User:Tohline/SSC/Stability/InstabilityOnsetOverview - Revision history

Tohline: /* Supporting Scratch Work */

2019-01-27T20:18:42Z

Supporting Scratch Work

← Older revision		Revision as of 20:18, 27 January 2019
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	[<font color="red">'''Note added on 1/4/2019:'''</font> The numerator of the first term on the RHS of this last expression used to be "n-3". This ~~type-setting~~ error has been fixed.]		[<font color="red">'''Note added on 1/4/2019:'''</font> The numerator of the first term on the RHS of this last expression used to be "n-3". This manuscript error has been fixed.]
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Tohline: /* Supporting Scratch Work */

2019-01-27T20:18:13Z

Supporting Scratch Work

← Older revision		Revision as of 20:18, 27 January 2019
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	[<font color="red">'''Note added on 1/4/2019:'''</font> The numerator of the first term on the RHS of this last expression used to be "n-3". ~~We have~~ fixed ~~this error~~.]		[<font color="red">'''Note added on 1/4/2019:'''</font> The numerator of the first term on the RHS of this last expression used to be "n-3". This type-setting error has been fixed.]
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Tohline: /* Supporting Scratch Work */

2019-01-04T23:36:06Z

Supporting Scratch Work

← Older revision		Revision as of 23:36, 4 January 2019
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	<math>~ \biggl(\frac{n-3}{n+1}\biggr) + \frac{n\sigma_c^2}{6(n+1)} \biggl[\frac{\xi}{\theta^'}\biggr]_{\xi_1} \, .</math>		<math>~ \biggl(\frac{3-n}{n+1}\biggr) + \frac{n\sigma_c^2}{6(n+1)} \biggl[\frac{\xi}{\theta^'}\biggr]_{\xi_1} \, .</math>
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			[<font color="red">'''Note added on 1/4/2019:'''</font> The numerator of the first term on the RHS of this last expression used to be "n-3". We have fixed this error.]
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	We have already demonstrated that this is the boundary condition used by [[User:Tohline/SSC/Stability/n3PolytropeLAWE#Schwarzschild_.281941.29\|Schwarzschild (1941)]] in his examination of radial oscillations in n = 3 polytropes. Presumably this also matches up with the boundary condition employed by [[User:Tohline/SSC/Stability/Polytropes#Boundary_Conditions\|HRW66]]. Notice that when n = 0, this final expression reduces to <math>~d\ln x/d\ln\xi = -3</math>, which is the surface boundary condition that we have been using to determine the eigenvectors of all pressure-truncated configurations.		We have already demonstrated that this is the boundary condition used by [[User:Tohline/SSC/Stability/n3PolytropeLAWE#Schwarzschild_.281941.29\|Schwarzschild (1941)]] in his examination of radial oscillations in n = 3 polytropes. Presumably this also matches up with the boundary condition employed by [[User:Tohline/SSC/Stability/Polytropes#Boundary_Conditions\|HRW66]]. Notice that when n = 0, this final expression reduces to <math>~d\ln x/d\ln\xi = -3</math>, which is the surface boundary condition that we have been using to determine the eigenvectors of all pressure-truncated configurations.

Tohline: /* Marginally Unstable Pressure-Truncated Gas Clouds */

2017-05-27T22:52:44Z

Marginally Unstable Pressure-Truncated Gas Clouds

← Older revision		Revision as of 22:52, 27 May 2017
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			{{LSU_HBook_header}}

	Additional details may be found [[User:Tohline/SSC/Stability/Isothermal#Radial_Oscillations_of_Pressure-Truncated_Isothermal_Spheres\|here]].		Additional details may be found [[User:Tohline/SSC/Stability/Isothermal#Radial_Oscillations_of_Pressure-Truncated_Isothermal_Spheres\|here]].

