User:Tohline/SSC/Stability/n1PolytropeLAWE - Revision history

Tohline: /* Second Attempt */

2019-04-16T19:31:18Z

Second Attempt

← Older revision		Revision as of 19:31, 16 April 2019
Line 4,706:		Line 4,706:


	Let's work through the analytic derivatives again. Keeping in mind that,		<span id="tagJanuary2019">
			Let's work through the analytic derivatives again. Keeping in mind that,</span>
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Following Up on the Brute-Force Trial Fit */

2019-02-23T22:26:47Z

Following Up on the Brute-Force Trial Fit

← Older revision		Revision as of 22:26, 23 February 2019
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Rightarrow~~~\biggl(\frac{\Lambda^4~~\eta^2~~}{2b_0}\biggr) \cdot</math>  LAWE		<math>~\Rightarrow~~~\biggl(\frac{\Lambda^4}{2b_0}\biggr) \cdot</math>  LAWE
	</td>		</td>
	<td align="center">		<td align="center">
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	<td align="left">		<td align="left">
	<math>~		<math>~
	g_\mathrm{F}~~^2 \eta~~^2 \biggl[ 3 - \Lambda^2		g_\mathrm{F}^2 \biggl[ 3 - \Lambda^2
	- (\Lambda + \Lambda^3)\cot(\Lambda-E)		- (\Lambda + \Lambda^3)\cot(\Lambda-E)
	- \Lambda^2\cot^2(\Lambda - E)		- \Lambda^2\cot^2(\Lambda - E)
Line 6,323:		Line 6,323:
	<math>~		<math>~
	+		+
	g_\mathrm{F} \Lambda \eta		\frac{g_\mathrm{F} \Lambda}{ \eta }
	\biggl[ \Lambda ^2 -2 + \Lambda\cot(\Lambda-E) + \Lambda^2\cot^2(\Lambda - E) \biggr]		\biggl[ \Lambda ^2 -2 + \Lambda\cot(\Lambda-E) + \Lambda^2\cot^2(\Lambda - E) \biggr]
	~-~ \biggl[\Lambda^2 [ 1 - \Lambda \cot(\Lambda - E) ] - \frac{a_0\Lambda^4}{b_0} \biggr]		~-~ \biggl(\frac{\Lambda}{\eta}\biggr)^2\biggl[ 1 - \Lambda \cot(\Lambda - E) - \frac{a_0\Lambda^2}{b_0} \biggr]
	</math>		</math>
	</td>		</td>
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	<math>~		<math>~
	+ \biggl[ \eta\cot(\eta-B) \biggr] \biggl\{		+ \biggl[ \eta\cot(\eta-B) \biggr] \biggl\{
	g_\mathrm{F} \Lambda \eta		\frac{g_\mathrm{F} \Lambda }{\eta }
	\biggl[ \Lambda ^2 -2 + \Lambda\cot(\Lambda-E) + \Lambda^2\cot^2(\Lambda - E) \biggr]		\biggl[ \Lambda ^2 -2 + \Lambda\cot(\Lambda-E) + \Lambda^2\cot^2(\Lambda - E) \biggr]
	~+~ \Lambda^2 [ 1 - \Lambda \cot(\Lambda - E) ] - \frac{a_0\Lambda^4}{b_0} \biggr\}		~+~ \biggl( \frac{\Lambda}{\eta}\biggr)^2 \biggl[ 1 - \Lambda \cot(\Lambda - E) - \frac{a_0\Lambda^2}{b_0 }\biggr] \biggr\}
	</math>		</math>
	</td>		</td>

