User:Tohline/Appendix/Ramblings/Nonlinar Oscillation - Revision history

Tohline: /* Layout */

2017-05-26T22:41:02Z

Layout

← Older revision		Revision as of 22:41, 26 May 2017
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{\tfrac{1}{2}\Delta r_\mathrm{SWS}}{<r_\mathrm{SWS}>} =		\frac{\tfrac{1}{2}\Delta r_\mathrm{SWS}}{\langle r_\mathrm{SWS} \rangle} =
	\frac{ \tfrac{1}{2} [r_2(m_\xi) - r_1(m_\xi) ]}{\tfrac{1}{2} [ r_2(m_\xi) + r_1(m_\xi)]}</math>		\frac{ \tfrac{1}{2} [r_2(m_\xi) - r_1(m_\xi) ]}{\tfrac{1}{2} [ r_2(m_\xi) + r_1(m_\xi)]}</math>
	</td>		</td>

Tohline: /* Analytic, Marginally Unstable Eigenfunction */

2017-05-26T22:39:30Z

Analytic, Marginally Unstable Eigenfunction

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	</table>		</table>
	</div>		</div>

			It is important to remember that, although the leading factor of this expression is <math>~\tfrac{2}{5}</math>, in general the overall amplitude of this eigenfunction can be set arbitrarily. In order to allow for this, we will introduce an overall scaling factor, <math>~A_0</math>, and write,

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

			<tr>
			<td align="right">
			<math>~x_P </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{2A_0}{5} \biggl[ \frac{10 -9~m_\xi^{2/3}}{4 -3~m_\xi^{2/3}} \biggr] \, .</math>
			</td>
			</tr>
			</table>
			</div>
			Then our originating expression for <math>~x_P</math> is retrieved by setting <math>~A_0 = 1</math>, in which case the amplitude of the eigenfunction is unity at the center <math>~(m_\xi = 0)</math> and it is <math>~\tfrac{2}{5}</math> at the surface <math>~(m_\xi = 1)</math>.

	====Delta Profiles====		====Delta Profiles====

Tohline: /* Identifying Equal-Mass Pairs */

2017-05-26T22:24:14Z

Identifying Equal-Mass Pairs

← Older revision		Revision as of 22:24, 26 May 2017
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	The mass-radius relationship for pressure-truncated, n = 5 polytropic configurations is displayed as a green solid curve, here on the right, in Figure 1. This sequence can be constructed either: (a) by choosing various values of the radius (between zero and the [[#extrema\|above-specified maximum radius]]) and, for each choice, determining the two corresponding values of the equilibrium mass from the pair of [[#Roots_of_Quadratic_Equation\|roots of the quadratic equation]]; or (b) by choosing various values of the mass (between zero and the [[#extrema\|above-specified maximum mass]]) and, for each choice, using the [[#Roots_of_Quartic_Equation\|two real roots of the quartic equation]] to determine the two corresponding values of the equilibrium radius. The green curve shown in Figure 1 is identical to the orange-dashed curve, labeled n = 5, that is nested among six other polytropic equilibrium sequences in [[User:Tohline/SSC/Structure/PolytropesEmbedded#Additional.2C_Numerically_Constructed_Polytropic_Configurations\|the righthand panel of Figure 3 in an accompanying discussion]].		The mass-radius relationship for pressure-truncated, n = 5 polytropic configurations is displayed as a green solid curve, here on the right, in Figure 1. This sequence can be constructed either: (a) by choosing various values of the radius (between zero and the [[#extrema\|above-specified maximum radius]]) and, for each choice, determining the two corresponding values of the equilibrium mass from the pair of [[#Roots_of_Quadratic_Equation\|roots of the quadratic equation]]; or (b) by choosing various values of the mass (between zero and the [[#extrema\|above-specified maximum mass]]) and, for each choice, using the [[#Roots_of_Quartic_Equation\|two real roots of the quartic equation]] to determine the two corresponding values of the equilibrium radius. The green curve shown in Figure 1 is identical to the orange-dashed curve, labeled n = 5, that is nested among six other polytropic equilibrium sequences in [[User:Tohline/SSC/Structure/PolytropesEmbedded#Additional.2C_Numerically_Constructed_Polytropic_Configurations\|the righthand panel of Figure 3 in an accompanying discussion]].

