User:Tohline/Appendix/Ramblings/Azimuthal Distortions - Revision history

Tohline: Type in cautionary introductory paragraph

2016-05-24T20:56:59Z

Type in cautionary introductory paragraph

← Older revision		Revision as of 20:56, 24 May 2016
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	<!-- __NOTOC__ will force TOC off -->		<!-- __NOTOC__ will force TOC off -->
	=Analyzing Azimuthal Distortions=		=Analyzing Azimuthal Distortions=

			<font color="red"><b>[Comment by J. E. Tohline on 24 May 2016]</b></font>   This chapter contains a set of technical notes and accompanying discussion that I put together several months ago as I was trying to gain a foundational understanding of the results of a large study of instabilities in self-gravitating tori published by the Imamura & Hadley collaboration. I have come to appreciate that some of the logic and interpretation of published results that are presented, below, has serious flaws. Therefore, anyone reading this should be quite cautious in deciding what subsections provide useful insight. I have written a separate chapter titled, "[[User:Tohline/Apps/ImamuraHadleyCollaboration#Characteristics_of_Unstable_Eigenvectors_in_Self-Gravitating_Tori\|Characteristics of Unstable Eigenvectors in Self-Gravitating Tori]]," that contains a much more trustworthy analysis of this very interesting problem.

			{{LSU_WorkInProgress}}


	{{LSU_HBook_header}}		{{LSU_HBook_header}}

Tohline: /* Adopted Notation */ Correct Kojima link

2016-03-19T00:01:52Z

Adopted Notation: Correct Kojima link

← Older revision		Revision as of 00:01, 19 March 2016
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	</div>		</div>

	Adopting [http://adsabs.harvard.edu/abs/1986PThPh..75.~~1464K~~ Kojima's (1986)] notation, that is, defining,		Adopting [http://adsabs.harvard.edu/abs/1986PThPh..75..251K Kojima's (1986)] notation, that is, defining,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Application to Specific Models */ Insert better model fit to E1

2016-02-15T21:38:06Z

Application to Specific Models: Insert better model fit to E1

← Older revision		Revision as of 21:38, 15 February 2016
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	<td align="center" colspan="2"><math>~\ln[f_2($A$1)]=+4.48864</math></td>		<td align="center" colspan="2"><math>~\ln[f_2($A$1)]=+4.48864</math></td>
	<td align="center" colspan="1" rowspan="5">		<td align="center" colspan="1" rowspan="5">
	[[File:~~ImamuraPaper2MyP1mirror~~.png\|380px\|right\|Empirically constructed P1]]		[[File:ImamuraPaper2MyP1better.png\|380px\|right\|Empirically constructed P1]]
	</td>		</td>
	</tr>		</tr>
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	<td align="center"><math>~2</math></td>		<td align="center"><math>~2</math></td>
	<td align="center"><math>~2.55</math></td>		<td align="center"><math>~2.55</math></td>
	<td align="center"><math>~0.~~048~~</math></td>		<td align="center"><math>~0.094</math></td>
	<td align="center"><math>~\tfrac{\pi}{2}</math></td>		<td align="center"><math>~\tfrac{\pi}{2}</math></td>
	</tr>		</tr>