	~~{{LSU_HBook_header}}~~

	==Isothermal==		==Isothermal==

Tohline: /* Marginally Unstable Pressure-Truncated Gas Clouds */

2017-05-26T21:40:27Z

Marginally Unstable Pressure-Truncated Gas Clouds

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	<font size="+2"><b>Abstract</b></font>		<font size="+2"><b>Abstract</b></font><p></p>Joel E. Tohline (April, 2017)
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Tohline: /* Marginally Unstable Pressure-Truncated Gas Clouds */

2017-04-28T04:15:47Z

Marginally Unstable Pressure-Truncated Gas Clouds

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	In a pair of papers that, to date, have not been widely cited, Yabushita ([http://adsabs.harvard.edu/abs/1974MNRAS.167...95Y 1974], [http://adsabs.harvard.edu/abs/1975MNRAS.172..441Y 1975]) proved that this is also precisely the configuration along the sequence of pressure-truncated isothermal spheres that marks the onset of dynamical instability. More specifically, drawing on traditional techniques for identifying the properties of radial pulsation modes in self-gravitating, isothermal fluids, Yabushita showed that an eigenvector prescribed analytically by the following eigenfrequency-eigenfunction pair:		In a pair of papers that, to date, have not been widely cited, Yabushita ([http://adsabs.harvard.edu/abs/1974MNRAS.167...95Y 1974], [http://adsabs.harvard.edu/abs/1975MNRAS.172..441Y 1975]) proved that this is also precisely the configuration along the sequence of pressure-truncated isothermal spheres that marks the onset of dynamical instability. More specifically, drawing on traditional techniques for identifying the properties of radial pulsation modes in self-gravitating, isothermal fluids, Yabushita showed that an eigenvector prescribed analytically by the following eigenfrequency-eigenfunction pair,
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	provides an exact definition of the fundamental mode of radial oscillation in the configuration that is associated with the Bonnor-Ebert limiting mass. Note that this expression for the eigenfunction has been given in terms of the dimensionless radial displacement, <math>~x \equiv \delta r/r_0</math>.		provides an exact definition of the fundamental mode of radial oscillation in the configuration that is associated with the Bonnor-Ebert limiting mass. Note that this expression for the eigenfunction has been given in terms of the dimensionless radial displacement, <math>~x \equiv \delta r/r_0</math>.

	[[File:MassVsRadiusCombined02.png\|left\|200px\|Equilibrium sequences of Pressure-Truncated Polytropes]]Building upon Yabushita's investigation, we have analyzed the properties of radial oscillation modes in pressure-truncated ''polytropic'' configurations having indexes, <math>~3 \le n < \infty</math>. As is illustrated here, on the left, for any polytropic index within this stated range, at least one extremum in the mass can be identified along the corresponding mass-radius equilibrium sequence. Separately, [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] and [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura (1981)] have shown that, along each such sequence, the configuration associated with the limiting mass (or, as the ~~case may be,~~ limiting pressure) occurs precisely where,		[[File:MassVsRadiusCombined02.png\|left\|200px\|Equilibrium sequences of Pressure-Truncated Polytropes]]Building upon Yabushita's investigation, we have analyzed the properties of radial oscillation modes in pressure-truncated ''polytropic'' configurations having indexes, <math>~3 \le n < \infty</math>. As is illustrated here, on the left, for any polytropic index within this stated range, at least one extremum in the mass can be identified along the corresponding mass-radius equilibrium sequence. Separately, [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] and [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura (1981)] have shown that, along each such sequence, the configuration associated with the limiting mass (correspondingly, the limiting pressure in a P-V diagram) occurs precisely where,
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	We have discovered that, along each ~~polytropic~~ sequence, the transition from stable to dynamically unstable configurations occurs precisely at the location of this limiting mass configuration. ~~More specifically~~, we have ~~shown~~ that, in each such transition model, the eigenvector associated with the fundamental mode of radial oscillation is prescribed analytically by the following eigenfrequency-eigenfunction pair:		We have discovered that, along each sequence, the transition from stable to dynamically unstable configurations occurs precisely at the location of this limiting mass configuration, if the adiabatic index governing the behavior of radial oscillations is tied to the polytropic index via the relation, <math>~\gamma = (n+1)/n</math>. Specifically, we have found that, in each such transition model, the eigenvector associated with the fundamental mode of radial oscillation is prescribed analytically by the following eigenfrequency-eigenfunction pair:
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Marginally Unstable Pressure-Truncated Gas Clouds */

