SSC/Stability/BiPolytropes - Revision history

Jet53man at 20:34, 31 October 2023

2023-10-31T20:34:37Z

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			=Organizational Index=
			The [[SSC/Stability/BiPolytropes/Index\|accompanying organizational index]] has helped us write this chapter succinctly.

	=In Search of Marginally Unstable (n<sub>c</sub>,n<sub>e</sub>) = (5,1) Bipolytropes=		=In Search of Marginally Unstable (n<sub>c</sub>,n<sub>e</sub>) = (5,1) Bipolytropes=

Jet53man: /* Supplemental Chapters */

2023-10-25T19:56:10Z

Supplemental Chapters

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	==Supplemental Chapters==		==Supplemental Chapters==
	<ol>		<ol>
			<li>Containes [[SSC/Stability/InstabilityOnsetOverview#Displacement_Functions_Summary\|Displacement Functions Summary]]</li>
	<li>[[SSC/Stability/BiPolytropes/PlannedApproach\|Earlier Planned Approach]]</li>		<li>[[SSC/Stability/BiPolytropes/PlannedApproach\|Earlier Planned Approach]]</li>
	<li>[[SSC/Stability/BiPolytropes/HeadScratching\|Headscratching]]</li>		<li>[[SSC/Stability/BiPolytropes/HeadScratching\|Headscratching]]</li>

Jet53man: /* New Material */

2023-10-24T00:07:16Z

New Material

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	</tr>		</tr>
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			The condition that the first variation of the energy vanishes, determines the state of equilibrium of the star and the positiveness of a second variation indicates stability.

	=Virial Analysis=		=Virial Analysis=

Jet53man: /* New Material */

2023-10-23T23:58:46Z

New Material

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 where: <math>\chi_\mathrm{eq}</math> is the dimensionless radius of the configuration when it is in equilibrium; <math>\mathfrak{s}_\mathrm{core}</math> and <math>\mathfrak{s}_\mathrm{env}</math> are the appropriately normalized thermal energy content of the <math>n = 5</math> core and of the <math>n = 1</math> envelope, respectively, in the configuration's equilibrium state; and <math>\mathfrak{w}</math> is the absolute value of the normalized total gravitational potential energy of the equilibrium configuration.  For every <math>(n_c, n_e) = (5, 1)</math> bipolytropic model, the values of these four terms can be obtained via [[SSC/Stability/BiPolytropes/PlannedApproach#Free_Energy_and_Its_Derivatives|our derived analytic expressions]]. And the value of <math>\mathfrak{g}</math> in an equilibrium state is obtained by setting the configuration's dimensionless radius, <math>\chi</math>, equal to <math>\chi_\mathrm{eq}</math>.
-This expression for <math>\mathfrak{g}</math> has been written in such a way that we can readily assess how the free energy varies while the configuration undergoes homologous (<math>\mathfrak{s}_\mathrm{core}, \mathfrak{s}_\mathrm{env}, \mathfrak{w}</math> all held fixed) expansion/contraction <math>\chi \gtrless \chi_\mathrm{eq}</math> about its equilibrium state.
+This expression for <math>\mathfrak{g}</math> has been written in such a way that we can readily assess how the free energy varies while the configuration undergoes homologous (<math>\mathfrak{s}_\mathrm{core}, \mathfrak{s}_\mathrm{env}, \mathfrak{w}</math> all held fixed) expansion/contraction <math>\chi \gtrless \chi_\mathrm{eq}</math> about its equilibrium state.  Specifically the ''first variation'' is,
 =Virial Analysis=