	Here we are interested in comparing the relative distribution of mass inside various pairs of models that have identical total masses. We therefore will focus on method "b". Specifically, given any value of the mass, <math>~M/M_\mathrm{SWS} < m_\mathrm{max}</math>, the roots of the quartic equation will give us the equilibrium radii of the two configurations that have the same, specified mass. As examples, the second column of Table 1 lists ten separate values of the normalized mass that lie within the region of parameter space that is identified by the black-dashed rectangle drawn in Figure 1, and columns ~~two~~ and ~~three~~ of Table 1 give values of the corresponding pair of equilibrium radii, <math>~\chi_\pm</math>.		Here we are interested in comparing the relative distribution of mass inside various pairs of models that have identical total masses. We therefore will focus on method "b". Specifically, given any value of the mass, <math>~M/M_\mathrm{SWS} < m_\mathrm{max}</math>, the roots of the quartic equation will give us the equilibrium radii of the two configurations that have the same, specified mass. As examples, the second column of Table 1 lists ten separate values of the normalized mass that lie within the region of parameter space that is identified by the black-dashed rectangle drawn in Figure 1; Column three lists the corresponding value of <math>~\mu</math>; and columns four and five of Table 1 give values of the corresponding pair of equilibrium radii, <math>~\chi_\pm</math>.

	In each case, from these two values of the dimensionless radius, we can, in turn, determine the corresponding pair of values of <math>~\ell_\pm</math> via the expression,		In each case, from these two values of the dimensionless radius, we can, in turn, determine the corresponding pair of values of <math>~\ell_\pm</math> via the expression,
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	</table>		</table>
	</div>		</div>
	This seems to work because, if we plug in a single value for <math>~\chi_\pm</math> — for example, the degenerate case of <math>~\chi_\pm = r_\mathrm{crit}</math> — we get the pair of values of <math>~\xi</math> along the equilibrium sequence where the equilibrium radius has this selected value. Specifically, when <math>~\chi_\pm = r_\mathrm{crit}</math>, we find that, <math>~\xi_\mathrm{high} = 3</math> and <math>~\xi_\mathrm{low} = 1</math>. Given that we are particularly interested in examining the region of parameter space that lies near the marginally unstable case — as identified by the black-dashed rectangle drawn in Figure 1 — columns ~~five~~ and ~~six~~ of Table 1 list only values of <math>~\xi_\pm</math> that correspond to the "high" roots.		This seems to work because, if we plug in a single value for <math>~\chi_\pm</math> — for example, the degenerate case of <math>~\chi_\pm = r_\mathrm{crit}</math> — we get the pair of values of <math>~\xi</math> along the equilibrium sequence where the equilibrium radius has this selected value. Specifically, when <math>~\chi_\pm = r_\mathrm{crit}</math>, we find that, <math>~\xi_\mathrm{high} = 3</math> and <math>~\xi_\mathrm{low} = 1</math>. Given that we are particularly interested in examining the region of parameter space that lies near the marginally unstable case — as identified by the black-dashed rectangle drawn in Figure 1 — columns six and seven of Table 1 list only values of <math>~\xi_\pm</math> that correspond to the "high" roots.

	<span id="Table1"> </span>		<span id="Table1"> </span>

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

2017-05-26T21:56:12Z

Radial Oscillations in Pressure-Truncated n = 5 Polytropes

← Older revision		Revision as of 21:56, 26 May 2017
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	<font size="+2"><b>Summary</b></font>		<font size="+2"><b>Summary</b></font><p></p>Joel E. Tohline (May, 2017)
	</td></tr>		</td></tr>
	<tr><td align="center">		<tr><td align="center">
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	In this expression the fractional mass, <math>~0 \le m_\xi \le 1</math>, has been employed as the Lagrangian radial coordinate. Note that, ''at'' the location of the (maximum-mass) extremum, <math>~{\tilde\xi}_i = 3</math> and, hence, <math>~{\tilde{C}}_i = 4</math> and <math>~m_\mathrm{tot} = m_\mathrm{max} \equiv [3^4\cdot 5^3/(2^{10}\pi)]^{1 / 2}</math>.		In this expression the fractional mass, <math>~0 \le m_\xi \le 1</math>, has been employed as the Lagrangian radial coordinate. Note that, ''at'' the location of the (maximum-mass) extremum, <math>~{\tilde\xi}_i = 3</math> and, hence, <math>~{\tilde{C}}_i = 4</math> and <math>~m_\mathrm{tot} = m_\mathrm{max} \equiv [3^4\cdot 5^3/(2^{10}\pi)]^{1 / 2}</math>.