Tohline: /* Following Up on the Brute-Force Trial Fit */

2019-02-22T23:20:57Z

Following Up on the Brute-Force Trial Fit

← Older revision		Revision as of 23:20, 22 February 2019
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	<table border="1" align="center" cellpadding="8">		<table border="1" align="center" cellpadding="8">
	<tr>		<tr>
	<td align="center" width="40%"><math>~B = \eta_i - \frac{\pi}{2} + \tan^{-1}f</math></td>		<td align="center" width="50%"><math>~B = \eta_i - \frac{\pi}{2} + \tan^{-1}f</math></td>
			<td align="center"><math>~E = \eta_i - \frac{5\pi}{4} + \tan^{-1}f_\alpha</math></td>
	</tr>		</tr>
	<tr>		<tr>
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	<td align="left">		<td align="left">
	<math>~\tan(B + \tfrac{\pi}{2} - \eta_i )</math>		<math>~\tan(B + \tfrac{\pi}{2} - \eta_i )</math>
			</td>
			</tr>
			</table>
			</td>
			<td align="center">
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>~\Rightarrow~~~\cot(\eta_i - E)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\tan[\tfrac{\pi}{2} - (\eta_i - E)]</math>
			</td>
			</tr>
			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\tan[\tfrac{\pi}{2} - (\tfrac{5\pi}{4} - \tan^{-1}f_\alpha)]</math>
			</td>
			</tr>
			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\tan( \tan^{-1}f_\alpha - \tfrac{3\pi}{4} )</math>
			</td>
			</tr>
			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~-\tan( \tfrac{3\pi}{4} - \tan^{-1}f_\alpha )</math>
			</td>
			</tr>
			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~-\cot( \tan^{-1}f_\alpha )</math>
			</td>
			</tr>
			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~-\frac{1}{f_\alpha}</math>
	</td>		</td>
	</tr>		</tr>
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	</tr>		</tr>
	</table>		</table>
			</td>
			<td align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			Hence …     <math>~\cot(\Lambda - E)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\tan[\tfrac{\pi}{2} - (\Lambda - E)]</math>
	</td>		</td>
	</tr>		</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\tan[\eta_i - \Lambda - \tfrac{3\pi}{4} + \tan^{-1}f_\alpha]</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{\tan(\eta_i - \Lambda - \tfrac{3\pi}{4}) + f_\alpha }{1 - f_\alpha \cdot \tan(\eta_i - \Lambda - \tfrac{3\pi}{4}) }</math>
			</td>
			</tr>
			</table>
			</td>
			</tr>
	<tr>		<tr>
	<td align="center">Also …    <math>~B = \eta_s - \pi</math></td>		<td align="center">Also …    <math>~B = \eta_s - \pi</math></td>
			<td align="center"><math>~</math></td>
	</tr>		</tr>
	<tr>		<tr>
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	</table>		</table>
	</td>		</td>
			<td align="center"><math>~</math></td>
	</tr>		</tr>
	</table>		</table>

Tohline at 22:04, 22 February 2019

2019-02-22T22:04:36Z

← Older revision		Revision as of 22:04, 22 February 2019
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	</td></tr></table>		</td></tr></table>
	Using a couple of separate Excel spreadsheets — FaulknerBipolytrope2.xlsx/mu100Mode0 and AnalyticTrialBipolytropeA.xlsx/Sheet2, both stored in a DropBox account under the folder Wiki_edits/Bipolytrope/LinearPerturbation — we used an inelegant and inefficient trial & error technique in search of an eigenfunction that had the same analytic ''form'' as the one represented above for <math>~x_P</math>, but that, when plotted, appeared to qualitatively match the numerically determined envelope eigenfunction. Then, on a whiteboard — see the photo, here on the right — we formulated a concise expression for a trial function that seemed to work pretty well. Our primary finding was that <math>~\alpha</math>, appearing as the argument to the <math>~\tan\alpha</math> function, needed to be shifted by something like <math>~-3\pi/4</math>.		Using a couple of separate Excel spreadsheets — FaulknerBipolytrope2.xlsx/mu100Mode0 and AnalyticTrialBipolytropeA.xlsx/Sheet2, both stored in a DropBox account under the folder Wiki_edits/Bipolytrope/LinearPerturbation — we used an inelegant and inefficient trial & error technique in search of an eigenfunction that had the same analytic ''form'' as the one represented above for <math>~x_P</math>, but that, when plotted, appeared to qualitatively match the numerically determined envelope eigenfunction. Then, on a whiteboard — see the photo, here on the right — we formulated a concise expression for a trial function that seemed to work pretty well. Our primary finding was that <math>~\alpha</math>, appearing as the argument to the <math>~\tan\alpha</math> function, needed to be shifted by something like <math>~-3\pi/4</math>.

			<div align="center">
			<font size="+3">
			<p> </p><p>THIS SPACE</p><p> INTENTIONALLY</p><p> LEFT BLANK</p>
			</font>
			</div>
	</td></tr></table>		</td></tr></table>

Tohline: /* Following Up on the Brute-Force Trial Fit */

2019-02-22T18:16:22Z

Following Up on the Brute-Force Trial Fit

← Older revision		Revision as of 18:16, 22 February 2019
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	</tr>		</tr>
	</table>		</table>


			----

			Here, we reference a [[User:Tohline/SSC/Stability/BiPolytropes#Attempt_1\|separate discussion of the bipolytrope's underlying equilibrium structure]]

