User:Tohline/Appendix/Ramblings/BiPolytrope51AnalyticStability - Revision history

Tohline: /* Attempt 4B */

2019-05-16T16:46:31Z

Attempt 4B

← Older revision		Revision as of 16:46, 16 May 2019
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	</tr>		</tr>
	</table>		</table>
	So the relevant LAWE becomes,		<span id="proof">So the relevant LAWE becomes,</span>

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

Tohline: /* Illustration */

2019-05-02T22:22:45Z

Illustration

← Older revision		Revision as of 22:22, 2 May 2019
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	The solid-purple, x<sub>P</sub>-''intercept'' curve that appears in the above figure/animation is defined by this function. Notice that this function never exceeds unity. This presumably means that if the tangent-based <math>~x_P</math> eigenfunction is the correct solution to the envelope's LAWE, then the dimensionless radius, <math>~\eta_s</math>, of the bipolytrope must never exceed unity.		The solid-purple, x<sub>P</sub>-''intercept'' curve that appears in the above figure/animation is defined by this function. Notice that this function never exceeds unity. This presumably means that if the tangent-based <math>~x_P</math> eigenfunction is the correct solution to the envelope's LAWE, then the dimensionless radius, <math>~\eta_s</math>, of the bipolytrope must never exceed unity.

			The following diagram — the original of which appears in our [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Parameter_Values\|accompanying discussion of the equilibrium properties of bipolytropic configurations having]] <math>~(n_c, n_e) = (5, 1)</math> — shows how <math>~\eta_i</math> (purple curve) and <math>~\eta_s</math> (green curve) vary with the interface location <math>~\xi_i</math> (ordinate). The solid yellow circular markers (with black edges) identify where the logarithmic derivative of the dimensionless displacement function, <math>~d\ln x_P/d\ln\eta</math>, equals negative one. If <math>~x_P</math> is the correct eigenfunction for the marginally unstable bipolytropic configuration, one of these yellow circular markers should coincide with the green curve, that is, it should be associated with the configuration's surface. Since the curve identified by the yellow circular markers does not appear to intersect the green curve, we conclude that we have not yet identified the correct eigenfunction.

			<div align="center">
			[[File:Bipolytrope51Boundaries03b.png\|500px\|Bipolytrope Properties]]
			</div>

	=====Is This Compatible With LAWE=====		=====Is This Compatible With LAWE=====

Tohline: /* Is This Compatible With LAWE */

2019-04-26T22:14:29Z

Is This Compatible With LAWE

← Older revision		Revision as of 22:14, 26 April 2019
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl[ 1 + \eta\tan(\eta-B_0 - \tfrac{\pi}{2} \pm m\pi) \biggr]</math>		<math>~\biggl[ 1 + \eta\tan(\eta-B_0 - \tfrac{\pi}{2} \pm m\pi) \biggr] \, ,</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
			in which case we have,

	<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>~
	+ \biggl[\~~cos~~(\eta - ~~B)+~~ \~~eta~~\~~sin(~~\~~eta-B~~) \biggr] \cdot \biggl[ \cos^2(\eta-B) - \eta^2 \biggr]		+ \biggl[ 1 + \eta\tan(\eta-B_0 - \tfrac{\pi}{2} \pm m\pi) \biggr] \cdot \biggl[ \cos^3(\eta-B) - \eta^2\cos(\eta-B) \biggr] \, .
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
			As we have just shown, above, in the context of a "<font color="red">'''Quick Check'''</font>", the expression on the RHS will go to zero if we adopt the transformation, <math>~[B_0 + \tfrac{\pi}{2} \mp m\pi] \rightarrow B</math>. Does this help shift the coordinate, <math>~\eta</math>?

