SSC/Stability/BiPolytropes/HeadScratching - Revision history

Jet53man: /* Planned Approach */

2023-10-15T21:30:45Z

Planned Approach

← Older revision		Revision as of 17:30, 15 October 2023
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	[[Image:~~TurningPoints51Bipolytropes~~.png\|300px\|center]]		[[Image:TurningPoints51BipolytropesLabels.png\|300px\|center]]
	</td>		</td>
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Jet53man: /* Central Boundary Condition */

2022-07-31T18:06:53Z

Central Boundary Condition

← Older revision		Revision as of 14:06, 31 July 2022
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	In order to kick-start the integration outward from the center of the configuration, we will following the procedure that has been detailed in an [[SSC/Stability/Polytropes#Numerical_Integration_from_the_Center.2C_Outward\|accompanying discussion]]. At the center of the configuration <math>(\xi_1 = 0)</math>, we label the fractional displacement function as <math>x_1</math> — value to be set later, perhaps in an effort to help secure the proper matching conditions at the interface — then we will draw on the [[Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement\|derived power-series expression]] to determine the value of the displacement function at the first radial grid line, <math>~\xi_2 = \Delta_\xi</math>, away from the center. Specifically, given that <math>n = 5, \gamma_g = 6/6</math>, and <math>\alpha_g = -1/3</math> in the core, we will set,		In order to kick-start the integration outward from the center of the configuration, we will following the procedure that has been detailed in an [[SSC/Stability/Polytropes#Numerical_Integration_from_the_Center.2C_Outward\|accompanying discussion]]. At the center of the configuration <math>(\xi_1 = 0)</math>, we label the fractional displacement function as <math>x_1</math> — value to be set later, perhaps in an effort to help secure the proper matching conditions at the interface — then we will draw on the [[Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement\|derived power-series expression]] to determine the value of the displacement function at the first radial grid line, <math>~\xi_2 = \Delta_\xi</math>, away from the center. Specifically, given that <math>n = 5, \gamma_g = 6/5</math>, and <math>\alpha_g = -1/3</math> in the core, we will set,

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Jet53man: /* Attempts to Identify Marginally Unstable (nc, ne) = (5, 1) Bipolytropes */

2022-07-20T22:40:29Z

Attempts to Identify Marginally Unstable (nc, ne) = (5, 1) Bipolytropes

← Older revision		Revision as of 18:40, 20 July 2022
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	<table border="0" cellpadding="5" align="right">		<table border="0" cellpadding="5" align="right">
	<tr>		<tr>
	<th align="center">[[File:DataFileButton02.png\|right\|60px\|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/~~qAndNuMaxAug21~~.xlsx --- worksheet = PlotSigma0]]Figure 3:  Conflicting Instability Regions</th>		<th align="center">[[File:DataFileButton02.png\|right\|60px\|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/qAndNuMax.xlsx --- worksheet = PlotSigma0]]Figure 3:  Conflicting Instability Regions</th>
	</tr>		</tr>
	<tr>		<tr>

Jet53man: /* Rethink Evolution and Stability */

2022-07-20T22:24:10Z

Rethink Evolution and Stability

← Older revision		Revision as of 18:24, 20 July 2022
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	<table border="0" align="left" cellpadding="8">		<table border="0" align="left" cellpadding="8">
	<tr>		<tr>
	<th align="center">Figure 4:   Mass vs. Radius</th>		<th align="center">[[File:DataFileButton02.png\|right\|60px\|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/qAndNuMaxAug21.xlsx --- worksheet = muratio0.25]]Figure 4:   Mass vs. Radius</th>
	<th align="center">Figure 5:   ν vs. q</th>		<th align="center">Figure 5:   ν vs. q</th>
	</tr>		</tr>

Jet53man: /* Attempts to Identify Marginally Unstable (nc, ne) = (5, 1) Bipolytropes */

