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

Tohline: /* From the Analysis of Taff and Van Horn (1974) */

2019-05-08T19:29:12Z

From the Analysis of Taff and Van Horn (1974)

← Older revision		Revision as of 19:29, 8 May 2019
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	</td>		</td>
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	[[File:TaffVanHorn1974Harmonic.gif\|450px\|~~Fundamental~~ mode animation]]<p></p>		[[File:TaffVanHorn1974Harmonic.gif\|450px\|First Harmonic mode animation]]<p></p>
	[[File:TaffVanHorn1974Fundamental.gif\|450px\|Fundamental mode animation]]		[[File:TaffVanHorn1974Fundamental.gif\|450px\|Fundamental mode animation]]
	</td>		</td>

Tohline: /* Linearized Wave Equation */

2019-01-29T20:46:54Z

Linearized Wave Equation

← Older revision		Revision as of 20:46, 29 January 2019
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	{{User:Tohline/Math/EQ_RadialPulsation01}}		{{User:Tohline/Math/EQ_RadialPulsation01}}

			[<b>[[User:Tohline/Appendix/References#P00\|<font color="red">P00</font>]]</b>], Vol. II, §3.7.1, p. 174, Eq. (3.144)
	</div>		</div>
	Here we review two published articles that have presented a partial analysis of radial modes of oscillation in pressure-truncated isothermal spheres. The analyses presented in both of these papers, effectively, employ this key wave equation, but the authors of these articles present it in different forms.		Here we review two published articles that have presented a partial analysis of radial modes of oscillation in pressure-truncated isothermal spheres. The analyses presented in both of these papers, effectively, employ this key wave equation, but the authors of these articles present it in different forms.

Tohline: /* From the Analysis of Taff and Van Horn (1974) */

2018-10-16T20:13:37Z

From the Analysis of Taff and Van Horn (1974)

← Older revision		Revision as of 20:13, 16 October 2018
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	<div align="center" id="Yabushita68">		<div align="center" id="Yabushita68">
			'''<font size="+1>Composite Display 1:</font>  Select Eigenfrequencies for Pressure-Truncated Isothermal Spheres'''

	<table border="1" cellpadding="5" width="75%">		<table border="1" cellpadding="5" width="75%">
	~~<tr>~~
	~~<th align="center" colspan="2"><font size="+1>Composite Display 1:</font>  Select Eigenfrequencies for Pressure-Truncated Isothermal Spheres</th>~~
	~~</tr>~~
	<tr>		<tr>
	<td align="center">		<td align="center">
	Data extracted from Tables I & II (p. 429) of [http://adsabs.harvard.edu/abs/1974MNRAS.168..427T L. G. Taff & H. M. Van Horn (1974~~, MNRAS, 168, 427-432~~)]<p></p>		Data extracted from Tables I & II (p. 429) of [http://adsabs.harvard.edu/abs/1974MNRAS.168..427T L. G. Taff & H. M. Van Horn (1974)]<p></p>
	"''Radial Pulsations of Finite Isothermal Gas Spheres''"<p></p>		"''Radial Pulsations of Finite Isothermal Gas Spheres''"<p></p>
	MNRAS, vol. ~~140~~, pp. ~~109~~-~~120~~ © Royal Astronomical Society		MNRAS, vol. 168, pp. 427 - 432 © Royal Astronomical Society
	</td>		</td>
	<td align="center">		<td align="center">
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	<div align="center" id="Yabushita68">		<div align="center" id="Yabushita68">
			'''<font size="+1>Composite Display 2:</font>  Select Eigenfunctions for Pressure-Truncated Isothermal Spheres'''

	<table border="1" cellpadding="5" width="75%">		<table border="1" cellpadding="5" width="75%">
	~~<tr>~~
	~~<th align="center" colspan="2"><font size="+1>Composite Display 2:</font>  Select Eigenfunctions for Pressure-Truncated Isothermal Spheres</th>~~
	~~</tr>~~
	<tr>		<tr>
	<td colspan="1" align="center">		<td colspan="1" align="center">
	Figure extracted from p. 430 of [http://adsabs.harvard.edu/abs/1974MNRAS.168..427T L. G. Taff & H. M. Van Horn (1974~~, MNRAS, 168, 427-432~~)]<p></p>		Figure & caption extracted from p. 430 of [http://adsabs.harvard.edu/abs/1974MNRAS.168..427T L. G. Taff & H. M. Van Horn (1974)]<p></p>
	"''Radial Pulsations of Finite Isothermal Gas Spheres''"<p></p>		"''Radial Pulsations of Finite Isothermal Gas Spheres''"<p></p>
	MNRAS, vol. ~~140~~, pp. ~~109~~-~~120~~ © Royal Astronomical Society		MNRAS, vol. 168, pp. 427 - 432 © Royal Astronomical Society
	</td>		</td>
	<td align="center">		<td align="center">

