User:Tohline/AxisymmetricConfigurations/SolutionStrategies - Revision history

Tohline: /* Specifying ~\dot\varphi(\varpi) in the Equilibrium Configuration */

2019-08-12T19:55:26Z

Specifying ~\dot\varphi(\varpi) in the Equilibrium Configuration

← Older revision		Revision as of 19:55, 12 August 2019
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	<sup>b</sup>Ou, S. & Tohline, J.E. [http://iopscience.iop.org/0004-637X/651/2/1068/pdf/0004-637X_651_2_1068.pdf 2006, ApJ, 651, 1068-1078]		<sup>b</sup>Ou, S. & Tohline, J.E. [http://iopscience.iop.org/0004-637X/651/2/1068/pdf/0004-637X_651_2_1068.pdf 2006, ApJ, 651, 1068-1078]
	(especially §2.1)<br />		(especially §2.1)<br />
	<sup>h</sup>The [[User:Tohline/Appendix/Ramblings/Hadley_and_Imamura_Supplementary_Database#See_Also\|Hadley & Imamura collaboration]] (Note that, as detailed [[User:Tohline/Appendix/Ramblings/Hadley_and_Imamura_Supplementary_Database#Simple_Rotation_Profiles\|elsewhere]], their definition of the power-law index, <math>~q</math>, is different from ours.)		<sup>h</sup>The [[User:Tohline/Appendix/Ramblings/Hadley_and_Imamura_Supplementary_Database#See_Also\|Hadley & Imamura collaboration]] (circa 2015)  [Note that, as detailed [[User:Tohline/Appendix/Ramblings/Hadley_and_Imamura_Supplementary_Database#Simple_Rotation_Profiles\|elsewhere]], their definition of the power-law index, <math>~q</math>, is different from ours.]