Tohline: /* Application to Specific Models */

2016-02-15T21:27:12Z

Application to Specific Models

← Older revision		Revision as of 21:27, 15 February 2016
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	<td align="left" colspan="5">		<td align="left" colspan="5">
	<sup>&Dagger;</sup>Note the following:		<sup>&Dagger;</sup>Note the following:
	* When the Paper II radial eigenfunction possesses more than one local minimum — and, in tandem, the Paper II phase function swings through more than <math>~2\pi</math> radians — we have set the parameter, <math>~r_+</math>, to a value less than <math>~r_\mathrm{outer}</math> because our empirically constructed eigenfunction is only designed to accommodate a single (the innermost) function minimum.		* When the Paper II radial eigenfunction possesses more than one local minimum — and, usually in tandem, the Paper II phase function swings through more than <math>~2\pi</math> radians — we have set the parameter, <math>~r_+</math>, to a value less than <math>~r_\mathrm{outer}</math> because our empirically constructed eigenfunction is only designed to accommodate a single (the innermost) function minimum.
	* Our empirical model's 5 adjustable free parameters are highlighted in blue.		* Our empirical model's 5 adjustable free parameters are highlighted in blue.
	* The undesirable kink that appears in our empirically constructed radial eigenfunction (to the left of the function minimum) arises because the exponent, <math>~p_\mathrm{green}</math>, is a fraction that is less than unity.
	* Before plotting our empirically constructed phase function, the tabulated phase shift, <math>~\phi_0</math>, has been added to both <math>~\phi_\mathrm{blue}</math> and <math>~\phi_\mathrm{green}</math> in an effort to match the overall orientation of the "constant phase locus" published in Paper II. (This is not considered to be an adjustable free parameter.)		* Before plotting our empirically constructed phase function, the tabulated phase shift, <math>~\phi_0</math>, has been added to both <math>~\phi_\mathrm{blue}</math> and <math>~\phi_\mathrm{green}</math> in an effort to match the overall orientation of the "constant phase locus" published in Paper II. (This is not considered to be an adjustable free parameter.)
	</td>		</td>

Tohline: /* Application to Specific Models */ Finish typing in numerical parameter values for Table 7

2016-02-15T21:24:29Z

Application to Specific Models: Finish typing in numerical parameter values for Table 7

← Older revision		Revision as of 21:24, 15 February 2016
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	<td align="center"><math>~1.35</math></td>		<td align="center"><math>~1.35</math></td>
	<td align="center"><math>~1.6</math></td>		<td align="center"><math>~1.6</math></td>
	<td align="center"><math>~0.~~005~~</math></td>		<td align="center"><math>~0.003135</math></td>
	</tr>		</tr>
	<tr>		<tr>
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	</tr>		</tr>
	<tr>		<tr>
	<td align="center"><math>~$A$77 = 1.~~085~~</math></td>		<td align="center"><math>~$A$100 = 1.00515</math></td>
	<td align="center"><math>~1.2</math></td>		<td align="center"><math>~1.0</math></td>
	<td align="center"><math>~$A$55 = 0.~~975~~</math></td>		<td align="center"><math>~$A$96 = 0.99261</math></td>
	<td align="center"><math>~0.75</math></td>		<td align="center"><math>~0.35</math></td>
	</tr>		</tr>
	<tr>		<tr>
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	</tr>		</tr>
	<tr>		<tr>
	<td align="center"><math>~$A$71 = 1.~~055~~</math></td>		<td align="center"><math>~$A$96 = 0.99261</math></td>
	<td align="center"><math>~-1.~~34494~~</math></td>		<td align="center"><math>~-3.06805</math></td>
	<td align="center" colspan="2"><math>~\ln[f_2($A$1)]=+5.~~19688~~</math></td>		<td align="center" colspan="2"><math>~\ln[f_2($A$1)]=+4.48864</math></td>
	<td align="center" colspan="1" rowspan="5">		<td align="center" colspan="1" rowspan="5">
	[[File:ImamuraPaper2MyP1mirror.png\|380px\|right\|Empirically constructed P1]]		[[File:ImamuraPaper2MyP1mirror.png\|380px\|right\|Empirically constructed P1]]
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	<tr>		<tr>
	<td align="center"><math>~2</math></td>		<td align="center"><math>~2</math></td>
	<td align="center"><math>~~~\tfrac{11 \pi}{8}~~</math></td>		<td align="center"><math>~2.55</math></td>
	<td align="center"><math>~3.0</math></td>		<td align="center"><math>~0.048</math></td>
	<td align="center"><math>~\tfrac{2\pi}{5}</math></td>		<td align="center"><math>~\tfrac{\pi}{2}</math></td>
	</tr>		</tr>
	<tr>		<tr>

Tohline: /* Attempts to Match P-modes and E-modes */ Finished descriptive text and start Table 7

2016-02-15T20:22:20Z

Attempts to Match P-modes and E-modes: Finished descriptive text and start Table 7

Show changes

Tohline: /* Attempts to Match P-modes and E-modes */ Switch labels on Tables 5 and 6

