SSC/Stability/Isothermal - Revision history

Jet53man: /* Behavior of First Logarithmic Derivative */

2023-11-06T00:52:25Z

Behavior of First Logarithmic Derivative

← Older revision		Revision as of 20:52, 5 November 2023
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{d\ln x}{d\ln \xi} = \frac{\xi}{x} \cdot \frac{dx}{d\xi}</math>		<math>\frac{d\ln x}{d\ln \xi} = \frac{\xi}{x} \cdot \frac{dx}{d\xi}</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	\biggl(\frac{3-n}{n-1}\biggr) \biggl[ 1		\biggl(\frac{3-n}{n-1}\biggr) \biggl[ 1
	+ \frac{n(\theta^')^2 }{\theta^{n+1}} + \frac{3\theta^' }{\xi \theta^{n}} \biggr]		+ \frac{n(\theta^')^2 }{\theta^{n+1}} + \frac{3\theta^' }{\xi \theta^{n}} \biggr]
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	- (n-3) \biggl[ 1		- (n-3) \biggl[ 1
	+ \frac{n(\theta^')^2 }{\theta^{n+1}} + \frac{3\theta^' }{\xi \theta^{n}} \biggr]		+ \frac{n(\theta^')^2 }{\theta^{n+1}} + \frac{3\theta^' }{\xi \theta^{n}} \biggr]
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Rightarrow ~~~ \biggl[ (n-1) + (n-3) \frac{\theta^' }{\xi \theta^{n}} \biggr] \frac{d\ln x}{d\ln \xi} </math>		<math>\Rightarrow ~~~ \biggl[ (n-1) + (n-3) \frac{\theta^' }{\xi \theta^{n}} \biggr] \frac{d\ln x}{d\ln \xi} </math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	- (n-3) \biggl[ 1		- (n-3) \biggl[ 1
	+ \frac{n(\theta^')^2 }{\theta^{n+1}} + \frac{3\theta^' }{\xi \theta^{n}} \biggr]		+ \frac{n(\theta^')^2 }{\theta^{n+1}} + \frac{3\theta^' }{\xi \theta^{n}} \biggr]
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Rightarrow ~~~ \biggl[ (n-1) \biggr] \frac{d\ln x}{d\ln \xi} </math>		<math>\Rightarrow ~~~ \biggl[ (n-1) \biggr] \frac{d\ln x}{d\ln \xi} </math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	- (n-3) \biggl\{ \biggl[ 1		- (n-3) \biggl\{ \biggl[ 1
	+ \frac{n(\theta^')^2 }{\theta^{n+1}} + \frac{3\theta^' }{\xi \theta^{n}} \biggr]		+ \frac{n(\theta^')^2 }{\theta^{n+1}} + \frac{3\theta^' }{\xi \theta^{n}} \biggr]
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	- (n-3) \biggl\{ \biggl[ 1		- (n-3) \biggl\{ \biggl[ 1
	+ \frac{n(\theta^')^2 }{\theta^{n+1}} \biggr]		+ \frac{n(\theta^')^2 }{\theta^{n+1}} \biggr]
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	</div>		</div>