	</td>		</td>

Tohline: /* Specifying ~\dot\varphi(\varpi) in the Equilibrium Configuration */

2019-08-12T19:51:31Z

Specifying ~\dot\varphi(\varpi) in the Equilibrium Configuration

← Older revision		Revision as of 19:51, 12 August 2019
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	<tr>		<tr>
	<td align="left" colspan="7">		<td align="left" colspan="7">
			<sup>f</sup>Maclaurin, C. 1742, ''A Treatise of Fluxions''<br />
			<sup>j</sup>Stoeckly, R. [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..208S/abstract 1965, ApJ, 142, 208 - 228]<br />
			<sup>k</sup>Ostriker, J. P. & Mark, J. W-K. [https://ui.adsabs.harvard.edu/abs/1973ApJ...180..159B/abstract 1968, ApJ, 151, 1075 - 1088]<br />
			<sup>ℓ</sup>Bodenheimer, P. & Ostriker, J. P. [https://ui.adsabs.harvard.edu/abs/1973ApJ...180..159B/abstract 1973, ApJ, 180, 159 - 169]<br />
			<sup>i</sup>Clement, M. J. [https://ui.adsabs.harvard.edu/abs/1979ApJ...230..230C/abstract 1979, ApJ, 230, 230 - 242]<br />
			<sup>e</sup>Hayashi, C., Narita, S. & Miyama, S.M. [http://adsabs.harvard.edu/abs/1982PThPh..68.1949H 1982, ''Progress of Theoretical Physics'', 68, 1949-1966]<br />
			<sup>g</sup>Papaloizou, J.C.B. & Pringle, J.E. [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P 1984, MNRAS, 208, 721-750]<br />
	<sup>a</sup>Hachisu, I. [http://adsabs.harvard.edu/abs/1986ApJS...61..479H 1986, ApJS, 61, 479-507]		<sup>a</sup>Hachisu, I. [http://adsabs.harvard.edu/abs/1986ApJS...61..479H 1986, ApJS, 61, 479-507]
	(especially §II.c)<br />		(especially §II.c)<br />
			<sup>d</sup>Tohline, J.E. & Hachisu, I. [http://adsabs.harvard.edu/abs/1990ApJ...361..394T 1990, ApJ, 361, 394-407]<br />
			<sup>c</sup>Woodward, J.W., Tohline, J.E. & Hachisu, I. [http://adsabs.harvard.edu/abs/1994ApJ...420..247W 1994, ApJ, 420, 247-267]<br />
			<sup>m</sup>Pickett, B. K., Durisen, R. H. & Davis, G. A. [https://ui.adsabs.harvard.edu/abs/1996ApJ...458..714P/abstract 1996, ApJ, 458, 714 - 738]<br />
	<sup>b</sup>Ou, S. & Tohline, J.E. [http://iopscience.iop.org/0004-637X/651/2/1068/pdf/0004-637X_651_2_1068.pdf 2006, ApJ, 651, 1068-1078]		<sup>b</sup>Ou, S. & Tohline, J.E. [http://iopscience.iop.org/0004-637X/651/2/1068/pdf/0004-637X_651_2_1068.pdf 2006, ApJ, 651, 1068-1078]
	(especially §2.1)<br />		(especially §2.1)<br />
	~~<sup>c</sup>Woodward, J.W., Tohline, J.E. & Hachisu, I. [http://adsabs.harvard.edu/abs/1994ApJ...420..247W 1994, ApJ, 420, 247-267]<br />~~		<sup>h</sup>The [[User:Tohline/Appendix/Ramblings/Hadley_and_Imamura_Supplementary_Database#See_Also\|Hadley & Imamura collaboration]] (Note that, as detailed [[User:Tohline/Appendix/Ramblings/Hadley_and_Imamura_Supplementary_Database#Simple_Rotation_Profiles\|elsewhere]], their definition of the power-law index, <math>~q</math>, is different from ours.)
	~~<sup>d</sup>Tohline, J.E. & Hachisu, I. [http://adsabs.harvard.edu/abs/1990ApJ...361..394T 1990, ApJ, 361, 394-407]<br />~~
	~~<sup>e</sup>Hayashi, C., Narita, S. & Miyama, S.M. [http://adsabs.harvard.edu/abs/1982PThPh..68.1949H 1982, ''Progress of Theoretical Physics'', 68, 1949-1966]<br />~~
	~~<sup>f</sup>Maclaurin, C. 1742, ''A Treatise of Fluxions''<br />~~
	~~<sup>g</sup>Papaloizou, J.C.B. & Pringle, J.E. [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P 1984, MNRAS, 208, 721-750]<br />~~
	<sup>h</sup>The [[User:Tohline/Appendix/Ramblings/Hadley_and_Imamura_Supplementary_Database#See_Also\|Hadley & Imamura collaboration]] (Note that, as detailed [[User:Tohline/Appendix/Ramblings/Hadley_and_Imamura_Supplementary_Database#Simple_Rotation_Profiles\|elsewhere]], their definition of the power-law index, <math>~q</math>, is different from ours.)~~<br />~~
	~~<sup>i</sup>Clement, M. J. [https://ui.adsabs.harvard.edu/abs/1979ApJ...230..230C/abstract 1979, ApJ, 230, 230 - 242]<br />~~
	~~<sup>j</sup>Stoeckly, R. [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..208S/abstract 1965, ApJ, 142, 208 - 228]<br />~~
	~~<sup>k</sup>Ostriker, J. P. & Mark, J. W-K. [https://ui.adsabs.harvard.edu/abs/1973ApJ...180..159B/abstract 1968, ApJ, 151, 1075 - 1088]<br />~~
	~~<sup>ℓ</sup>Bodenheimer, P. & Ostriker, J. P. [https://ui.adsabs.harvard.edu/abs/1973ApJ...180..159B/abstract 1973, ApJ, 180, 159 - 169]<br />~~
	~~<sup>m</sup>Pickett, B. K., Durisen, R. H. & Davis, G. A. [https://ui.adsabs.harvard.edu/abs/1996ApJ...458..714P/abstract 1996, ApJ, 458, 714 - 738]~~

	</td>		</td>

Tohline: /* Specifying ~\dot\varphi(\varpi) in the Equilibrium Configuration */

2019-08-12T19:44:37Z

Specifying ~\dot\varphi(\varpi) in the Equilibrium Configuration

← Older revision		Revision as of 19:44, 12 August 2019
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	</td>		</td>
	<td align="center" colspan="5">See [[#Uniform-Density_Initially_.28n.27_.3D_0.29\|discussion below]] of specific angular momentum distribution, <math>~h[m(\varpi)]</math></td>		<td align="center" colspan="5">See [[#Uniform-Density_Initially_.28n.27_.3D_0.29\|discussion below]] of specific angular momentum distribution, <math>~h[m(\varpi)]</math></td>
	<td align="center">j,k,&~~ell~~;,m</td>		<td align="center">j,k,ℓ,m</td>
	</tr>		</tr>

