SSC/Stability/InstabilityOnsetOverview - Revision history

Joel2: /* Focus on n = 1 Configurations */

2025-10-03T18:11:21Z

Focus on n = 1 Configurations

← Older revision		Revision as of 14:11, 3 October 2025
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	and the logarithmic derivative of the displacement function,		and the logarithmic derivative of the displacement function,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{d\ln x_P}{d\ln\xi} \biggr\|_{n=1}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			( 1- \tilde\xi \cot\tilde\xi )^{-1}
			\biggl[ -2 + \tilde\xi \cot\tilde\xi + \frac{{\tilde\xi}^2}{\sin^2 \tilde\xi} \biggr]\, .
			</math>
			</td>
			</tr>
			</table>
			<!--
	<table border="1" cellpadding="8" align="center">		<table border="1" cellpadding="8" align="center">
	<tr><td align="center">		<tr><td align="center">
	<font color="red">~~Incorrect~~ Expression!</font>		<font color="red">Originally Typed Expression:</font>
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	<math>~		<math>~
	( 1- \tilde\xi \cot\tilde\xi )^{-1}		( 1- \tilde\xi \cot\tilde\xi )^{-1}
	( \tilde\xi \cot\tilde\xi + \tilde\xi^2 + \tilde\xi^2 \cot^2\tilde\xi -2)\, .		( \tilde\xi \cot\tilde\xi + \tilde\xi^2 + \tilde\xi^2 \cot^2 \tilde\xi -2)\, .
	</math>		</math>
	</td>		</td>
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	----		----

	<font color="red">Expression ~~Corrected~~ (9/30/2025)</font>		<font color="red">Preferable Expression (9/30/2025)</font>
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	</td></tr>		</td></tr>
	</table>		</table>
			-->
	If the polytropic configuration is truncated by the pressure, <math>P_e</math>, of a hot, tenuous external medium, then the solution to the LAWE is subject to the outer boundary condition,		If the polytropic configuration is truncated by the pressure, <math>P_e</math>, of a hot, tenuous external medium, then the solution to the LAWE is subject to the outer boundary condition,
	<div align="center">		<div align="center">

Joel2: /* Focus on n = 1 Configurations */

2025-09-30T22:55:12Z

Focus on n = 1 Configurations

@@ Line 1,400: / Line 1,400: @@
 and the logarithmic derivative of the displacement function,
 <table border="0" cellpadding="5" align="center">
@@ Line 1,418: / Line 1,421: @@
 </table>
 If the polytropic configuration is truncated by the pressure, <math>P_e</math>, of a hot, tenuous external medium, then the solution to the LAWE is subject to the outer boundary condition,

Joel2: /* Focus on n = 1 Configurations */

2025-09-30T17:59:35Z

Focus on n = 1 Configurations

← Older revision		Revision as of 13:59, 30 September 2025
Line 1,353:		Line 1,353:

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

Line 1,393:		Line 1,392:
	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{3}{\xi^2}\biggl[ 1- \xi \cot\xi \biggr] \, .		\frac{3}{\xi^2}\biggl[ 1- \xi \cot\xi \biggr] \, ;
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	</~~div~~>
			and the logarithmic derivative of the displacement function,

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

			<tr>
			<td align="right">
			<math>~\frac{d\ln x_P}{d\ln\xi} \biggr\|_{n=1}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			( 1- \tilde\xi \cot\tilde\xi )^{-1}
			( \tilde\xi \cot\tilde\xi + \tilde\xi^2 + \tilde\xi^2 \cot^2\tilde\xi -2)\, .
			</math>
			</td>
			</tr>
			</table>


	If the polytropic configuration is truncated by the pressure, <math>P_e</math>, of a hot, tenuous external medium, then the solution to the LAWE is subject to the outer boundary condition,		If the polytropic configuration is truncated by the pressure, <math>P_e</math>, of a hot, tenuous external medium, then the solution to the LAWE is subject to the outer boundary condition,
Line 1,407:		Line 1,426:
	<div align="center">		<div align="center">
	<math>- \frac{d\ln x}{d\ln \xi} = \biggl(\frac{3-n}{n+1}\biggr) + \frac{n\sigma_c^2}{6(n+1)} \biggl[\frac{\xi}{\theta^'}\biggr] </math>         at         <math>\xi = \xi_\mathrm{surf} \, .</math><br />		<math>- \frac{d\ln x}{d\ln \xi} = \biggl(\frac{3-n}{n+1}\biggr) + \frac{n\sigma_c^2}{6(n+1)} \biggl[\frac{\xi}{\theta^'}\biggr] </math>         at         <math>\xi = \xi_\mathrm{surf} \, .</math><br />


	</div>		</div>

Joel2: /* Focus on n = 1 Configurations */

2025-09-30T17:53:35Z

Focus on n = 1 Configurations

← Older revision		Revision as of 13:53, 30 September 2025
Line 1,393:		Line 1,393:
	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{3}{\xi^2}\biggl[ 1- \xi \cot\xi \biggr] ~~= 1 + \frac{\xi^2}{15} + \frac{2\xi^4}{315} + \frac{\xi^6}{1575} + \cdots~~ \, .		\frac{3}{\xi^2}\biggl[ 1- \xi \cot\xi \biggr] \, .
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
			</div>

