Appendix/Ramblings/51AnalyticStabilitySynopsis - Revision history

Jet53man: /* Review of Attempt 4B */

2022-07-09T15:44:14Z

Review of Attempt 4B

← Older revision		Revision as of 11:44, 9 July 2022
Line 476:		Line 476:
	</tr>		</tr>
	</table>		</table>

			===Playing Around===

			Evidently, for our chosen example "Amodel2", <math>d\ln Q/d\ln\eta = - 7/100</math> exactly. How can this be?

	=See Also=		=See Also=

	{{ SGFfooter }}		{{ SGFfooter }}

Jet53man: /* Transition at Interface */

2022-07-08T17:09:30Z

Transition at Interface

@@ Line 408: / Line 408: @@
 <math>
 Q_i^{-1} \biggl[\eta_i^2 + (1-Q_i)^2 + Q_i - 1 \biggr] - 2
 </math>
    </td>

Jet53man: /* Transition at Interface */

2022-07-08T16:52:46Z

Transition at Interface

← Older revision		Revision as of 12:52, 8 July 2022
Line 273:		Line 273:

	===Transition at Interface===		===Transition at Interface===

			<table border="1" align="center" width="60%" cellpadding="5"><tr><td align="left">
			Here, as a numerical example, we will adopt the parameters that are relevant to [[SSC/Structure/BiPolytropes/AnalyzeStepFunction#Model_Amodel2\|Amodel2 from an associated discussion]]. For example, <math>(\mu_e/\mu_c) = 0.31</math> and <math>\xi_i = 9.0149598</math>.
			</td></tr></table>

	Under [[Appendix/Ramblings/BiPolytrope51AnalyticStability#Attempt_1\|"Attempt 1" of our accompanying discussion]], we have shown that, at the core/envelope interface (note the following mappings:   <math>b \rightarrow 3c_0</math> and <math>B \rightarrow b_0</math>),		Under [[Appendix/Ramblings/BiPolytrope51AnalyticStability#Attempt_1\|"Attempt 1" of our accompanying discussion]], we have shown that, at the core/envelope interface (note the following mappings:   <math>b \rightarrow 3c_0</math> and <math>B \rightarrow b_0</math>),
Line 297:		Line 301:
	</td>		</td>
	<td align="left" colspan="3">		<td align="left" colspan="3">
	<math>\biggl(\frac{\mu_e}{\mu_c}\biggr) \biggl[\frac{3\xi_i^2 }{3 + \xi_i^2}\biggr] \, ;</math>		<math>\biggl(\frac{\mu_e}{\mu_c}\biggr) \biggl[\frac{3\xi_i^2 }{3 + \xi_i^2}\biggr] = 0.8968919 \, ;</math>
	</td>		</td>
	</tr>		</tr>
Line 309:		Line 313:
	</td>		</td>
	<td align="left" colspan="3">		<td align="left" colspan="3">
	<math>3^{1 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr) \xi_i \biggl[1 + \frac{\xi^2}{3} \biggr]^{-1} </math>		<math>3^{1 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr) \xi_i \biggl[1 + \frac{\xi^2}{3} \biggr]^{-1} = 0.1723205</math>
	</td>		</td>
	</tr>		</tr>
Line 357:		Line 361:
	<td align="left">		<td align="left">
	<math>		<math>
	\frac{3}{5}\biggl(\frac{\mu_e}{\mu_c}\biggr) \biggl[\frac{15-\xi_i^2}{3+\xi_i^2}\biggr] \, .		\frac{3}{5}\biggl(\frac{\mu_e}{\mu_c}\biggr) \biggl[\frac{15-\xi_i^2}{3+\xi_i^2}\biggr] \, ;
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>b_0</math>
			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			\eta_i - \frac{\pi}{2} + \tan^{-1}\biggl[\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}} \biggr] = -0.8592701 \, .
	</math>		</math>
	</td>		</td>