	Now, as we have [[#From_Yabushita.27s_.281992.29_Analysis\|discussed above]], ~~[http://adsabs.harvard.edu/abs/1992Ap%26SS.193..173Y Yabushita (1992)]~~ has argued that for pressure-truncated configurations, the appropriate surface boundary condition should be,		Now, as we have [[#From_Yabushita.27s_.281992.29_Analysis\|discussed above]], {{ Yabushita92 }} has argued that for pressure-truncated configurations, the appropriate surface boundary condition should be,
	<div align="center">		<div align="center">
	<math>\frac{d\ln x}{d\ln\xi} = -3 \, .</math>		<math>\frac{d\ln x}{d\ln\xi} = -3 \, .</math>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~3 (n-1) </math>		<math>3 (n-1) </math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	(n-3) \biggl[ 1		(n-3) \biggl[ 1
	+ \frac{n(\theta^')^2 }{\theta^{n+1}} \biggr]		+ \frac{n(\theta^')^2 }{\theta^{n+1}} \biggr]
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Rightarrow~~~		<math>\Rightarrow~~~
	(n-3) \biggl[ \frac{n(\theta^')^2 }{\theta^{n+1}} \biggr]		(n-3) \biggl[ \frac{n(\theta^')^2 }{\theta^{n+1}} \biggr]
	</math>		</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	2n		2n
	</math>		</math>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Rightarrow~~~		<math>\Rightarrow~~~
	\frac{(\theta^')^2 }{\theta^{n+1}}		\frac{(\theta^')^2 }{\theta^{n+1}}
	</math>		</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	\frac{2 }{(n-3) } \, .		\frac{2 }{(n-3) } \, .
	</math>		</math>
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	</table>		</table>
	</div>		</div>
	According to [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Derivation\|~~Horedt (1970)~~]], this is precisely the functional condition that pinpoints the <math>~P_e</math>-max turning point(s) along the equilibrium sequence in a <math>~P-V</math> diagram, for all pressure-truncated, polytropic spheres with <math>~n \ge 3</math>. And, according to [[SSC/Structure/PolytropesEmbedded#Other_Limits\|~~Kimura (1981)~~]], this is precisely the functional condition that pinpoints the maximum mass turning point(s) along equlibrium sequences as displayed in a mass-radius diagram.		According to {{ Horedt70 }} — see our [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Derivation\|accompanying discussion]] — this is precisely the functional condition that pinpoints the <math>~P_e</math>-max turning point(s) along the equilibrium sequence in a <math>~P-V</math> diagram, for all pressure-truncated, polytropic spheres with <math>~n \ge 3</math>. And, according to {{ Kimura81b }} — again, see our [[SSC/Structure/PolytropesEmbedded#Other_Limits\|accompanying discussion]] — this is precisely the functional condition that pinpoints the maximum mass turning point(s) along equlibrium sequences as displayed in a mass-radius diagram.

	<!-- CHRISTY OR COX BOUNDARY CONDITIONS ...		<!-- CHRISTY OR COX BOUNDARY CONDITIONS ...
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	</div>		</div>
	-->		-->

	===Behavior Approaching Center===		===Behavior Approaching Center===
	<span id="CentralValue">It would also be good to know</span> what the limiting value of this displacement function is at the center of the polytropic configuration. Drawing on some of our already-developed [[Appendix/Ramblings/PowerSeriesExpressions#Approximate_Power-Series_Expressions\|power-series expressions]], we have,		<span id="CentralValue">It would also be good to know</span> what the limiting value of this displacement function is at the center of the polytropic configuration. Drawing on some of our already-developed [[Appendix/Ramblings/PowerSeriesExpressions#Approximate_Power-Series_Expressions\|power-series expressions]], we have,

Jet53man: /* Derivation of Polytropic Displacement Function */

2023-11-06T00:45:05Z

Derivation of Polytropic Displacement Function

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Jet53man: /* Verification of Analytic Solution */

2023-11-06T00:40:01Z

Verification of Analytic Solution

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Jet53man: /* Verification of Analytic Solution */

2023-11-06T00:36:27Z

Verification of Analytic Solution

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Jet53man: /* Evolving from 1968 and 1974, to 1975 */

2023-11-06T00:31:27Z

Evolving from 1968 and 1974, to 1975

← Older revision		Revision as of 20:31, 5 November 2023
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	{{ Yabushita75full }}, on the other hand, adopted a more conventional approach and couched his discussions of radial oscillations and stability in terms of the search for solutions to, what we have referred to as, the (standard) [[#IsothermalLAWE\|Isothermal LAWE]], involving the radial displacement function, <math>x \equiv \delta r/r_0</math>. In order to facilitate a transition from his earlier work, we demonstrate, here, the relationship between the perturbation variable, <math>g</math>, and the displacement function, <math>x</math>.		{{ Yabushita75full }}, on the other hand, adopted a more conventional approach and couched his discussions of radial oscillations and stability in terms of the search for solutions to, what we have referred to as, the (standard) [[#IsothermalLAWE\|Isothermal LAWE]], involving the radial displacement function, <math>x \equiv \delta r/r_0</math>. In order to facilitate a transition from his earlier work, we demonstrate, here, the relationship between the perturbation variable, <math>g</math>, and the displacement function, <math>x</math>.