			If the polytropic configuration is truncated by the pressure, <math>P_e</math>, of a hot, tenuous external medium, then the solution to the LAWE is subject to the outer boundary condition,
			<div align="center">
			<math>-\frac{d\ln x}{d\ln\xi} = 3</math>         at         <math>\xi = \tilde\xi \, .</math>
			</div>
			But, for ''isolated'' polytropes — see the [[#Supporting_Scratch_Work\|supporting derivation]] — the sought-after solution is subject to the more conventional boundary condition,
			<div align="center">
			<math>- \frac{d\ln x}{d\ln \xi} = \biggl(\frac{3-n}{n+1}\biggr) + \frac{n\sigma_c^2}{6(n+1)} \biggl[\frac{\xi}{\theta^'}\biggr] </math>         at         <math>\xi = \xi_\mathrm{surf} \, .</math><br />
	</div>		</div>

Joel2: /* Focus on n = 1 Configurations */

2025-09-30T17:49:15Z

Focus on n = 1 Configurations

← Older revision		Revision as of 13:49, 30 September 2025
Line 1,347:		Line 1,347:

	====Focus on n = 1 Configurations====		====Focus on n = 1 Configurations====
			[[SSC/Structure/Polytropes#Primary_E-Type_Solution_2\|Specifically for n = 1 configurations]],
			<div align="center">
			<math>~\Theta_H = \frac{\sin\xi}{\xi} \, .</math>
			</div>

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

			<tr>
			<td align="right">
			<math>~x_P\biggr\|_{n=1}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			-3 \biggl[ \biggl( \frac{1}{\xi \theta}\biggr) \frac{d\theta}{d\xi}\biggr]_{n=1}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{3}{\xi} \biggl( \frac{\xi}{\sin\xi}\biggr) \biggl[\frac{\sin\xi}{\xi^2} - \frac{\cos\xi}{\xi} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{3}{\xi^2}\biggl[ 1- \xi \cot\xi \biggr] = 1 + \frac{\xi^2}{15} + \frac{2\xi^4}{315} + \frac{\xi^6}{1575} + \cdots \, .
			</math>
			</td>
			</tr>
			</table>
			</div>

	==Supporting Scratch Work==		==Supporting Scratch Work==

Joel2: /* Polytropic */

2025-09-30T17:45:58Z

Polytropic

← Older revision		Revision as of 13:45, 30 September 2025
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	Presumably the same logic — and ultimate consequence — applies to all other equilibrium, pressure-truncated polytropic sequences that have <math>~n < 3</math>.		Presumably the same logic — and ultimate consequence — applies to all other equilibrium, pressure-truncated polytropic sequences that have <math>~n < 3</math>.

			====Focus on n = 1 Configurations====

	==Supporting Scratch Work==		==Supporting Scratch Work==

Joel2: /* Configurations Having an Index Less Than Three */

2025-07-13T18:38:22Z

Configurations Having an Index Less Than Three

← Older revision		Revision as of 14:38, 13 July 2025
Line 1,319:		Line 1,319:
	</div>		</div>

	In an [[SSC/Stability/n1PolytropeLAWE#Succinct_Demonstration\|accompanying discussion]] we show that this displacement function precisely satisfies the n = 1, polytropic LAWE when <math>~\sigma_c^2 = 0</math>.		In an [[SSC/Stability/n1PolytropeLAWE#Succinct_Demonstration\|accompanying discussion]] <font color="red"><b>[<== Very Useful Link]</b></font> we show that this displacement function precisely satisfies the n = 1, polytropic LAWE when <math>~\sigma_c^2 = 0</math>.

	Does this mean that at least one configuration along the equilibrium sequence of pressure-truncated, n = 1 polytropes — see [[SSC/Structure/PolytropesEmbedded#Additional.2C_Numerically_Constructed_Polytropic_Configurations\|the right panel of Figure 3 in an accompanying discussion]] — is marginally [dynamically] unstable? The answer is, "No," because, when it is evaluated at the surface <math>~(\tilde\xi)</math> of the truncated configuration, the logarithmic derivative of the displacement function ,		Does this mean that at least one configuration along the equilibrium sequence of pressure-truncated, n = 1 polytropes — see [[SSC/Structure/PolytropesEmbedded#Additional.2C_Numerically_Constructed_Polytropic_Configurations\|the right panel of Figure 3 in an accompanying discussion]] — is marginally [dynamically] unstable? The answer is, "No," because, when it is evaluated at the surface <math>~(\tilde\xi)</math> of the truncated configuration, the logarithmic derivative of the displacement function ,

Joel2: Created page with "FORCETOC =Marginally Unstable Pressure-Truncated Gas Clouds=

Abstract
Joel E. Tohline (April, 2017)

Pressure-Truncated Isothermal Equilibrium SequenceWhen an isotherm..."
2023-12-18T19:04:51Z

Created page with "FORCETOC  =Marginally Unstable Pressure-Truncated Gas Clouds= <table border="1" width="90%" align="center" cellpadding="15"> <tr><td align="center"> <table border="0" cellpadding="8"> <tr><td align="center"> <font size="+2"><b>Abstract</b></font><br />Joel E. Tohline (April, 2017) </td></tr> <tr><td align="left"> right|200px|center|Pressure-Truncated Isothermal Equilibrium SequenceWhen an isotherm..."
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