SSC/Virial/PolytropesEmbedded/SecondEffortAgain - Revision history

Jet53man: /* Part III */

2021-10-28T21:41:50Z

Part III

← Older revision		Revision as of 17:41, 28 October 2021
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	From our above, detailed analysis of the mass-radius relation for pressure-truncated polytropes, we concluded that configurations along "Stahler's" equilibrium sequence become dynamically unstable at a point that does not coincide with the maximum-mass configuration. Instead, the onset of dynamical instability is associated with the critical point on the mass-radius relation that arises from the free-energy-based virial theorem. In drawing this conclusion, we have implicitly assumed that the proper way to analyze an equilibrium configuration's stability is to vary its radius while, not only holding its mass, specific entropy, and surface pressure <math>~(P_e)</math> constant, but also assuming that the configuration's structural form factors are invariable.		From our above, detailed analysis of the mass-radius relation for pressure-truncated polytropes, we concluded that configurations along "Stahler's" equilibrium sequence become dynamically unstable at a point that does not coincide with the maximum-mass configuration. Instead, the onset of dynamical instability is associated with the critical point on the mass-radius relation that arises from the free-energy-based virial theorem. In drawing this conclusion, we have implicitly assumed that the proper way to analyze an equilibrium configuration's stability is to vary its radius while, not only holding its mass, specific entropy, and surface pressure <math>~(P_e)</math> constant, but also assuming that the configuration's structural form factors are invariable.

			This seems like a reasonable assumption, given that we're asking how a configuration's characteristics will vary ''dynamically'' when perturbed about an equilibrium state. While oscillating about an equilibrium state, it seems more reasonable to assume that the system will expand and contract in a nearly homologous fashion than that its internal structure will readily readjust to produce a different ''and'' desirable set of form factors. In support of this argument, we point to the paper by {{ GW80full }} which explicitly derives a self-similar solution for the ''homologous'' collapse of stellar cores that can be modeled as <math>~n=3</math> polytropes; an associated [[Apps/GoldreichWeber80#Homologously_Collapsing_Stellar_Cores\|chapter of this H_Book details the Goldreich & Weber derivation]]. Goldreich & Weber use [[SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29\|linear perturbation techniques]] to analyze the stability of their homologously collapsing configurations. In §IV of their paper, they describe the eigenvalues and eigenfunctions that result from this analysis. They discovered, for example, that "the lowest radial mode can be found analytically ... [and it] corresponds to a homologous perturbation of the entire core." Our assumption that the structural form factors remain constant when pressure-truncated polytropic configurations undergo radial size variations therefore appears not to be unreasonable. (Based on the Goldreich & Weber discussion, we should also look at the published work of {{ Schwarzschild41full }}, who has evaluated radial modes, and of {{ Cowling41full }}, who has obtained eigenvalues of some low-order nonradial modes.)
	This seems like a reasonable assumption, given that we're asking how a configuration's characteristics will vary ''dynamically'' when perturbed about an equilibrium state. While oscillating about an equilibrium state, it seems more reasonable to assume that the system will expand and contract in a nearly homologous fashion than that its internal structure will readily readjust to produce a different ''and'' desirable set of form factors. In support of this argument, we point to the paper by {{ GW80full }} which explicitly derives a self-similar solution for the ''homologous'' collapse of stellar cores that can be modeled as <math>~n=3</math> polytropes; an associated [[Apps/GoldreichWeber80#Homologously_Collapsing_Stellar_Cores\|chapter of this H_Book details the Goldreich & Weber derivation]]. Goldreich & Weber use [[SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29\|linear perturbation techniques]] to analyze the stability of their homologously collapsing configurations. In §IV of their paper, they describe the eigenvalues and eigenfunctions that result from this analysis. They discovered, for example, that "the lowest radial mode can be found analytically ... [and it] corresponds to a homologous perturbation of the entire core." Our assumption that the structural form factors remain constant when pressure-truncated polytropic configurations undergo radial size variations therefore appears not to be unreasonable. (Based on the Goldreich & Weber discussion, we should also look at the published work of {{ Schwarzschild41full }}, who has evaluated radial modes, and of ~~[http://adsabs.harvard.edu/abs/1941MNRAS.101..367C Cowling (1941, MNRAS, 101, 367)]~~, who has obtained eigenvalues of some low-order nonradial modes.)


	In addition, it would seem that a certain amount of dissipation would be required for the system to readjust to new structural form factors. In order to test this underlying assumption, following [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich & Weber (1980)], it would be desirable to carry out a full-blown perturbation analysis that involves looking for, for example, the eigenvector associated with the system's fundamental radial mode of pulsation. Ideally, we should be using the structural form factors associated with this pulsation-mode eigenfunction in our free-energy analysis of stability. Better yet, the ''sign'' of the eigenfrequency associated with the system's pulsation-mode eigenvector should signal whether the system is dynamically stable or unstable.		In addition, it would seem that a certain amount of dissipation would be required for the system to readjust to new structural form factors. In order to test this underlying assumption, following [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich & Weber (1980)], it would be desirable to carry out a full-blown perturbation analysis that involves looking for, for example, the eigenvector associated with the system's fundamental radial mode of pulsation. Ideally, we should be using the structural form factors associated with this pulsation-mode eigenfunction in our free-energy analysis of stability. Better yet, the ''sign'' of the eigenfrequency associated with the system's pulsation-mode eigenvector should signal whether the system is dynamically stable or unstable.