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	<sup>j</sup>Stoeckly, R. [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..208S/abstract 1965, ApJ, 142, 208 - 228]<br />		<sup>j</sup>Stoeckly, R. [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..208S/abstract 1965, ApJ, 142, 208 - 228]<br />
	<sup>k</sup>Ostriker, J. P. & Mark, J. W-K. [https://ui.adsabs.harvard.edu/abs/1973ApJ...180..159B/abstract 1968, ApJ, 151, 1075 - 1088]<br />		<sup>k</sup>Ostriker, J. P. & Mark, J. W-K. [https://ui.adsabs.harvard.edu/abs/1973ApJ...180..159B/abstract 1968, ApJ, 151, 1075 - 1088]<br />
	<sup>l</sup>Bodenheimer, P. & Ostriker, J. P. [https://ui.adsabs.harvard.edu/abs/1973ApJ...180..159B/abstract 1973, ApJ, 180, 159 - 169]<br />		<sup>ℓ</sup>Bodenheimer, P. & Ostriker, J. P. [https://ui.adsabs.harvard.edu/abs/1973ApJ...180..159B/abstract 1973, ApJ, 180, 159 - 169]<br />
	<sup>m</sup>Pickett, B. K., Durisen, R. H. & Davis, G. A. [https://ui.adsabs.harvard.edu/abs/1996ApJ...458..714P/abstract 1996, ApJ, 458, 714 - 738]		<sup>m</sup>Pickett, B. K., Durisen, R. H. & Davis, G. A. [https://ui.adsabs.harvard.edu/abs/1996ApJ...458..714P/abstract 1996, ApJ, 458, 714 - 738]

Tohline: /* Specifying ~\dot\varphi(\varpi) in the Equilibrium Configuration */

2019-08-12T19:40:16Z

Specifying ~\dot\varphi(\varpi) in the Equilibrium Configuration

← Older revision		Revision as of 19:40, 12 August 2019
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	<sup>h</sup>The [[User:Tohline/Appendix/Ramblings/Hadley_and_Imamura_Supplementary_Database#See_Also\|Hadley & Imamura collaboration]] (Note that, as detailed [[User:Tohline/Appendix/Ramblings/Hadley_and_Imamura_Supplementary_Database#Simple_Rotation_Profiles\|elsewhere]], their definition of the power-law index, <math>~q</math>, is different from ours.)<br />		<sup>h</sup>The [[User:Tohline/Appendix/Ramblings/Hadley_and_Imamura_Supplementary_Database#See_Also\|Hadley & Imamura collaboration]] (Note that, as detailed [[User:Tohline/Appendix/Ramblings/Hadley_and_Imamura_Supplementary_Database#Simple_Rotation_Profiles\|elsewhere]], their definition of the power-law index, <math>~q</math>, is different from ours.)<br />
	<sup>i</sup>Clement, M. J. [https://ui.adsabs.harvard.edu/abs/1979ApJ...230..230C/abstract 1979, ApJ, 230, 230 - 242]<br />		<sup>i</sup>Clement, M. J. [https://ui.adsabs.harvard.edu/abs/1979ApJ...230..230C/abstract 1979, ApJ, 230, 230 - 242]<br />
	<sup>k</sup>[https://ui.adsabs.harvard.edu/abs/~~1973ApJ~~...~~180~~..~~159B~~/abstract Ostriker & Mark (1968)], ~~§III.d~~, ~~Eq. (50)~~<br />		<sup>j</sup>Stoeckly, R. [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..208S/abstract 1965, ApJ, 142, 208 - 228]<br />
	<sup>l</sup>[https://ui.adsabs.harvard.edu/abs/1973ApJ...180..159B/abstract ~~Bodenheimer & Ostriker (~~1973)], ~~§II~~, ~~Eq. (4)~~<br />		<sup>k</sup>Ostriker, J. P. & Mark, J. W-K. [https://ui.adsabs.harvard.edu/abs/1973ApJ...180..159B/abstract 1968, ApJ, 151, 1075 - 1088]<br />
	<sup>m</sup>[https://ui.adsabs.harvard.edu/abs/1996ApJ...458..714P/abstract ~~Pickett~~, ~~Durisen & Davis (1996)]~~, ~~§2.1~~, ~~Figure 1~~		<sup>l</sup>Bodenheimer, P. & Ostriker, J. P. [https://ui.adsabs.harvard.edu/abs/1973ApJ...180..159B/abstract 1973, ApJ, 180, 159 - 169]<br />
			<sup>m</sup>Pickett, B. K., Durisen, R. H. & Davis, G. A. [https://ui.adsabs.harvard.edu/abs/1996ApJ...458..714P/abstract 1996, ApJ, 458, 714 - 738]

