Appendix/Ramblings/AzimuthalDistortions - Revision history

Jet53man: /* Adopted Notation */

2021-10-03T21:29:43Z

Adopted Notation

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	''The Stability of Thick, Self-gravitating Disks in Protostellar Systems''<br />		''The Stability of Thick, Self-gravitating Disks in Protostellar Systems''<br />
	ApJ, vol. 420, pp. 247-267 © [http://aas.org/ American Astronomical Society]<br />[https://doi.org/10.1086/173556 https://doi.org/10.1086/173556]<br />		ApJ, vol. 420, pp. 247-267 © [http://aas.org/ American Astronomical Society]<br />[https://doi.org/10.1086/173556 https://doi.org/10.1086/173556]<br />
	[[File:PermissionsRectYellow.png\|75px\|link=Appendix/~~Permissions~~#WTH94]]		[[File:PermissionsRectYellow.png\|75px\|link=Appendix/CopyrightPermissions#WTH94]]
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Jet53man: /* Adopted Notation */

2021-10-03T21:23:12Z

Adopted Notation

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	'''Four panels extracted<sup>&dagger;</sup> from Figure 2, p. 252 of<br />[http://adsabs.harvard.edu/abs/1994ApJ...420..247W J. W. Woodward, J. E. Tohline & I. Hachisu (1994)]'''<br />		'''Four panels extracted<sup>&dagger;</sup> from Figure 2, p. 252 of<br />[http://adsabs.harvard.edu/abs/1994ApJ...420..247W J. W. Woodward, J. E. Tohline & I. Hachisu (1994)]'''<br />
	''The Stability of Thick, Self-gravitating Disks in Protostellar Systems''<br />		''The Stability of Thick, Self-gravitating Disks in Protostellar Systems''<br />
	ApJ, vol. 420, pp. 247-267 © [http://aas.org/ American Astronomical Society]<br />[https://doi.org/ https://doi.org/]<br />		ApJ, vol. 420, pp. 247-267 © [http://aas.org/ American Astronomical Society]<br />[https://doi.org/10.1086/173556 https://doi.org/10.1086/173556]<br />
	[[File:PermissionsRectYellow.png\|75px\|link=Appendix/Permissions#WTH94]]		[[File:PermissionsRectYellow.png\|75px\|link=Appendix/Permissions#WTH94]]
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Jet53man: /* Adopted Notation */

2021-10-03T21:21:02Z

Adopted Notation

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	</td>		</td>
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	<tr><td align="center">		<tr><td align="center" bgcolor="lightgreen">
	Four panels extracted<sup>&dagger;</sup> from ~~figure~~ 2, p. 252 of [http://adsabs.harvard.edu/abs/1994ApJ...420..247W J. W. Woodward, J. E. Tohline & I. Hachisu (1994)]<~~p><~~/p>		'''Four panels extracted<sup>&dagger;</sup> from Figure 2, p. 252 of<br />[http://adsabs.harvard.edu/abs/1994ApJ...420..247W J. W. Woodward, J. E. Tohline & I. Hachisu (1994)]'''<br />
	"''The Stability of Thick, Self-gravitating Disks in Protostellar Systems''~~"<p>~~</p>		''The Stability of Thick, Self-gravitating Disks in Protostellar Systems''<br />
	ApJ, vol. 420, pp. 247-267 © [http://aas.org/ American Astronomical Society]		ApJ, vol. 420, pp. 247-267 © [http://aas.org/ American Astronomical Society]<br />[https://doi.org/ https://doi.org/]<br />
			[[File:PermissionsRectYellow.png\|75px\|link=Appendix/Permissions#WTH94]]
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Jet53man: /* Attempts to Match P-modes and E-modes */