Jet53man: /* Part III */

2021-10-28T21:28:58Z

Part III

← Older revision		Revision as of 17:28, 28 October 2021
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	This seems like a reasonable assumption, given that we're asking how a configuration's characteristics will vary ''dynamically'' when perturbed about an equilibrium state. While oscillating about an equilibrium state, it seems more reasonable to assume that the system will expand and contract in a nearly homologous fashion than that its internal structure will readily readjust to produce a different ''and'' desirable set of form factors. In support of this argument, we point to the paper by ~~[http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich & Weber (1980)]~~ which explicitly derives a self-similar solution for the ''homologous'' collapse of stellar cores that can be modeled as <math>~n=3</math> polytropes; an associated [[Apps/GoldreichWeber80#Homologously_Collapsing_Stellar_Cores\|chapter of this H_Book details the Goldreich & Weber derivation]]. Goldreich & Weber use [[SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29\|linear perturbation techniques]] to analyze the stability of their homologously collapsing configurations. In §IV of their paper, they describe the eigenvalues and eigenfunctions that result from this analysis. They discovered, for example, that "the lowest radial mode can be found analytically ... [and it] corresponds to a homologous perturbation of the entire core." Our assumption that the structural form factors remain constant when pressure-truncated polytropic configurations undergo radial size variations therefore appears not to be unreasonable. (Based on the Goldreich & Weber discussion, we should also look at the published work of {{ Schwarzschild41full }}, who has evaluated radial modes, and of [http://adsabs.harvard.edu/abs/1941MNRAS.101..367C Cowling (1941, MNRAS, 101, 367)], who has obtained eigenvalues of some low-order nonradial modes.)		This seems like a reasonable assumption, given that we're asking how a configuration's characteristics will vary ''dynamically'' when perturbed about an equilibrium state. While oscillating about an equilibrium state, it seems more reasonable to assume that the system will expand and contract in a nearly homologous fashion than that its internal structure will readily readjust to produce a different ''and'' desirable set of form factors. In support of this argument, we point to the paper by {{ GW80full }} which explicitly derives a self-similar solution for the ''homologous'' collapse of stellar cores that can be modeled as <math>~n=3</math> polytropes; an associated [[Apps/GoldreichWeber80#Homologously_Collapsing_Stellar_Cores\|chapter of this H_Book details the Goldreich & Weber derivation]]. Goldreich & Weber use [[SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29\|linear perturbation techniques]] to analyze the stability of their homologously collapsing configurations. In §IV of their paper, they describe the eigenvalues and eigenfunctions that result from this analysis. They discovered, for example, that "the lowest radial mode can be found analytically ... [and it] corresponds to a homologous perturbation of the entire core." Our assumption that the structural form factors remain constant when pressure-truncated polytropic configurations undergo radial size variations therefore appears not to be unreasonable. (Based on the Goldreich & Weber discussion, we should also look at the published work of {{ Schwarzschild41full }}, who has evaluated radial modes, and of [http://adsabs.harvard.edu/abs/1941MNRAS.101..367C Cowling (1941, MNRAS, 101, 367)], who has obtained eigenvalues of some low-order nonradial modes.)

Jet53man: /* Relating and Reconciling Two Mass-Radius Relationships for n = 4 Polytropes */

2021-10-28T21:09:11Z

Relating and Reconciling Two Mass-Radius Relationships for n = 4 Polytropes

← Older revision		Revision as of 17:09, 28 October 2021
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	[For the record we note that, throughout the structure of an <math>~n=4</math> polytrope, <math>~\mathfrak{b}_{n=4}</math> is a number of order unity. Its value is never less than <math>~3^{1/4}</math>, which pertains to the center of the configuration; its maximum value of <math>\approx 5.098</math> occurs at <math>~\tilde\xi \approx 4.0</math>; and <math>~\mathfrak{b}_{n=4} \approx 3.946</math> at its (zero pressure) surface, <math>~\tilde\xi = \xi_1 \approx 14.97</math>. A plot showing the variation with <math>~P_e</math> of the closely allied parameter, <math>~\mathcal{B}\|_{n=4} = (4\pi)^{1/4} \mathfrak{b}_{n=4}</math> is presented in the righthand panel of the [[SSC/Virial/PolytropesSummary#Summary\|above parameter summary figure]].]		[For the record we note that, throughout the structure of an <math>n=4</math> polytrope, <math>\mathfrak{b}_{n=4}</math> is a number of order unity. Its value is never less than <math>3^{1/4}</math>, which pertains to the center of the configuration; its maximum value of <math>\approx 5.098</math> occurs at <math>\tilde\xi \approx 4.0</math>; and <math>\mathfrak{b}_{n=4} \approx 3.946</math> at its (zero pressure) surface, <math>\tilde\xi = \xi_1 \approx 14.97</math>. A plot showing the variation with <math>P_e</math> of the closely allied parameter, <math>\mathcal{B}\|_{n=4} = (4\pi)^{1/4} \mathfrak{b}_{n=4}</math> is presented in the righthand panel of the [[SSC/Virial/PolytropesSummary#Summary\|above parameter summary figure]].]

