been plotted on top of the analytically derived solution.		We have also numerically constructed an eigenfunction that matches our [[Appendix/Ramblings/AdditionalAnalyticallySpecifiedEigenvectors00Bipolytropes#Illustration22\|accompanying analytic Illustration22]]. In Figure 3, the numerically derived solution has been plotted on top of the analytically derived solution.
	<div align="center" id="Figure3">		<div align="center" id="Figure3">
	<table border="1" align="center" cellpadding="5">		<table border="1" align="center" cellpadding="5">
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	</table>		</table>
	</div>		</div>


	====Motivated by Analytic31====		====Motivated by Analytic31====

Jet53man at 20:07, 12 April 2022

2022-04-12T20:07:07Z

← Older revision		Revision as of 16:07, 12 April 2022
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	<!-- __NOTOC__ will force TOC off -->		<!-- __NOTOC__ will force TOC off -->
	=Numerically Determined Eigenvectors of a Zero-Zero Bipolytrope=		=Numerically Determined Eigenvectors of a Zero-Zero Bipolytrope=
	Here we build on the analytic foundation summarized in an [[~~User:Tohline/~~SSC/Stability/~~BiPolytrope0_0~~#Radial_Oscillations_of_a_Zero-Zero_Bipolytrope\|accompanying chapter]] and attempt to ''numerically'' construct a variety of eigenvectors that describe radial oscillations of bipolytropes for which, <math>~(n_c, n_e) = (0,0)</math>.		Here we build on the analytic foundation summarized in an [[SSC/Stability/BiPolytrope00#Radial_Oscillations_of_a_Zero-Zero_Bipolytrope\|accompanying chapter]] and attempt to ''numerically'' construct a variety of eigenvectors that describe radial oscillations of bipolytropes for which, <math>~(n_c, n_e) = (0,0)</math>.

	==Setup==		==Setup==
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	<!-- THIS SHOWS HOW THE ABOVE LAWE WAS DERIVED ...		<!-- THIS SHOWS HOW THE ABOVE LAWE WAS DERIVED ...
	For the [[~~User:Tohline/~~SSC/Stability/~~BiPolytrope0_0Details~~#EnvelopeLAWE\|envelope]] we have,		For the [[SSC/Stability/BiPolytrope00Details#EnvelopeLAWE\|envelope]] we have,

	<div align="center">		<div align="center">
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	<tr>		<tr>
	<td align="left">		<td align="left">
	Same analytic, <math>~(\ell,j) = (2,1)</math> eigenfunction as [[~~User:Tohline/~~Appendix/Ramblings/~~Additional_Analytically_Specified_Eigenvectors_for_Zero-Zero_Bipolytropes~~#Illustration21\|here]], but renormalized to unity at the center; numerical label provides value of <math>~\sigma_c^2</math>.		Same analytic, <math>~(\ell,j) = (2,1)</math> eigenfunction as [[Appendix/Ramblings/AdditionalAnalyticallySpecifiedEigenvectors00Bipolytropes#Illustration21\|here]], but renormalized to unity at the center; numerical label provides value of <math>~\sigma_c^2</math>.
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	</span>		</span>
	Evidently, [[~~User:Tohline/~~SSC/Stability/~~BiPolytrope0_0~~#Figure1\|one analytic solution]] with quantum numbers, <math>~(\ell,j) = (2,1)</math>, shown again here on the right, is available for a zero-zero bipolytrope that has the following properties:		Evidently, [[SSC/Stability/BiPolytrope00#Figure1\|one analytic solution]] with quantum numbers, <math>~(\ell,j) = (2,1)</math>, shown again here on the right, is available for a zero-zero bipolytrope that has the following properties:
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	* It seems unlikely that any of our analytically derived eigenfunctions will happen to satisfy the specified boundary condition. The analytic functions have nevertheless proven to be useful in the sense that they provide a terrific check for the computational algorithm that, then, can be used to identify physically meaningful solutions numerically.		* It seems unlikely that any of our analytically derived eigenfunctions will happen to satisfy the specified boundary condition. The analytic functions have nevertheless proven to be useful in the sense that they provide a terrific check for the computational algorithm that, then, can be used to identify physically meaningful solutions numerically.

	* Tie this new (numerical) technique into the [[~~User:Tohline/~~SSC/Stability/~~BiPolytrope0_0CompareApproaches~~#Comparing_Stability_Analyses_of_Zero-Zero_Bipolytropes\|associated discussion]] of the relationship between stability determinations via LAWEs and global free-energy considerations.		* Tie this new (numerical) technique into the [[SSC/Stability/BiPolytrope00CompareApproaches#Comparing_Stability_Analyses_of_Zero-Zero_Bipolytropes\|associated discussion]] of the relationship between stability determinations via LAWEs and global free-energy considerations.

	* The Figure 2 animation displays numerically determined solutions, <math>~x(r/R)</math>, to the combined/matching LAWEs only for ''non-negative'' values of <math>~\sigma_c^2</math>; these should be relevant only to ''stable'' radial oscillations. We should also see what can be learned from solutions associated with ''negative'' values of <math>~\sigma_c^2</math>, which may be relevant to unstable radial modes of oscillation.		* The Figure 2 animation displays numerically determined solutions, <math>~x(r/R)</math>, to the combined/matching LAWEs only for ''non-negative'' values of <math>~\sigma_c^2</math>; these should be relevant only to ''stable'' radial oscillations. We should also see what can be learned from solutions associated with ''negative'' values of <math>~\sigma_c^2</math>, which may be relevant to unstable radial modes of oscillation.

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