	</td>		</td>

Tohline: /* Specifying ~\dot\varphi(\varpi) in the Equilibrium Configuration */

2019-08-12T19:30:14Z

Specifying ~\dot\varphi(\varpi) in the Equilibrium Configuration

← Older revision		Revision as of 19:30, 12 August 2019
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	</td>		</td>
	<td align="center" colspan="5">See [[#Uniform-Density_Initially_.28n.27_.3D_0.29\|discussion below]] of specific angular momentum distribution, <math>~h[m(\varpi)]</math></td>		<td align="center" colspan="5">See [[#Uniform-Density_Initially_.28n.27_.3D_0.29\|discussion below]] of specific angular momentum distribution, <math>~h[m(\varpi)]</math></td>
	<td align="center">j,k,&ell;</td>		<td align="center">j,k,&ell;,m</td>
	</tr>		</tr>

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	<sup>h</sup>The [[User:Tohline/Appendix/Ramblings/Hadley_and_Imamura_Supplementary_Database#See_Also\|Hadley & Imamura collaboration]] (Note that, as detailed [[User:Tohline/Appendix/Ramblings/Hadley_and_Imamura_Supplementary_Database#Simple_Rotation_Profiles\|elsewhere]], their definition of the power-law index, <math>~q</math>, is different from ours.)<br />		<sup>h</sup>The [[User:Tohline/Appendix/Ramblings/Hadley_and_Imamura_Supplementary_Database#See_Also\|Hadley & Imamura collaboration]] (Note that, as detailed [[User:Tohline/Appendix/Ramblings/Hadley_and_Imamura_Supplementary_Database#Simple_Rotation_Profiles\|elsewhere]], their definition of the power-law index, <math>~q</math>, is different from ours.)<br />
	<sup>i</sup>Clement, M. J. [https://ui.adsabs.harvard.edu/abs/1979ApJ...230..230C/abstract 1979, ApJ, 230, 230 - 242]<br />		<sup>i</sup>Clement, M. J. [https://ui.adsabs.harvard.edu/abs/1979ApJ...230..230C/abstract 1979, ApJ, 230, 230 - 242]<br />
			<sup>k</sup>[https://ui.adsabs.harvard.edu/abs/1973ApJ...180..159B/abstract Ostriker & Mark (1968)], §III.d, Eq. (50)<br />
			<sup>l</sup>[https://ui.adsabs.harvard.edu/abs/1973ApJ...180..159B/abstract Bodenheimer & Ostriker (1973)], §II, Eq. (4)<br />
			<sup>m</sup>[https://ui.adsabs.harvard.edu/abs/1996ApJ...458..714P/abstract Pickett, Durisen & Davis (1996)], §2.1, Figure 1

	</td>		</td>
	</tr>		</tr>

Tohline: /* Specifying ~\dot\varphi(\varpi) in the Equilibrium Configuration */

2019-08-12T19:23:46Z

Specifying ~\dot\varphi(\varpi) in the Equilibrium Configuration

← Older revision		Revision as of 19:23, 12 August 2019
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	</td>		</td>
	</tr>		</tr>


	<tr>		<tr>
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	</td>		</td>
	</tr>		</tr>


	<tr>		<tr>
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	a,b,i		a,b,i
	</td>		</td>
			</tr>