	In both panels of the following figure, the blue curve displays the mass-radius relation for pressure-truncated <math>~n=4</math> polytropes, <math>~\mathcal{Y}(\mathcal{X})</math>, that is generated by Stahler's pair of parametric equations. The coordinates of discrete points along the curve have been determined from the tabular data provided on p. 399 of ~~[http://adsabs.harvard.edu/abs/1986Ap%26SS.126..357H Horedt (1986, ApJS, vol. 126)]]~~ while Excel has been used to generate the "smooth," continuous blue curve connecting the points; this set of points and accompanying blue curve are identical in both figure panels. In both figure panels, a set of discrete, triangle-shaped points traces the mass-radius relation, <math>~\mathcal{Y}(\mathcal{X})</math>, that is obtained via the virial theorem, assuming that the coefficient, <math>~\mathfrak{b}_{n=4}</math>, is constant along the sequence. The "green" sequence in the lefthand panel results from setting <math>~\mathfrak{b}_{n=4} = 3.4205</math>, which is the value of the constant that results from Horedt's tabulated data if the configuration is truncated at <math>~\tilde\xi = 1.4</math>; the "orange" sequence in the righthand panel results from setting <math>~\mathfrak{b}_{n=4} = 4.8926</math>, which is the value of the constant that results from Horedt's tabulated data if the configuration is truncated at <math>~\tilde\xi = 2.8</math>.		In both panels of the following figure, the blue curve displays the mass-radius relation for pressure-truncated <math>n=4</math> polytropes, <math>\mathcal{Y}(\mathcal{X})</math>, that is generated by Stahler's pair of parametric equations. The coordinates of discrete points along the curve have been determined from the tabular data provided on p. 399 of {{ Horedt86full }} while Excel has been used to generate the "smooth," continuous blue curve connecting the points; this set of points and accompanying blue curve are identical in both figure panels. In both figure panels, a set of discrete, triangle-shaped points traces the mass-radius relation, <math>\mathcal{Y}(\mathcal{X})</math>, that is obtained via the virial theorem, assuming that the coefficient, <math>\mathfrak{b}_{n=4}</math>, is constant along the sequence. The "green" sequence in the lefthand panel results from setting <math>\mathfrak{b}_{n=4} = 3.4205</math>, which is the value of the constant that results from Horedt's tabulated data if the configuration is truncated at <math>\tilde\xi = 1.4</math>; the "orange" sequence in the righthand panel results from setting <math>\mathfrak{b}_{n=4} = 4.8926</math>, which is the value of the constant that results from Horedt's tabulated data if the configuration is truncated at <math>\tilde\xi = 2.8</math>.

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	According to ~~[http://adsabs.harvard.edu/abs/1986Ap%26SS.126..357H Horedt's (1986)]] tabulated data~~, the surface of an isolated <math>~(P_e = 0)</math>, spherically symmetric, <math>~n=4</math> polytrope occurs at the dimensionless (Lane-Emden) radius, <math>~\xi_1 = 14.9715463</math>. In both panels of the above figure, this ''isolated'' configuration is identified by the discrete (blue diamond) point at the origin, that is, at <math>~(\mathcal{X}, \mathcal{Y}) = (0, 0)</math>. As we begin to examine pressure-truncated models and <math>~\tilde\xi</math> is steadily decreased from <math>~\xi_1</math>, the mass-radius coordinate of equilibrium configurations "moves" away from the origin, upward along the upper branch of the displayed (blue) mass-radius relation. A maximum mass of <math>~\mathcal{Y} \approx 2.042</math> (corresponding to a radius of <math>~\mathcal{X} \approx 0.4585</math>) is reached ''from the left'' as <math>~\tilde\xi</math> drops to a value of approximately <math>~3.4</math>. As <math>~\tilde\xi</math> continues to decrease, the mass-radius coordinates of equilibrium configurations move along the lower branch of the displayed (blue) curve, reaching a maximum radius at <math>~(\mathcal{X}, \mathcal{Y}) \approx (0.555, 1.554)</math> — corresponding to <math>~\tilde\xi \approx 2.0</math> — then decreasing in radius until, once again, the origin is reached, but this time because <math>~\tilde\xi</math> drops to zero.		According to {{ Horedt86 }}, the surface of an isolated <math>(P_e = 0)</math>, spherically symmetric, <math>n=4</math> polytrope occurs at the dimensionless (Lane-Emden) radius, <math>\xi_1 = 14.9715463</math>. In both panels of the above figure, this ''isolated'' configuration is identified by the discrete (blue diamond) point at the origin, that is, at <math>(\mathcal{X}, \mathcal{Y}) = (0, 0)</math>. As we begin to examine pressure-truncated models and <math>\tilde\xi</math> is steadily decreased from <math>\xi_1</math>, the mass-radius coordinate of equilibrium configurations "moves" away from the origin, upward along the upper branch of the displayed (blue) mass-radius relation. A maximum mass of <math>\mathcal{Y} \approx 2.042</math> (corresponding to a radius of <math>\mathcal{X} \approx 0.4585</math>) is reached ''from the left'' as <math>\tilde\xi</math> drops to a value of approximately <math>3.4</math>. As <math>\tilde\xi</math> continues to decrease, the mass-radius coordinates of equilibrium configurations move along the lower branch of the displayed (blue) curve, reaching a maximum radius at <math>(\mathcal{X}, \mathcal{Y}) \approx (0.555, 1.554)</math> — corresponding to <math>\tilde\xi \approx 2.0</math> — then decreasing in radius until, once again, the origin is reached, but this time because <math>~\tilde\xi</math> drops to zero.

			If we set <math>\mathfrak{b}_{n=4} = 3.4205</math> (corresponding to a choice of <math>\tilde\xi = 1.4</math>), the virial theorem mass-radius relation maps onto the "Stahler" mass-radius coordinate plane as depicted by the set of green, triangle-shaped points in the lefthand panel of the above figure. While the (green) curve corresponding to this relation does not overlay the blue mass-radius relation, the two curves do intersect. They intersect precisely at the coordinate location along the blue curve (emphasized by the black filled circle) corresponding to a detailed force-balanced model having <math>~\tilde\xi = 1.4</math>. In an analogous fashion, in the righthand panel of the figure, the curve delineated by the set of orange triangle-shaped points shows how the virial theorem mass-radius relation maps onto the "Stahler" mass-radius coordinate plane when we set <math>~\mathfrak{b}_{n=4} = 4.8926</math> (corresponding to a choice of <math>~\tilde\xi = 2.8</math>); it intersects the blue mass-radius relation precisely at the coordinate location, <math>~(\mathcal{X}, \mathcal{Y}) \approx (0.5108, 1.965)</math> — again, emphasized by a black filled circle — that corresponds to a detailed force-balanced model having <math>~\tilde\xi = 2.8</math>. Hence, the two relations give the same mass-radius coordinates when the value of <math>~\mathfrak{b}_{n=4}</math> that is plugged into the virial theorem matches the value of <math>~\mathfrak{b}_{n=4}</math> that reflects the structural form factor that is properly associated with a detailed force-balanced model.