	Following the [[SSC/PerspectiveReconciliation#Approach_by_Ledoux_and_Walraven\|derivation of the linearized wave equation]] presented, as by {{ LW58full }} — see, also, equation <font color="green">①</font> in ~~an [https://www.vistrails.org/images/ImageOfDerivations07GoodLedouxWalraven.png~~ associated derivation diagram] — the linearized continuity equation may be written as,		[[File:ImageOfDerivations07GoodLedouxWalraven.png\|250px\|right\|LW58 derivation]]Following the [[SSC/PerspectiveReconciliation#Approach_by_Ledoux_and_Walraven\|derivation of the linearized wave equation]] presented, as by {{ LW58full }} — see, also, equation <font color="green">①</font> in the associated derivation diagram shown here, on the right — the linearized continuity equation may be written as,
	<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>~\frac{\delta\rho}{\rho_c} </math>		<math>\frac{\delta\rho}{\rho_c} </math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~- \frac{1}{r^2}\frac{d}{d r} \biggl[ r^3 \biggl(\frac{\rho}{\rho_c}\biggr) \frac{\delta r}{r_0} \biggr] </math>		<math>- \frac{1}{r^2}\frac{d}{d r} \biggl[ r^3 \biggl(\frac{\rho}{\rho_c}\biggr) \frac{\delta r}{r_0} \biggr] </math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~- \frac{1}{\xi^2}\frac{d}{d \xi} \biggl[ \xi^3 \biggl(\frac{\rho}{\rho_c}\biggr) x \biggr]\, .</math>		<math>- \frac{1}{\xi^2}\frac{d}{d \xi} \biggl[ \xi^3 \biggl(\frac{\rho}{\rho_c}\biggr) x \biggr]\, .</math>
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{1}{\xi^2} \frac{dg}{d\xi}</math>		<math>\frac{1}{\xi^2} \frac{dg}{d\xi}</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~- \frac{1}{\xi^2}\frac{d}{d \xi} \biggl[ \xi^3 \biggl(\frac{\rho}{\rho_c}\biggr) x \biggr] </math>		<math>- \frac{1}{\xi^2}\frac{d}{d \xi} \biggl[ \xi^3 \biggl(\frac{\rho}{\rho_c}\biggr) x \biggr] </math>
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Rightarrow~~~ g</math>		<math>\Rightarrow~~~ g</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~- \xi^3 \biggl(\frac{\rho}{\rho_c}\biggr) x </math>		<math>- \xi^3 \biggl(\frac{\rho}{\rho_c}\biggr) x </math>
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Rightarrow~~~ x</math>		<math>\Rightarrow~~~ x</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~- \frac{g}{\xi^3}~ e^{\psi} \, . </math>		<math>- \frac{g}{\xi^3}~ e^{\psi} \, . </math>
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~x_Y</math>		<math>x_Y</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~- \frac{g_Y}{\xi^3}~ e^{\psi} </math>		<math>- \frac{g_Y}{\xi^3}~ e^{\psi} </math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~- \biggl[\frac{1}{\xi} \frac{d\psi}{d\xi} - e^{-\psi}\biggr]e^{\psi} \, .</math>		<math>- \biggl[\frac{1}{\xi} \frac{d\psi}{d\xi} - e^{-\psi}\biggr]e^{\psi} \, .</math>
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Yabushita (1975) */

2023-11-06T00:19:34Z

Yabushita (1975)