	[[File:A16Annotated.png\|right\|~~250px~~\|center\|Pressure-Truncated n = 5 Equilibrium Sequence]]<font color="maroon">Static Configurations of Equal Mass Near an Extremum:</font>   In the figure displayed here, on the right, two models with the same total mass and lying near the sequence extremum have been identified. We use the (always real and positive) parameter,		[[File:A16Annotated.png\|right\|225px\|center\|Pressure-Truncated n = 5 Equilibrium Sequence]]<font color="maroon">Static Configurations of Equal Mass Near an Extremum:</font>   In the figure displayed here, on the right, two models with the same total mass and lying near the sequence extremum have been identified. We use the (always real and positive) parameter,
	<div align="center">		<div align="center">
	<math>~\mu \equiv \biggl[ 1 - \biggl( \frac{m_\mathrm{tot}}{m_\mathrm{max}} \biggr)^2 \biggr]^{1 / 2} \, ,</math>		<math>~\mu \equiv \biggl[ 1 - \biggl( \frac{m_\mathrm{tot}}{m_\mathrm{max}} \biggr)^2 \biggr]^{1 / 2} \, ,</math>
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	where, <math>~A_0</math> is an arbitrary, overall scaling coefficient. This is what [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74] refer to as the "eigenfunction of the neutral mode."		where, <math>~A_0</math> is an arbitrary, overall scaling coefficient. This is what [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74] refer to as the "eigenfunction of the neutral mode."

	[[File:Displacement5.gif\|~~right~~\|~~250px~~\|center\|Animated Comparison of Displacement Functions]]<font color="maroon">Comparison:</font>  By plotting one function on top of the other, in the righthand panel of [[#Fig2\|Figure 2, below]] — see, also, the animation displayed here on the ~~right~~ — we illustrate that, as conjectured by [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74], the nonlinear fractional displacement function, <math>~x_\mathrm{BKB74} </math>, does very accurately represent the exact eigenfunction of the neutral mode for small values of the parameter, <math>~\mu</math>. The validity of this conjecture can be demonstrated even more quantitatively by expanding the function, <math>~x_\mathrm{BKB74} </math>, as a power series in <math>~\mu</math>. To lowest order, we find that,		[[File:Displacement5.gif\|left\|225px\|center\|Animated Comparison of Displacement Functions]]<font color="maroon">Comparison:</font>  By plotting one function on top of the other, in the righthand panel of [[#Fig2\|Figure 2, below]] — see, also, the animation displayed here on the left — we illustrate that, as conjectured by [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74], the nonlinear fractional displacement function, <math>~x_\mathrm{BKB74} </math>, does very accurately represent the exact eigenfunction of the neutral mode for small values of the parameter, <math>~\mu</math>. The validity of this conjecture can be demonstrated even more quantitatively by expanding the function, <math>~x_\mathrm{BKB74} </math>, as a power series in <math>~\mu</math>. To lowest order, we find that,

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

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

2017-05-26T20:36:47Z

Radial Oscillations in Pressure-Truncated n = 5 Polytropes

← Older revision		Revision as of 20:36, 26 May 2017
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	where, <math>~A_0</math> is an arbitrary, overall scaling coefficient. This is what [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74] refer to as the "eigenfunction of the neutral mode."		where, <math>~A_0</math> is an arbitrary, overall scaling coefficient. This is what [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74] refer to as the "eigenfunction of the neutral mode."