	If we set <math>~\mathfrak{b}_{n=4} = 3.4205</math> (corresponding to a choice of <math>\tilde\xi = 1.4</math>), the virial theorem mass-radius relation maps onto the "Stahler" mass-radius coordinate plane as depicted by the set of green, triangle-shaped points in the lefthand panel of the above figure. While the (green) curve corresponding to this relation does not overlay the blue mass-radius relation, the two curves do intersect. They intersect precisely at the coordinate location along the blue curve (emphasized by the black filled circle) corresponding to a detailed force-balanced model having <math>~\tilde\xi = 1.4</math>. In an analogous fashion, in the righthand panel of the figure, the curve delineated by the set of orange triangle-shaped points shows how the virial theorem mass-radius relation maps onto the "Stahler" mass-radius coordinate plane when we set <math>~\mathfrak{b}_{n=4} = 4.8926</math> (corresponding to a choice of <math>~\tilde\xi = 2.8</math>); it intersects the blue mass-radius relation precisely at the coordinate location, <math>~(\mathcal{X}, \mathcal{Y}) \approx (0.5108, 1.965)</math> — again, emphasized by a black filled circle — that corresponds to a detailed force-balanced model having <math>~\tilde\xi = 2.8</math>. Hence, the two relations give the same mass-radius coordinates when the value of <math>~\mathfrak{b}_{n=4}</math> that is plugged into the virial theorem matches the value of <math>~\mathfrak{b}_{n=4}</math> that reflects the structural form factor that is properly associated with a detailed force-balanced model.		When we mapped the virial theorem mass-radius relation onto Stahler's mass-radius coordinate plane using a value of <math>\mathfrak{b}_{n=4} = 4.8926</math> (as traced by the orange triangle-shaped points in the righthand panel of the above figure), we expected it to intersect the blue curve at the point along the blue sequence where <math>\tilde\xi = 2.8</math>, for the reason just discussed. After constructing the plot, it became clear that the two curves also intersect at the coordinate location, <math>(\mathcal{X}, \mathcal{Y}) \approx (0.255, 1.67)</math> — also highlighted by a black filled circle — that corresponds to a detailed force-balanced model having <math>\tilde\xi \approx 6.0</math>. This makes it clear that it is the equality of the structural form factors, not the equality of the dimensionless (Lane-Emden) radius, <math>\tilde\xi</math>, that assures precise agreement between the two different mass-radius expressions.


	When we mapped the virial theorem mass-radius relation onto Stahler's mass-radius coordinate plane using a value of <math>~\mathfrak{b}_{n=4} = 4.8926</math> (as traced by the orange triangle-shaped points in the righthand panel of the above figure), we expected it to intersect the blue curve at the point along the blue sequence where <math>~\tilde\xi = 2.8</math>, for the reason just discussed. After constructing the plot, it became clear that the two curves also intersect at the coordinate location, <math>~(\mathcal{X}, \mathcal{Y}) \approx (0.255, 1.67)</math> — also highlighted by a black filled circle — that corresponds to a detailed force-balanced model having <math>~\tilde\xi \approx 6.0</math>. This makes it clear that it is the equality of the structural form factors, not the equality of the dimensionless (Lane-Emden) radius, <math>~\tilde\xi</math>, that assures precise agreement between the two different mass-radius expressions.


	As is detailed in our [[SSC/Virial/PolytropesSummary#Stability\|above discussion of the dynamical stability of pressure-truncated polytropes]], an examination of free-energy variations can not only assist us in identifying the properties of equilibrium configurations (via a free-energy derivation of the virial theorem) but also in determining which of these configurations are dynamically stable and which are dynamically unstable. We showed that, for a certain range of polytropic indexes, there is a critical point along the corresponding model sequence where the transition from stability to instability occurs. As has been detailed in our [[SSC/Virial/PolytropesSummary#Try_Polytropic_Index_of_4\|above groundwork derivations]], for <math>~n = 4</math> polytropic structures, the critical point is identified by the dimensionless parameters,		As is detailed in our [[SSC/Virial/PolytropesSummary#Stability\|above discussion of the dynamical stability of pressure-truncated polytropes]], an examination of free-energy variations can not only assist us in identifying the properties of equilibrium configurations (via a free-energy derivation of the virial theorem) but also in determining which of these configurations are dynamically stable and which are dynamically unstable. We showed that, for a certain range of polytropic indexes, there is a critical point along the corresponding model sequence where the transition from stability to instability occurs. As has been detailed in our [[SSC/Virial/PolytropesSummary#Try_Polytropic_Index_of_4\|above groundwork derivations]], for <math>~n = 4</math> polytropic structures, the critical point is identified by the dimensionless parameters,

Jet53man: /* Plotting Concise Mass-Radius Relation */

2021-10-28T20:13:04Z

Plotting Concise Mass-Radius Relation

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 </div>
-is plotted for seven different values of the polytropic index, <math>~n</math>, as indicated, in the lefthand diagram of the following composite figure.  For comparison, the ''schematic'' diagram displayed on the righthand side of the figure is a reproduction of Figure 17 from Appendix B of [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)].  It seems that our derived, analytically prescribable, mass-radius relationship &#8212; which is, in essence, a statement of the scalar virial theorem &#8212; embodies most of the attributes of the mass-radius relationship for pressure-truncated polytropes that were already understood, and conveyed schematically, by Stahler in 1983.
+is plotted for seven different values of the polytropic index, <math>~n</math>, as indicated, in the lefthand diagram of the following composite figure.  For comparison, the ''schematic'' diagram displayed on the righthand side of the figure is a reproduction of Figure 17 from Appendix B of {{ Stahler83 }}.  It seems that our derived, analytically prescribable, mass-radius relationship &#8212; which is, in essence, a statement of the scalar virial theorem &#8212; embodies most of the attributes of the mass-radius relationship for pressure-truncated polytropes that were already understood, and conveyed schematically, by Stahler in 1983.
+<table border="1" width="100%" cellpadding="3" align="center">
+<tr>
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 Let's do this again using the mass-radius relation as written explicitly in terms of the normalizations, <math>~M_\mathrm{SWS}</math> and <math>~R_\mathrm{SWS}</math>.  The relevant, generic nonlinear equation is,