2017-04-27T23:16:20Z

Marginally Unstable Pressure-Truncated Gas Clouds

← Older revision		Revision as of 23:16, 27 April 2017
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	provides an exact definition of the fundamental mode of radial oscillation in the configuration that is associated with the Bonnor-Ebert limiting mass. Note that this expression for the eigenfunction has been given in terms of the dimensionless radial displacement, <math>~x \equiv \delta r/r_0</math>.		provides an exact definition of the fundamental mode of radial oscillation in the configuration that is associated with the Bonnor-Ebert limiting mass. Note that this expression for the eigenfunction has been given in terms of the dimensionless radial displacement, <math>~x \equiv \delta r/r_0</math>.

	[[File:MassVsRadiusCombined02.png\|left\|~~250px~~\|Equilibrium sequences of Pressure-Truncated Polytropes]]Building upon Yabushita's investigation, we have analyzed the properties of radial oscillation modes in pressure-truncated ''polytropic'' configurations having indexes, <math>~3 \le n < \infty</math>. As is illustrated here, on the left, for any polytropic index within this stated range, at least one extremum in the mass can be identified along the corresponding mass-radius equilibrium sequence. Separately, [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] and [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura (1981)] have shown that, along each such sequence, the configuration associated with the ~~"maximum~~ mass~~" turning point~~ occurs precisely where,		[[File:MassVsRadiusCombined02.png\|left\|200px\|Equilibrium sequences of Pressure-Truncated Polytropes]]Building upon Yabushita's investigation, we have analyzed the properties of radial oscillation modes in pressure-truncated ''polytropic'' configurations having indexes, <math>~3 \le n < \infty</math>. As is illustrated here, on the left, for any polytropic index within this stated range, at least one extremum in the mass can be identified along the corresponding mass-radius equilibrium sequence. Separately, [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] and [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura (1981)] have shown that, along each such sequence, the configuration associated with the limiting mass (or, as the case may be, limiting pressure) occurs precisely where,
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	We have discovered that, along each polytropic sequence, the transition from stable to dynamically unstable configurations occurs precisely at the location of this ~~"maximum~~ mass" configuration. ~~Specifically~~, we ~~show~~ that, in each such transition model, the eigenvector associated with the fundamental mode of radial oscillation is prescribed analytically by the following eigenfrequency-eigenfunction pair:		We have discovered that, along each polytropic sequence, the transition from stable to dynamically unstable configurations occurs precisely at the location of this limiting mass configuration. More specifically, we have shown that, in each such transition model, the eigenvector associated with the fundamental mode of radial oscillation is prescribed analytically by the following eigenfrequency-eigenfunction pair:
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Marginally Unstable Pressure-Truncated Gas Clouds */

2017-04-27T23:09:05Z

Marginally Unstable Pressure-Truncated Gas Clouds

← Older revision		Revision as of 23:09, 27 April 2017
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	provides an exact definition of the fundamental mode of radial oscillation in the configuration that is associated with the Bonnor-Ebert limiting mass. Note that this expression for the eigenfunction has been given in terms of the dimensionless radial displacement, <math>~x \equiv \delta r/r_0</math>.		provides an exact definition of the fundamental mode of radial oscillation in the configuration that is associated with the Bonnor-Ebert limiting mass. Note that this expression for the eigenfunction has been given in terms of the dimensionless radial displacement, <math>~x \equiv \delta r/r_0</math>.