2016-02-15T02:47:55Z

Attempts to Match P-modes and E-modes: Switch labels on Tables 5 and 6

Show changes

Tohline: /* Attempts to Match P-modes and E-modes */ Insert E2 "table 6" and model images

2016-02-15T02:26:42Z

Attempts to Match P-modes and E-modes: Insert E2 "table 6" and model images

← Older revision		Revision as of 02:26, 15 February 2016
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	<td align="center" colspan="2"><math>~\ln[f_2($A$1)]=+5.19688</math></td>		<td align="center" colspan="2"><math>~\ln[f_2($A$1)]=+5.19688</math></td>
	<td align="center" colspan="1" rowspan="5">		<td align="center" colspan="1" rowspan="5">
	[[File:ImamuraPaper2MyE3mirror.png\|380px\|right\|My E3]]		[[File:ImamuraPaper2MyE3mirror.png\|380px\|right\|Empirically constructed E3]]
	</td>		</td>
	</tr>		</tr>
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	</table>		</table>
	END COMMENTING OUT OLD TABLE 5 -->		END COMMENTING OUT OLD TABLE 5 -->


			<table border="1" align="center" cellpadding="5" width="75%">
			<tr><td align="center" colspan="5" bgcolor="yellow"><font size="+1"><b>Table 6:</b></font>  Model E2<sup>&Dagger;</sup></td></tr>
			<tr>
			<td align="center" colspan="4" bgcolor="white"><font size="+1"><b>Model Parameters</b></font></td>
			<td align="center" colspan="1" rowspan="1" bgcolor="white">Results from [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II (Hadley et al. 2014)] Simulation</td>
			</tr>
			<tr>
			<td align="center"><math>~r_-</math></td>
			<td align="center"><math>~r_+</math></td>
			<td align="center"><math>~r_\mathrm{outer}</math></td>
			<td align="center"><math>~\delta r</math></td>
			<td align="center" colspan="1" rowspan="6">
			[[File:ImamuraPaper2Fig4.png\|400px\|right\|Figure 4b from Paper II]]
			</td>
			</tr>
			<tr>
			<td align="center"><math>~0.60</math></td>
			<td align="center"><math>~2.0</math></td>
			<td align="center"><math>~3.0</math></td>
			<td align="center"><math>~0.007</math></td>
			</tr>
			<tr>
			<td bgcolor="gray" colspan="4"> </td>
			</tr>
			<tr>
			<td align="center"><math>~r_\mathrm{blue}</math></td>
			<td align="center"><math>~p_\mathrm{blue}</math></td>
			<td align="center"><math>~r_\mathrm{green}</math></td>
			<td align="center"><math>~p_\mathrm{green}</math></td>
			</tr>
			<tr>
			<td align="center"><math>~$A$68 = 1.006</math></td>
			<td align="center"><math>~1.2</math></td>
			<td align="center"><math>~$A$65 = 0.985</math></td>
			<td align="center"><math>~0.50</math></td>
			</tr>
			<tr>
			<td bgcolor="gray" colspan="4"> </td>
			</tr>
			<tr>
			<td align="center"><math>~r_\mathrm{min}</math></td>
			<td align="center"><math>~[f_\ln]_\mathrm{min}</math></td>
			<td align="center" colspan="2"><math>~[f_\ln]_\mathrm{max}</math></td>
			<td align="center" colspan="1" bgcolor="white">Our Empirically Constructed Eigenvector</td>
			</tr>
			<tr>
			<td align="center"><math>~$A$65 = 0.985</math></td>
			<td align="center"><math>~-3.49047</math></td>
			<td align="center" colspan="2"><math>~\ln[f_2($A$1)]=+4.85166</math></td>
			<td align="center" colspan="1" rowspan="5">
			[[File:ImamuraPaper2MyE2mirror.png\|380px\|right\|Empirically constructed E2]]
			</td>
			</tr>
			<tr>
			<td bgcolor="gray" colspan="4"> </td>
			</tr>
			<tr>
			<td align="center"><math>~m</math></td>
			<td align="center"><math>~\aleph_2</math></td>
			<td align="center"><math>~A_0</math></td>
			<td align="center"><math>~\phi_0</math></td>
			</tr>
			<tr>
			<td align="center"><math>~2</math></td>
			<td align="center"><math>~\pi</math></td>
			<td align="center"><math>~0.25</math></td>
			<td align="center"><math>~\tfrac{9\pi}{16}</math></td>
			</tr>
			<tr>
			<td bgcolor="gray" colspan="4"> </td>
			</tr>
			<tr>
			<td align="left" colspan="5">
			<sup>&Dagger;</sup>Note the following:
			* When the Paper II radial eigenfunction possesses more than one local minimum — and, in tandem, the Paper II phase function swings through more than <math>~2\pi</math> radians — we have set the parameter, <math>~r_+</math>, to a value less than <math>~r_\mathrm{outer}</math> because our empirically constructed eigenfunction is only designed to accommodate a single (the innermost) function minimum.
			* The undesirable kink that appears in our empirically constructed radial eigenfunction (to the left of the function minimum) arises because the exponent, <math>~p_\mathrm{green}</math>, is a fraction that is less than unity.
			* Before plotting our empirically constructed phase function, the tabulated phase shift, <math>~\phi_0</math>, has been added to both <math>~\phi_\mathrm{blue}</math> and <math>~\phi_\mathrm{blue}</math> in an effort to match the overall orientation of the "constant phase locus" published in Paper II.
			</td>
			</tr>