Jet53man: /* Detailed Force-Balanced Solution */

2021-10-28T19:53:01Z

Detailed Force-Balanced Solution

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	===Detailed Force-Balanced Solution===		===Detailed Force-Balanced Solution===
	As has been summarized in our [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation\|accompanying review]] of detailed force-balanced models of pressure-truncated polytropes, {{ Stahler83 }} found that a spherical configuration's equilibrium radius is related to its mass through the following pair of parametric equations:		As has been summarized in our [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation\|accompanying review]] of detailed force-balanced models of pressure-truncated polytropes, {{ Stahler83 }} — hereafter, {{ Stahler83hereafter }} — found that a spherical configuration's equilibrium radius is related to its mass through the following pair of parametric equations:
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Jet53man: /* Other Model Sequences */

2021-10-28T18:04:04Z

Other Model Sequences

← Older revision		Revision as of 14:04, 28 October 2021
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	===Relating and Reconciling Two Mass-Radius Relationships for n = 4 Polytropes===		===Relating and Reconciling Two Mass-Radius Relationships for n = 4 Polytropes===

	For pressure-truncated <math>~n=4</math> polytropes, ~~[http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)]~~ did not identify a polynomial relationship between the mass and radius of equilibrium configurations. However, from his analysis of detailed force-balance models ([[~~User:Tohline/~~SSC/Virial/PolytropesSummary#Detailed_Force-Balanced_Solution_2\|summarized above]]), we appreciate that the governing pair of parametric relations is,		For pressure-truncated <math>n=4</math> polytropes, {{ Stahler83 }} did not identify a polynomial relationship between the mass and radius of equilibrium configurations. However, from his analysis of detailed force-balance models ([[SSC/Virial/PolytropesSummary#Detailed_Force-Balanced_Solution_2\|summarized above]]), we appreciate that the governing pair of parametric relations is,
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	On the other hand, the polynomial that results from plugging <math>~n=4</math> into the [[~~User:Tohline/~~SSC/Virial/PolytropesSummary#ConciseVirialXY\|general mass-radius relation that is obtained via the virial theorem]] is,		On the other hand, the polynomial that results from plugging <math>~n=4</math> into the [[SSC/Virial/PolytropesSummary#ConciseVirialXY\|general mass-radius relation that is obtained via the virial theorem]] is,

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	</div>		</div>

	[For the record we note that, throughout the structure of an <math>~n=4</math> polytrope, <math>~\mathfrak{b}_{n=4}</math> is a number of order unity. Its value is never less than <math>~3^{1/4}</math>, which pertains to the center of the configuration; its maximum value of <math>\approx 5.098</math> occurs at <math>~\tilde\xi \approx 4.0</math>; and <math>~\mathfrak{b}_{n=4} \approx 3.946</math> at its (zero pressure) surface, <math>~\tilde\xi = \xi_1 \approx 14.97</math>. A plot showing the variation with <math>~P_e</math> of the closely allied parameter, <math>~\mathcal{B}\|_{n=4} = (4\pi)^{1/4} \mathfrak{b}_{n=4}</math> is presented in the righthand panel of the [[~~User:Tohline/~~SSC/Virial/PolytropesSummary#Summary\|above parameter summary figure]].]		[For the record we note that, throughout the structure of an <math>~n=4</math> polytrope, <math>~\mathfrak{b}_{n=4}</math> is a number of order unity. Its value is never less than <math>~3^{1/4}</math>, which pertains to the center of the configuration; its maximum value of <math>\approx 5.098</math> occurs at <math>~\tilde\xi \approx 4.0</math>; and <math>~\mathfrak{b}_{n=4} \approx 3.946</math> at its (zero pressure) surface, <math>~\tilde\xi = \xi_1 \approx 14.97</math>. A plot showing the variation with <math>~P_e</math> of the closely allied parameter, <math>~\mathcal{B}\|_{n=4} = (4\pi)^{1/4} \mathfrak{b}_{n=4}</math> is presented in the righthand panel of the [[SSC/Virial/PolytropesSummary#Summary\|above parameter summary figure]].]

