SSC/Stability/n3PolytropeLAWE - Revision history

Jet53man: /* Schwarzschild (1941) */

2021-08-22T20:57:52Z

Schwarzschild (1941)

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	Each sharp valley in this semi-log plot highlights the location of a node in the corresponding eigenfunction, that is, it identifies where <math>~x(\xi)</math> crosses through zero.		Each sharp valley in this semi-log plot highlights the location of a node in the corresponding eigenfunction, that is, it identifies where <math>~x(\xi)</math> crosses through zero.

			<span id="Fig1"> </span>
	<table border="1" cellpadding="5" align="center">		<table border="1" cellpadding="5" align="center">
	<tr>		<tr>

Jet53man: /* Related Discussions */

2021-07-31T22:16:54Z

Related Discussions

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	=Related Discussions=		=Related Discussions=
	* Radial Oscillations of [[SSC/UniformDensity#The_Stability_of_Uniform-Density_Spheres\|Uniform-density sphere]]		* Radial Oscillations of [[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres\|Uniform-density sphere]]
	* Radial Oscillations of Isolated Polytropes		* Radial Oscillations of Isolated Polytropes
	** [[SSC/Stability/Polytropes#Radial_Oscillations_of_Polytropic_Spheres\|Setup]]		** [[SSC/Stability/Polytropes#Radial_Oscillations_of_Polytropic_Spheres\|Setup]]
	** n = 1:  [[SSC/Stability/n1PolytropeLAWE\|Attempt at Formulating an Analytic Solution]]		** n = 1:  [[SSC/Stability/n1PolytropeLAWE\|Attempt at Formulating an Analytic Solution]]
	** n = 3:  [[SSC/Stability/n3PolytropeLAWE\|Numerical Solution]] to compare with [http://adsabs.harvard.edu/abs/1941ApJ....94..245S M. Schwarzschild (1941)]		** n = 3:  [[SSC/Stability/n3PolytropeLAWE#Numerical_Integration\|Numerical Solution]] to compare with [http://adsabs.harvard.edu/abs/1941ApJ....94..245S M. Schwarzschild (1941)]
	** n = 5:  [[SSC/Stability/n5PolytropeLAWE\|Attempt at Formulating an Analytic Solution]]		** n = 5:  [[SSC/Stability/n5PolytropeLAWE\|Attempt at Formulating an Analytic Solution]]

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Jet53man: /* Related Discussions */

2021-07-31T22:14:45Z

Related Discussions

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	=Related Discussions=		=Related Discussions=
	* Radial Oscillations of [[~~User:Tohline/~~SSC/UniformDensity#The_Stability_of_Uniform-Density_Spheres\|Uniform-density sphere]]		* Radial Oscillations of [[SSC/UniformDensity#The_Stability_of_Uniform-Density_Spheres\|Uniform-density sphere]]
	* Radial Oscillations of Isolated Polytropes		* Radial Oscillations of Isolated Polytropes
	** [[~~User:Tohline/~~SSC/Stability/Polytropes#Radial_Oscillations_of_Polytropic_Spheres\|Setup]]		** [[SSC/Stability/Polytropes#Radial_Oscillations_of_Polytropic_Spheres\|Setup]]
	** n = 1:  [[~~User:Tohline/~~SSC/Stability/n1PolytropeLAWE\|Attempt at Formulating an Analytic Solution]]		** n = 1:  [[SSC/Stability/n1PolytropeLAWE\|Attempt at Formulating an Analytic Solution]]
	** n = 3:  [[~~User:Tohline/~~SSC/Stability/n3PolytropeLAWE\|Numerical Solution]] to compare with [http://adsabs.harvard.edu/abs/1941ApJ....94..245S M. Schwarzschild (1941)]		** n = 3:  [[SSC/Stability/n3PolytropeLAWE\|Numerical Solution]] to compare with [http://adsabs.harvard.edu/abs/1941ApJ....94..245S M. Schwarzschild (1941)]
	** n = 5:  [[~~User:Tohline/~~SSC/Stability/n5PolytropeLAWE\|Attempt at Formulating an Analytic Solution]]		** n = 5:  [[SSC/Stability/n5PolytropeLAWE\|Attempt at Formulating an Analytic Solution]]

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Jet53man: /* Analytic Inquiry */

2021-07-31T22:12:06Z

Analytic Inquiry

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	=Analytic Inquiry=		=Analytic Inquiry=