			</table>

	=See Also=		=See Also=

Tohline: /* Attempts to Match P-modes and E-modes */ Replace empirically constructed image with "mirror" image

2016-02-15T01:23:16Z

Attempts to Match P-modes and E-modes: Replace empirically constructed image with "mirror" image

← Older revision		Revision as of 01:23, 15 February 2016
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	<td align="center"><math>~\delta r</math></td>		<td align="center"><math>~\delta r</math></td>
	<td align="center" colspan="1" rowspan="6">		<td align="center" colspan="1" rowspan="6">
	[[File:ImamuraPaper2Fig4ModelE3.png\|400px\|Figure 4a from Paper II]]		[[File:ImamuraPaper2Fig4ModelE3.png\|400px\|right\|Figure 4a from Paper II]]
	</td>		</td>
	</tr>		</tr>
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	<td align="center" colspan="2"><math>~\ln[f_2($A$1)]=+5.19688</math></td>		<td align="center" colspan="2"><math>~\ln[f_2($A$1)]=+5.19688</math></td>
	<td align="center" colspan="1" rowspan="5">		<td align="center" colspan="1" rowspan="5">
	[[File:~~ImamuraPaper2MyE3~~.png\|~~400px~~\|My E3]]		[[File:ImamuraPaper2MyE3mirror.png\|380px\|right\|My E3]]
	</td>		</td>
	</tr>		</tr>
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	<td align="center"><math>~\aleph_2</math></td>		<td align="center"><math>~\aleph_2</math></td>
	<td align="center"><math>~A_0</math></td>		<td align="center"><math>~A_0</math></td>
	<td>~~ ~~</td>		<td align="center"><math>~\phi_0</math></td>
	</tr>		</tr>
	<tr>		<tr>
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	<td align="center"><math>~\tfrac{11 \pi}{8}</math></td>		<td align="center"><math>~\tfrac{11 \pi}{8}</math></td>
	<td align="center"><math>~3.0</math></td>		<td align="center"><math>~3.0</math></td>
	<td>~~ ~~</td>		<td align="center"><math>~\tfrac{2\pi}{5}</math></td>
	</tr>		</tr>
	<tr>		<tr>
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	<sup>&Dagger;</sup>Note the following:		<sup>&Dagger;</sup>Note the following:
	* When the Paper II radial eigenfunction possesses more than one local minimum — and, in tandem, the Paper II phase function swings through more than <math>~2\pi</math> radians — we have set the parameter, <math>~r_+</math>, to a value less than <math>~r_\mathrm{outer}</math> because our empirically constructed eigenfunction is only designed to accommodate a single (the innermost) function minimum.		* When the Paper II radial eigenfunction possesses more than one local minimum — and, in tandem, the Paper II phase function swings through more than <math>~2\pi</math> radians — we have set the parameter, <math>~r_+</math>, to a value less than <math>~r_\mathrm{outer}</math> because our empirically constructed eigenfunction is only designed to accommodate a single (the innermost) function minimum.
	* The undesirable kink that appears in our empirically constructed radial eigenfunction, to the left of the function minimum, arises because the exponent, <math>~p_\mathrm{green}</math>, is a fraction that is less than unity.		* The undesirable kink that appears in our empirically constructed radial eigenfunction (to the left of the function minimum) arises because the exponent, <math>~p_\mathrm{green}</math>, is a fraction that is less than unity.
			* Before plotting our empirically constructed phase function, the tabulated phase shift, <math>~\phi_0</math>, has been added to both <math>~\phi_\mathrm{blue}</math> and <math>~\phi_\mathrm{blue}</math> in an effort to match the overall orientation of the "constant phase locus" published in Paper II.
	</td>		</td>
	</tr>		</tr>

