Appendix/Ramblings/NonlinarOscillation - Revision history

Jet53man: /* Determining the Relevant Values of ~\tilde\xi */

2022-05-10T16:25:34Z

Determining the Relevant Values of ~\tilde\xi

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Jet53man: /* Configuration Pairing */

2022-05-09T23:55:24Z

Configuration Pairing

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Jet53man: /* Two Example Eigenfunctions */

2022-05-09T23:48:43Z

Two Example Eigenfunctions

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	===Two Example Eigenfunctions===		===Two Example Eigenfunctions===

	The following figure is fundamentally a reproduction of [[SSC/FreeEnergy/~~PowerPoint~~#Figure3\|Figure 3 from an accompanying discussion]]. It presents the "Case M" equilibrium sequence from both an order-of-magnitude analysis (marked by light-blue squares) and a detailed force-balance analysis (light-green triangles). The dark green circular dot identifies the configuration at the pressure maximum of the sequence — <math>~P_\mathrm{max}/P_\mathrm{norm} = 160.867</math> — and the red circular dot identifies the location along the sequence where the transition from stable to dynamically unstable configurations occurs — <math>~P_e/P_\mathrm{norm} = 124.447</math>. (All pressures have been normalized to <math>~P_\mathrm{max}</math> in the figure.)		The following figure is fundamentally a reproduction of [[SSC/FreeEnergy/Powerpoint#Figure3\|Figure 3 from an accompanying discussion]]. It presents the "Case M" equilibrium sequence from both an order-of-magnitude analysis (marked by light-blue squares) and a detailed force-balance analysis (light-green triangles). The dark green circular dot identifies the configuration at the pressure maximum of the sequence — <math>P_\mathrm{max}/P_\mathrm{norm} = 160.867</math> — and the red circular dot identifies the location along the sequence where the transition from stable to dynamically unstable configurations occurs — <math>P_e/P_\mathrm{norm} = 124.447</math>. (All pressures have been normalized to <math>P_\mathrm{max}</math> in the figure.)

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	At any <math>~P_e</math> between these two limiting values, a pair of ''stable'' equilibrium configurations exist; approximately twenty example pairings are listed in Table 1. The horizontal, black dashed line in the figure has been drawn at <math>~P_e/P_\mathrm{norm} = 158.664</math>. The pair of equilibrium configurations associated with this pressure is identified graphically by the two points at which this dashed line intersects the detailed force-balance equilibrium sequence; as is detailed in the second row of Table 1, the configurations correspond to models having <math>~\tilde\xi = 2.83986</math> (right intersection) and <math>~\tilde\xi = 3.17241</math> (left intersection). The left-hand panel of Figure 2 shows how the Lagrangian radial coordinate varies with mass, <math>~r_\xi(m_\xi)</math>, throughout the interior of these two equilibrium configurations (the locus of green and orange dots, respectively); for reference, the profile of the configuration at <math>~P_\mathrm{max}</math> is presented as well (locus of black dots). The right-hand panel of Figure 2 shows the same paired configuration profiles, but ''relative'' to the profile of the configuration at <math>~P_\mathrm{max}</math>.		At any <math>P_e</math> between these two limiting values, a pair of ''stable'' equilibrium configurations exist; approximately twenty example pairings are listed in Table 1. The horizontal, black dashed line in the figure has been drawn at <math>P_e/P_\mathrm{norm} = 158.664</math>. The pair of equilibrium configurations associated with this pressure is identified graphically by the two points at which this dashed line intersects the detailed force-balance equilibrium sequence; as is detailed in the second row of Table 1, the configurations correspond to models having <math>\tilde\xi = 2.83986</math> (right intersection) and <math>\tilde\xi = 3.17241</math> (left intersection). The left-hand panel of Figure 2 shows how the Lagrangian radial coordinate varies with mass, <math>r_\xi(m_\xi)</math>, throughout the interior of these two equilibrium configurations (the locus of green and orange dots, respectively); for reference, the profile of the configuration at <math>P_\mathrm{max}</math> is presented as well (locus of black dots). The right-hand panel of Figure 2 shows the same paired configuration profiles, but ''relative'' to the profile of the configuration at <math>P_\mathrm{max}</math>.