Tohline: /* P-V Diagram */

2018-10-16T20:01:18Z

P-V Diagram

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	<td align="left">		<td align="left">
	Bonnor's (1956) equilibrium sequence, but as generated from our [[User:Tohline/SSC/Structure/IsothermalSphere#Our_Numerical_Integration\|numerical integration of the isothermal Lane-Emden equation]]; in our plot, we have adopted [~~[User~~:~~Tohline~~/~~SSC~~/~~Structure~~/~~BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|~~Whitworth's normalizations]], <math>~P_\mathrm{rf}</math> and <math>~V_\mathrm{rf}</math>.		Bonnor's (1956) equilibrium sequence, but as generated from our [[User:Tohline/SSC/Structure/IsothermalSphere#Our_Numerical_Integration\|numerical integration of the isothermal Lane-Emden equation]]; in our plot, we have adopted [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth's (1981)] normalizations, <math>~P_\mathrm{rf}</math> and <math>~V_\mathrm{rf}</math>.
	</td>		</td>
	</tr>		</tr>

Tohline: /* P-V Diagram */

2018-10-16T19:57:48Z

P-V Diagram

← Older revision		Revision as of 19:57, 16 October 2018
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	====P-V Diagram====		====P-V Diagram====

	As we have discussed in a [[User:Tohline/SSC/Structure/BonnorEbert#P-V_Diagram\|separate chapter that focuses on the structural properties of pressure-truncated Isothermal spheres]], [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor (1956)] examined the ''sequence'' of equilibrium models that is generated by varying the truncation radius over the range, <math>~0 < \xi_e < \infty</math>. In a diagram that shows how <math>~P_e(\xi_e)</math> varies with equilibrium volume, <math>~V(\xi_e) \propto R^3</math>, Bonnor noticed that there is a pressure above which no equilibrium configurations exist. The original P-V diagram published by [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor (1956)] has been reproduced here, in the left-hand panel of our Figure 1. The right-hand panel of Figure 1 shows the same equilibrium sequence, as generated from our [[User:Tohline/SSC/Structure/IsothermalSphere#Our_Numerical_Integration\|numerical integration of the isothermal Lane-Emden equation]]; ~~in our plot,~~ we have adopted [[User:Tohline/SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Whitworth's normalizations]], <math>~P_\mathrm{rf}</math> and <math>~V_\mathrm{rf}</math>.		As we have discussed in a [[User:Tohline/SSC/Structure/BonnorEbert#P-V_Diagram\|separate chapter that focuses on the structural properties of pressure-truncated Isothermal spheres]], [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor (1956)] examined the ''sequence'' of equilibrium models that is generated by varying the truncation radius over the range, <math>~0 < \xi_e < \infty</math>. In a diagram that shows how <math>~P_e(\xi_e)</math> varies with equilibrium volume, <math>~V(\xi_e) \propto R^3</math>, Bonnor noticed that there is a pressure above which no equilibrium configurations exist. The original P-V diagram published by [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor (1956)] has been reproduced here, in the left-hand panel of our Figure 1. The right-hand panel of Figure 1 shows the same equilibrium sequence, as generated from our [[User:Tohline/SSC/Structure/IsothermalSphere#Our_Numerical_Integration\|numerical integration of the isothermal Lane-Emden equation]]; we have adopted [[User:Tohline/SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Whitworth's normalizations]], <math>~P_\mathrm{rf}</math> and <math>~V_\mathrm{rf}</math>.