2022-07-20T22:19:30Z

Attempts to Identify Marginally Unstable (nc, ne) = (5, 1) Bipolytropes

← Older revision		Revision as of 18:19, 20 July 2022
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	<table border="0" cellpadding="5" align="right">		<table border="0" cellpadding="5" align="right">
	<tr>		<tr>
	<th align="center">Figure 3:  Conflicting Instability Regions</th>		<th align="center">[[File:DataFileButton02.png\|right\|60px\|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/qAndNuMaxAug21.xlsx --- worksheet = PlotSigma0]]Figure 3:  Conflicting Instability Regions</th>
	</tr>		</tr>
	<tr>		<tr>

Jet53man: /* Planned Approach */

2022-07-20T22:17:00Z

Planned Approach

← Older revision		Revision as of 18:17, 20 July 2022
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	<th colspan="2" align="center">Figure 2: Equilibrium Sequences of Bipolytropes <br />with <math>~(n_c,n_e) = (5,1)</math> and Various <math>~\mu_e/\mu_c</math></th>		<th colspan="2" align="center">[[File:DataFileButton02.png\|right\|60px\|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/qAndNuMaxAug21.xlsx --- worksheet = PlotSigma0]]Figure 2: Equilibrium Sequences of Bipolytropes <br />with <math>~(n_c,n_e) = (5,1)</math> and Various <math>~\mu_e/\mu_c</math></th>
	</tr>		</tr>
	<tr>		<tr>

Jet53man: /* Attempts to Identify Marginally Unstable (nc, ne) = (5, 1) Bipolytropes */

2022-07-19T22:09:41Z

Attempts to Identify Marginally Unstable (nc, ne) = (5, 1) Bipolytropes

← Older revision		Revision as of 18:09, 19 July 2022
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	<li>Solving the Relevant LAWE:		<li>Solving the Relevant LAWE:
	<ul>		<ul>
	<li>[[SSC/Stability/BiPolytropes#Review_of_the_Analysis_by_Murphy_.26_Fiedler_.281985b.29\|Review of the Analysis by ~~Murphy & Fiedler (1985b)~~]]</li>		<li>[[SSC/Stability/BiPolytropes#Review_of_the_Analysis_by_Murphy_.26_Fiedler_.281985b.29\|Review of the Analysis by]] {{ MF85b }}</li>
	<li>[[SSC/Stability/BiPolytropes#Radial_Oscillations_of_.28nc.2C_ne.29_.3D_.285.2C_1.29_Models\|Radial Oscillations of 51 Models]]   <math>\Leftarrow</math>   [[SSC/Stability/BiPolytropes#Interface\|Interface]] probably not handled correctly here.</li>		<li>[[SSC/Stability/BiPolytropes#Radial_Oscillations_of_.28nc.2C_ne.29_.3D_.285.2C_1.29_Models\|Radial Oscillations of 51 Models]]   <math>\Leftarrow</math>   [[SSC/Stability/BiPolytropes#Interface\|Interface]] probably not handled correctly here.</li>
	<li>[[SSC/Stability/BiPolytropes#Variational_Principle\|Variational Principle]]</li>		<li>[[SSC/Stability/BiPolytropes#Variational_Principle\|Variational Principle]]</li>