	<font color="maroon">Comparison:</font>  By plotting one function on top of the other, in the righthand panel of [[#Fig2\|Figure 2, below]], we illustrate that, as conjectured by [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74], the nonlinear fractional displacement function, <math>~x_\mathrm{BKB74} </math>, does very accurately represent the exact eigenfunction of the neutral mode for small values of the parameter, <math>~\mu</math>. The validity of this conjecture can be demonstrated even more quantitatively by expanding the function, <math>~x_\mathrm{BKB74} </math>, as a power series in <math>~\mu</math>. To lowest order, we find that,		[[File:Displacement5.gif\|right\|250px\|center\|Animated Comparison of Displacement Functions]]<font color="maroon">Comparison:</font>  By plotting one function on top of the other, in the righthand panel of [[#Fig2\|Figure 2, below]] — see, also, the animation displayed here on the right — we illustrate that, as conjectured by [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74], the nonlinear fractional displacement function, <math>~x_\mathrm{BKB74} </math>, does very accurately represent the exact eigenfunction of the neutral mode for small values of the parameter, <math>~\mu</math>. The validity of this conjecture can be demonstrated even more quantitatively by expanding the function, <math>~x_\mathrm{BKB74} </math>, as a power series in <math>~\mu</math>. To lowest order, we find that,

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


			{{LSU_HBook_header}}


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	* [http://adsabs.harvard.edu/abs/1974A%26A....31..391B Bisnovatyi-Kogan & Blinnikov (1974)] — see especially their §6 — as we have briefly reviewed in an [[User:Tohline/H_Book#BKB74pt2\|introductory paragraph]]		* [http://adsabs.harvard.edu/abs/1974A%26A....31..391B Bisnovatyi-Kogan & Blinnikov (1974)] — see especially their §6 — as we have briefly reviewed in an [[User:Tohline/H_Book#BKB74pt2\|introductory paragraph]]
	* [http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983)], as we have briefly reviewed in a [[User:Tohline/SSC/Stability/InstabilityOnsetOverview#Stahler832\|separate chapter]]		* [http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983)], as we have briefly reviewed in a [[User:Tohline/SSC/Stability/InstabilityOnsetOverview#Stahler832\|separate chapter]]


	~~{{LSU_HBook_header}}~~

	==Revised Attack==		==Revised Attack==

Tohline: /* Abstract */

2017-05-26T20:14:10Z

Abstract

← Older revision		Revision as of 20:14, 26 May 2017
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	=Radial Oscillations in Pressure-Truncated n = 5 Polytropes=		=Radial Oscillations in Pressure-Truncated n = 5 Polytropes=
	~~==Abstract==~~

	<table border="1" width="90%" align="center" cellpadding="15">		<table border="1" width="90%" align="center" cellpadding="15">
	<tr><td align="center">		<tr><td align="center">
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	where, <math>~A_0</math> is an arbitrary, overall scaling coefficient. This is what [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74] refer to as the "eigenfunction of the neutral mode."		where, <math>~A_0</math> is an arbitrary, overall scaling coefficient. This is what [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74] refer to as the "eigenfunction of the neutral mode."

	<font color="maroon">Comparison:</font>  By plotting one function on top of the other, in the righthand panel of [[#Fig2\|Figure 2, below]], we illustrate that, as conjectured by [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74], the nonlinear fractional displacement function, <math>~x_\mathrm{~~B-KB74~~} </math>, does very accurately represent the exact eigenfunction of the neutral mode for small values of the parameter, <math>~\mu</math>. The ~~virility~~ of this conjecture can be demonstrated even more quantitatively by expanding the function, <math>~x_\mathrm{BKB74} </math>, as a power series in <math>~\mu</math>. To lowest order, we find that,		<font color="maroon">Comparison:</font>  By plotting one function on top of the other, in the righthand panel of [[#Fig2\|Figure 2, below]], we illustrate that, as conjectured by [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74], the nonlinear fractional displacement function, <math>~x_\mathrm{BKB74} </math>, does very accurately represent the exact eigenfunction of the neutral mode for small values of the parameter, <math>~\mu</math>. The validity of this conjecture can be demonstrated even more quantitatively by expanding the function, <math>~x_\mathrm{BKB74} </math>, as a power series in <math>~\mu</math>. To lowest order, we find that,