	=See Also=		=See Also=

Tohline: /* Is This Compatible With LAWE */

2019-04-26T21:46:07Z

Is This Compatible With LAWE

← Older revision		Revision as of 21:46, 26 April 2019
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Rightarrow ~~~\frac{\eta^4\cos^3(\eta-B)}{6c_1} </math>  LAWE		<math>~\Rightarrow ~~~\biggl[ \frac{\eta^4\cos^3(\eta-B)}{6c_1} \biggr] \cdot</math>  LAWE
	</td>		</td>
	<td align="center">		<td align="center">
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	+ \eta^2 \cos(\eta-B) + \eta^3 \sin(\eta-B)		+ \eta^2 \cos(\eta-B) + \eta^3 \sin(\eta-B)
	+ Q_0\biggl[ \cos^3(\eta-B) - \eta^2\cos(\eta - B) \biggr]		+ Q_0\biggl[ \cos^3(\eta-B) - \eta^2\cos(\eta - B) \biggr]
			</math>
			</td>
			</tr>
			</table>

			<table border="1" width="85%" cellpadding="10" align="center"><tr><td align="left">
			<font color="red">'''Quick Check:'''</font>   Now, if we set <math>~Q_0 = 1 + \eta\tan(\eta-B)</math>, these RHS terms should sum to zero. Let's check.

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

			<tr>
			<td align="right">
			<math>~\biggl[ \frac{\eta^4\cos^3(\eta-B)}{6c_1} \biggr] \cdot</math>  LAWE
			</td>
			<td align="center">
			<math>~\rightarrow</math>
			</td>
			<td align="left">
			<math>~
			-\cos^3(\eta-B)
			- \eta \sin(\eta-B)\cos^2(\eta-B)
			+ \eta^2 \cos(\eta-B) + \eta^3 \sin(\eta-B)
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			+ \biggl[\cos(\eta - B)+ \eta\sin(\eta-B) \biggr] \cdot \biggl[ \cos^2(\eta-B) - \eta^2 \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			0 \, .
			</math>
			</td>
			</tr>
			</table>

			Excellent!
			</td></tr></table>


			Now, let's plug in the expression for the ''structural'' <math>~Q_0</math>. Specifically, we want,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~Q_0</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~\biggl[ 1 - \eta\cot(\eta-B_0) \biggr]</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~\biggl[ 1 + \eta\tan(\eta-B_0 - \tfrac{\pi}{2} \pm m\pi) \biggr]</math>
			</td>
			</tr>
			</table>


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

			<tr>
			<td align="right">
			<math>~\biggl[ \frac{\eta^4\cos^3(\eta-B)}{6c_1} \biggr] \cdot</math>  LAWE
			</td>
			<td align="center">
			<math>~\rightarrow</math>
			</td>
			<td align="left">
			<math>~
			-\cos^3(\eta-B)
			- \eta \sin(\eta-B)\cos^2(\eta-B)
			+ \eta^2 \cos(\eta-B) + \eta^3 \sin(\eta-B)
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			+ \biggl[\cos(\eta - B)+ \eta\sin(\eta-B) \biggr] \cdot \biggl[ \cos^2(\eta-B) - \eta^2 \biggr]
	</math>		</math>
	</td>		</td>

Tohline: /* Is This Compatible With LAWE */

2019-04-26T21:18:28Z

Is This Compatible With LAWE

← Older revision		Revision as of 21:18, 26 April 2019
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	<td align="left">		<td align="left">
	<math>~		<math>~
	- Q_0\biggl\{ \cos(\eta - B)~~\biggl[\eta^2~~ -2\cos^2(\eta-B) - \eta\sin(\eta-B)\cos(\eta-B)\biggr]		- Q_0\biggl\{ \biggl[\eta^2\cos(\eta - B) -2\cos^3(\eta-B) - \eta\sin(\eta-B)\cos^2(\eta-B)\biggr]
	+\cos^3(\eta - B) \~~biggl[1 +~~ \eta\~~tan~~(\eta-B)\biggr]		+ \biggl[\cos^3(\eta - B) + \eta\sin(\eta-B)\cos^2(\eta - B)\biggr]
	\biggr\}		\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			-\cos^3(\eta-B)
			- \eta \sin(\eta-B)\cos^2(\eta-B)
			+ \eta^2 \cos(\eta-B) + \eta^3 \sin(\eta-B)
			+ Q_0\biggl[ \cos^3(\eta-B) - \eta^2\cos(\eta - B) \biggr]
	</math>		</math>
	</td>		</td>