← Older revision		Revision as of 20:19, 5 November 2023
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	===Yabushita (1975)===		===Yabushita (1975)===
	====Evolving from 1968 and 1974, to 1975====		====Evolving from 1968 and 1974, to 1975====
	As we have [[#Yabushita_.281968.29\|discussed, above]], ~~[http://adsabs.harvard.edu/abs/1968MNRAS.140..109Y Yabushita (1968)]~~ couched his stability analysis in terms of the search for a solution to, what we have labeled as, the [[#Yabushita68LAWE\|Yabushita68 Isothermal LAWE]]. In taking this approach, he introduced (a rather unconventional) [[#gDefinition\|perturbation variable]],		As we have [[#Yabushita_.281968.29\|discussed, above]], {{ Yabushita68 }} couched his stability analysis in terms of the search for a solution to, what we have labeled as, the [[#Yabushita68LAWE\|Yabushita68 Isothermal LAWE]]. In taking this approach, he introduced (a rather unconventional) [[#gDefinition\|perturbation variable]],
	<div align="center">		<div align="center">
	<math>~g \equiv \xi^2 \frac{d}{d\xi}\biggl( \frac{\delta\Phi}{c_s^2}\biggr) \, .</math>		<math>g \equiv \xi^2 \frac{d}{d\xi}\biggl( \frac{\delta\Phi}{c_s^2}\biggr) \, .</math>
	</div>		</div>
	He pointed out that, defined in this way, the density variation, <math>~\delta\rho/\rho_c</math>, can be determined from <math>~g</math> via the differential relation,		He pointed out that, defined in this way, the density variation, <math>\delta\rho/\rho_c</math>, can be determined from <math>g</math> via the differential relation,
	<div align="center">		<div align="center">
	<math>~\frac{\delta \rho}{\rho_c} = \frac{1}{\xi^2} \frac{dg}{d\xi} \, .</math>		<math>\frac{\delta \rho}{\rho_c} = \frac{1}{\xi^2} \frac{dg}{d\xi} \, .</math>
	</div>		</div>
	It was in this context that ~~[http://adsabs.harvard.edu/abs/1974MNRAS.167…95Y Yabushita (1974)]~~ discovered that the eigenvector for the fundamental mode of radial oscillation in the configuration that has the Bonnor-Ebert limiting mass, can be specified analytically. Specifically, the eigenfrequency is zero — that is, the configuration is marginally [dynamically] unstable — and the eigenfunction is, precisely,		It was in this context that {{ Yabushita74 }} discovered that the eigenvector for the fundamental mode of radial oscillation in the configuration that has the [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Bonnor-Ebert limiting mass]], can be specified analytically. Specifically, the eigenfrequency is zero — that is, the configuration is marginally [dynamically] unstable — and the eigenfunction is, precisely,
	<div align="center">		<div align="center">
	<math>g_Y = M - \xi^3 e^{-\psi} = \xi^2 \frac{d\psi}{d\xi} - \xi^3 e^{-\psi} \, .</math>		<math>g_Y = M - \xi^3 e^{-\psi} = \xi^2 \frac{d\psi}{d\xi} - \xi^3 e^{-\psi} \, .</math>
	</div>		</div>

	~~[http://adsabs.harvard.edu/abs/1975MNRAS.172..441Y Yabushita (1975)]~~, on the other hand, adopted a more conventional approach and couched his discussions of radial oscillations and stability in terms of the search for solutions to, what we have referred to as, the (standard) [[#IsothermalLAWE\|Isothermal LAWE]], involving the radial displacement function, <math>~x \equiv \delta r/r_0</math>. In order to facilitate a transition from his earlier work, we demonstrate, here, the relationship between the perturbation variable, <math>~g</math>, and the displacement function, <math>~x</math>.		{{ Yabushita75full }}, on the other hand, adopted a more conventional approach and couched his discussions of radial oscillations and stability in terms of the search for solutions to, what we have referred to as, the (standard) [[#IsothermalLAWE\|Isothermal LAWE]], involving the radial displacement function, <math>x \equiv \delta r/r_0</math>. In order to facilitate a transition from his earlier work, we demonstrate, here, the relationship between the perturbation variable, <math>g</math>, and the displacement function, <math>x</math>.