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

Tohline: /* Abstract */

2017-05-26T20:09:40Z

Abstract

← Older revision		Revision as of 20:09, 26 May 2017
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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 [http://adsabs.harvard.edu/abs/1974A%26A....31..391B G. S. Bisnovatyi-Kogan & S. I. Blinnikov (1974)]
	<p></p>		<p></p>
	<p></p>		<p></p>
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	It is clear, therefore, that the (nonlinear, static-model) eigenfunction referenced in the above [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74] quote is definable analytically for pressure-truncated, n = 5 polytropic configurations. In what follows we have found it useful to reference, as well, the corresponding nonlinear ''fractional'' displacement function,		It is clear, therefore, that the (nonlinear, static-model) eigenfunction referenced in the above [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74] quote is definable analytically for pressure-truncated, n = 5 polytropic configurations. In what follows we have found it useful to reference, as well, the corresponding nonlinear ''fractional'' displacement function,
	<div align="center">		<div align="center">
	<math>~x_\mathrm{~~B-KB74~~} = \frac{\Delta r}{r} \equiv \frac{\mathfrak{x}}{2\langle r \rangle} = \frac{r_2(m_\xi) - r_1(m_\xi)}{r_2(m_\xi) + r_1(m_\xi)} \, .</math>		<math>~x_\mathrm{BKB74} = \frac{\Delta r}{r} \equiv \frac{\mathfrak{x}}{2\langle r \rangle} = \frac{r_2(m_\xi) - r_1(m_\xi)}{r_2(m_\xi) + r_1(m_\xi)} \, .</math>
	</div>		</div>

	<font color="maroon">Eigenfunction of the Neutral Mode:</font>  In a [[User:Tohline/SSC/Stability/InstabilityOnsetOverview#Analyses_of_Radial_Oscillations\|separate chapter]], we have proven that the fundamental mode of radial oscillation associated with the maximum-mass model along the equilibrium sequence has an eigenfrequency that is precisely zero — that is, the model at the extremum of the sequence is marginally unstable [dynamically] — and it has a radial eigenfunction that is ~~precisely~~ defined by the expression,		<font color="maroon">Eigenfunction of the Neutral Mode:</font>  In a [[User:Tohline/SSC/Stability/InstabilityOnsetOverview#Analyses_of_Radial_Oscillations\|separate chapter]], we have proven that the fundamental mode of radial oscillation associated with the maximum-mass model along the equilibrium sequence has an eigenfrequency that is precisely zero — that is, the model at the extremum of the sequence is marginally unstable [dynamically] — and it has a radial eigenfunction that is defined by 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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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{2}{5} \biggl[ \frac{10 -9~m_\xi^{2/3}}{4 -3~m_\xi^{2/3}} \biggr] \, .</math>		<math>~\frac{2A_0}{5} \biggl[ \frac{10 -9~m_\xi^{2/3}}{4 -3~m_\xi^{2/3}} \biggr] \, ,</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	</div>		</div>
	This is what [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74] refer to as the "eigenfunction of the neutral mode."		where, <math>~A_0</math> is an arbitrary, overall scaling coefficient. This is what [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74] refer to as the "eigenfunction of the neutral mode."

	<font color="maroon">Comparison:</font>  By plotting one function on top of the other, we illustrate that, as conjectured by [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74], the nonlinear fractional displacement function, <math>~x_\mathrm{B-KB74} </math>, does very accurately represent the exact eigenfunction of the neutral mode~~, <math>~x_P </math>,~~ for small values of the parameter, <math>~\mu</math>. The virility of this conjecture can be demonstrated quantitatively by expanding the function, <math>~x_\mathrm{~~B-KB74~~} </math>, as a power series in <math>~\mu</math>. To lowest order~~(s)~~ we find that,		<font color="maroon">Comparison:</font>  By plotting one function on top of the other, in the righthand panel of [[#Fig2\|Figure 2, below]], we illustrate that, as conjectured by [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74], the nonlinear fractional displacement function, <math>~x_\mathrm{B-KB74} </math>, does very accurately represent the exact eigenfunction of the neutral mode for small values of the parameter, <math>~\mu</math>. The virility of this conjecture can be demonstrated even more quantitatively by expanding the function, <math>~x_\mathrm{BKB74} </math>, as a power series in <math>~\mu</math>. To lowest order, we find that,