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	MNRAS, vol. 116, pp. 351 - 359 © Royal Astronomical Society		MNRAS, vol. 116, pp. 351 - 359 © Royal Astronomical Society
	</td>		</td>
	<th align="~~center~~">~~Panel B~~</th>		<td align="left">
			Bonnor's (1956) equilibrium sequence, but as generated from our [[User:Tohline/SSC/Structure/IsothermalSphere#Our_Numerical_Integration\|numerical integration of the isothermal Lane-Emden equation]]; in our plot, we have adopted [[User:Tohline/SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Whitworth's normalizations]], <math>~P_\mathrm{rf}</math> and <math>~V_\mathrm{rf}</math>.
			</td>
	</tr>		</tr>
	<tr>		<tr>

Tohline: /* P-V Diagram */

2018-10-16T19:53:37Z

P-V Diagram

← Older revision		Revision as of 19:53, 16 October 2018
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	<table border="1" align="center" cellpadding="8">		<table border="1" align="center" cellpadding="8">
	<tr>		<tr>
	<td align="~~left" colspan="2~~">~~Temporary~~</td>		<td align="center">
	</tr>		Figure extracted from p. 355 of [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B W. B. Bonnor (1956)]<p></p>
	~~<tr>~~		"''Boyle's Law and Gravitational Instability''"<p></p>
	~~<th align="center">Panel A~~</th>		MNRAS, vol. 116, pp. 351 - 359 © Royal Astronomical Society
			</td>
	<th align="center">Panel B</th>		<th align="center">Panel B</th>
	</tr>		</tr>
	<tr>		<tr>
	<td align="center">		<td align="center">
	[[File:~~Bonnor1956Fig1reproduced~~.~~jpg~~\|400px\|center\|Bonnor (1956, MNRAS, 116, 351)]]		[[File:Bonnor1956Fig1reproducedB.png\|400px\|center\|Bonnor (1956, MNRAS, 116, 351)]]
	</td>		</td>
	<td align="center">		<td align="center">

Tohline: /* P-V Diagram */

2018-10-16T19:39:42Z

P-V Diagram

← Older revision		Revision as of 19:39, 16 October 2018
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	As we have discussed in a [[User:Tohline/SSC/Structure/BonnorEbert#P-V_Diagram\|separate chapter that focuses on the structural properties of pressure-truncated Isothermal spheres]], [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor (1956)] examined the ''sequence'' of equilibrium models that is generated by varying the truncation radius over the range, <math>~0 < \xi_e < \infty</math>. In a diagram that shows how <math>~P_e(\xi_e)</math> varies with equilibrium volume, <math>~V(\xi_e) \propto R^3</math>, Bonnor noticed that there is a pressure above which no equilibrium configurations exist. The original P-V diagram published by [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor (1956)] has been reproduced here, in the left-hand panel of our Figure 1. The right-hand panel of Figure 1 shows the same equilibrium sequence, as generated from our [[User:Tohline/SSC/Structure/IsothermalSphere#Our_Numerical_Integration\|numerical integration of the isothermal Lane-Emden equation]]; in our plot, we have adopted [[User:Tohline/SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Whitworth's normalizations]], <math>~P_\mathrm{rf}</math> and <math>~V_\mathrm{rf}</math>.		As we have discussed in a [[User:Tohline/SSC/Structure/BonnorEbert#P-V_Diagram\|separate chapter that focuses on the structural properties of pressure-truncated Isothermal spheres]], [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor (1956)] examined the ''sequence'' of equilibrium models that is generated by varying the truncation radius over the range, <math>~0 < \xi_e < \infty</math>. In a diagram that shows how <math>~P_e(\xi_e)</math> varies with equilibrium volume, <math>~V(\xi_e) \propto R^3</math>, Bonnor noticed that there is a pressure above which no equilibrium configurations exist. The original P-V diagram published by [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor (1956)] has been reproduced here, in the left-hand panel of our Figure 1. The right-hand panel of Figure 1 shows the same equilibrium sequence, as generated from our [[User:Tohline/SSC/Structure/IsothermalSphere#Our_Numerical_Integration\|numerical integration of the isothermal Lane-Emden equation]]; in our plot, we have adopted [[User:Tohline/SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Whitworth's normalizations]], <math>~P_\mathrm{rf}</math> and <math>~V_\mathrm{rf}</math>.


	<div align="center">		<div align="center">
			<b>Figure 1:   Bonnor's P-V Diagram</b> (see [[User:Tohline/SSC/Structure/BonnorEbert#Fig1\|accompanying discussion]] for details)

	<table border="1" align="center" cellpadding="8">		<table border="1" align="center" cellpadding="8">
	<tr>		<tr>
	<td align="~~center~~" colspan="2">		<td align="left" colspan="2">Temporary</td>
	<b>~~Figure 1:   Bonnor's P-V Diagram~~</b><br />~~(see [[User:Tohline/SSC/Structure/BonnorEbert#Fig1\|related discussion]])~~		</tr>
	</td>		<tr>
			<th align="center">Panel A</th>
			<th align="center">Panel B</th>
	</tr>		</tr>
	<tr>		<tr>

