User:Tohline/Appendix/Ramblings/BiPolytrope51ContinueSearch - Revision history

Tohline: /* Applied to the Envelope */

2019-05-19T16:04:27Z

Applied to the Envelope

← Older revision		Revision as of 16:04, 19 May 2019
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	<math>~		<math>~
	\frac{1}{\eta^3 \sin^2(\eta-B)} \biggl[\eta^2 + \eta\sin(\eta-B)\cos(\eta-B) -2\sin^2(\eta-B)\biggr]		\frac{1}{\eta^3 \sin^2(\eta-B)} \biggl[\eta^2 + \eta\sin(\eta-B)\cos(\eta-B) -2\sin^2(\eta-B)\biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{\eta\sin^2(\eta - B)} + \frac{\cot(\eta-B)}{\eta^2} - \frac{2}{\eta^3}
	\, ;</math>		\, ;</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{~~1}{\eta^2~~} ~~\frac~~{d~~^2Q_~~\eta}{d\~~eta^2}~~		\frac{d}{d\eta}\biggl[\frac{1}{\eta\sin^2(\eta - B)} + \frac{\cot(\eta-B)}{\eta^2} - \frac{2}{\eta^3} \biggr] </math>
	- \frac{2}{\eta^~~3} \frac{dQ_\eta}{d~~\eta}
	+ \frac{~~6Q_~~\eta}{\eta^4}
	- \frac{2}{\eta^3} \~~frac{dQ_\eta}{d\eta}~~
	</math>
	</td>		</td>
	</tr>		</tr>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{1}{\eta^2} \~~biggl[2~~		\frac{6}{\eta^4} - \frac{2\cot(\eta-B)}{\eta^3} - \frac{2}{\eta^2\sin^2(\eta-B)} - \frac{2\cos(\eta-B)}{\eta \sin^3(\eta-B)}
	-2~~\eta~~ \cot(\eta - B)		\, ,
	~~+ 2~~\~~cot~~^2(\eta ~~- B)~~
	-2\~~eta \cot~~^3(\eta - B)
	~~\biggr]~~
	- \frac{~~4}{~~\~~eta^3} \biggl[ \eta -\cot~~(\eta - B)
	+\eta\~~cot~~^2(\eta - B)
	~~\biggr]~~
	~~+ \frac{6Q_\eta}{\eta^4}~~ \, ,
	</math>		</math>
	</td>		</td>

Tohline: /* Applied to the Envelope */

2019-05-19T14:59:04Z

Applied to the Envelope

← Older revision		Revision as of 14:59, 19 May 2019
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	\biggr]		\biggr]
	- \frac{2Q_\eta}{\eta^3}		- \frac{2Q_\eta}{\eta^3}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{\eta^3 \sin^2(\eta-B)} \biggl[\eta^2 + \eta\sin(\eta-B)\cos(\eta-B) -2\sin^2(\eta-B)\biggr]
	\, ;</math>		\, ;</math>
	</td>		</td>
	</tr>		</tr>

	<tr>		<tr>
	<td align="right">		<td align="right">

Tohline: /* Key Differential Equation */

2019-05-18T21:36:07Z

Key Differential Equation

← Older revision		Revision as of 21:36, 18 May 2019
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	</tr>		</tr>
	</table>		</table>
	where, <math>~\alpha_g \equiv (3 - 4/\gamma_g)</math>.		where, <math>~\alpha_g \equiv (3 - 4/\gamma_g)</math>. Alternatively — see, for example, our [[User:Tohline/SSC/Stability/Polytropes#Numerical_Integration_from_the_Center.2C_Outward\|introductory discussion]] — for polytropic configurations we may write,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~0 </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{d^2x}{d\xi^2} + \biggl[4 - (n+1) \biggl(- \frac{d\ln \theta}{d\ln \xi} \biggr)\biggr] \frac{1}{\xi} \frac{dx}{d\xi} +
			\biggl\{ \frac{(n+1)}{\theta} \biggl[ \frac{\sigma_c^2}{6\gamma_g}\biggr]
			-
			(n+1) \biggl(- \frac{d\ln \theta}{d\ln \xi} \biggr) \frac{\alpha_g}{\xi^2 } \biggr\} x \, .</math>
			</td>
			</tr>
			</table>