2021-10-03T00:23:16Z

Attempts to Match P-modes and E-modes

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	</table>		</table>

	[[Appendix/Ramblings/~~To_Hadley_and_Imamura~~#Recognition_.233\|As in our separate summary discussion]], our two-piece radial eigenfunction is defined by the expressions,		[[Appendix/Ramblings/ToHadleyAndImamura#Recognition_.233\|As in our separate summary discussion]], our two-piece radial eigenfunction is defined by the expressions,

	<div align="center">		<div align="center">

Jet53man: /* Attempts to Match P-modes and E-modes */

2021-10-03T00:22:02Z

Attempts to Match P-modes and E-modes

← Older revision		Revision as of 20:22, 2 October 2021
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~$A$N</math>		<math>~\$A\$N</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</tr>		</tr>
	<tr>		<tr>
	<td align="center"><math>~$A$68 = 1.006</math></td>		<td align="center"><math>~\$A\$68 = 1.006</math></td>
	<td align="center"><math>~1.2</math></td>		<td align="center"><math>~1.2</math></td>
	<td align="center"><math>~$A$65 = 0.985</math></td>		<td align="center"><math>~\$A\$65 = 0.985</math></td>
	<td align="center"><math>~0.50</math></td>		<td align="center"><math>~0.50</math></td>
	</tr>		</tr>
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	</tr>		</tr>
	<tr>		<tr>
	<td align="center"><math>~$A$65 = 0.985</math></td>		<td align="center"><math>~\$A\$65 = 0.985</math></td>
	<td align="center"><math>~-3.49047</math></td>		<td align="center"><math>~-3.49047</math></td>
	<td align="center" colspan="2"><math>~\ln[f_2($A$1)]=+4.85166</math></td>		<td align="center" colspan="2"><math>~\ln[f_2(\$A\$1)]=+4.85166</math></td>
	<td align="center" colspan="1" rowspan="5">		<td align="center" colspan="1" rowspan="5">
	[[File:ImamuraPaper2MyE2mirror.png\|380px\|right\|Empirically constructed E2]]		[[File:ImamuraPaper2MyE2mirror.png\|380px\|right\|Empirically constructed E2]]
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	</tr>		</tr>
	<tr>		<tr>
	<td align="center"><math>~$A$77 = 1.085</math></td>		<td align="center"><math>~\$A\$77 = 1.085</math></td>
	<td align="center"><math>~1.2</math></td>		<td align="center"><math>~1.2</math></td>
	<td align="center"><math>~$A$55 = 0.975</math></td>		<td align="center"><math>~\$A\$55 = 0.975</math></td>
	<td align="center"><math>~0.75</math></td>		<td align="center"><math>~0.75</math></td>
	</tr>		</tr>
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	</tr>		</tr>
	<tr>		<tr>
	<td align="center"><math>~$A$71 = 1.055</math></td>		<td align="center"><math>~\$A\$71 = 1.055</math></td>
	<td align="center"><math>~-1.34494</math></td>		<td align="center"><math>~-1.34494</math></td>
	<td align="center" colspan="2"><math>~\ln[f_2($A$1)]=+5.19688</math></td>		<td align="center" colspan="2"><math>~\ln[f_2(\$A\$1)]=+5.19688</math></td>
	<td align="center" colspan="1" rowspan="5">		<td align="center" colspan="1" rowspan="5">
	[[File:ImamuraPaper2MyE3mirror.png\|380px\|right\|Empirically constructed E3]]		[[File:ImamuraPaper2MyE3mirror.png\|380px\|right\|Empirically constructed E3]]
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	<td align="center"><math>~r_\mathrm{min}</math></td>		<td align="center"><math>~r_\mathrm{min}</math></td>
	<td align="center"><math>~[f_\ln]_\mathrm{min}</math></td>		<td align="center"><math>~[f_\ln]_\mathrm{min}</math></td>
	<td align="center"><math>~[f_\ln]_\mathrm{max}=\ln[f_2($A$1)]</math></td>		<td align="center"><math>~[f_\ln]_\mathrm{max}=\ln[f_2(\$A\$1)]</math></td>
	<td align="center"><math>~m</math></td>		<td align="center"><math>~m</math></td>
	<td align="center"><math>~\aleph_2</math></td>		<td align="center"><math>~\aleph_2</math></td>
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	<td align="center"><math>~1.7</math></td>		<td align="center"><math>~1.7</math></td>
	<td align="center"><math>~0.005</math></td>		<td align="center"><math>~0.005</math></td>
	<td align="center"><math>~$A$77 = 1.085</math></td>		<td align="center"><math>~\$A\$77 = 1.085</math></td>
	<td align="center"><math>~1.2</math></td>		<td align="center"><math>~1.2</math></td>
	<td align="center"><math>~$A$55 = 0.975</math></td>		<td align="center"><math>~\$A\$55 = 0.975</math></td>
	<td align="center"><math>~0.75</math></td>		<td align="center"><math>~0.75</math></td>
	<td align="center"><math>~$A$71 = 1.055</math></td>		<td align="center"><math>~\$A\$71 = 1.055</math></td>
	<td align="center"><math>~-1.34494</math></td>		<td align="center"><math>~-1.34494</math></td>
	<td align="center"><math>~+5.19688</math></td>		<td align="center"><math>~+5.19688</math></td>
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	</tr>		</tr>
	<tr>		<tr>
	<td align="center"><math>~$A$100 = 1.00515</math></td>		<td align="center"><math>~\$A\$100 = 1.00515</math></td>
	<td align="center"><math>~1.0</math></td>		<td align="center"><math>~1.0</math></td>
	<td align="center"><math>~$A$96 = 0.99261</math></td>		<td align="center"><math>~\$A\$96 = 0.99261</math></td>
	<td align="center"><math>~0.35</math></td>		<td align="center"><math>~0.35</math></td>
	</tr>		</tr>
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	</tr>		</tr>
	<tr>		<tr>
	<td align="center"><math>~$A$96 = 0.99261</math></td>		<td align="center"><math>~\$A\$96 = 0.99261</math></td>
	<td align="center"><math>~-3.06805</math></td>		<td align="center"><math>~-3.06805</math></td>
	<td align="center" colspan="2"><math>~\ln[f_2($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:ImamuraPaper2MyP1better.png\|380px\|right\|Empirically constructed P1]]		[[File:ImamuraPaper2MyP1better.png\|380px\|right\|Empirically constructed P1]]