	<font color="red"><b>NOTE</b> (from J. E. Tohline in April, 2017):</font>   The following subsections present some exploratory ideas that were pursued while I was searching for analytic solutions to the polytropic LAWE. For the most part this material has been superseded by a [[~~User:Tohline/~~SSC/Stability/InstabilityOnsetOverview#Polytropic\|separate discussion]] in which we describe the desired analytic solution, which we discovered in March, 2017.		<font color="red"><b>NOTE</b> (from J. E. Tohline in April, 2017):</font>   The following subsections present some exploratory ideas that were pursued while I was searching for analytic solutions to the polytropic LAWE. For the most part this material has been superseded by a [[SSC/Stability/InstabilityOnsetOverview#Polytropic\|separate discussion]] in which we describe the desired analytic solution, which we discovered in March, 2017.

	==Fundamental-Mode, Homentropic Oscillations==		==Fundamental-Mode, Homentropic Oscillations==
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	The LAWE, [[#Background\|presented above]], that is relevant to polytropic spheres, may be rewritten as,		The LAWE, [[#Background\|presented above]], that is relevant to polytropic spheres, may be rewritten as,
	<div align="center">		<div align="center">
	{{ ~~User:Tohline/~~Math/EQ_RadialPulsation02 }}		{{ Math/EQ_RadialPulsation02 }}
	</div>		</div>

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	===Homologous Collapse===		===Homologous Collapse===

	If we examine only an <math>~n=3</math> polytropic configuration, then the last term disappears. This means that ''in this very special case'', a perfectly valid solution to the LAWE is <math>~x = \mathrm{constant}</math>. This is presumably the eigenfunction that Schwarzschild deduced; the fundamental-mode "oscillations" are perfectly homologous. Given that the model is marginally unstable, an ensuing dynamical collapse will presumably begin in a perfectly homologous fashion. This is precisely the type of "free-fall" collapse that was discussed and modeled by [[~~User:Tohline/~~Apps/GoldreichWeber80#Homologous_Solution\|Goldreich & Weber (1980)]].		If we examine only an <math>~n=3</math> polytropic configuration, then the last term disappears. This means that ''in this very special case'', a perfectly valid solution to the LAWE is <math>~x = \mathrm{constant}</math>. This is presumably the eigenfunction that Schwarzschild deduced; the fundamental-mode "oscillations" are perfectly homologous. Given that the model is marginally unstable, an ensuing dynamical collapse will presumably begin in a perfectly homologous fashion. This is precisely the type of "free-fall" collapse that was discussed and modeled by [[Apps/GoldreichWeber80#Homologous_Solution\|Goldreich & Weber (1980)]].

	===Another Potential Option===		===Another Potential Option===
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	</div>		</div>

	Referring to the [[~~User:Tohline/~~Appendix/Ramblings/PowerSeriesExpressions#PolytropicLaneEmden\|power-series expansion of the polytropic Lane-Emden function]], <math>~\theta(\xi)</math>, about the configuration's center, we see that the product, <math>~\xi\theta</math>, goes to zero as the first power of <math>~\xi</math>. This means that the right-hand side of this last differential equation blows up at the center. This, therefore, does not appear to provide a physically viable avenue by which to identify an alternative fundamental-mode eigenfunction.		Referring to the [[Appendix/Ramblings/PowerSeriesExpressions#PolytropicLaneEmden\|power-series expansion of the polytropic Lane-Emden function]], <math>~\theta(\xi)</math>, about the configuration's center, we see that the product, <math>~\xi\theta</math>, goes to zero as the first power of <math>~\xi</math>. This means that the right-hand side of this last differential equation blows up at the center. This, therefore, does not appear to provide a physically viable avenue by which to identify an alternative fundamental-mode eigenfunction.