	===Applied to the Core===		===Applied to the Core===

Tohline: /* Applied to the Core */

2019-05-18T20:01:32Z

Applied to the Core

← Older revision		Revision as of 20:01, 18 May 2019
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Rightarrow ~~~ 0</math>		<math>~\Rightarrow ~~~ \biggl( \frac{3}{4\pi} \biggr) \cdot 0</math>
	</td>		</td>
	<td align="center">		<td align="center">

Tohline: /* Continue Search for Marginally Unstable (5,1) Bipolytropes */

2019-05-16T17:48:53Z

Continue Search for Marginally Unstable (5,1) Bipolytropes

← Older revision		Revision as of 17:48, 16 May 2019
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	\frac{[\eta^2 - 2\sin^2(\eta-B) + \eta \sin(\eta-B) \cos(\eta - B) ] }{[\sin^2(\eta-B) - \eta \sin(\eta-B) \cos(\eta-B)]} \, .		\frac{[\eta^2 - 2\sin^2(\eta-B) + \eta \sin(\eta-B) \cos(\eta - B) ] }{[\sin^2(\eta-B) - \eta \sin(\eta-B) \cos(\eta-B)]} \, .
	</math>		</math>
			</td>
			</tr>
			</table>

			==Interface Matching==
			According to our [[User:Tohline/SSC/Stability/BiPolytropes#Interface_Conditions\|accompanying discussion]] of the interface matching condition — as we presently understand it — the proper eigenfunction will exhibit a discontinuity in the slope of the dimensionless displacement function such that,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{d\ln x_\mathrm{env}}{d\ln \eta} \biggr\|_{\eta=\eta_i}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~3\biggl(\frac{\gamma_c}{\gamma_e} -1\biggr) + \frac{\gamma_c}{\gamma_e} \biggl( \frac{d\ln x_\mathrm{core}}{d\ln \xi} \biggr)_{\xi=\xi_i} </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{3}{5}\biggl[ \biggl( \frac{d\ln x_\mathrm{core}}{d\ln \xi} \biggr)_{\xi=\xi_i} -2 \biggr] \, .</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Applied to the Envelope */

2019-05-16T17:18:23Z

Applied to the Envelope

← Older revision		Revision as of 17:18, 16 May 2019
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	<td align="left">		<td align="left">
	<math>~		<math>~
	~~\biggl\{~~ \frac{~~\eta~~}{Q_\eta} \biggl[\eta -\cot(\eta - B)		\frac{1}{Q_\eta} \biggl[\eta^2 - \eta \cot(\eta - B)
	+\eta\cot^2(\eta - B)		+\eta^2 \cot^2(\eta - B)
	\biggr]		\biggr]
	- 2 \~~biggr~~\}		- 2
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{[\eta^2 - 2 + \eta \cot(\eta - B)+\eta^2 \cot^2(\eta - B) ] }{[1 - \eta\cot(\eta-B)]}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{[\eta^2 - 2\sin^2(\eta-B) + \eta \sin(\eta-B) \cos(\eta - B) ] }{[\sin^2(\eta-B) - \eta \sin(\eta-B) \cos(\eta-B)]} \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Applied to the Envelope */