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	The horizontal, black dot-dash line in Figure 1 has been drawn at <math>~P_e/P_\mathrm{norm} = 124.447</math>. The pair of equilibrium configurations associated with this pressure is identified graphically by the two points at which this dot-dash line intersects the detailed force-balance equilibrium sequence; as is detailed in the last row of Table 1, the configurations correspond to models having <math>~\tilde\xi = 2.83986</math> (right intersection) and <math>~\tilde\xi = 3.17241</math> (left intersection), which is the configuration that marks the onset of a dynamical instability. The left-hand panel of Figure 3 shows how the Lagrangian radial coordinate varies with mass, <math>~r_\xi(m_\xi)</math>, throughout the interior of these two ''paired'' equilibrium configurations (the locus of green and orange dots, respectively); again, for reference, the profile of the configuration at <math>~P_\mathrm{max}</math> is presented as well (locus of black dots). The right-hand panel of Figure 3 shows the same paired configuration profiles, but ''relative'' to the profile of the configuration at <math>~P_\mathrm{max}</math>.		The horizontal, black dot-dash line in Figure 1 has been drawn at <math>P_e/P_\mathrm{norm} = 124.447</math>. The pair of equilibrium configurations associated with this pressure is identified graphically by the two points at which this dot-dash line intersects the detailed force-balance equilibrium sequence; as is detailed in the last row of Table 1, the configurations correspond to models having <math>\tilde\xi = 2.83986</math> (right intersection) and <math>\tilde\xi = 3.17241</math> (left intersection), which is the configuration that marks the onset of a dynamical instability. The left-hand panel of Figure 3 shows how the Lagrangian radial coordinate varies with mass, <math>r_\xi(m_\xi)</math>, throughout the interior of these two ''paired'' equilibrium configurations (the locus of green and orange dots, respectively); again, for reference, the profile of the configuration at <math>P_\mathrm{max}</math> is presented as well (locus of black dots). The right-hand panel of Figure 3 shows the same paired configuration profiles, but ''relative'' to the profile of the configuration at <math>P_\mathrm{max}</math>.

Jet53man: /* Equilibrium Structural Properties */

2022-05-09T23:38:34Z

Equilibrium Structural Properties

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Jet53man: /* Analytic Solution */

2022-05-09T23:33:07Z

Analytic Solution

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Jet53man: /* Approximation Near the Limiting Mass */

2022-05-09T23:26:57Z

Approximation Near the Limiting Mass

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Jet53man: /* Radial Oscillations in Pressure-Truncated n = 5 Polytropes */

2022-05-09T22:55:57Z

Radial Oscillations in Pressure-Truncated n = 5 Polytropes

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Jet53man: /* Radial Oscillations in Pressure-Truncated n = 5 Polytropes */

2022-05-09T22:24:26Z

Radial Oscillations in Pressure-Truncated n = 5 Polytropes

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	--- Adapted from §6 of ~~[http://adsabs.harvard.edu/abs/1974A%26A....31..391B G. S. Bisnovatyi~~-~~Kogan & S. I. Blinnikov (1974)]~~		--- Adapted from §6 of {{ B-KB74 }}
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	In this Chapter, we focus on pressure-truncated, n = 5 polytropic configurations and demonstrate, entirely via analytical means, that the above-referenced conjecture by ~~[http://adsabs.harvard.edu/abs/1974A%26A....31..391B G. S. Bisnovatyi~~-~~Kogan & S. I. Blinnikov (1974)]~~ (hereafter, B-~~KB74~~) is correct.		In this Chapter, we focus on pressure-truncated, n = 5 polytropic configurations and demonstrate, entirely via analytical means, that the above-referenced conjecture by {{ B-KB74 }} (hereafter, {{ B-KB74hereafter }}) is correct.

	<font color="maroon">Internal Structural Profile of Individual Static Configurations:</font>  When an n = 5 polytropic configuration is embedded in a hot, tenuous external medium of pressure, <math>~P_e</math>, its equilibrium radius will vary with its mass as displayed in [[#Fig1\|Figure 1]]; the upper portion of this analytically defined equilibrium sequence is shown immediately below, on the right. The location along this sequence of a particular equilibrium (static) configuration — tagged by the subscript index, ''i'' — is uniquely identified by the value of a single model parameter:  the dimensionless truncation radius, <math>~{\tilde\xi}_i</math>. The model's normalized total mass is,		<font color="maroon">Internal Structural Profile of Individual Static Configurations:</font>  When an n = 5 polytropic configuration is embedded in a hot, tenuous external medium of pressure, <math>~P_e</math>, its equilibrium radius will vary with its mass as displayed in [[#Fig1\|Figure 1]]; the upper portion of this analytically defined equilibrium sequence is shown immediately below, on the right. The location along this sequence of a particular equilibrium (static) configuration — tagged by the subscript index, ''i'' — is uniquely identified by the value of a single model parameter:  the dimensionless truncation radius, <math>~{\tilde\xi}_i</math>. The model's normalized total mass is,

Jet53man: /* Radial Oscillations in Pressure-Truncated n = 5 Polytropes */

2021-08-07T23:41:29Z

Radial Oscillations in Pressure-Truncated n = 5 Polytropes

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	<font size="+2"><b>Summary</b></font><~~p><~~/p>Joel E. Tohline (May, 2017)		<font size="+2"><b>Summary</b></font><br />Joel E. Tohline (May, 2017)
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Jet53man at 23:12, 31 July 2021

2021-07-31T23:12:46Z

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