	<div align="center">		<div align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~x_\mathrm{~~B-KB74~~} </math>		<math>~x_\mathrm{BKB74} </math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~~~\approx~~</math>		<math>~=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>~
	~~\biggl\{~~ \biggl( \frac{1}{2 \cdot 3} \biggr)^{1 / 2}\mu		\biggl( \frac{1}{2 \cdot 3} \biggr)^{1 / 2}\mu \biggl[ \frac{10 -9~m_\xi^{2/3}}{4 -3~m_\xi^{2/3}} \biggr] + \mathcal{O}(\mu^2)
	+ \biggl[ \frac{5\~~cdot 23}~~{(2 ~~\cdot~~ 3~~)^{7 / 2~~}} ~~\biggr] \mu^3~~		\, .
	~~\biggr\}~~
	~~\biggl\~~{
	(3-2\~~beta~~^2~~) + \biggl[ \biggl(\frac{5~~}~~{18~~}\biggr~~) \mu^2~~
	+ \~~biggl( \frac~~{~~293~~}~~{2^3 \cdot 3^5} \biggr)~~ \mu^4
	~~\biggr] (8\beta^4 - 15~~ )
	~~\biggr\}~~ \, .
	</math>		</math>
	</td>		</td>
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	</table>		</table>
	</div>		</div>
			Especially after setting <math>~A_0 = [5^2/(2^3\cdot 3)]^{1 / 2}\mu</math>, it is clear that <math>~x_\mathrm{BKB74} </math> more and more accurately matches <math>~x_P</math>, as <math>~\mu \rightarrow 0</math>.
	</td></tr>		</td></tr>
	</table>		</table>

Tohline: /* Abstract */

2017-05-26T04:59:07Z

Abstract

← Older revision		Revision as of 04:59, 26 May 2017
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	<math>~\mathfrak{x} = r_2(m_\xi) - r_1(m_\xi) \, .</math>		<math>~\mathfrak{x} = r_2(m_\xi) - r_1(m_\xi) \, .</math>
	</div>		</div>
	Is is clear, therefore, that the (nonlinear, static-model) eigenfunction referenced in the above [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74] quote is definable analytically for pressure-truncated, n = 5 polytropic configurations. In what follows we have found it useful to reference, as well, the corresponding nonlinear ''fractional'' displacement function,		It is clear, therefore, that the (nonlinear, static-model) eigenfunction referenced in the above [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74] quote is definable analytically for pressure-truncated, n = 5 polytropic configurations. In what follows we have found it useful to reference, as well, the corresponding nonlinear ''fractional'' displacement function,
	<div align="center">		<div align="center">
	<math>~x_\mathrm{~~nonlinear~~} = \frac{\Delta r}{r} \equiv \frac{\mathfrak{x}}{2\langle r \rangle} = \frac{r_2(m_\xi) - r_1(m_\xi)}{r_2(m_\xi) + r_1(m_\xi)} \, .</math>		<math>~x_\mathrm{B-KB74} = \frac{\Delta r}{r} \equiv \frac{\mathfrak{x}}{2\langle r \rangle} = \frac{r_2(m_\xi) - r_1(m_\xi)}{r_2(m_\xi) + r_1(m_\xi)} \, .</math>
	</div>		</div>
	~~Expanding this function as a power series in terms of the parameter, <math>~\mu</math> — which will be small for model pairs near the extremum — we find that, to lowest orders,~~