Jet53man: /* New Material */

2023-10-23T23:50:18Z

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	</table>		</table>

	where: <math>\chi_\mathrm{eq}</math> is the dimensionless radius of the configuration when it is in equilibrium; <math>\mathfrak{s}_\mathrm{core}</math> and <math>\mathfrak{s}_\mathrm{env}</math> are the appropriately normalized thermal energy content of the <math>n = 5</math> core and of the <math>n = 1</math> envelope, respectively, in the configuration's equilibrium state; and <math>\mathfrak{w}</math> is the absolute value of the total gravitational potential energy of the equilibrium configuration. For every <math>(n_c, n_e) = (5, 1)</math> bipolytropic model, the values of these four terms can be obtained via our derived analytic expressions. And the value of <math>\mathfrak{g}</math> in an equilibrium state is obtained by setting the configuration's dimensionless radius, <math>\chi</math>, equal to <math>\chi_\mathrm{eq}</math>. ~~More generally, the~~ expression for <math>\mathfrak{g}</math> is written in such a way that we can readily assess how the free energy varies ~~when~~ the configuration undergoes homologous expansion/contraction <math>\chi \gtrless \chi_\mathrm{eq}</math> about its equilibrium state.		where: <math>\chi_\mathrm{eq}</math> is the dimensionless radius of the configuration when it is in equilibrium; <math>\mathfrak{s}_\mathrm{core}</math> and <math>\mathfrak{s}_\mathrm{env}</math> are the appropriately normalized thermal energy content of the <math>n = 5</math> core and of the <math>n = 1</math> envelope, respectively, in the configuration's equilibrium state; and <math>\mathfrak{w}</math> is the absolute value of the normalized total gravitational potential energy of the equilibrium configuration. For every <math>(n_c, n_e) = (5, 1)</math> bipolytropic model, the values of these four terms can be obtained via [[SSC/Stability/BiPolytropes/PlannedApproach#Free_Energy_and_Its_Derivatives\|our derived analytic expressions]]. And the value of <math>\mathfrak{g}</math> in an equilibrium state is obtained by setting the configuration's dimensionless radius, <math>\chi</math>, equal to <math>\chi_\mathrm{eq}</math>.

			This expression for <math>\mathfrak{g}</math> has been written in such a way that we can readily assess how the free energy varies while the configuration undergoes homologous (<math>\mathfrak{s}_\mathrm{core}, \mathfrak{s}_\mathrm{env}, \mathfrak{w}</math> all held fixed) expansion/contraction <math>\chi \gtrless \chi_\mathrm{eq}</math> about its equilibrium state.

	=Virial Analysis=		=Virial Analysis=

Jet53man: /* New Material */

2023-10-22T23:00:50Z

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	</table>		</table>

	where: <math>\chi_\mathrm{eq}</math> is the dimensionless radius of the configuration when it is in equilibrium; <math>\mathfrak{s}_\mathrm{core}</math> and <math>\mathfrak{s}_\mathrm{env}</math> are the appropriately normalized thermal energy content of the <math>n = 5</math> core and of the <math>n = 1</math> envelope, respectively, in the configuration's equilibrium state; and <math>\mathfrak{w}</math> is the absolute value of the total gravitational potential energy of the equilibrium configuration. ~~The~~ value of <math>\mathfrak{g}</math> in an equilibrium state is obtained by setting the configuration's dimensionless radius, <math>\chi</math>, equal to <math>\chi_\mathrm{eq}</math>. More generally, the expression is written in such a way that we can readily assess how the free energy varies when the configuration undergoes homologous expansion/contraction <math>\chi \gtrless \chi_\mathrm{eq}</math> about its equilibrium state.		where: <math>\chi_\mathrm{eq}</math> is the dimensionless radius of the configuration when it is in equilibrium; <math>\mathfrak{s}_\mathrm{core}</math> and <math>\mathfrak{s}_\mathrm{env}</math> are the appropriately normalized thermal energy content of the <math>n = 5</math> core and of the <math>n = 1</math> envelope, respectively, in the configuration's equilibrium state; and <math>\mathfrak{w}</math> is the absolute value of the total gravitational potential energy of the equilibrium configuration. For every <math>(n_c, n_e) = (5, 1)</math> bipolytropic model, the values of these four terms can be obtained via our derived analytic expressions. And the value of <math>\mathfrak{g}</math> in an equilibrium state is obtained by setting the configuration's dimensionless radius, <math>\chi</math>, equal to <math>\chi_\mathrm{eq}</math>. More generally, the expression for <math>\mathfrak{g}</math> is written in such a way that we can readily assess how the free energy varies when the configuration undergoes homologous expansion/contraction <math>\chi \gtrless \chi_\mathrm{eq}</math> about its equilibrium state.