Jet53man: /* Discussion */

2021-10-03T00:19:14Z

Discussion

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	<tr>		<tr>
	<td align="center" colspan="4">		<td align="center" colspan="4">
	<math>~r_\mathrm{blue} = $A$113 = 1.107635</math>		<math>~r_\mathrm{blue} = \$A\$113 = 1.107635</math>
	</td>		</td>
	</tr>		</tr>
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	<td align="center" rowspan="6">		<td align="center" rowspan="6">
	<math>~$A$102 = 1.05929</math><p> </p>		<math>~\$A\$102 = 1.05929</math><p> </p>
	<p> </p>		<p> </p>
	<p> </p>		<p> </p>

Jet53man: /* Specific Application to HI11's Figure 16 */

2021-10-02T22:51:55Z

Specific Application to HI11's Figure 16

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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~$A$N</math>		<math>~\$A\$N</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<td align="center" rowspan="7">		<td align="center" rowspan="7">
	<math>~$A$2 = 0.61979</math><p> </p>		<math>~\$A\$2 = 0.61979</math><p> </p>
	<p> </p>		<p> </p>
	<p> </p>		<p> </p>

Jet53man at 20:35, 19 September 2021

2021-09-19T20:35:45Z

← Older revision		Revision as of 16:35, 19 September 2021
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	=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.		<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, "[[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.

	{{ SGFworkInProgress }}		{{ SGFworkInProgress }}
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	whose relative amplitude — with a radial structure as specified inside the curly braces — is undergoing a uniform exponential growth but is otherwise unchanging.		whose relative amplitude — with a radial structure as specified inside the curly braces — is undergoing a uniform exponential growth but is otherwise unchanging.