	==Play With Form of LAWE==		==Play With Form of LAWE==
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	</div>		</div>

	may be rewritten entirely as an expression that relates the logarithmic derivatives of <math>~x, \xi,</math> and <math>~\theta</math>. Multiplying through by <math>~\xi^2/x</math>, then drawing on a differential relation that has been [[~~User:Tohline/~~SSC/Stability/~~BiPolytrope0_0Details~~#Idea_Involving_Logarithmic_Derivatives\|derived in a separate context]], namely,		may be rewritten entirely as an expression that relates the logarithmic derivatives of <math>~x, \xi,</math> and <math>~\theta</math>. Multiplying through by <math>~\xi^2/x</math>, then drawing on a differential relation that has been [[SSC/Stability/BiPolytrope00Details#Idea_Involving_Logarithmic_Derivatives\|derived in a separate context]], namely,
	<div align="center">		<div align="center">
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	===Behavior for Known n=5 Solution===		===Behavior for Known n=5 Solution===
	We know that the [[~~User:Tohline/~~SSC/Stability/n5PolytropeLAWE#Eureka_Moment\|FMMUM for pressure-truncated, n = 5 polytropic configurations]] takes the form,		We know that the [[SSC/Stability/n5PolytropeLAWE#Eureka_Moment\|FMMUM for pressure-truncated, n = 5 polytropic configurations]] takes the form,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	Now, from the polytropic Lane-Emden equation, we also know that,		Now, from the polytropic Lane-Emden equation, we also know that,
	<div align="center">		<div align="center">
	{{ ~~User:Tohline/~~Math/EQ_SSLaneEmden01 }}		{{ Math/EQ_SSLaneEmden01 }}
	</div>		</div>
	That is,		That is,

Jet53man: /* Radial Oscillations of n = 3 Polytropic Spheres */

2021-07-31T22:08:29Z

Radial Oscillations of n = 3 Polytropic Spheres

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	<font color="#770000">'''Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />		<font color="#770000">'''Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />

	{{~~User:Tohline/~~Math/EQ_RadialPulsation01}}		{{Math/EQ_RadialPulsation01}}
	</div>		</div>

	whose solution gives eigenfunctions that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations. Because this widely used form of the radial pulsation equation is not dimensionless but, rather, has units of inverse length-squared, we have found it useful to also recast it in the following [[~~User:Tohline/~~SSC/Perturbations#Dimensionless_Expression\|dimensionless form]]:		whose solution gives eigenfunctions that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations. Because this widely used form of the radial pulsation equation is not dimensionless but, rather, has units of inverse length-squared, we have found it useful to also recast it in the following [[SSC/Perturbations#Dimensionless_Expression\|dimensionless form]]:
	<div align="center">		<div align="center">
	<math>		<math>
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	</div>		</div>

	In a [[~~User:Tohline/~~SSC/Stability/Polytropes#Radial_Oscillations_of_Polytropic_Spheres\|separate discussion]], we showed that specifically for isolated, ''polytropic'' configurations, this linear adiabatic wave equation (LAWE) can be rewritten as,		In a [[SSC/Stability/Polytropes#Radial_Oscillations_of_Polytropic_Spheres\|separate discussion]], we showed that specifically for isolated, ''polytropic'' configurations, this linear adiabatic wave equation (LAWE) can be rewritten as,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	===Schwarzschild (1941)===		===Schwarzschild (1941)===
	We can directly compare our results with [http://adsabs.harvard.edu/abs/1941ApJ....94..245S Schwarzschild's (1941)] published work on "Overtone Pulsations for the Standard [Stellar] Model." To begin with, it is straightforward to demonstrate that the last form of the LAWE, provided above, matches equation (2) from [[~~User:Tohline/~~SSC/Perturbations#Schwarzschild_.281941.29\|Schwarzschild (1941)]], if <math>~n</math> is set to 3 — see the boxed-in excerpt, immediately below. Note as well that Schwarzschild's dimensionless oscillation frequency — defined in his equation (1) and which we will label, <math>~\omega_\mathrm{Sch}</math> — is related to our dimensionless frequency via the expression,		We can directly compare our results with [http://adsabs.harvard.edu/abs/1941ApJ....94..245S Schwarzschild's (1941)] published work on "Overtone Pulsations for the Standard [Stellar] Model." To begin with, it is straightforward to demonstrate that the last form of the LAWE, provided above, matches equation (2) from [[SSC/Perturbations#Schwarzschild_.281941.29\|Schwarzschild (1941)]], if <math>~n</math> is set to 3 — see the boxed-in excerpt, immediately below. Note as well that Schwarzschild's dimensionless oscillation frequency — defined in his equation (1) and which we will label, <math>~\omega_\mathrm{Sch}</math> — is related to our dimensionless frequency via the expression,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</div>		</div>