			<tr>
			<td align="center">
			<math>~n'</math><br />
			<font color="maroon">Sequences</font>
			</td>
			<td align="center" colspan="5">See [[#Uniform-Density_Initially_.28n.27_.3D_0.29\|discussion below]] of specific angular momentum distribution, <math>~h[m(\varpi)]</math></td>
			<td align="center">j,k,&ell;</td>
	</tr>		</tr>

Tohline: /* Centrally Condensed Initially (n' > 0) */

2019-08-12T19:10:00Z

Centrally Condensed Initially (n' > 0)

← Older revision		Revision as of 19:10, 12 August 2019
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	[from K. Braly's (1969) unpublished undergraduate thesis, Princeton University]		[from K. Braly's (1969) unpublished undergraduate thesis, Princeton University]
	</td>		</td>
	<td align="center">Pickett, Durisen & Davis (1996)</td>		<td align="center">
			Figure & caption extracted from p. 715 of
			[https://ui.adsabs.harvard.edu/abs/1996ApJ...458..714P/abstract Pickett, Durisen & Davis (1996)]<br />
			<i>"The Dynamic Stability of Rotating Protostars and Protostellar Disks<br />
			I. The Effects of the Angular Momentum Distribution"</i><br />
			ApJ, vol. 458, pp. 714 -738 © American Astronomical Society
			</td>
	</tr>		</tr>
	<tr>		<tr>

Tohline: /* Centrally Condensed Initially (n' > 0) */

2019-08-12T16:55:12Z

Centrally Condensed Initially (n' > 0)

← Older revision		Revision as of 16:55, 12 August 2019
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	[from K. Braly's (1969) unpublished undergraduate thesis, Princeton University]		[from K. Braly's (1969) unpublished undergraduate thesis, Princeton University]
	</td>		</td>
			<td align="center">Pickett, Durisen & Davis (1996)</td>
	</tr>		</tr>
	<tr>		<tr>
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	<td align="center"><math>~\alpha_2 = \frac{1}{q_1}</math></td>		<td align="center"><math>~\alpha_2 = \frac{1}{q_1}</math></td>
	<td align="center"><math>~\alpha_3 = \frac{1}{q_2}</math></td>		<td align="center"><math>~\alpha_3 = \frac{1}{q_2}</math></td>
			<td align="center" rowspan="10">[[File:PickettDurisenDavis96Fig1.png\|400px]]</td>
	</tr>		</tr>
	<tr>		<tr>

Tohline: /* Centrally Condensed Initially (n' > 0) */

2019-08-12T16:30:49Z

Centrally Condensed Initially (n' > 0)

← Older revision		Revision as of 16:30, 12 August 2019
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	<td align="center">+0.235287</td>		<td align="center">+0.235287</td>
	<td align="center"><math>~\tfrac{2}{9}</math></td>		<td align="center"><math>~\tfrac{2}{9}</math></td>
			</tr>
			<tr>
			<td align="center" colspan="6">
			<b>Coefficients for <math>~h(m)</math> Expression</b><br />
			used by [https://ui.adsabs.harvard.edu/abs/1968ApJ...151.1089O/abstract Ostriker & Bodenheimer (1968)], p. 1090, Eq. (6)
			</td>
			</tr>
			<tr>
			<td align="center"><math>~\tfrac{3}{2}</math></td>
			<td align="center">+4.8239</td>
			<td align="center">+1.8744</td>
			<td align="center">-6.6983</td>
			<td align="center">+0.5622</td>
			<td align="center"><math>~\tfrac{1}{3}</math></td>
	</tr>		</tr>
	</table>		</table>

Tohline: /* Uniform-Density Initially (n' = 0) */

2019-08-12T16:09:56Z

Uniform-Density Initially (n' = 0)

← Older revision		Revision as of 16:09, 12 August 2019
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{5J}{2M\varpi^2}\biggl[ 1 - ~~\biggl(~~1 - ~~\frac{M_\varpi~~(\varpi)~~}{M}\biggr)~~^{2 / 3} \biggr] \, ,		\frac{5J}{2M\varpi^2}\biggl\{ 1 - [1 - m(\varpi) ]^{2 / 3} \biggr\} \, ,
	</math>		</math>
	</td>		</td>