Tohline: /* Taff and Van Horn (1974) */

2018-10-16T19:33:16Z

Taff and Van Horn (1974)

← Older revision		Revision as of 19:33, 16 October 2018
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	Aside from the leading (constant) coefficient, this expression is identical to the linearized wave equation that [http://adsabs.harvard.edu/abs/1974MNRAS.168..427T Taff & Van Horn (1974)] used to examine the radial pulsation modes of pressure-truncated isothermal spheres; their governing relation is displayed in the following, boxed-in image:		Aside from the leading (constant) coefficient, this expression is identical to the linearized wave equation that [http://adsabs.harvard.edu/abs/1974MNRAS.168..427T Taff & Van Horn (1974)] used to examine the radial pulsation modes of pressure-truncated isothermal spheres; their governing relation is displayed in the following, boxed-in image:

	<div align="center" id="~~Yabushita68~~">		<div align="center" id="TVH74">
	<table border="1" cellpadding="5" width="80%">		<table border="1" cellpadding="5" width="80%">
	<tr><td align="center">		<tr><td align="center">
	Equation extracted from p. 427 of [http://adsabs.harvard.edu/abs/1974MNRAS.168..427T L. G. Taff & H. M. Van Horn (1974~~, MNRAS, 168, 427-432~~)]<p></p>		Equation extracted from p. 427 of [http://adsabs.harvard.edu/abs/1974MNRAS.168..427T L. G. Taff & H. M. Van Horn (1974)]<p></p>
	"''Radial Pulsations of Finite Isothermal Gas Spheres''"<p></p>		"''Radial Pulsations of Finite Isothermal Gas Spheres''"<p></p>
	MNRAS, vol. ~~140~~, pp. ~~109~~-~~120~~ © Royal Astronomical Society		MNRAS, vol. 168, pp. 427 - 432 © Royal Astronomical Society
	</td></tr>		</td></tr>
	<tr>		<tr>

Tohline: /* Yabushita (1968) */

2018-10-16T19:30:20Z

Yabushita (1968)

← Older revision		Revision as of 19:30, 16 October 2018
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	<table border="1" cellpadding="5" width="80%">		<table border="1" cellpadding="5" width="80%">
	<tr><td align="center">		<tr><td align="center">
	Equation extracted from p. 111 of [http://adsabs.harvard.edu/abs/1968MNRAS.140..109Y S. Yabushita (1968~~, MNRAS, 140, 109~~)]<p></p>		Equation extracted from p. 111 of [http://adsabs.harvard.edu/abs/1968MNRAS.140..109Y S. Yabushita (1968)]<p></p>
	"''Jeans's Type Gravitational Instability of Finite Isothermal Gas Spheres''"<p></p>		"''Jeans's Type Gravitational Instability of Finite Isothermal Gas Spheres''"<p></p>
	MNRAS, vol. 140, pp. 109-120 © Royal Astronomical Society		MNRAS, vol. 140, pp. 109-120 © Royal Astronomical Society

Tohline: /* Review */

2018-10-16T19:29:32Z

Review

← Older revision		Revision as of 19:29, 16 October 2018
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	<table border="1" cellpadding="5" width="80%">		<table border="1" cellpadding="5" width="80%">
	<tr><td align="center">		<tr><td align="center">
	Equations extracted<sup>&dagger;</sup> from p. 110 of [http://adsabs.harvard.edu/abs/1968MNRAS.140..109Y S. Yabushita (1968~~, MNRAS, 140, 109~~)]<p></p>		Equations extracted<sup>&dagger;</sup> from p. 110 of [http://adsabs.harvard.edu/abs/1968MNRAS.140..109Y S. Yabushita (1968)]<p></p>
	"''Jeans's Type Gravitational Instability of Finite Isothermal Gas Spheres''"<p></p>		"''Jeans's Type Gravitational Instability of Finite Isothermal Gas Spheres''"<p></p>
	MNRAS, vol. 140, pp. 109-120 © Royal Astronomical Society		MNRAS, vol. 140, pp. 109-120 © Royal Astronomical Society
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	</tr>		</tr>
	<tr><td align="left">		<tr><td align="left">
	<sup>&dagger;</sup>~~Mathematical expressions displayed here with layout~~ modified from the original publication.		<sup>&dagger;</sup>Layout of equations has been modified from the original publication.
	</td></tr>		</td></tr>
	</table>		</table>
	</div>		</div>


	====P-V Diagram====		====P-V Diagram====