Jet53man: /* Review of Attempt 4B */

2022-07-08T15:47:40Z

Review of Attempt 4B

← Older revision		Revision as of 11:47, 8 July 2022
Line 279:		Line 279:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>\eta_i </math>		<math>\eta_i \cot(\eta_i - b_0)</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 285:		Line 285:
	</td>		</td>
	<td align="left" colspan="3">		<td align="left" colspan="3">
	<math>~~3^{~~1 ~~/ 2}~~ \biggl(\frac{\mu_e}{\mu_c}\biggr) ~~\xi_i~~ \biggl[~~1 +~~ \frac{\xi^2}{3} \biggr]~~^{-1}~~ </math>		<math>1 - \biggl(\frac{\mu_e}{\mu_c}\biggr) \biggl[\frac{3\xi_i^2 }{3 + \xi_i^2}\biggr] </math>
	</td>		</td>
	</tr>		</tr>
Line 291:		Line 291:
	<tr>		<tr>
	<td align="right">		<td align="right">
	~~ ~~		<math>\Rightarrow ~~~ Q_i</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 297:		Line 297:
	</td>		</td>
	<td align="left" colspan="3">		<td align="left" colspan="3">
	<math>\biggl(\frac{\mu_e}{\mu_c}\biggr) \biggl[\frac{3^~~{3 /~~ 2~~}\xi_i~~}{3 + \xi^2} \biggr] \, ;</math>		<math>\biggl(\frac{\mu_e}{\mu_c}\biggr) \biggl[\frac{3\xi_i^2 }{3 + \xi_i^2}\biggr] \, ;</math>
	</td>		</td>
	</tr>		</tr>
Line 303:		Line 303:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>\eta_i ~~\cot(\eta_i - b_0)~~</math>		<math>\eta_i </math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 309:		Line 309:
	</td>		</td>
	<td align="left" colspan="3">		<td align="left" colspan="3">
	<math>1 - \biggl(\frac{\mu_e}{\mu_c}\biggr) \biggl[\frac{3\~~xi_i~~^2 }{3 ~~+ \xi_i^2~~}\biggr] </math>		<math>3^{1 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr) \xi_i \biggl[1 + \frac{\xi^2}{3} \biggr]^{-1} </math>
	</td>		</td>
	</tr>		</tr>
Line 315:		Line 315:
	<tr>		<tr>
	<td align="right">		<td align="right">
	~~<math>\Rightarrow ~~~ Q_i</math>~~
	</td>		</td>
	<td align="center">		<td align="center">
Line 321:		Line 321:
	</td>		</td>
	<td align="left" colspan="3">		<td align="left" colspan="3">
	<math>\biggl(\frac{\mu_e}{\mu_c}\biggr) \biggl[\frac{3\xi_i^2 }{3 + \~~xi_i~~^2}\biggr] \, ;</math>		<math>
			3^{1 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr) \biggl[\frac{3\xi_i}{3 + \xi^2} \biggr]
			=
			\frac{3^{1 / 2}Q_i}{\xi_i}
			\, ;
			</math>
	</td>		</td>
	</tr>		</tr>
Line 353:		Line 358:
	<math>		<math>
	\frac{3}{5}\biggl(\frac{\mu_e}{\mu_c}\biggr) \biggl[\frac{15-\xi_i^2}{3+\xi_i^2}\biggr] \, .		\frac{3}{5}\biggl(\frac{\mu_e}{\mu_c}\biggr) \biggl[\frac{15-\xi_i^2}{3+\xi_i^2}\biggr] \, .
			</math>
			</td>
			</tr>
			</table>

			As viewed from the perspective of the envelope, then,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>\biggl[ \frac{d\ln x_P}{d\ln\eta} \biggr]_i</math>
			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			\biggl[ \frac{d\ln Q}{d\ln \eta}\biggr]_i - 2
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			Q_i^{-1} \biggl[\eta_i^2 + (1-Q_i)^2 + Q_i - 1 \biggr] - 2
	</math>		</math>
	</td>		</td>

Jet53man: /* Transition at Interface */

2022-07-08T15:34:09Z

Transition at Interface

@@ Line 276: / Line 276: @@
 Under [[Appendix/Ramblings/BiPolytrope51AnalyticStability#Attempt_1|"Attempt 1" of our accompanying discussion]], we have shown that, at the core/envelope interface (note the following mappings: &nbsp; <math>b \rightarrow 3c_0</math> and <math>B \rightarrow b_0</math>),
 <table border="0" cellpadding="5" align="center">
 <tr>

Jet53man at 23:34, 7 July 2022

2022-07-07T23:34:24Z

Show changes

Jet53man: /* Transition at Interface */

2022-07-07T23:02:47Z

Transition at Interface

@@ Line 190: / Line 190: @@
 ===Transition at Interface===
 =See Also=
 {{ SGFfooter }}