	In both panels of the following figure, the blue curve displays the mass-radius relation for pressure-truncated <math>~n=4</math> polytropes, <math>~\mathcal{Y}(\mathcal{X})</math>, that is generated by Stahler's pair of parametric equations. The coordinates of discrete points along the curve have been determined from the tabular data provided on p. 399 of [http://adsabs.harvard.edu/abs/1986Ap%26SS.126..357H Horedt (1986, ApJS, vol. 126)]] while Excel has been used to generate the "smooth," continuous blue curve connecting the points; this set of points and accompanying blue curve are identical in both figure panels. In both figure panels, a set of discrete, triangle-shaped points traces the mass-radius relation, <math>~\mathcal{Y}(\mathcal{X})</math>, that is obtained via the virial theorem, assuming that the coefficient, <math>~\mathfrak{b}_{n=4}</math>, is constant along the sequence. The "green" sequence in the lefthand panel results from setting <math>~\mathfrak{b}_{n=4} = 3.4205</math>, which is the value of the constant that results from Horedt's tabulated data if the configuration is truncated at <math>~\tilde\xi = 1.4</math>; the "orange" sequence in the righthand panel results from setting <math>~\mathfrak{b}_{n=4} = 4.8926</math>, which is the value of the constant that results from Horedt's tabulated data if the configuration is truncated at <math>~\tilde\xi = 2.8</math>.		In both panels of the following figure, the blue curve displays the mass-radius relation for pressure-truncated <math>~n=4</math> polytropes, <math>~\mathcal{Y}(\mathcal{X})</math>, that is generated by Stahler's pair of parametric equations. The coordinates of discrete points along the curve have been determined from the tabular data provided on p. 399 of [http://adsabs.harvard.edu/abs/1986Ap%26SS.126..357H Horedt (1986, ApJS, vol. 126)]] while Excel has been used to generate the "smooth," continuous blue curve connecting the points; this set of points and accompanying blue curve are identical in both figure panels. In both figure panels, a set of discrete, triangle-shaped points traces the mass-radius relation, <math>~\mathcal{Y}(\mathcal{X})</math>, that is obtained via the virial theorem, assuming that the coefficient, <math>~\mathfrak{b}_{n=4}</math>, is constant along the sequence. The "green" sequence in the lefthand panel results from setting <math>~\mathfrak{b}_{n=4} = 3.4205</math>, which is the value of the constant that results from Horedt's tabulated data if the configuration is truncated at <math>~\tilde\xi = 1.4</math>; the "orange" sequence in the righthand panel results from setting <math>~\mathfrak{b}_{n=4} = 4.8926</math>, which is the value of the constant that results from Horedt's tabulated data if the configuration is truncated at <math>~\tilde\xi = 2.8</math>.
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	If we set <math>~\mathfrak{b}_{n=4} = 3.4205</math> (corresponding to a choice of <math>~\tilde\xi = 1.4</math>), the virial theorem mass-radius relation maps onto the "Stahler" mass-radius coordinate plane as depicted by the set of green, triangle-shaped points in the lefthand panel of the above figure. While the (green) curve corresponding to this relation does not overlay the blue mass-radius relation, the two curves do intersect. They intersect precisely at the coordinate location along the blue curve (emphasized by the black filled circle) corresponding to a detailed force-balanced model having <math>~\tilde\xi = 1.4</math>. In an analogous fashion, in the righthand panel of the figure, the curve delineated by the set of orange triangle-shaped points shows how the virial theorem mass-radius relation maps onto the "Stahler" mass-radius coordinate plane when we set <math>~\mathfrak{b}_{n=4} = 4.8926</math> (corresponding to a choice of <math>~\tilde\xi = 2.8</math>); it intersects the blue mass-radius relation precisely at the coordinate location, <math>~(\mathcal{X}, \mathcal{Y}) \approx (0.5108, 1.965)</math> — again, emphasized by a black filled circle — that corresponds to a detailed force-balanced model having <math>~\tilde\xi = 2.8</math>. Hence, the two relations give the same mass-radius coordinates when the value of <math>~\mathfrak{b}_{n=4}</math> that is plugged into the virial theorem matches the value of <math>~\mathfrak{b}_{n=4}</math> that reflects the structural form factor that is properly associated with a detailed force-balanced model.		If we set <math>~\mathfrak{b}_{n=4} = 3.4205</math> (corresponding to a choice of <math>\tilde\xi = 1.4</math>), the virial theorem mass-radius relation maps onto the "Stahler" mass-radius coordinate plane as depicted by the set of green, triangle-shaped points in the lefthand panel of the above figure. While the (green) curve corresponding to this relation does not overlay the blue mass-radius relation, the two curves do intersect. They intersect precisely at the coordinate location along the blue curve (emphasized by the black filled circle) corresponding to a detailed force-balanced model having <math>~\tilde\xi = 1.4</math>. In an analogous fashion, in the righthand panel of the figure, the curve delineated by the set of orange triangle-shaped points shows how the virial theorem mass-radius relation maps onto the "Stahler" mass-radius coordinate plane when we set <math>~\mathfrak{b}_{n=4} = 4.8926</math> (corresponding to a choice of <math>~\tilde\xi = 2.8</math>); it intersects the blue mass-radius relation precisely at the coordinate location, <math>~(\mathcal{X}, \mathcal{Y}) \approx (0.5108, 1.965)</math> — again, emphasized by a black filled circle — that corresponds to a detailed force-balanced model having <math>~\tilde\xi = 2.8</math>. Hence, the two relations give the same mass-radius coordinates when the value of <math>~\mathfrak{b}_{n=4}</math> that is plugged into the virial theorem matches the value of <math>~\mathfrak{b}_{n=4}</math> that reflects the structural form factor that is properly associated with a detailed force-balanced model.