	Drawing from our [[~~User:Tohline/~~SSC/Perturbations#Ensure_Finite-Amplitude_Fluctuations\|discussion of the historical treatment of boundary conditions]], we presume that Schwarzschild imposed the following constraint at the surface:		Drawing from our [[SSC/Perturbations#Ensure_Finite-Amplitude_Fluctuations\|discussion of the historical treatment of boundary conditions]], we presume that Schwarzschild imposed the following constraint at the surface:

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

	Recognizing from an [[~~User:Tohline/~~SSC/Structure/Polytropes#Horedt2004\|accompanying tabulation]] that, for <math>~n=3</math> polytropes,		Recognizing from an [[SSC/Structure/Polytropes#Horedt2004\|accompanying tabulation]] that, for <math>~n=3</math> polytropes,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<tr>		<tr>
	<th align="center"><font size="+1">Figure 1:</font>		<th align="center"><font size="+1">Figure 1:</font>
	Schwarzschild's Eigenfunctions for an n = 3 Polytrope with α = 0.4 (γ = 20/13)		Schwarzschild's Eigenfunctions for an n = 3 Polytrope with <math>\alpha = 0.4 ~(\gamma = 20/13)</math>
	</th>		</th>
	</tr>		</tr>
	<tr>		<tr>
	<td align="center">[[File:~~Schwarzschild1941Combined3small~~.png\|800 px\|Schwarzschild (1941) eigenfunctions]]</td>		<td align="center">[[File:Schwarzschild1941Combined3.png\|800 px\|Schwarzschild (1941) eigenfunctions]]</td>
	</tr>		</tr>
	</table>		</table>
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	===From the Core to the Surface===		===From the Core to the Surface===

	Here we use the [[~~User:Tohline/~~SSC/Stability/Polytropes#Numerical_Integration_from_the_Center.2C_Outward\|finite-difference algorithm described separately]] to integrate the discretized LAWE from the center of the polytropic configuration, outward to its surface, which in this case — see, for example, [[~~User:Tohline/~~SSC/Structure/Polytropes#Horedt2004\|p. 77 of Horedt (2004)]] — is located at the polytropic-coordinate location,		Here we use the [[SSC/Stability/Polytropes#Numerical_Integration_from_the_Center.2C_Outward\|finite-difference algorithm described separately]] to integrate the discretized LAWE from the center of the polytropic configuration, outward to its surface, which in this case — see, for example, [[SSC/Structure/Polytropes#Horedt2004\|p. 77 of Horedt (2004)]] — is located at the polytropic-coordinate location,
	<div align="center">		<div align="center">
	<math>~\xi_\mathrm{max} = 6.89684862 \, .</math>		<math>~\xi_\mathrm{max} = 6.89684862 \, .</math>
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	</li>		</li>
	<li>Set the eigenfunction to unity at the center <math>~(\xi_0 = 0)</math> of the configuration, that is, set <math>~x_0 = 1</math>.</li>		<li>Set the eigenfunction to unity at the center <math>~(\xi_0 = 0)</math> of the configuration, that is, set <math>~x_0 = 1</math>.</li>
	<li>Determine the value of the eigenfunction at the first grid ''line'' away from the center — having coordinate location, <math>~\xi_1 = \Delta_\xi </math> — via the [[~~User:Tohline/~~Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement\|derived power-series expression]],		<li>Determine the value of the eigenfunction at the first grid ''line'' away from the center — having coordinate location, <math>~\xi_1 = \Delta_\xi </math> — via the [[Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement\|derived power-series expression]],
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	Each frame of the Figure 2 animation also displays, as smooth solid curves, the radial eigenfunctions that Schwarzschild (1941) obtained for the fundamental mode (blue curve) and the first three overtone modes (green, purple, & orange curves, repectively) for his model with <math>~\alpha = 0.4</math>. These are the same curves that appear in the left-hand panel of Figure 1, but here the displacement amplitude has been renormalized such that <math>~x_0 = 1</math>, and, along the horizontal axis, the radial location is marked in terms of the fractional radius, <math>~r/R \equiv \xi/\xi_\mathrm{max}</math>. In our examination of this model, as we approached each specific value of a modal eigenfrequency identified by Schwarzschild — see the frequencies highlighted in pink in our Table 1 — we fine-tuned our choice of the eigenfrequency in order to find a displacement function whose surface ''amplitude'' matched, to a high level of precision, the surface amplitude associated with Schwarzschild's corresponding published eigenfunction. The column of our Table 2 whose heading is "Match Schwarzschild" identifies — to at least 10 digits precision — the frequency choice that was required in order for these surface ''amplitudes'' to match in each case.		Each frame of the Figure 2 animation also displays, as smooth solid curves, the radial eigenfunctions that Schwarzschild (1941) obtained for the fundamental mode (blue curve) and the first three overtone modes (green, purple, & orange curves, repectively) for his model with <math>~\alpha = 0.4</math>. These are the same curves that appear in the left-hand panel of Figure 1, but here the displacement amplitude has been renormalized such that <math>~x_0 = 1</math>, and, along the horizontal axis, the radial location is marked in terms of the fractional radius, <math>~r/R \equiv \xi/\xi_\mathrm{max}</math>. In our examination of this model, as we approached each specific value of a modal eigenfrequency identified by Schwarzschild — see the frequencies highlighted in pink in our Table 1 — we fine-tuned our choice of the eigenfrequency in order to find a displacement function whose surface ''amplitude'' matched, to a high level of precision, the surface amplitude associated with Schwarzschild's corresponding published eigenfunction. The column of our Table 2 whose heading is "Match Schwarzschild" identifies — to at least 10 digits precision — the frequency choice that was required in order for these surface ''amplitudes'' to match in each case.