Tohline: /* Is This Compatible With LAWE */

2019-04-26T21:12:39Z

Is This Compatible With LAWE

← Older revision		Revision as of 21:12, 26 April 2019
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	+ \biggl[ 2 - Q_0 \biggr]\cos(\eta - B)\biggl[\eta^2 -2\cos^2(\eta-B) - \eta\sin(\eta-B)\cos(\eta-B)\biggr]		+ \biggl[ 2 - Q_0 \biggr]\cos(\eta - B)\biggl[\eta^2 -2\cos^2(\eta-B) - \eta\sin(\eta-B)\cos(\eta-B)\biggr]
	- Q_0 \cos^3(\eta - B) \biggl[1 + \eta\tan(\eta-B)\biggr]		- Q_0 \cos^3(\eta - B) \biggl[1 + \eta\tan(\eta-B)\biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			3\cos^3(\eta-B)
			+ \eta \sin(\eta-B)\cos^2(\eta-B)
			- \eta^2 \cos(\eta-B) + \eta^3 \sin(\eta-B)
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			-4\cos^3(\eta-B) - 2\eta\sin(\eta-B)\cos^2(\eta-B) + 2\eta^2\cos(\eta - B)
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			- Q_0\biggl\{ \cos(\eta - B)\biggl[\eta^2 -2\cos^2(\eta-B) - \eta\sin(\eta-B)\cos(\eta-B)\biggr]
			+\cos^3(\eta - B) \biggl[1 + \eta\tan(\eta-B)\biggr]
			\biggr\}
	</math>		</math>
	</td>		</td>

Tohline: /* Is This Compatible With LAWE */

2019-04-26T21:08:10Z

Is This Compatible With LAWE

← Older revision		Revision as of 21:08, 26 April 2019
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	+ \biggl[ 2 - Q_0 \biggr] \frac{2}{\eta}\cdot \frac{3c_1}{\eta^3\cos^2(\eta-B)} \biggl[\eta^2 -2\cos^2(\eta-B) - \eta\sin(\eta-B)\cos(\eta-B)\biggr]		+ \biggl[ 2 - Q_0 \biggr] \frac{2}{\eta}\cdot \frac{3c_1}{\eta^3\cos^2(\eta-B)} \biggl[\eta^2 -2\cos^2(\eta-B) - \eta\sin(\eta-B)\cos(\eta-B)\biggr]
	- \frac{2 Q_0}{\eta^2} \cdot \frac{3c_1 }{\eta^2}\biggl[1 + \eta\tan(\eta-B)\biggr]		- \frac{2 Q_0}{\eta^2} \cdot \frac{3c_1 }{\eta^2}\biggl[1 + \eta\tan(\eta-B)\biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~\frac{\eta^4\cos^3(\eta-B)}{6c_1} </math>  LAWE
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[ 3\cos^3(\eta-B)
			+ \eta \sin(\eta-B)\cos^2(\eta-B)
			- \eta^2 \cos(\eta-B) + \eta^3 \sin(\eta-B)
			\biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			+ \biggl[ 2 - Q_0 \biggr]\cos(\eta - B)\biggl[\eta^2 -2\cos^2(\eta-B) - \eta\sin(\eta-B)\cos(\eta-B)\biggr]
			- Q_0 \cos^3(\eta - B) \biggl[1 + \eta\tan(\eta-B)\biggr]
	</math>		</math>
	</td>		</td>