	Following the [[SSC/PerspectiveReconciliation#Approach_by_Ledoux_and_Walraven\|derivation of the linearized wave equation presented, as by ~~Ledoux & Walraven]]~~ — see, also, equation <font color="green">①</font> in an [https://www.vistrails.org/images/ImageOfDerivations07GoodLedouxWalraven.png associated derivation diagram] — the linearized continuity equation may be written as,		Following the [[SSC/PerspectiveReconciliation#Approach_by_Ledoux_and_Walraven\|derivation of the linearized wave equation]] presented, as by {{ LW58full }} — see, also, equation <font color="green">①</font> in an [https://www.vistrails.org/images/ImageOfDerivations07GoodLedouxWalraven.png associated derivation diagram] — the linearized continuity equation may be written as,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man: /* Stability Analysis */

2023-11-05T20:01:37Z

Stability Analysis

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Jet53man: /* Review */

2023-11-05T19:47:10Z

Review

← Older revision		Revision as of 15:47, 5 November 2023
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	</table>		</table>
	</div>		</div>
	The relativistic version of these expressions is provided as equation (2.1) of ~~[http://adsabs.harvard.edu/abs/1974MNRAS.167…95Y Yabushita (1974)]~~. Tracking back, directly to the statement of,		The relativistic version of these expressions is provided as equation (2.1) of {{ Yabushita74 }}. Tracking back, directly to the statement of,

	<div align="center">		<div align="center">
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	{{Math/EQ_SSmassConservation01}}		{{Math/EQ_SSmassConservation01}}
	</div>		</div>
	we straightforwardly recognize that (see also ~~Yabushita~~'s equation 2.2),		we straightforwardly recognize that (see also {{ Yabushita74hereafter }}'s equation 2.2),
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man: /* From the Analysis of Taff & van Horn (1974) */

2023-11-04T21:35:54Z

From the Analysis of Taff & van Horn (1974)

← Older revision		Revision as of 17:35, 4 November 2023
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	For three of their models — the ones having <math>x_0 = 3, 6, \And 9</math> — {{ TVH74 }} also determined an eigenvalue for the first harmonic mode of oscillation; see the seventh column of numbers, labeled <math>\lambda_1^2</math>, that appears in the digital image of their "Table II" that we have extracted from their paper and presented in Composite Display 1, above. Via our own [[#Our_Numerical_Integration\|numerical analysis]], we have determined an eigenvalue for the first harmonic mode of oscillation in all nine of the models. These results are presented in the right-hand panel of Composite Display 1, under the heading, "First Harmonic." Again, the agreement between our numerically determined eigenvalues (see the three highlighted by pink, rectangular boxes) and the ones reported by ~~Taff & Van Horn~~ is excellent.		For three of their models — the ones having <math>x_0 = 3, 6, \And ~9</math> — {{ TVH74 }} also determined an eigenvalue for the first harmonic mode of oscillation; see the seventh column of numbers, labeled <math>\lambda_1^2</math>, that appears in the digital image of their "Table II" that we have extracted from their paper and presented in Composite Display 1, above. Via our own [[#Our_Numerical_Integration\|numerical analysis]], we have determined an eigenvalue for the first harmonic mode of oscillation in all nine of the models. These results are presented in the right-hand panel of Composite Display 1, under the heading, "First Harmonic." Again, the agreement between our numerically determined eigenvalues (see the three highlighted by pink, rectangular boxes) and the ones reported by {{ TVH74hereafter }} is excellent.