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	As is detailed in our [[~~User:Tohline/~~SSC/Virial/PolytropesSummary#Stability\|above discussion of the dynamical stability of pressure-truncated polytropes]], an examination of free-energy variations can not only assist us in identifying the properties of equilibrium configurations (via a free-energy derivation of the virial theorem) but also in determining which of these configurations are dynamically stable and which are dynamically unstable. We showed that, for a certain range of polytropic indexes, there is a critical point along the corresponding model sequence where the transition from stability to instability occurs. As has been detailed in our [[~~User:Tohline/~~SSC/Virial/PolytropesSummary#Try_Polytropic_Index_of_4\|above groundwork derivations]], for <math>~n = 4</math> polytropic structures, the critical point is identified by the dimensionless parameters,		As is detailed in our [[SSC/Virial/PolytropesSummary#Stability\|above discussion of the dynamical stability of pressure-truncated polytropes]], an examination of free-energy variations can not only assist us in identifying the properties of equilibrium configurations (via a free-energy derivation of the virial theorem) but also in determining which of these configurations are dynamically stable and which are dynamically unstable. We showed that, for a certain range of polytropic indexes, there is a critical point along the corresponding model sequence where the transition from stability to instability occurs. As has been detailed in our [[SSC/Virial/PolytropesSummary#Try_Polytropic_Index_of_4\|above groundwork derivations]], for <math>~n = 4</math> polytropic structures, the critical point is identified by the dimensionless parameters,
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	In the context of the above figure, independent of the chosen value of <math>~\mathfrak{b}_{n=4}</math>, this critical point always corresponds to the maximum mass that occurs along the mass-radius relationship established via the virial theorem. In both panels of the figure, a horizontal red-dotted line has been drawn tangent to this critical point and identifies the corresponding critical value of <math>~\mathcal{Y}</math>; a vertical red-dashed line drawn through this same point helps identify the corresponding critical value of <math>~\mathcal{X}</math>. We have deduced (details of the derivation not shown) that, for pressure-truncated <math>~n=4</math> polytropes, the coordinates of this critical point in Stahler's <math>~\mathcal{X}-\mathcal{Y}</math> plane depends on the choice of <math>~\mathfrak{b}_{n=4}</math> as follows:		In the context of the above figure, independent of the chosen value of <math>~\mathfrak{b}_{n=4}</math>, this critical point always corresponds to the maximum mass that occurs along the mass-radius relationship established via the virial theorem. In both panels of the figure, a horizontal red-dotted line has been drawn tangent to this critical point and identifies the corresponding critical value of <math>\mathcal{Y}</math>; a vertical red-dashed line drawn through this same point helps identify the corresponding critical value of <math>\mathcal{X}</math>. We have deduced (details of the derivation not shown) that, for pressure-truncated <math>n=4</math> polytropes, the coordinates of this critical point in Stahler's <math>\mathcal{X}-\mathcal{Y}</math> plane depends on the choice of <math>~\mathfrak{b}_{n=4}</math> as follows:
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	In practice, for a given plot of the type displayed in the above figure — that is, for a given choice of the structural parameter, <math>\mathfrak{b}_{n=4}</math> — it only makes sense to compare the location of this critical point to the location of points that have been highlighted by a filled black circle, that is, points that identify the intersection between the two mass-radius relations. If, in a given figure panel, a filled black circle lies to the right of the vertical dashed line, the equilibrium configuration corresponding to that black circle is dynamically stable. On the other hand, if the filled black circle lies to the left of the vertical dashed line, its corresponding equilibrium configuration is dynamically unstable. We conclude, therefore, that the equilibrium configuration marked by a filled black circle in the lefthand panel of the above figure is ''stable''; however, both configurations identified by filled black circles in the righthand panel are ''unstable''.		In practice, for a given plot of the type displayed in the above figure — that is, for a given choice of the structural parameter, <math>\mathfrak{b}_{n=4}</math> — it only makes sense to compare the location of this critical point to the location of points that have been highlighted by a filled black circle, that is, points that identify the intersection between the two mass-radius relations. If, in a given figure panel, a filled black circle lies to the right of the vertical dashed line, the equilibrium configuration corresponding to that black circle is dynamically stable. On the other hand, if the filled black circle lies to the left of the vertical dashed line, its corresponding equilibrium configuration is dynamically unstable. We conclude, therefore, that the equilibrium configuration marked by a filled black circle in the lefthand panel of the above figure is ''stable''; however, both configurations identified by filled black circles in the righthand panel are ''unstable''.

	It is significant that the critical point identified by our free-energy-based stability analysis does not correspond to the equilibrium configuration having the largest mass along "Stahler's" (blue) equilibrium model sequence. One might naively expect that a configuration of maximum mass along the blue curve is the relevant demarcation point and that, correspondingly, all models along this sequence that fall "to the right" of this maximum-mass point are stable. But the righthand panel of our above figure contradicts this expectation. While both of the black filled circles in the righthand panel of the above figure lie to the left of the vertical dashed line and therefore, as just concluded, are both unstable, one of the two configurations lies ''to the right'' of the maximum-mass point along the blue "Stahler" sequence. This finding is related to [[~~User:Tohline/~~SSC/Virial/PolytropesSummary#Curiosity\|the curiosity raised earlier]] in our discussion of the structural properties of pressure-truncated, <math>~n=4</math> polytropes.		It is significant that the critical point identified by our free-energy-based stability analysis does not correspond to the equilibrium configuration having the largest mass along "Stahler's" (blue) equilibrium model sequence. One might naively expect that a configuration of maximum mass along the blue curve is the relevant demarcation point and that, correspondingly, all models along this sequence that fall "to the right" of this maximum-mass point are stable. But the righthand panel of our above figure contradicts this expectation. While both of the black filled circles in the righthand panel of the above figure lie to the left of the vertical dashed line and therefore, as just concluded, are both unstable, one of the two configurations lies ''to the right'' of the maximum-mass point along the blue "Stahler" sequence. This finding is related to [[SSC/Virial/PolytropesSummary#Curiosity\|the curiosity raised earlier]] in our discussion of the structural properties of pressure-truncated, <math>~n=4</math> polytropes.

	===Relating and Reconciling Two Mass-Radius Relationships for n = 3 Polytropes===		===Relating and Reconciling Two Mass-Radius Relationships for n = 3 Polytropes===

	For pressure-truncated <math>~n=3</math> polytropes, ~~[http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)]~~ did not identify a polynomial relationship between the mass and radius of equilibrium configurations. However, from his analysis of detailed force-balance models ([[~~User:Tohline/~~SSC/Virial/PolytropesSummary#Detailed_Force-Balanced_Solution_2\|summarized above]]), we appreciate that the governing pair of parametric relations is,		For pressure-truncated <math>n=3</math> polytropes, {{ Stahler83 }} did not identify a polynomial relationship between the mass and radius of equilibrium configurations. However, from his analysis of detailed force-balance models ([[SSC/Virial/PolytropesSummary#Detailed_Force-Balanced_Solution_2\|summarized above]]), we appreciate that the governing pair of parametric relations is,
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	On the other hand, the polynomial that results from plugging <math>~n=3</math> into the [[~~User:Tohline/~~SSC/Virial/PolytropesSummary#ConciseVirialXY\|general mass-radius relation that is obtained via the virial theorem]] is,		On the other hand, the polynomial that results from plugging <math>~n=3</math> into the [[SSC/Virial/PolytropesSummary#ConciseVirialXY\|general mass-radius relation that is obtained via the virial theorem]] is,