Tohline: /* Attempt 5 */

2019-04-26T21:03:11Z

Attempt 5

← Older revision		Revision as of 21:03, 26 April 2019
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{3c_1}{\eta^4\cos^3(\eta-B)} \biggl[ 6\cos^3(\eta-B)		<math>~
			\frac{3c_1}{\eta^4\cos^3(\eta-B)} \biggl[ 6\cos^3(\eta-B)
	+ 2\eta \sin(\eta-B)\cos^2(\eta-B)		+ 2\eta \sin(\eta-B)\cos^2(\eta-B)
	- 2 \eta^2 \cos(\eta-B) + 2\eta^3 \sin(\eta-B)		- 2 \eta^2 \cos(\eta-B) + 2\eta^3 \sin(\eta-B)
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	<math>~		<math>~
	\frac{d^2x_P}{d\eta^2} + \biggl[ 2 - Q_0 \biggr] \frac{2}{\eta}\cdot \frac{dx_P}{d\eta} -2 Q_0 \cdot \frac{x_P}{\eta^2}		\frac{d^2x_P}{d\eta^2} + \biggl[ 2 - Q_0 \biggr] \frac{2}{\eta}\cdot \frac{dx_P}{d\eta} -2 Q_0 \cdot \frac{x_P}{\eta^2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~</math>  LAWE
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{3c_1}{\eta^4\cos^3(\eta-B)} \biggl[ 6\cos^3(\eta-B)
			+ 2\eta \sin(\eta-B)\cos^2(\eta-B)
			- 2 \eta^2 \cos(\eta-B) + 2\eta^3 \sin(\eta-B)
			\biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			+ \biggl[ 2 - Q_0 \biggr] \frac{2}{\eta}\cdot \frac{3c_1}{\eta^3\cos^2(\eta-B)} \biggl[\eta^2 -2\cos^2(\eta-B) - \eta\sin(\eta-B)\cos(\eta-B)\biggr]
			- \frac{2 Q_0}{\eta^2} \cdot \frac{3c_1 }{\eta^2}\biggl[1 + \eta\tan(\eta-B)\biggr]
	</math>		</math>
	</td>		</td>

Tohline: /* Is This Compatible With LAWE */

2019-04-26T20:56:12Z

Is This Compatible With LAWE

← Older revision		Revision as of 20:56, 26 April 2019
Line 4,293:		Line 4,293:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~Q</math>		<math>~Q_0</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~- \biggl[ \frac{1}{\sin(\eta - B)} \biggr]\biggl[ \eta\cos(\eta-B) - \sin(\eta-B) \biggr]</math>		<math>~- \biggl[ \frac{1}{\sin(\eta - B_0)} \biggr]\biggl[ \eta\cos(\eta-B_0) - \sin(\eta-B_0) \biggr]</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl[1 - \eta\cot(\eta-B) \biggr] \, .</math>		<math>~\biggl[1 - \eta\cot(\eta-B_0) \biggr] \, .</math>
			</td>
			</tr>
			</table>

			Hence, after recognizing that for this specific case, <math>~\alpha_g = +1</math>, we have,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			LAWE
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{d^2x_P}{d\eta^2} + \biggl[ 2 - Q_0 \biggr] \frac{2}{\eta}\cdot \frac{dx_P}{d\eta} -2 Q_0 \cdot \frac{x_P}{\eta^2}
			</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* New Strategy */

2019-04-26T20:49:53Z

New Strategy

← Older revision		Revision as of 20:49, 26 April 2019
Line 3,924:		Line 3,924:
	+ \eta^2 \biggl[ 2\cos(\eta-B) - \cos^3(\eta-B)+ \sin^2(\eta-B)\cos(\eta-B)\biggr]		+ \eta^2 \biggl[ 2\cos(\eta-B) - \cos^3(\eta-B)+ \sin^2(\eta-B)\cos(\eta-B)\biggr]
	\biggr\}		\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{3c_1}{\eta^4\cos^3(\eta-B)} \biggl\{6\cos^3(\eta-B)
			+ \eta^3 \biggl[2\sin(\eta-B)\biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			+ \eta \biggl[ 4\sin(\eta-B)\cos^2(\eta-B) - \sin(\eta-B)\cos^2(\eta-B) - 4\sin(\eta-B)\cos^2(\eta-B) +3 \sin(\eta-B)\cos^2(\eta-B) \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

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

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{3c_1}{\eta^4\cos^3(\eta-B)} \biggl[ 6\cos^3(\eta-B)
			+ 2\eta \sin(\eta-B)\cos^2(\eta-B)
			- 2 \eta^2 \cos(\eta-B) + 2\eta^3 \sin(\eta-B)
			\biggr] \, .
	</math>		</math>
	</td>		</td>