	<span id="CompositeDisplay2">The right-hand column of Composite Display 2, below,</span> presents a pair of animations that display how our numerically derived displacement function, <math>~x(\xi)</math>, varies with radius — from the center of the isothermal sphere, out to <math>~\xi = 9</math> — for a variety of values of the square of the eigenfrequency, <math>~\lambda_0^2</math>; each frame of both animations is tagged by the relevant value of <math>~\lambda_0^2</math>. The segment of the <math>~x(\xi)</math> curve that has been drawn in blue identifies the ''eigenfunction'' that corresponds to the specified value of the eigenfrequency. In each frame, the radial location at which the blue segment terminates simultaneously identifies the truncation radius of the relevant isothermal sphere, and the radius at which the boundary condition, <math>~d\ln x/d\ln\xi = -3</math>, has been enforced. In every frame, the <math>~x(\xi)</math> function has been normalized such that the displacement amplitude is unity at the truncated configuration's surface.		<span id="CompositeDisplay2">The right-hand column of Composite Display 2, below,</span> presents a pair of animations that display how our numerically derived displacement function, <math>x(\xi)</math>, varies with radius — from the center of the isothermal sphere, out to <math>\xi = 9</math> — for a variety of values of the square of the eigenfrequency, <math>\lambda_0^2</math>; each frame of both animations is tagged by the relevant value of <math>\lambda_0^2</math>. The segment of the <math>x(\xi)</math> curve that has been drawn in blue identifies the ''eigenfunction'' that corresponds to the specified value of the eigenfrequency. In each frame, the radial location at which the blue segment terminates simultaneously identifies the truncation radius of the relevant isothermal sphere, and the radius at which the boundary condition, <math>d\ln x/d\ln\xi = -3</math>, has been enforced. In every frame, the <math>x(\xi)</math> function has been normalized such that the displacement amplitude is unity at the truncated configuration's surface.


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	<tr>		<tr>
	<td colspan="1" align="center">		<td colspan="1" align="center">
	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>~~		Figure & caption extracted from p. 430 of <br />{{ TVH74figure }}
	~~"''Radial Pulsations of Finite Isothermal Gas Spheres''"<p>~~</p>
	~~MNRAS, vol. 168, pp. 427 - 432 © Royal Astronomical Society~~
	</td>		</td>
	<td align="center">		<td align="center">
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	</div>		</div>

	The blue curve segments in the bottom animation identify ''fundamental-mode'' eigenfunctions; most significantly, no radial nodes appear between the center and the surface of the truncated configuration. The ones that terminate at <math>~\xi_e = 3, 6 \And 9</math> appear to be identical to the three fundamental-mode eigenfunctions with corresponding values of the truncation radius that appear in Figure 1a of ~~[http://adsabs.harvard.edu/abs/1974MNRAS.168..427T Taff & Van Horn (1974)]~~ — see the bottom half of the image that has been extracted from the ~~Taff & Van Horn~~ publication and presented here in the left-hand column of our Composite Display 2.		The blue curve segments in the bottom animation identify ''fundamental-mode'' eigenfunctions; most significantly, no radial nodes appear between the center and the surface of the truncated configuration. The ones that terminate at <math>\xi_e = 3, 6, \And ~9</math> appear to be identical to the three fundamental-mode eigenfunctions with corresponding values of the truncation radius that appear in Figure 1a of {{ TVH74 }} — see the bottom half of the image that has been extracted from the {{ TVH74hereafter }} publication and presented here in the left-hand column of our Composite Display 2.

	The blue curve segments in the ''top'' animation identify "first harmonic" eigenfunctions; for each of the displayed eigenfunctions, one radial node exists between the center and the surface of the truncated configuration. The ones that terminate at <math>~\xi_e = 3, 6, \And 9</math>, appear to be identical to the three first harmonic eigenfunctions with corresponding values of the truncation radius that appear in Figure 1b of ~~[http://adsabs.harvard.edu/abs/1974MNRAS.168..427T Taff & Van Horn (1974)]~~ — see the top half of the image that has been extracted from the ~~Taff & Van Horn~~ publication and presented here in the left-hand column of our Composite Display 2.		The blue curve segments in the ''top'' animation identify "first harmonic" eigenfunctions; for each of the displayed eigenfunctions, one radial node exists between the center and the surface of the truncated configuration. The ones that terminate at <math>\xi_e = 3, 6, \And ~9</math>, appear to be identical to the three first harmonic eigenfunctions with corresponding values of the truncation radius that appear in Figure 1b of {{ TVH74 }} — see the top half of the image that has been extracted from the {{ TVH74hereafter }} publication and presented here in the left-hand column of our Composite Display 2.