			<font color="maroon">Eigenfunction of the Neutral Mode:</font>  In a [[User:Tohline/SSC/Stability/InstabilityOnsetOverview#Analyses_of_Radial_Oscillations\|separate chapter]], we have proven that the fundamental mode of radial oscillation associated with the maximum-mass model along the equilibrium sequence has an eigenfrequency that is precisely zero — that is, the model at the extremum of the sequence is marginally unstable [dynamically] — and it has a radial eigenfunction that is precisely defined by 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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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~x_\~~mathrm~~{~~nonlinear~~} </math>		<math>~x_P \equiv \frac{\delta r}{r}</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~~~\approx~~</math>		<math>~=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>~\frac{2}{5} \biggl[ \frac{10 -9~m_\xi^{2/3}}{4 -3~m_\xi^{2/3}} \biggr] \, .</math>
	~~\biggl\{ \biggl(~~ \frac~~{1}~~{2 ~~\cdot 3~~} ~~\biggr)^~~{~~1 / 2~~}~~\mu~~
	+ \biggl[ \frac{5\~~cdot 23}~~{(2 ~~\cdot~~ 3~~)^{7 / 2~~}} ~~\biggr] \mu^3~~
	~~\biggr\}~~
	~~\biggl\~~{
	(3-2\~~beta~~^~~2) + \biggl[ \biggl(\frac{5}~~{~~18}\biggr) \mu^~~2
	~~+ \biggl( \frac{293~~}~~{2^3 \cdot 3^5~~} ~~\biggr) \mu^4~~
	\biggr] ~~(8\beta^4 - 15 )~~
	~~\biggr\}~~ \, .
	</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	</div>		</div>
			This is what [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74] refer to as the "eigenfunction of the neutral mode."

			<font color="maroon">Comparison:</font>  By plotting one function on top of the other, we illustrate that, as conjectured by [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74], the nonlinear fractional displacement function, <math>~x_\mathrm{B-KB74} </math>, does very accurately represent the exact eigenfunction of the neutral mode, <math>~x_P </math>, for small values of the parameter, <math>~\mu</math>. The virility of this conjecture can be demonstrated quantitatively by expanding the function, <math>~x_\mathrm{B-KB74} </math>, as a power series in <math>~\mu</math>. To lowest order(s) we find that,

	<font color="maroon">Eigenfunction of the Neutral Mode:</font>  In a [[User:Tohline/SSC/Stability/InstabilityOnsetOverview#Analyses_of_Radial_Oscillations\|separate chapter]], we have proven that the fundamental mode of radial oscillation associated with the maximum-mass model along the equilibrium sequence has an eigenfrequency that is precisely zero — that is, the model at the extremum of the sequence is [dynamically] marginally unstable — and it has a radial eigenfunction that is precisely defined by 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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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~~~x_P~~ \~~equiv \frac{\delta r}~~{r}</math>		<math>~x_\mathrm{B-KB74} </math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>~\approx</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{2}{5} \biggl[ \frac{~~10 -9~m_~~\xi^{2/3}}{~~4 -3~m_~~\xi^{2/3}} \biggr] \, .</math>		<math>~
			\biggl\{ \biggl( \frac{1}{2 \cdot 3} \biggr)^{1 / 2}\mu
			+ \biggl[ \frac{5\cdot 23}{(2 \cdot 3)^{7 / 2}} \biggr] \mu^3
			\biggr\}
			\biggl\{
			(3-2\beta^2) + \biggl[ \biggl(\frac{5}{18}\biggr) \mu^2
			+ \biggl( \frac{293}{2^3 \cdot 3^5} \biggr) \mu^4
			\biggr] (8\beta^4 - 15 )
			\biggr\} \, .
			</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	</div>		</div>
	~~This is what [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74] refer to as the "eigenfunction of the neutral mode."~~
	</td></tr>		</td></tr>
	</table>		</table>

Tohline: /* Abstract */

2017-05-25T21:24:13Z

Abstract

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	<div align="center">		<div align="center">
	<math>~x_\mathrm{nonlinear} = \frac{\Delta r}{r} \equiv \frac{\mathfrak{x}}{2\langle r \rangle} = \frac{r_2(m_\xi) - r_1(m_\xi)}{r_2(m_\xi) + r_1(m_\xi)} \, .</math>		<math>~x_\mathrm{nonlinear} = \frac{\Delta r}{r} \equiv \frac{\mathfrak{x}}{2\langle r \rangle} = \frac{r_2(m_\xi) - r_1(m_\xi)}{r_2(m_\xi) + r_1(m_\xi)} \, .</math>
			</div>
			Expanding this function as a power series in terms of the parameter, <math>~\mu</math> — which will be small for model pairs near the extremum — we find that, to lowest orders,