Tohline: /* Attempts to Match P-modes and E-modes */ Rearrange and elaborate within Table 5

2016-02-15T00:52:16Z

Attempts to Match P-modes and E-modes: Rearrange and elaborate within Table 5

← Older revision		Revision as of 00:52, 15 February 2016
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	<table border="1" align="center" cellpadding="5" width="75%">		<table border="1" align="center" cellpadding="5" width="75%">
	<tr><td align="center" colspan="5"><font size="+1"><b>Table 5:</b></font>  ~~Empirical Fits~~<sup>&Dagger;</sup></td></tr>		<tr><td align="center" colspan="5" bgcolor="yellow"><font size="+1"><b>Table 5:</b></font>  Model E3<sup>&Dagger;</sup></td></tr>
	<tr>		<tr>
	<td align="center" colspan="4" bgcolor="~~yellow~~"><font size="+1"><b>Model~~:   E3~~</b></font></td>		<td align="center" colspan="4" bgcolor="white"><font size="+1"><b>Model Parameters</b></font></td>
	<td align="center" colspan="1" rowspan="13">		<td align="center" colspan="1" rowspan="1" bgcolor="white">Results from [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II (Hadley et al. 2014)] Simulation</td>
	[[~~File~~:~~ImamuraPaper2Fig4ModelE3~~.~~png\|400px\|Model E3]]<p><~~/p>
	~~[[File:ImamuraPaper2MyE3~~.~~png\|400px\|My E3~~]]
	</td>
	</tr>		</tr>
	<tr>		<tr>
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	<td align="center"><math>~r_\mathrm{outer}</math></td>		<td align="center"><math>~r_\mathrm{outer}</math></td>
	<td align="center"><math>~\delta r</math></td>		<td align="center"><math>~\delta r</math></td>
			<td align="center" colspan="1" rowspan="6">
			[[File:ImamuraPaper2Fig4ModelE3.png\|400px\|Figure 4a from Paper II]]
			</td>
	</tr>		</tr>
	<tr>		<tr>
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	<td align="center"><math>~[f_\ln]_\mathrm{min}</math></td>		<td align="center"><math>~[f_\ln]_\mathrm{min}</math></td>
	<td align="center" colspan="2"><math>~[f_\ln]_\mathrm{max}</math></td>		<td align="center" colspan="2"><math>~[f_\ln]_\mathrm{max}</math></td>
			<td align="center" colspan="1" bgcolor="white">Our Empirically Constructed Eigenvector</td>
	</tr>		</tr>
	<tr>		<tr>
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	<td align="center"><math>~-1.34494</math></td>		<td align="center"><math>~-1.34494</math></td>
	<td align="center" colspan="2"><math>~\ln[f_2($A$1)]=+5.19688</math></td>		<td align="center" colspan="2"><math>~\ln[f_2($A$1)]=+5.19688</math></td>
			<td align="center" colspan="1" rowspan="5">
			[[File:ImamuraPaper2MyE3.png\|400px\|My E3]]
			</td>
	</tr>		</tr>
	<tr>		<tr>