	It is gratifying to see that our resulting frequencies match well the values published by Schwarzschild (as highlighted in pink, above). But this does not satisfactorily explain why, among the entire range of displacement functions displayed (in red) in the Figure 2 animation, Schwarzschild labeled these specific ones as the eigenmodes. As we shall now demonstrate, his eigenmode identifications resulted from the imposition of a specific, physically justified constraint on the ''slope'', rather than the value, of the displacement function at the surface of the configuration. (See also our separate brief answer to the question, "[[~~User:Tohline/~~SSC/Perturbations#What_Makes_This_an_Eigenvalue_Problem.3F\|What makes this an eigenvalue problem?]]".)		It is gratifying to see that our resulting frequencies match well the values published by Schwarzschild (as highlighted in pink, above). But this does not satisfactorily explain why, among the entire range of displacement functions displayed (in red) in the Figure 2 animation, Schwarzschild labeled these specific ones as the eigenmodes. As we shall now demonstrate, his eigenmode identifications resulted from the imposition of a specific, physically justified constraint on the ''slope'', rather than the value, of the displacement function at the surface of the configuration. (See also our separate brief answer to the question, "[[SSC/Perturbations#What_Makes_This_an_Eigenvalue_Problem.3F\|What makes this an eigenvalue problem?]]".)

	===Surface Boundary Condition===		===Surface Boundary Condition===

Jet53man: /* Radial Oscillations of n = 3 Polytropic Spheres */

2021-07-31T21:56:34Z

Radial Oscillations of n = 3 Polytropic Spheres

← Older revision		Revision as of 17:56, 31 July 2021
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	=Radial Oscillations of n = 3 Polytropic Spheres=		=Radial Oscillations of n = 3 Polytropic Spheres=

			{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
			\|-
			! style="height: 125px; width: 125px; background-color:white;" \|
			<font size="-1">[[H_BookTiledMenu#MoreStabilityAnalyses\|<b>Isolated<br />n = 3<br />Polytrope</b>]]</font>
			\|}
	==Background==		==Background==
			<br />
			<br />
			<br />
			<br />

	===Our Formulation of the Problem===		===Our Formulation of the Problem===

Jet53man: Created page with "FORCETOC =Radial Oscillations of n = 3 Polytropic Spheres= ==Background== ===Our Formulation of the Problem=== In an SSC/Pertur..."

2021-07-31T21:47:24Z

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SSC/Stability/n3PolytropeLAWE - Revision history

Jet53man: /* Schwarzschild (1941) */

Jet53man: /* Related Discussions */

Jet53man: /* Related Discussions */

Jet53man: /* Analytic Inquiry */

Jet53man: /* Radial Oscillations of n = 3 Polytropic Spheres */

Jet53man: /* Radial Oscillations of n = 3 Polytropic Spheres */

Jet53man: Created page with "__FORCETOC__ =Radial Oscillations of n = 3 Polytropic Spheres= ==Background== ===Our Formulation of the Problem=== In an SSC/Pertur..."

Jet53man: Created page with "FORCETOC =Radial Oscillations of n = 3 Polytropic Spheres= ==Background== ===Our Formulation of the Problem=== In an SSC/Pertur..."