	===From Yabushita's (1992) Analysis===		===From Yabushita's (1992) Analysis===

Jet53man: /* From the Analysis of Taff & van Horn (1974) */

2023-11-04T20:51:40Z

From the Analysis of Taff & van Horn (1974)

← Older revision		Revision as of 16:51, 4 November 2023
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	==Previously Published Eigenvalues and Eigenfunctions==		==Previously Published Eigenvalues and Eigenfunctions==
	===From the Analysis of Taff & van Horn (1974)===		===From the Analysis of Taff & van Horn (1974)===
	The left-hand column of Composite Display 1, below, contains a pair of images that present properties of the eigenvectors that resulted from the ~~[http://adsabs.harvard.edu/abs/1974MNRAS.168..427T Taff & Van Horn (1974)]~~ analysis of radial oscillations in pressure-truncated isothermal spheres, assuming that the configurations remain isothermal — that is, adopting an adiabatic exponent of <math>~\Gamma_1 = 1</math> — during the oscillations. In the image titled, "Table I", that we have extracted from their paper, the first column of numbers identifies values of nine adopted truncation radii in the range, <math>~2 \le x_0 \le 10</math>, while the second column lists the corresponding value of <math>~\lambda_0^2</math> that were determined via their numerical analysis. For three values of the truncation radius — <math>~x_0 = 6, 7, \And 8</math> — the third column lists the values of <math>~\lambda_0^2</math> that had been previously reported by ~~[http://adsabs.harvard.edu/abs/1968MNRAS.140..109Y Yabushita (1968)]~~.		The left-hand column of Composite Display 1, below, contains a pair of images that present properties of the eigenvectors that resulted from the {{ TVH74full }} analysis of radial oscillations in pressure-truncated isothermal spheres, assuming that the configurations remain isothermal — that is, adopting an adiabatic exponent of <math>~\Gamma_1 = 1</math> — during the oscillations. In the image titled, "Table I", that we have extracted from their paper, the first column of numbers identifies values of nine adopted truncation radii in the range, <math>~2 \le x_0 \le 10</math>, while the second column lists the corresponding value of <math>~\lambda_0^2</math> that were determined via their numerical analysis. For three values of the truncation radius — <math>~x_0 = 6, 7, \And 8</math> — the third column lists the values of <math>~\lambda_0^2</math> that had been previously reported by {{ Yabushita68full }}.

	In the right-hand column of Composite Display 1, we have detailed some results from our own [[#Our_Numerical_Integration\|numerical analysis]] of the same set of nine configurations that were studied by ~~[http://adsabs.harvard.edu/abs/1974MNRAS.168..427T Taff & Van Horn (1974)]~~ . In the second column of our table that has the heading, "Fundamental Mode," we have listed the value of <math>~\mathfrak{F}</math> that was required in our analysis in order to generate an eigenfunction whose logarithmic derivative at each configuration's surface was precisely negative three (to five significant digits). In order to facilitate quantitative comparison with the work of ~~Taff & Van Horn~~, the third column of our table lists for each model the corresponding value of <math>~\lambda_0^2</math>; recalling that <math>~\gamma = 1</math> and <math>~\alpha = -1</math>, each of these values was determined via the relation,		In the right-hand column of Composite Display 1, we have detailed some results from our own [[#Our_Numerical_Integration\|numerical analysis]] of the same set of nine configurations that were studied by {{ TVH74 }}. In the second column of our table that has the heading, "Fundamental Mode," we have listed the value of <math>~\mathfrak{F}</math> that was required in our analysis in order to generate an eigenfunction whose logarithmic derivative at each configuration's surface was precisely negative three (to five significant digits). In order to facilitate quantitative comparison with the work of {{ TVH74 }}, the third column of our table lists for each model the corresponding value of <math>\lambda_0^2</math>; recalling that <math>\gamma = 1</math> and <math>\alpha = -1</math>, each of these values was determined via the relation,
	<div align="center">		<div align="center">
	<math>~\lambda_0^2 = \frac{\gamma(\mathfrak{F}+2\alpha)}{6} = \frac{(\mathfrak{F}-2)}{6} \, .</math>		<math>~\lambda_0^2 = \frac{\gamma(\mathfrak{F}+2\alpha)}{6} = \frac{(\mathfrak{F}-2)}{6} \, .</math>
	</div>		</div>
	The agreement between our numerically determined fundamental-mode eigenvalues (highlighted by pink, rectangular boxes) and the ones reported by ~~Taff & Van Horn~~ is excellent, across the board. Notice that <math>~\lambda_0^2</math> is negative for the models having <math>~x_0 = 7, 8, 9, \And 10</math>, which indicates that these four models are dynamically unstable.		The agreement between our numerically determined fundamental-mode eigenvalues (highlighted by pink, rectangular boxes) and the ones reported by {{ TVH74 }} is excellent, across the board. Notice that <math>\lambda_0^2</math> is negative for the models having <math>x_0 = 7, 8, 9, \And 10</math>, which indicates that these four models are dynamically unstable.