	[[File:MassVsRadiusCombined02.png\|left\|~~200px~~\|Equilibrium sequences of Pressure-Truncated Polytropes]]Building upon Yabushita's investigation, we have analyzed the properties of radial oscillation modes in pressure-truncated ''polytropic'' configurations having indexes, <math>~3 \le n < \infty</math>. As is illustrated here, on the left, for any polytropic index within this stated range, at least one extremum in the mass can be identified along the corresponding mass-radius equilibrium sequence. Separately, [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] and [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura (1981)] have shown that, along each such sequence, the configuration associated with the "maximum mass" turning point occurs precisely where,		[[File:MassVsRadiusCombined02.png\|left\|250px\|Equilibrium sequences of Pressure-Truncated Polytropes]]Building upon Yabushita's investigation, we have analyzed the properties of radial oscillation modes in pressure-truncated ''polytropic'' configurations having indexes, <math>~3 \le n < \infty</math>. As is illustrated here, on the left, for any polytropic index within this stated range, at least one extremum in the mass can be identified along the corresponding mass-radius equilibrium sequence. Separately, [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] and [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura (1981)] have shown that, along each such sequence, the configuration associated with the "maximum mass" turning point occurs precisely where,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Marginally Unstable Pressure-Truncated Gas Clouds */

2017-04-27T23:07:45Z

Marginally Unstable Pressure-Truncated Gas Clouds

← Older revision		Revision as of 23:07, 27 April 2017
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			We have discovered that, along each polytropic sequence, the transition from stable to dynamically unstable configurations occurs precisely at the location of this "maximum mass" configuration. Specifically, we show that, in each such transition model, the eigenvector associated with the fundamental mode of radial oscillation is prescribed analytically by the following eigenfrequency-eigenfunction pair:
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			<math>~\sigma_c^2 = 0</math>
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			and
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			<math>~x = \frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{1}{\xi \theta^{n}}\biggr) \frac{d\theta}{d\xi}\biggr] \, .</math>
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Tohline: /* Marginally Unstable Pressure-Truncated Gas Clouds */

2017-04-27T22:13:20Z

Marginally Unstable Pressure-Truncated Gas Clouds

← Older revision		Revision as of 22:13, 27 April 2017
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	provides an exact definition of the fundamental mode of radial oscillation in the configuration that is associated with the Bonnor-Ebert limiting mass. Note that this expression for the eigenfunction has been given in terms of the dimensionless radial displacement, <math>~x \equiv \delta r/r_0</math>.		provides an exact definition of the fundamental mode of radial oscillation in the configuration that is associated with the Bonnor-Ebert limiting mass. Note that this expression for the eigenfunction has been given in terms of the dimensionless radial displacement, <math>~x \equiv \delta r/r_0</math>.

	[[File:MassVsRadiusCombined02.png\|left\|200px\|Equilibrium sequences of Pressure-Truncated Polytropes]]Building upon Yabushita's investigation, we have analyzed the properties of radial oscillation modes in pressure-truncated ''polytropic'' configurations having indexes, <math>~3 \le n < \infty</math>. As is illustrated here, on the left, for any polytropic index within this stated range, at least one extremum in the mass can be identified along the corresponding mass-radius equilibrium sequence.		[[File:MassVsRadiusCombined02.png\|left\|200px\|Equilibrium sequences of Pressure-Truncated Polytropes]]Building upon Yabushita's investigation, we have analyzed the properties of radial oscillation modes in pressure-truncated ''polytropic'' configurations having indexes, <math>~3 \le n < \infty</math>. As is illustrated here, on the left, for any polytropic index within this stated range, at least one extremum in the mass can be identified along the corresponding mass-radius equilibrium sequence. Separately, [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] and [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura (1981)] have shown that, along each such sequence, the configuration associated with the "maximum mass" turning point occurs precisely where,
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			<math>~
			\biggl[ \frac{1}{\theta^{n+1}} \biggl(\frac{d\theta}{d\xi}\biggr)^2\biggr]_{\tilde\xi}
			</math>
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			<math>~=</math>
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			<math>~\frac{2}{n-3} \, .</math>
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