Jet53man: /* Planned Approach */

2022-07-19T22:07:27Z

Planned Approach

← Older revision		Revision as of 18:07, 19 July 2022
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	Ideally we would like to answer the just-stated "principal question" using purely analytic techniques. But, to date, we have been unable to fully address the relevant issues analytically, even in what would be expected to be the simplest case:   [[SSC/Stability/BiPolytrope00#Radial_Oscillations_of_a_Zero-Zero_Bipolytrope\|bipolytropic models that have <math>~(n_c,n_e) = (0, 0)</math>]]. Instead, we will streamline the investigation a bit and proceed — at least initially — using a blend of techniques. We will investigate the relative stability of bipolytropic models having <math>~(n_c,n_e) = (5,1) </math> whose ''equilibrium structures'' are completely defined analytically; then the eigenvectors describing radial modes of oscillation will be determined, one at a time, by solving the relevant LAWE(s) numerically. We are optimistic that this can be successfully accomplished because we have had experience numerically integrating the LAWE that governs the oscillation of:		Ideally we would like to answer the just-stated "principal question" using purely analytic techniques. But, to date, we have been unable to fully address the relevant issues analytically, even in what would be expected to be the simplest case:   [[SSC/Stability/BiPolytrope00#Radial_Oscillations_of_a_Zero-Zero_Bipolytrope\|bipolytropic models that have <math>~(n_c,n_e) = (0, 0)</math>]]. Instead, we will streamline the investigation a bit and proceed — at least initially — using a blend of techniques. We will investigate the relative stability of bipolytropic models having <math>~(n_c,n_e) = (5,1) </math> whose ''equilibrium structures'' are completely defined analytically; then the eigenvectors describing radial modes of oscillation will be determined, one at a time, by solving the relevant LAWE(s) numerically. We are optimistic that this can be successfully accomplished because we have had experience numerically integrating the LAWE that governs the oscillation of:
	* [[SSC/Stability/n3PolytropeLAWE#Radial_Oscillations_of_n_.3D_3_Polytropic_Spheres\|Isolated n = 3 polytropes]] — including a quantitative comparison against ~~[http://adsabs.harvard.edu/abs/1941ApJ....94..245S Schwarzschild's (1941)]~~ published work;		* [[SSC/Stability/n3PolytropeLAWE#Radial_Oscillations_of_n_.3D_3_Polytropic_Spheres\|Isolated n = 3 polytropes]] — including a quantitative comparison against the published work of {{ Schwarzschild41full }};
	* [[SSC/Stability/Isothermal#Radial_Oscillations_of_Pressure-Truncated_Isothermal_Spheres\|Pressure-truncated isothermal spheres]] — including a quantitative comparison against the published analysis of ~~[http://adsabs.harvard.edu/abs/1974MNRAS.168..427T Taff & Van Horn (1974)]~~; and		* [[SSC/Stability/Isothermal#Radial_Oscillations_of_Pressure-Truncated_Isothermal_Spheres\|Pressure-truncated isothermal spheres]] — including a quantitative comparison against the published analysis of {{ TVH74full }}; and
	* [[SSC/Stability/n5PolytropeLAWE#Radial_Oscillations_of_n_.3D_5_Polytropic_Spheres\|Pressure-truncated n = 5 polytropes]].		* [[SSC/Stability/n5PolytropeLAWE#Radial_Oscillations_of_n_.3D_5_Polytropic_Spheres\|Pressure-truncated n = 5 polytropes]].

	A key reference throughout this investigation will be the paper by ~~[http://adsabs.harvard.edu/abs/1985PASAu...6..222M J. O. Murphy & R. Fiedler (1985b, Proc. Astr. Soc. of Australia, 6, 222)]~~. They studied ''Radial Pulsations and Vibrational Stability of a Sequence of Two Zone Polytropic Stellar Models.'' Specifically, their underlying equilibrium models were bipolytropes that have <math>~(n_c,n_e) = (1, 5)</math>. In an [[SSC/Structure/BiPolytropes/Analytic15#BiPolytrope_with_nc_.3D_1_and_ne_.3D_5\|accompanying chapter]], we describe in detail how ~~Murphy & Fiedler~~ obtained these equilibrium bipolytropic structures and detail some of their equilibrium properties.		A key reference throughout this investigation will be the paper by {{ MF85bfull }}. They studied ''Radial Pulsations and Vibrational Stability of a Sequence of Two Zone Polytropic Stellar Models.'' Specifically, their underlying equilibrium models were bipolytropes that have <math>~(n_c,n_e) = (1, 5)</math>. In an [[SSC/Structure/BiPolytropes/Analytic15#BiPolytrope_with_nc_.3D_1_and_ne_.3D_5\|accompanying chapter]], we describe in detail how {{ MF85b }} obtained these equilibrium bipolytropic structures and detail some of their equilibrium properties.