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

Jet53man at 17:57, 28 October 2021

2021-10-28T17:57:49Z

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Jet53man: /* Discussion */

2021-10-28T17:51:03Z

Discussion

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	==Discussion==		==Discussion==
	A spherically symmetric, self-gravitating gas cloud whose effective adiabatic exponent is <math>~\gamma < 4/3</math> — equivalently, <math>~n > 3</math> — cannot exist in a dynamically stable equilibrium state, in isolation. Such clouds can be stabilized, however, if they are embedded in a hot, tenuous external medium and effectively confined by an external pressure, <math>~P_e</math>. The pressure-truncated, <math>~n = 5 ~(\gamma = 6/5)</math> polytropic configurations being discussed here provide examples of such embedded clouds. A direct analogy can be drawn between this discussion and discussions of pressure-truncated isothermal <math>~(\gamma = 1; n = \infty)</math> clouds — see, for example, our review of isothermal cloud structures in the context of [[~~User:Tohline/~~SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Bonnor-Ebert spheres]].		A spherically symmetric, self-gravitating gas cloud whose effective adiabatic exponent is <math>\gamma < 4/3</math> — equivalently, <math>n > 3</math> — cannot exist in a dynamically stable equilibrium state, in isolation. Such clouds can be stabilized, however, if they are embedded in a hot, tenuous external medium and effectively confined by an external pressure, <math>~P_e</math>. The pressure-truncated, <math>n = 5 (\gamma = 6/5)</math> polytropic configurations being discussed here provide examples of such embedded clouds. A direct analogy can be drawn between this discussion and discussions of pressure-truncated isothermal <math>~(\gamma = 1; n = \infty)</math> clouds — see, for example, our review of isothermal cloud structures in the context of [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Bonnor-Ebert spheres]].

	===Part I: Physical Significance of the Two Curves===		===Part I: Physical Significance of the Two Curves===

Jet53man: /* Part III */

2021-10-28T17:48:41Z

Part III

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	{{~~LSU_WorkInProgress~~}}		{{ SGFworkInProgress }}

	==Other Model Sequences==		==Other Model Sequences==

Jet53man: /* Part III */

2021-10-28T17:48:17Z

Part III

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	This seems like a reasonable assumption, given that we're asking how a configuration's characteristics will vary ''dynamically'' when perturbed about an equilibrium state. While oscillating about an equilibrium state, it seems more reasonable to assume that the system will expand and contract in a nearly homologous fashion than that its internal structure will readily readjust to produce a different ''and'' desirable set of form factors. In support of this argument, we point to the paper by [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich & Weber (1980)] which explicitly derives a self-similar solution for the ''homologous'' collapse of stellar cores that can be modeled as <math>~n=3</math> polytropes; an associated [[~~User:Tohline/~~Apps/GoldreichWeber80#Homologously_Collapsing_Stellar_Cores\|chapter of this H_Book details the Goldreich & Weber derivation]]. Goldreich & Weber use [[~~User:Tohline/~~SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29\|linear perturbation techniques]] to analyze the stability of their homologously collapsing configurations. In §IV of their paper, they describe the eigenvalues and eigenfunctions that result from this analysis. They discovered, for example, that "the lowest radial mode can be found analytically ... [and it] corresponds to a homologous perturbation of the entire core." Our assumption that the structural form factors remain constant when pressure-truncated polytropic configurations undergo radial size variations therefore appears not to be unreasonable. (Based on the Goldreich & Weber discussion, we should also look at the published work of ~~[http://adsabs.harvard.edu/abs/1941ApJ....94..245S Schwarzschild (1941, ApJ, 94, 245)]~~, who has evaluated radial modes, and of [http://adsabs.harvard.edu/abs/1941MNRAS.101..367C Cowling (1941, MNRAS, 101, 367)], who has obtained eigenvalues of some low-order nonradial modes.)		This seems like a reasonable assumption, given that we're asking how a configuration's characteristics will vary ''dynamically'' when perturbed about an equilibrium state. While oscillating about an equilibrium state, it seems more reasonable to assume that the system will expand and contract in a nearly homologous fashion than that its internal structure will readily readjust to produce a different ''and'' desirable set of form factors. In support of this argument, we point to the paper by [http://adsabs.harvard.edu/abs/1980ApJ...238..991G Goldreich & Weber (1980)] which explicitly derives a self-similar solution for the ''homologous'' collapse of stellar cores that can be modeled as <math>~n=3</math> polytropes; an associated [[Apps/GoldreichWeber80#Homologously_Collapsing_Stellar_Cores\|chapter of this H_Book details the Goldreich & Weber derivation]]. Goldreich & Weber use [[SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29\|linear perturbation techniques]] to analyze the stability of their homologously collapsing configurations. In §IV of their paper, they describe the eigenvalues and eigenfunctions that result from this analysis. They discovered, for example, that "the lowest radial mode can be found analytically ... [and it] corresponds to a homologous perturbation of the entire core." Our assumption that the structural form factors remain constant when pressure-truncated polytropic configurations undergo radial size variations therefore appears not to be unreasonable. (Based on the Goldreich & Weber discussion, we should also look at the published work of {{ Schwarzschild41full }}, who has evaluated radial modes, and of [http://adsabs.harvard.edu/abs/1941MNRAS.101..367C Cowling (1941, MNRAS, 101, 367)], who has obtained eigenvalues of some low-order nonradial modes.)