2019-05-16T17:01:22Z

Applied to the Envelope

← Older revision		Revision as of 17:01, 16 May 2019
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	</tr>		</tr>
	</table>		</table>
	~~Notice that, if~~ we set <math>~\sigma_c^2 = 0</math> and <math>~\gamma_g = \gamma_e = 2 ~~\Rightarrow ~~ \alpha_g = +1</math>, the envelope LAWE simplifies to the form,		If we set <math>~\sigma_c^2 = 0</math> and <math>~\gamma_g = \gamma_e = 2 ~~\Rightarrow ~~ \alpha_g = +1</math>, the envelope LAWE simplifies to the form,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{1}{\eta^2}\biggl[\eta -\cot(\eta - ~~b_0~~)		\frac{1}{\eta^2}\biggl[\eta -\cot(\eta - B)
	+\eta\cot^2(\eta - ~~b_0~~)		+\eta\cot^2(\eta - B)
	\biggr]		\biggr]
	- \frac{2Q_\eta}{\eta^3}		- \frac{2Q_\eta}{\eta^3}
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	<math>~		<math>~
	\frac{1}{\eta^2} \biggl[2		\frac{1}{\eta^2} \biggl[2
	-2\eta \cot(\eta - ~~b_0~~)		-2\eta \cot(\eta - B)
	+ 2\cot^2(\eta - ~~b_0~~)		+ 2\cot^2(\eta - B)
	-2\eta \cot^3(\eta - ~~b_0~~)		-2\eta \cot^3(\eta - B)
	\biggr]		\biggr]
	- \frac{4}{\eta^3} \biggl[ \eta -\cot(\eta - ~~b_0~~)		- \frac{4}{\eta^3} \biggl[ \eta -\cot(\eta - B)
	+\eta\cot^2(\eta - ~~b_0~~)		+\eta\cot^2(\eta - B)
	\biggr]		\biggr]
	+ \frac{6Q_\eta}{\eta^4} \, ,		+ \frac{6Q_\eta}{\eta^4} \, ,
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	</tr>		</tr>
	</table>		</table>
	and [[User:Tohline/Appendix/Ramblings/BiPolytrope51AnalyticStability#proof\|it can be shown]] that the simplified ''envelope'' LAWE is perfectly satisfied.		and [[User:Tohline/Appendix/Ramblings/BiPolytrope51AnalyticStability#proof\|it can be shown]] that the simplified ''envelope'' LAWE is perfectly satisfied. Notice that, with this adopted segment of the eigenfunction for the envelope, we have,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{d\ln x_P}{d\ln\eta}\biggr\|_\mathrm{env} = \frac{\eta^3}{3c_0 Q_\eta}\cdot \frac{dx_P}{d\eta}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{\eta^3}{Q_\eta} \biggl\{ \frac{1}{\eta^2}\biggl[\eta -\cot(\eta - B)
			+\eta\cot^2(\eta - B)
			\biggr]
			- \frac{2Q_\eta}{\eta^3} \biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl\{ \frac{\eta}{Q_\eta} \biggl[\eta -\cot(\eta - B)
			+\eta\cot^2(\eta - B)
			\biggr]
			- 2 \biggr\}
			</math>
			</td>
			</tr>
			</table>

	=See Also=		=See Also=

Tohline: /* Applied to the Envelope */

2019-05-16T16:48:43Z

Applied to the Envelope

← Older revision		Revision as of 16:48, 16 May 2019
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	</tr>		</tr>
	</table>		</table>
			Notice that, if we set <math>~\sigma_c^2 = 0</math> and <math>~\gamma_g = \gamma_e = 2 ~~\Rightarrow ~~ \alpha_g = +1</math>, the envelope LAWE simplifies to the form,

			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>~ \biggl(\frac{r^*}{\eta}\biggr)^2 \cdot~ \mathrm{LAWE}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{d^2x}{d\eta^2}
			+ \biggl\{ 4 - 2Q_\eta
			\biggr\} \frac{1}{\eta} \frac{dx}{d\eta}
			- \biggl\{ \frac{2Q_\eta}{\eta^2} \biggr\} x \, .
			</math>
			</td>
			</tr>
			</table>


	In [[User:Tohline/Appendix/Ramblings/BiPolytrope51AnalyticStability#Attempt_4B\|yet another ''Ramblings Appendix'' derivation]] we have explored a trial dimensionless displacement for the envelope of the form,		In [[User:Tohline/Appendix/Ramblings/BiPolytrope51AnalyticStability#Attempt_4B\|yet another ''Ramblings Appendix'' derivation]] we have explored a trial dimensionless displacement for the envelope of the form,
Line 537:		Line 558:
	+\eta\cot^2(\eta - b_0)		+\eta\cot^2(\eta - b_0)
	\biggr]		\biggr]
	+ \frac{6Q_\eta}{\eta^4} \, .		+ \frac{6Q_\eta}{\eta^4} \, ,
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
			and [[User:Tohline/Appendix/Ramblings/BiPolytrope51AnalyticStability#proof\|it can be shown]] that the simplified ''envelope'' LAWE is perfectly satisfied.