	=Virial Analysis=		=Virial Analysis=

Jet53man: /* New Material */

2023-10-22T22:53:47Z

New Material

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	</table>		</table>

	where: <math>\chi_\mathrm{eq}</math> is the dimensionless radius of the configuration when it is in equilibrium; <math>\mathfrak{s}_\mathrm{core}</math> and <math>\mathfrak{s}_\mathrm{env}</math> are the appropriately normalized thermal energy content of the <math>n = 5</math> core and of the <math>n = 1</math> envelope, respectively; and <math>\mathfrak{w}</math> is the absolute value of the total gravitational potential energy of the equilibrium configuration. ~~This expression is written in such a way that the~~ value of <math>\mathfrak{g}</math> in an equilibrium state is obtained by setting the configuration's dimensionless radius, <math>\chi</math>, equal to <math>\chi_\mathrm{eq}</math>		where: <math>\chi_\mathrm{eq}</math> is the dimensionless radius of the configuration when it is in equilibrium; <math>\mathfrak{s}_\mathrm{core}</math> and <math>\mathfrak{s}_\mathrm{env}</math> are the appropriately normalized thermal energy content of the <math>n = 5</math> core and of the <math>n = 1</math> envelope, respectively, in the configuration's equilibrium state; and <math>\mathfrak{w}</math> is the absolute value of the total gravitational potential energy of the equilibrium configuration. The value of <math>\mathfrak{g}</math> in an equilibrium state is obtained by setting the configuration's dimensionless radius, <math>\chi</math>, equal to <math>\chi_\mathrm{eq}</math>. More generally, the expression is written in such a way that we can readily assess how the free energy varies when the configuration undergoes homologous expansion/contraction <math>\chi \gtrless \chi_\mathrm{eq}</math> about its equilibrium state.

	=Virial Analysis=		=Virial Analysis=

Jet53man: /* New Material */

2023-10-22T22:32:39Z

New Material

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	==New Material==		==New Material==
			Drawing from our [[SSC/Structure/BiPolytropes/FreeEnergy51#Free_Energy_of_BiPolytrope_with\|accompanying detailed discussion]] — see also an [[SSC/Stability/BiPolytropes/PlannedApproach#Free_Energy_and_Its_Derivatives\|accompanying summary]] — the normalized free-energy associated with each of our spherically symmetric, bipolytropic configurations is given by the expression,

			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>\mathfrak{g} \equiv \frac{\mathfrak{G}}{E_\mathrm{norm}}</math>
			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			-\biggl(\frac{\chi_\mathrm{eq}}{\chi}\biggr)\mathfrak{w}
			+ \biggl[ \frac{2}{3(\gamma_c - 1)} \biggr] \biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{3 - 3\gamma_c}\mathfrak{s}_\mathrm{core}
			+ \biggl[ \frac{2}{3(\gamma_e - 1)} \biggr] \biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{3 - 3\gamma_e}\mathfrak{s}_\mathrm{env}
			\, ,
			</math>
			</td>
			</tr>
			</table>

			where: <math>\chi_\mathrm{eq}</math> is the dimensionless radius of the configuration when it is in equilibrium; <math>\mathfrak{s}_\mathrm{core}</math> and <math>\mathfrak{s}_\mathrm{env}</math> are the appropriately normalized thermal energy content of the <math>n = 5</math> core and of the <math>n = 1</math> envelope, respectively; and <math>\mathfrak{w}</math> is the absolute value of the total gravitational potential energy of the equilibrium configuration. This expression is written in such a way that the value of <math>\mathfrak{g}</math> in an equilibrium state is obtained by setting the configuration's dimensionless radius, <math>\chi</math>, equal to <math>\chi_\mathrm{eq}</math>

	=Virial Analysis=		=Virial Analysis=

Jet53man: /* Third Approach: Free-Energy Analysis */

2023-10-22T21:50:27Z

Third Approach: Free-Energy Analysis

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	=Third Approach: Free-Energy Analysis=		=Third Approach: Free-Energy Analysis=