			<table border="1" align="center" cellpadding="8">
			<tr>
			<td align="center" width="40%"><math>~B = \eta_i - \frac{\pi}{2} + \tan^{-1}f</math></td>
			</tr>
			<tr>
			<td align="center">
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>~\Rightarrow~~~\cot(\eta_i - B)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\tan[\tfrac{\pi}{2} - (\eta_i - B)]</math>
			</td>
			</tr>
			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\tan[\tfrac{\pi}{2} - (\tfrac{\pi}{2} - \tan^{-1}f)]</math>
			</td>
			</tr>
			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~f</math>
			</td>
			</tr>
			<tr>
			<td align="right">
			<math>~\Rightarrow~~~f</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\tan(B + \tfrac{\pi}{2} - \eta_i )</math>
			</td>
			</tr>
			</table>
			</td>
			</tr>
			<tr>
			<td align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			Hence …     <math>~\cot(\eta - B)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\tan[\tfrac{\pi}{2} - (\eta - B)]</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\tan[\tfrac{\pi}{2} - \eta + \eta_i - \tfrac{\pi}{2} + \tan^{-1}f]</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\tan[\eta_i - \eta + \tan^{-1}f]</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{ \tan(\eta_i-\eta) + f }{ 1 - f \cdot \tan(\eta_i - \eta)}</math>
			</td>
			</tr>
			</table>
			</td>
			</tr>
			<tr>
			<td align="center">Also …    <math>~B = \eta_s - \pi</math></td>
			</tr>
			<tr>
			<td align="center">
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~ f = \cot(\eta_i - B)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\cot(\eta_i - \eta_s + \pi)</math>
			</td>
			</tr>
			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~ \frac{1}{f}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\tan(\eta_i - \eta_s + \pi)</math>
			</td>
			</tr>
			</table>
			</td>
			</tr>
			</table>


			----

	Let's examine the first and second derivatives of this trial eigenfunction, recognizing that,		Let's examine the first and second derivatives of this trial eigenfunction, recognizing that,

Tohline: /* Third Attempt */

2019-02-22T17:26:58Z

Third Attempt

← Older revision		Revision as of 17:26, 22 February 2019
Line 5,639:		Line 5,639:

	====Brute-Force Trial Fit====		====Brute-Force Trial Fit====
			<table align="left" width="100%" cellpadding="0"><tr><td align="left">
	<table border="0" align="right"><tr><td align="center">		<table border="0" align="right"><tr><td align="center">
	[[File:BruteForceWhiteBoardsmall.png\|500px\|Photo of white board with steps showing development of trial eigenfunction. This should be paired with an Excel spreadsheet.]]		[[File:BruteForceWhiteBoardsmall.png\|500px\|Photo of white board with steps showing development of trial eigenfunction. This should be paired with an Excel spreadsheet.]]
	</td></tr></table>		</td></tr></table>
	Using a couple of separate Excel spreadsheets — FaulknerBipolytrope2.xlsx/mu100Mode0 and AnalyticTrialBipolytropeA.xlsx/Sheet2, both stored in a DropBox account under the folder Wiki_edits/Bipolytrope/LinearPerturbation — we used an inelegant and inefficient trial & error technique in search of an eigenfunction had the same analytic ''form'' as the one represented above for <math>~x_P</math>, but that, when plotted, appeared to qualitatively match the numerically determined envelope eigenfunction. Then, on a whiteboard — see the photo, here on the right — we formulated a concise expression for a trial function that seemed to work pretty well. Our primary finding was that <math>~\alpha</math>, appearing as the argument to the <math>~\tan\alpha</math> function, needed to be shifted by something like <math>~-3\pi/4</math>.		Using a couple of separate Excel spreadsheets — FaulknerBipolytrope2.xlsx/mu100Mode0 and AnalyticTrialBipolytropeA.xlsx/Sheet2, both stored in a DropBox account under the folder Wiki_edits/Bipolytrope/LinearPerturbation — we used an inelegant and inefficient trial & error technique in search of an eigenfunction that had the same analytic ''form'' as the one represented above for <math>~x_P</math>, but that, when plotted, appeared to qualitatively match the numerically determined envelope eigenfunction. Then, on a whiteboard — see the photo, here on the right — we formulated a concise expression for a trial function that seemed to work pretty well. Our primary finding was that <math>~\alpha</math>, appearing as the argument to the <math>~\tan\alpha</math> function, needed to be shifted by something like <math>~-3\pi/4</math>.
			</td></tr></table>

	====Following Up on the Brute-Force Trial Fit====		====Following Up on the Brute-Force Trial Fit====

Tohline: /* Brute-Force Trial Fit */

2019-02-22T17:20:58Z

Brute-Force Trial Fit

← Older revision		Revision as of 17:20, 22 February 2019
Line 5,639:		Line 5,639:

	====Brute-Force Trial Fit====		====Brute-Force Trial Fit====
	<~~div~~ align="center">		<table border="0" align="right"><tr><td align="center">
	[[File:BruteForceWhiteBoardsmall.png\|Photo of white board with steps showing development of trial eigenfunction. This should be paired with an Excel spreadsheet.]]		[[File:BruteForceWhiteBoardsmall.png\|500px\|Photo of white board with steps showing development of trial eigenfunction. This should be paired with an Excel spreadsheet.]]
	</~~div~~>		</td></tr></table>
			Using a couple of separate Excel spreadsheets — FaulknerBipolytrope2.xlsx/mu100Mode0 and AnalyticTrialBipolytropeA.xlsx/Sheet2, both stored in a DropBox account under the folder Wiki_edits/Bipolytrope/LinearPerturbation — we used an inelegant and inefficient trial & error technique in search of an eigenfunction had the same analytic ''form'' as the one represented above for <math>~x_P</math>, but that, when plotted, appeared to qualitatively match the numerically determined envelope eigenfunction. Then, on a whiteboard — see the photo, here on the right — we formulated a concise expression for a trial function that seemed to work pretty well. Our primary finding was that <math>~\alpha</math>, appearing as the argument to the <math>~\tan\alpha</math> function, needed to be shifted by something like <math>~-3\pi/4</math>.

	====Following Up on the Brute-Force Trial Fit====		====Following Up on the Brute-Force Trial Fit====

Tohline: /* Third Attempt */

2019-02-22T01:46:06Z

Third Attempt

← Older revision		Revision as of 01:46, 22 February 2019
Line 5,637:		Line 5,637:

	Okay. Now let's determine at what value of <math>~\eta</math> the logarithmic derivative of <math>~x_P</math> goes to negative one.		Okay. Now let's determine at what value of <math>~\eta</math> the logarithmic derivative of <math>~x_P</math> goes to negative one.

			====Brute-Force Trial Fit====
			<div align="center">
			[[File:BruteForceWhiteBoardsmall.png\|Photo of white board with steps showing development of trial eigenfunction. This should be paired with an Excel spreadsheet.]]
			</div>


	====Following Up on the Brute-Force Trial Fit====		====Following Up on the Brute-Force Trial Fit====

Tohline: /* Following Up on the Brute-Force Trial Fit */

2019-02-21T22:35:44Z

Following Up on the Brute-Force Trial Fit

← Older revision		Revision as of 22:35, 21 February 2019
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	</td>		</td>
	</tr>		</tr>
			</table>
			<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
			<math>~\Rightarrow~~~\biggl(\frac{\Lambda^4\eta^2}{2b_0}\biggr) \cdot</math>  LAWE
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~~~\frac{2b_0}{\Lambda^4\eta^2} \biggl\{~~		<math>~
	g_\mathrm{F}^2 \eta^2 \biggl[ 3 - \Lambda^2		g_\mathrm{F}^2 \eta^2 \biggl[ 3 - \Lambda^2
	- (\Lambda + \Lambda^3)\cot(\Lambda-E)		- (\Lambda + \Lambda^3)\cot(\Lambda-E)
Line 6,061:		Line 6,063:
	<td align="left">		<td align="left">
	<math>~		<math>~
	+ \biggl[ \eta\cot(\eta-B) \biggr] \~~cdot~~		+ \biggl[ \eta\cot(\eta-B) \biggr] \biggl\{
	g_\mathrm{F} \Lambda \eta		g_\mathrm{F} \Lambda \eta
	\biggl[ \Lambda ^2 -2 + \Lambda\cot(\Lambda-E) + \Lambda^2\cot^2(\Lambda - E) \biggr]		\biggl[ \Lambda ^2 -2 + \Lambda\cot(\Lambda-E) + \Lambda^2\cot^2(\Lambda - E) \biggr]
	~~</math>~~		~+~ \Lambda^2 [ 1 - \Lambda \cot(\Lambda - E) ] - \frac{a_0\Lambda^4}{b_0} \biggr\}
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~ ~~
	~~</td>~~
	~~<td align="center">~~
	~~ ~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~+~ ~~\biggl[\eta\cot(\eta-B) \biggr]~~
	~~\biggl[~~\Lambda^2 [ 1 - \Lambda \cot(\Lambda - E) ] - \frac{a_0\Lambda^4}{b_0} ~~\biggr]~~
	\biggr\}
	</math>		</math>
	</td>		</td>

Tohline: /* Following Up on the Brute-Force Trial Fit */

2019-02-21T18:04:33Z

Following Up on the Brute-Force Trial Fit

Show changes