Jet53man: /* LAWE */

2022-07-07T22:48:50Z

LAWE

← Older revision		Revision as of 18:48, 7 July 2022
Line 97:		Line 97:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\sigma_c^2 = 0</math>		<math>\sigma_c^2 = 0</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 103:		Line 103:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~x_P </math>		<math>x_P </math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~ \equiv \frac{3c_0 (n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{1}{\eta \phi^{n}}\biggr) \frac{d\phi}{d\eta}\biggr] </math>		<math>\equiv \frac{3c_0 (n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{1}{\eta \phi^{n}}\biggr) \frac{d\phi}{d\eta}\biggr] </math>
	</td>		</td>
	</tr>		</tr>
Line 114:		Line 114:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~= -\biggl( \frac{3c_0}{\eta \phi}\biggr) \frac{d\phi}{d\eta} = \frac{3c_0}{\eta^2} \cdot Q \, , </math>		<math>= -\biggl( \frac{3c_0}{\eta \phi}\biggr) \frac{d\phi}{d\eta} = \frac{3c_0}{\eta^2} \cdot Q \, , </math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>

	satisfies the ~~governing LAWE precisely, namely,~~		precisely satisfies the

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

			<tr>
			<td align="center" colspan="3"><font color="maroon"><b>Governing LAWE</b></font></td>
			</tr>
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~0</math>		<math>0</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	\frac{d^2x_P}{d\eta^2} + \biggl[4 - 2Q\biggr]\frac{1}{\eta}\cdot \frac{dx_P}{d\eta} - 2Q\cdot \frac{x_P}{\eta^2} \, .		\frac{d^2x_P}{d\eta^2} + \biggl[4 - 2Q\biggr]\frac{1}{\eta}\cdot \frac{dx_P}{d\eta} - 2Q\cdot \frac{x_P}{\eta^2} \, .
	</math>		</math>
Line 137:		Line 140:
	</tr>		</tr>
	</table>		</table>

			Note for later use that,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>\frac{d\ln x_P}{d\ln\eta} = \frac{\eta}{x_P} \cdot \frac{d}{d\eta}\biggl[ \frac{3c_0}{\eta^2} \cdot Q \biggr]</math>
			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			3c_0\eta \biggl[ \frac{\eta^2}{3c_0\cdot Q} \biggr] \cdot \frac{d}{d\eta}\biggl[ \frac{Q}{\eta^2} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			\biggl[ \frac{\eta^3}{Q} \biggr] \cdot \biggl[ \frac{1}{\eta^2} \frac{dQ}{d\eta} - \frac{2Q}{\eta^3}\biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			\biggl[ \frac{d\ln Q}{d\ln \eta} - 2\biggr] \, .
			</math>
			</td>
			</tr>
			</table>

	While it is rather amazing that we have been able to identify this analytic solution to the LAWE, the solution seems troubling because it blows up at the surface, where, <math>\eta_s - b_0 = \pi</math>. We will ignore this undesired behavior for the time being.		While it is rather amazing that we have been able to identify this analytic solution to the LAWE, the solution seems troubling because it blows up at the surface, where, <math>\eta_s - b_0 = \pi</math>. We will ignore this undesired behavior for the time being.

			===Transition at Interface===

	=See Also=		=See Also=

	{{ SGFfooter }}		{{ SGFfooter }}

Jet53man at 22:31, 7 July 2022

2022-07-07T22:31:46Z

@@ Line 5: / Line 5: @@
 The ideas that are captured in this chapter have arisen after a review of [[Appendix/Ramblings/BiPolytrope51AnalyticStability|a previous hunt for the desired analytic eigenvector]] and as an extension of our [[SSC/Structure/BiPolytropes/Analytic51Renormalize|accompanying "renormalization" of the Analytic51 bipolytrope]].
 =See Also=
 {{ SGFfooter }}

Jet53man: Created page with "FORCETOC =More Focused Search for Analytic EigenVector of (5,1) Bipolytropes= The ideas that are captured in this chapter have arisen after a review of a previous hunt for the desired analytic eigenvector and as an extension of our SSC/Structure/BiPolytropes/Analytic51Renormalize|accompanying "renormalization" of the A..."

2022-07-07T22:10:18Z

Created page with "__FORCETOC__   =More Focused Search for Analytic EigenVector of (5,1) Bipolytropes= The ideas that are captured in this chapter have arisen after a review of a previous hunt for the desired analytic eigenvector and as an extension of our SSC/Structure/BiPolytropes/Analytic51Renormalize|accompanying "renormalization" of the A..."

New page

__FORCETOC__ 

=More Focused Search for Analytic EigenVector of (5,1) Bipolytropes=

The ideas that are captured in this chapter have arisen after a review of [[Appendix/Ramblings/BiPolytrope51AnalyticStability|a previous hunt for the desired analytic eigenvector]] and as an extension of our [[SSC/Structure/BiPolytropes/Analytic51Renormalize|accompanying "renormalization" of the Analytic51 bipolytrope]].

=See Also=

{{ SGFfooter }}