	=See Also=		=See Also=

Tohline: /* Applied to the Envelope */

2019-05-16T16:34:45Z

Applied to the Envelope

← Older revision		Revision as of 16:34, 16 May 2019
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~= \frac{3c_0}{\eta^2} \cdot Q \, .</math>		<math>~= \frac{3c_0}{\eta^2} \cdot Q_\eta \, .</math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{1}{\eta^2} \frac{dQ}{d\eta} - \frac{2Q}{\eta^3} </math>		<math>~\frac{1}{\eta^2} \frac{dQ_\eta}{d\eta} - \frac{2Q_\eta}{\eta^3} </math>
	</td>		</td>
	</tr>		</tr>
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	+\eta\cot^2(\eta - b_0)		+\eta\cot^2(\eta - b_0)
	\biggr]		\biggr]
	- \frac{2Q}{\eta^3}		- \frac{2Q_\eta}{\eta^3}
	\, ;</math>		\, ;</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{1}{\eta^2} \frac{d^2Q}{d\eta^2}		\frac{1}{\eta^2} \frac{d^2Q_\eta}{d\eta^2}
	- \frac{2}{\eta^3} \frac{dQ}{d\eta}		- \frac{2}{\eta^3} \frac{dQ_\eta}{d\eta}
	+ \frac{6Q}{\eta^4}		+ \frac{6Q_\eta}{\eta^4}
	- \frac{2}{\eta^3} \frac{dQ}{d\eta}		- \frac{2}{\eta^3} \frac{dQ_\eta}{d\eta}
	</math>		</math>
	</td>		</td>
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	+\eta\cot^2(\eta - b_0)		+\eta\cot^2(\eta - b_0)
	\biggr]		\biggr]
	+ \frac{6Q}{\eta^4} \, .		+ \frac{6Q_\eta}{\eta^4} \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Applied to the Envelope */

2019-05-16T16:30:47Z

Applied to the Envelope

← Older revision		Revision as of 16:30, 16 May 2019
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	<td align="left">		<td align="left">
	<math>~- \frac{d\ln \phi}{d\ln\eta} = \biggl[1 - \eta\cot(\eta-B) \biggr] \, .</math>		<math>~- \frac{d\ln \phi}{d\ln\eta} = \biggl[1 - \eta\cot(\eta-B) \biggr] \, .</math>
			</td>
			</tr>
			</table>

			In [[User:Tohline/Appendix/Ramblings/BiPolytrope51AnalyticStability#Attempt_4B\|yet another ''Ramblings Appendix'' derivation]] we have explored a trial dimensionless displacement for the envelope of the form,
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>~x_P\biggr\|_\mathrm{env} </math>
			</td>
			<td align="left">
			<math>~= \frac{3c_0}{\eta^2} \cdot Q \, .</math>
			</td>
			</tr>
			</table>
			In this case,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{1}{3c_0}\cdot \frac{dx_P}{d\eta}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{1}{\eta^2} \frac{dQ}{d\eta} - \frac{2Q}{\eta^3} </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{\eta^2}\biggl[\eta -\cot(\eta - b_0)
			+\eta\cot^2(\eta - b_0)
			\biggr]
			- \frac{2Q}{\eta^3}
			\, ;</math>
			</td>
			</tr>
			<tr>
			<td align="right">
			<math>~\frac{1}{3c_0}\cdot \frac{d^2x_P}{d\eta^2}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{\eta^2} \frac{d^2Q}{d\eta^2}
			- \frac{2}{\eta^3} \frac{dQ}{d\eta}
			+ \frac{6Q}{\eta^4}
			- \frac{2}{\eta^3} \frac{dQ}{d\eta}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{\eta^2} \biggl[2
			-2\eta \cot(\eta - b_0)
			+ 2\cot^2(\eta - b_0)
			-2\eta \cot^3(\eta - b_0)
			\biggr]
			- \frac{4}{\eta^3} \biggl[ \eta -\cot(\eta - b_0)
			+\eta\cot^2(\eta - b_0)
			\biggr]
			+ \frac{6Q}{\eta^4} \, .
			</math>
	</td>		</td>
	</tr>		</tr>