	Drawing from figure 2 of [http://adsabs.harvard.edu/abs/1994ApJ...420..247W WTH94], our Figure 1, immediately below, illustrates how the behavior of each factor in this expression can reveal itself during a numerical simulation that follows the time-evolutionary development of an unstable, nonaxisymmetric eigenmode. The initial model for this depicted evolution (model O3 from Table 1 of [http://adsabs.harvard.edu/abs/1994ApJ...420..247W WTH94]) is a zero-mass — that is, it is a [[~~User:Tohline/~~Apps/PapaloizouPringleTori#Massless_Polytropic_Tori\|Papaloizou-Pringle like torus]] — with [[~~User:Tohline/~~SR#Barotropic_Structure\|polytropic index]],<math>~n = 3</math>, and a rotation-law profile defined by [[~~User:Tohline/~~AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential\|uniform specific angular momentum]].		Drawing from figure 2 of [http://adsabs.harvard.edu/abs/1994ApJ...420..247W WTH94], our Figure 1, immediately below, illustrates how the behavior of each factor in this expression can reveal itself during a numerical simulation that follows the time-evolutionary development of an unstable, nonaxisymmetric eigenmode. The initial model for this depicted evolution (model O3 from Table 1 of [http://adsabs.harvard.edu/abs/1994ApJ...420..247W WTH94]) is a zero-mass — that is, it is a [[Apps/PapaloizouPringleTori#Massless_Polytropic_Tori\|Papaloizou-Pringle like torus]] — with [[SR#Barotropic_Structure\|polytropic index]],<math>~n = 3</math>, and a rotation-law profile defined by [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential\|uniform specific angular momentum]].
	* The top-left panel shows how, at any radial location, the phase angle, <math>~\phi_1/(2\pi)</math>, for the <math>~m=1</math> eigenmode, varies with time, <math>~t/t_\mathrm{rot}</math>, where, <math>~t_\mathrm{rot} \equiv 2\pi/\Omega_0</math> is the rotation period at the density maximum;		* The top-left panel shows how, at any radial location, the phase angle, <math>~\phi_1/(2\pi)</math>, for the <math>~m=1</math> eigenmode, varies with time, <math>~t/t_\mathrm{rot}</math>, where, <math>~t_\mathrm{rot} \equiv 2\pi/\Omega_0</math> is the rotation period at the density maximum;
	* Using a semi-log plot, the top-right panel shows the exponential growth of the amplitude of three separate modes: The dominant unstable mode, displaying the largest amplitude, is <math>~m = 1</math>.		* Using a semi-log plot, the top-right panel shows the exponential growth of the amplitude of three separate modes: The dominant unstable mode, displaying the largest amplitude, is <math>~m = 1</math>.
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	===Summary===		===Summary===
	While studying the series of three papers that was published recently by the [[#See_Also\|Imamura & Hadley collaboration]], I was particularly drawn to the pair of plots presented in Figure 6 — and, again, in the top portion of Figure 13 — of [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H HI11]. This pair of plots has been reprinted here, without modification, as our Figure 2. As in the bottom two panels of our Figure 1, the curves delineated by the blue dots in this pair of [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H HI11] plots display (on the left) the shape of the eigenfunction, <math>~f_1(\varpi)</math>, and (on the right) the "constant phase locus," <math>~\phi_1(\varpi)</math>, for an unstable, <math>~m=1</math> mode. In this case, the initial model for the depicted evolution is the equilibrium model from Table 2 of [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H HI11] that has <math>~T/\|W\| = 0.253</math>; it is a fully self-gravitating torus with [[~~User:Tohline/~~SR#Barotropic_Structure\|polytropic index]], <math>~n = 3/2</math>, and a rotation-law profile defined by a [[~~User:Tohline/~~AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential\|"Keplerian" angular velocity profile]].		While studying the series of three papers that was published recently by the [[#See_Also\|Imamura & Hadley collaboration]], I was particularly drawn to the pair of plots presented in Figure 6 — and, again, in the top portion of Figure 13 — of [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H HI11]. This pair of plots has been reprinted here, without modification, as our Figure 2. As in the bottom two panels of our Figure 1, the curves delineated by the blue dots in this pair of [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H HI11] plots display (on the left) the shape of the eigenfunction, <math>~f_1(\varpi)</math>, and (on the right) the "constant phase locus," <math>~\phi_1(\varpi)</math>, for an unstable, <math>~m=1</math> mode. In this case, the initial model for the depicted evolution is the equilibrium model from Table 2 of [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H HI11] that has <math>~T/\|W\| = 0.253</math>; it is a fully self-gravitating torus with [[SR#Barotropic_Structure\|polytropic index]], <math>~n = 3/2</math>, and a rotation-law profile defined by a [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential\|"Keplerian" angular velocity profile]].