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	For three of their models — the ones having <math>~x_0 = 3, 6, \And 9</math> — ~~[http://adsabs.harvard.edu/abs/1974MNRAS.168..427T Taff & Van Horn (1974)]~~ also determined an eigenvalue for the first harmonic mode of oscillation; see the seventh column of numbers, labeled <math>~\lambda_1^2</math>, that appears in the digital image of their "Table II" that we have extracted from their paper and presented in Composite Display 1, above. Via our own [[#Our_Numerical_Integration\|numerical analysis]], we have determined an eigenvalue for the first harmonic mode of oscillation in all nine of the models. These results are presented in the right-hand panel of Composite Display 1, under the heading, "First Harmonic." Again, the agreement between our numerically determined eigenvalues (see the three highlighted by pink, rectangular boxes) and the ones reported by Taff & Van Horn is excellent.		For three of their models — the ones having <math>x_0 = 3, 6, \And 9</math> — {{ TVH74 }} also determined an eigenvalue for the first harmonic mode of oscillation; see the seventh column of numbers, labeled <math>\lambda_1^2</math>, that appears in the digital image of their "Table II" that we have extracted from their paper and presented in Composite Display 1, above. Via our own [[#Our_Numerical_Integration\|numerical analysis]], we have determined an eigenvalue for the first harmonic mode of oscillation in all nine of the models. These results are presented in the right-hand panel of Composite Display 1, under the heading, "First Harmonic." Again, the agreement between our numerically determined eigenvalues (see the three highlighted by pink, rectangular boxes) and the ones reported by Taff & Van Horn is excellent.

	<span id="CompositeDisplay2">The right-hand column of Composite Display 2, below,</span> presents a pair of animations that display how our numerically derived displacement function, <math>~x(\xi)</math>, varies with radius — from the center of the isothermal sphere, out to <math>~\xi = 9</math> — for a variety of values of the square of the eigenfrequency, <math>~\lambda_0^2</math>; each frame of both animations is tagged by the relevant value of <math>~\lambda_0^2</math>. The segment of the <math>~x(\xi)</math> curve that has been drawn in blue identifies the ''eigenfunction'' that corresponds to the specified value of the eigenfrequency. In each frame, the radial location at which the blue segment terminates simultaneously identifies the truncation radius of the relevant isothermal sphere, and the radius at which the boundary condition, <math>~d\ln x/d\ln\xi = -3</math>, has been enforced. In every frame, the <math>~x(\xi)</math> function has been normalized such that the displacement amplitude is unity at the truncated configuration's surface.		<span id="CompositeDisplay2">The right-hand column of Composite Display 2, below,</span> presents a pair of animations that display how our numerically derived displacement function, <math>~x(\xi)</math>, varies with radius — from the center of the isothermal sphere, out to <math>~\xi = 9</math> — for a variety of values of the square of the eigenfrequency, <math>~\lambda_0^2</math>; each frame of both animations is tagged by the relevant value of <math>~\lambda_0^2</math>. The segment of the <math>~x(\xi)</math> curve that has been drawn in blue identifies the ''eigenfunction'' that corresponds to the specified value of the eigenfrequency. In each frame, the radial location at which the blue segment terminates simultaneously identifies the truncation radius of the relevant isothermal sphere, and the radius at which the boundary condition, <math>~d\ln x/d\ln\xi = -3</math>, has been enforced. In every frame, the <math>~x(\xi)</math> function has been normalized such that the displacement amplitude is unity at the truncated configuration's surface.