			==Old Material==
	<table border="0" align="right" cellpadding="5"><tr><td align="center">[[File:VirialStability01.png\|350px\|Virial Stability01]]</td></tr></table>		<table border="0" align="right" cellpadding="5"><tr><td align="center">[[File:VirialStability01.png\|350px\|Virial Stability01]]</td></tr></table>
	The following set of menu tiles include links to chapters where this approach has been applied to: (a) uniform-density configurations, (b) pressure-truncated isothermal spheres, (c) an isolated n = 3 polytrope, (d) pressure-truncated n = 5 configurations, and (e) bipolytropes having <math>(n_c, n_e) = (1, 5)</math>.		The following set of menu tiles include links to chapters where this approach has been applied to: (a) uniform-density configurations, (b) pressure-truncated isothermal spheres, (c) an isolated n = 3 polytrope, (d) pressure-truncated n = 5 configurations, and (e) bipolytropes having <math>(n_c, n_e) = (1, 5)</math>.
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	** Choose a maximum-mass model along the bipolytropic sequence that has, for example, <math>\mu_e/\mu_c = 1/4</math>. Hopefully, we will be able to identify precisely (analytically) where this maximum-mass model lies along the sequence. <font color="red">'''Yes!'''</font> Our [[SSC/Structure/BiPolytropes/Analytic51#Limiting_Mass\|earlier analysis]] does provide an analytic prescription of the model that sits at the maximum-mass location along the chosen sequence.		** Choose a maximum-mass model along the bipolytropic sequence that has, for example, <math>\mu_e/\mu_c = 1/4</math>. Hopefully, we will be able to identify precisely (analytically) where this maximum-mass model lies along the sequence. <font color="red">'''Yes!'''</font> Our [[SSC/Structure/BiPolytropes/Analytic51#Limiting_Mass\|earlier analysis]] does provide an analytic prescription of the model that sits at the maximum-mass location along the chosen sequence.
	** Solve the relevant eigenvalue problem for this specific model, initially for <math>(\gamma_c, \gamma_e) = (6/5, 2)</math> and initially for the fundamental mode of oscillation.		** Solve the relevant eigenvalue problem for this specific model, initially for <math>(\gamma_c, \gamma_e) = (6/5, 2)</math> and initially for the fundamental mode of oscillation.

			==New Material==

	=Virial Analysis=		=Virial Analysis=

Jet53man: /* Equilibrium Models */

2023-10-22T21:47:52Z

Equilibrium Models

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	=Equilibrium Models=		=Equilibrium Models=

			The original text contained in this section has been commented out.
			<!--
	The equilibrium sequences displayed in Figure 1 — for pressure-truncated polytropes — and in Figure 2 — for <math>(n_c, n_e) = (5, 1)</math> bipolytropes — were constructed by solving the second-order ODE that governs detailed force-balance throughout spherically symmetric configurations that obey polytropic equations of state. In Figure 1, the sequences labeled <math>n = 1</math> and <math>n = 5</math> can be specified entirely through analytical expressions; these are well-known analytic solutions to the Lane-Emden equation. In Figure 2, ''all seven equilibrium sequences'' are specifiable analytically; the location(s) of ''turning points'' along each sequence (when they exist) are also completely specifiable analytically.		The equilibrium sequences displayed in Figure 1 — for pressure-truncated polytropes — and in Figure 2 — for <math>(n_c, n_e) = (5, 1)</math> bipolytropes — were constructed by solving the second-order ODE that governs detailed force-balance throughout spherically symmetric configurations that obey polytropic equations of state. In Figure 1, the sequences labeled <math>n = 1</math> and <math>n = 5</math> can be specified entirely through analytical expressions; these are well-known analytic solutions to the Lane-Emden equation. In Figure 2, ''all seven equilibrium sequences'' are specifiable analytically; the location(s) of ''turning points'' along each sequence (when they exist) are also completely specifiable analytically.

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	In line with the '''<font color="red">principal question</font>''' stated above, we are particularly interested in examining the relative stability of models that are associated with the "maximum-mass turning points" along bipolytropic sequences for which <math>\mu_e/\mu_c \le \tfrac{1}{3}</math> — see the solid green circular markers in Figure 2.		In line with the '''<font color="red">principal question</font>''' stated above, we are particularly interested in examining the relative stability of models that are associated with the "maximum-mass turning points" along bipolytropic sequences for which <math>\mu_e/\mu_c \le \tfrac{1}{3}</math> — see the solid green circular markers in Figure 2.
			-->

	=Third Approach: Free-Energy Analysis=		=Third Approach: Free-Energy Analysis=