	<div align="center">		<div align="center">
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	</tr>		</tr>
	<tr>		<tr>
	<td align="left" colspan="10"><sup>&dagger;</sup>In all three papers from the [[#See_Also\|Imamura & Hadley collaboration]], <math>~q = 2</math> means, "[[~~User:Tohline/~~AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential\|uniform specific angular momentum]]."</td>		<td align="left" colspan="10"><sup>&dagger;</sup>In all three papers from the [[#See_Also\|Imamura & Hadley collaboration]], <math>~q = 2</math> means, "[[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential\|uniform specific angular momentum]]."</td>
	</tr>		</tr>
	</table>		</table>

	[[~~User:Tohline/~~Appendix/Ramblings/To_Hadley_and_Imamura#Recognition_.233\|As in our separate summary discussion]], our two-piece radial eigenfunction is defined by the expressions,		[[Appendix/Ramblings/To_Hadley_and_Imamura#Recognition_.233\|As in our separate summary discussion]], our two-piece radial eigenfunction is defined by the expressions,

	<div align="center">		<div align="center">
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	</div>		</div>

	Over the inner ("blue") region of the torus, we will continue to use the phase-function prescription that has been described [[~~User:Tohline/~~Appendix/Ramblings/To_Hadley_and_Imamura#Constant_Phase_Loci\|in our separate summary discussion]], namely,		Over the inner ("blue") region of the torus, we will continue to use the phase-function prescription that has been described [[Appendix/Ramblings/To_Hadley_and_Imamura#Constant_Phase_Loci\|in our separate summary discussion]], namely,

	<div align="center">		<div align="center">

Jet53man at 23:19, 22 July 2021

2021-07-22T23:19:06Z

← Older revision		Revision as of 19:19, 22 July 2021
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			{{ SGFworkInProgress }}
	{{~~LSU_WorkInProgress~~}}

	==Attempts to Match P-modes and E-modes==		==Attempts to Match P-modes and E-modes==

Jet53man: Created page with "FORCETOC =Analyzing Azimuthal Distortions= [Comment by J. E. Tohline on 24 May 2016] This cha..."

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Appendix/Ramblings/AzimuthalDistortions - Revision history

Jet53man: /* Adopted Notation */

Jet53man: /* Adopted Notation */

Jet53man: /* Adopted Notation */

Jet53man: /* Attempts to Match P-modes and E-modes */

Jet53man: /* Attempts to Match P-modes and E-modes */

Jet53man: /* Discussion */

Jet53man: /* Specific Application to HI11's Figure 16 */

Jet53man at 20:35, 19 September 2021

Jet53man at 23:19, 22 July 2021

Jet53man: Created page with "__FORCETOC__ =Analyzing Azimuthal Distortions= [Comment by J. E. Tohline on 24 May 2016] This cha..."

Jet53man: Created page with "FORCETOC =Analyzing Azimuthal Distortions= [Comment by J. E. Tohline on 24 May 2016] This cha..."