	Here are the steps we initially plan to take:		Here are the steps we initially plan to take:

	* Governing LAWEs:		* Governing LAWEs:
	** Identify the relevant LAWEs that govern the behavior of radial oscillations in the <math>~n_c = 5</math> core and, separately, in the <math>~n_e = 1</math> envelope. Check these LAWE specifications against the published work of ~~[http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)]~~.		** Identify the relevant LAWEs that govern the behavior of radial oscillations in the <math>~n_c = 5</math> core and, separately, in the <math>~n_e = 1</math> envelope. Check these LAWE specifications against the published work of {{ MF85b }}.
	** Determine the matching conditions that must be satisfied across the core/envelope interface. Be sure to take into account the critical interface ''jump'' conditions spelled out by ~~[http://adsabs.harvard.edu/abs/1958HDP....51..353L P. Ledoux & Th. Walraven (1958)]~~, as we have already discussed in the context of an [[SSC/Stability/BiPolytrope00#Radial_Oscillations_of_a_Zero-Zero_Bipolytrope\|analysis of radial oscillations in zero-zero bipolytropes]].		** Determine the matching conditions that must be satisfied across the core/envelope interface. Be sure to take into account the critical interface ''jump'' conditions spelled out by {{ LW58full }}, as we have already discussed in the context of an [[SSC/Stability/BiPolytrope00#Radial_Oscillations_of_a_Zero-Zero_Bipolytrope\|analysis of radial oscillations in zero-zero bipolytropes]].
	* Determine what surface boundary condition should be imposed on physically relevant LAWE solutions, i.e., on the physically relevant radial-oscillation eigenvectors.		* Determine what surface boundary condition should be imposed on physically relevant LAWE solutions, i.e., on the physically relevant radial-oscillation eigenvectors.
	* Initial Analysis:		* Initial Analysis:

Jet53man: /* Through the Envelope */

2021-08-25T20:21:15Z

Through the Envelope

← Older revision		Revision as of 16:21, 25 August 2021
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	<math>		<math>
	\frac{x_+ }{2 \Delta_\eta}		\frac{x_{J+1} }{2 \Delta_\eta}
	-		-
	\frac{1}{2 \Delta_\eta} \biggl[ x_- \biggr]_\mathrm{project} \, .		\frac{1}{2 \Delta_\eta} \biggl[ x_- \biggr]_\mathrm{project} \, .

Jet53man: /* Through the Envelope */

2021-08-25T20:18:02Z

Through the Envelope

← Older revision		Revision as of 16:18, 25 August 2021
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	<math>\biggl\{		<math>\biggl[
	1 - \frac{\mathcal{H}\Delta_\eta}{2\eta}		1 - \frac{\mathcal{H}\Delta_\eta}{2\eta}
	\biggr\}\biggl[x_-\biggr]_\mathrm{project}		\biggr]_J\biggl[x_-\biggr]_\mathrm{project}
	</math>		</math>
	</td>		</td>
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	\biggl(\frac{\sigma_c^2}{6}\biggr)\frac{\Delta_\eta^2 }{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}		\biggl(\frac{\sigma_c^2}{6}\biggr)\frac{\Delta_\eta^2 }{\theta_i^5}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}
	\biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]		\biggl[ \frac{\eta}{A\sin(\eta-B)}\biggr]
	\biggr\} ~~x_i~~		\biggr\}_J x_J
	-		-
	\biggl[1+ \frac{\mathcal{H}\Delta_\eta}{2\eta}\biggr] x_+ \, .		\biggl[1+ \frac{\mathcal{H}\Delta_\eta}{2\eta}\biggr]_J x_{J+1} \, .
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
			Then we conclude that,

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	<math>~~x_i~~' = \frac{dx}{d\eta}</math>		<math>[x_\mathrm{env}^']_J \equiv \biggl[ \frac{dx_\mathrm{env}}{d\eta} \biggr]_J</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~~\approx~~</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>		<math>
	\frac{x_+ - ~~x_-~~}{2 \Delta_\eta} \, ;		\frac{x_+ }{2 \Delta_\eta}
			-
			\frac{1}{2 \Delta_\eta} \biggl[ x_- \biggr]_\mathrm{project} \, .
	</math>		</math>
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~~
	~~x_i'' = \frac{d^2x}{d\eta^2}~~
	~~</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>\approx</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>\frac{x_+ - 2x_i + x_-}{\Delta_\eta^2} \, .</math>~~
	</td>		</td>
	</tr>		</tr>