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

			<tr>
			<td align="right">
			<math>~x_\mathrm{nonlinear} </math>
			</td>
			<td align="center">
			<math>~\approx</math>
			</td>
			<td align="left">
			<math>~
			\biggl\{ \biggl( \frac{1}{2 \cdot 3} \biggr)^{1 / 2}\mu
			+ \biggl[ \frac{5\cdot 23}{(2 \cdot 3)^{7 / 2}} \biggr] \mu^3
			\biggr\}
			\biggl\{
			(3-2\beta^2) + \biggl[ \biggl(\frac{5}{18}\biggr) \mu^2
			+ \biggl( \frac{293}{2^3 \cdot 3^5} \biggr) \mu^4
			\biggr] (8\beta^4 - 15 )
			\biggr\} \, .
			</math>
			</td>
			</tr>
			</table>
	</div>		</div>

Tohline: /* Abstract */

2017-05-25T20:17:16Z

Abstract

← Older revision		Revision as of 20:17, 25 May 2017
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	<font size="+2"><b>~~Abstract~~</b></font>		<font size="+2"><b>Summary</b></font>
	</td></tr>		</td></tr>
	<tr><td align="center">		<tr><td align="center">
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	</table>		</table>
	</div>		</div>
	In this expression the fractional mass, <math>~0 \le m_\xi \le 1</math>, has been employed as the Lagrangian radial coordinate. Note that, ''at'' the location of the (maximum-mass) extremum, <math>~{\tilde\xi}_i = 3</math> and, hence, <math>~{\tilde{C}}_i = 4</math> and <math>~m_\mathrm{tot} = [3^4\cdot 5^3/(2^{10}\pi)]^{1 / 2}</math>.		In this expression the fractional mass, <math>~0 \le m_\xi \le 1</math>, has been employed as the Lagrangian radial coordinate. Note that, ''at'' the location of the (maximum-mass) extremum, <math>~{\tilde\xi}_i = 3</math> and, hence, <math>~{\tilde{C}}_i = 4</math> and <math>~m_\mathrm{tot} = m_\mathrm{max} \equiv [3^4\cdot 5^3/(2^{10}\pi)]^{1 / 2}</math>.

	[[File:A16Annotated.png\|right\|250px\|center\|Pressure-Truncated n = 5 Equilibrium Sequence]]<font color="maroon">Static Configurations of Equal Mass Near an Extremum:</font>   In the figure displayed here, on the right, two models with the same total mass and lying near the sequence extremum have been identified. Model 1 (Model 2) necessarily has a dimensionless truncation radius that is greater (less) than 3; as we demonstrate below, the precise values of the <math>~({\tilde\xi}_1, {\tilde\xi}_2)</math> parameter pair that are associated with a given choice of ~~the configuration mass~~ can be determined analytically as roots of the quartic equation that, itself, defines the mass-radius equilibrium sequence.		[[File:A16Annotated.png\|right\|250px\|center\|Pressure-Truncated n = 5 Equilibrium Sequence]]<font color="maroon">Static Configurations of Equal Mass Near an Extremum:</font>   In the figure displayed here, on the right, two models with the same total mass and lying near the sequence extremum have been identified. We use the (always real and positive) parameter,
			<div align="center">
			<math>~\mu \equiv \biggl[ 1 - \biggl( \frac{m_\mathrm{tot}}{m_\mathrm{max}} \biggr)^2 \biggr]^{1 / 2} \, ,</math>
			</div>
			to quantify how close the mass of a given pair of models is to the limiting mass. As depicted here, Model 1 (Model 2) necessarily has a dimensionless truncation radius that is greater (less) than 3; as we demonstrate below, the precise values of the <math>~({\tilde\xi}_1, {\tilde\xi}_2)</math> parameter pair that are associated with a given choice of <math>~\mu</math> can be determined analytically as roots of the quartic equation that, itself, defines the mass-radius equilibrium sequence.

	<font color="maroon">The Nonlinear Displacement Function, <math>~\mathfrak{x}</math>:</font>  The (nonlinear) displacement that will map the radial location of every mass shell within Model 1 into the corresponding radial location of every mass shell within Model 2 is given precisely by the expression,		<font color="maroon">The Nonlinear Displacement Function, <math>~\mathfrak{x}</math>:</font>  The (nonlinear) displacement that will map the radial location of every mass shell within Model 1 into the corresponding radial location of every mass shell within Model 2 is given precisely by the expression,