User:Tohline/Apps/MaclaurinSpheroidSequence - Revision history

Tohline: /* Angular Velocity or T/|W| vs. Angular Momentum */

2020-07-30T23:12:14Z

Angular Velocity or T/|W| vs. Angular Momentum

← Older revision		Revision as of 23:12, 30 July 2020
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	===Angular Velocity or T/\|W\| ''vs.'' Angular Momentum===		===Angular Velocity or T/\|W\| ''vs.'' Angular Momentum===

	~~Figure~~ 5 ~~shows~~ how the square of the angular velocity ~~varies~~ with the square of the angular momentum for models along the Maclaurin spheroid sequence. In generating ~~this plot~~, following the lead of [https://ui.adsabs.harvard.edu/abs/1983PThPh..69.1131E/abstract Eriguchi & Hachisu (1983)], we have normalized the square of the angular velocity by <math>~4\pi G \rho</math> — a factor of four larger than the normalization used in EFE — and we have adopted a slightly different angular-momentum normalization, namely,		Figures 5 and 6, respectively, show how the square of the angular velocity and how the energy ratio, τ, vary with the square of the angular momentum for models along the Maclaurin spheroid sequence. In generating these plots, following the lead of [https://ui.adsabs.harvard.edu/abs/1983PThPh..69.1131E/abstract Eriguchi & Hachisu (1983)], we have normalized the square of the angular velocity by <math>~4\pi G \rho</math> — a factor of four larger than the normalization used in EFE — and we have adopted a slightly different angular-momentum normalization, namely,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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

	As above, the small solid-green square marker identifies the location along ~~the sequence~~ where the system with the maximum angular velocity resides:		As above, the small solid-green square marker identifies the location along both sequences where the system with the maximum angular velocity resides:
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Angular Velocity or T/|W| vs. Angular Momentum */

2020-07-30T23:03:28Z

Angular Velocity or T/|W| vs. Angular Momentum

← Older revision		Revision as of 23:03, 30 July 2020
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\biggl[ j^2, \frac{\omega_0^2}{4\pi G \rho} \biggr]</math>		<math>~\biggl[ j^2, \frac{\omega_0^2}{4\pi G \rho}, \tau \biggr]</math>
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	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl[ 0.010105, 0.112333 \biggr] \, .</math>		<math>~\biggl[ 0.010105, 0.112333, 0.237894 \biggr] \, .</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Angular Velocity or T/|W| vs. Angular Momentum */

2020-07-30T23:00:53Z

Angular Velocity or T/|W| vs. Angular Momentum

← Older revision		Revision as of 23:00, 30 July 2020
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	</tr>		</tr>
	</table>		</table>

			<table border="0" align="center" cellpadding="5"><tr><td align="center">
	<table border="0" align="center" cellpadding="3">		<table border="0" align="center" cellpadding="3">
	<tr>		<tr>
	<td align="center">'''Figure 5'''</td>		<td align="center">'''Figure 5'''</td>
			<td align="center" rowspan="3">          </td>
	<td align="center">'''Figure 6'''</td>		<td align="center">'''Figure 6'''</td>
	</tr>		</tr>
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	</tr>		</tr>
	</table>		</table>
			</td></tr></table>

	As above, the small solid-green square marker identifies the location along the sequence where the system with the maximum angular velocity resides:		As above, the small solid-green square marker identifies the location along the sequence where the system with the maximum angular velocity resides:

Tohline: /* Energy Ratio, T/|W| */

2020-07-30T22:59:02Z

Energy Ratio, T/|W|

← Older revision		Revision as of 22:59, 30 July 2020
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	Building on an [[User:Tohline/Apps/MaclaurinSpheroids#Gravitational_Potential\|accompanying discussion of the structure of Maclaurin spheroids]], Table 2 — shown just above, on the right — lists the limiting values of several key functions. Note, in particular, that as the eccentricity varies smoothly from zero (spherical configuration) to unity (infinitesimally thin disk), the energy ratio, <math>~\tau</math>, varies smoothly from zero to one-half. In his examination of the Maclaurin spheroid sequence, [[User:Tohline/Appendix/References#T78\|Tassoul (1978)]] chose to use this energy ratio as the ''order parameter'', rather than the eccentricity.		Building on an [[User:Tohline/Apps/MaclaurinSpheroids#Gravitational_Potential\|accompanying discussion of the structure of Maclaurin spheroids]], Table 2 — shown just above, on the right — lists the limiting values of several key functions. Note, in particular, that as the eccentricity varies smoothly from zero (spherical configuration) to unity (infinitesimally thin disk), the energy ratio, <math>~\tau</math>, varies smoothly from zero to one-half. In his examination of the Maclaurin spheroid sequence, [[User:Tohline/Appendix/References#T78\|Tassoul (1978)]] chose to use this energy ratio as the ''order parameter'', rather than the eccentricity.

	<table border="0" align="center"><tr><td align="center">		<table border="1" align="center"><tr><td align="center">
	<table border="0" align="center" cellpadding="3">		<table border="0" align="center" cellpadding="3">
	<tr>		<tr>
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	Solid black curve also may be found in:		Solid black curve also may be found in:
	<div align="center">		<div align="center">
	Fig. 4.2 (p. 88) & Fig. 10.1 (p. 236) of [<b>[[User:Tohline/Appendix/References#T78\|<font color="red">T78</font>]]</b>];		Fig. 4.2 (p. 88) & Fig. 10.1 (p. 236) of [<b>[[User:Tohline/Appendix/References#T78\|<font color="red">T78</font>]]</b>]
	</div>		</div>
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	This solid black curve also appears in:		This solid black curve also appears in:
	<div align="center">		<div align="center">
	Fig. 4.2 (p. 88) & Fig. 10.12 (p. 237) of [<b>[[User:Tohline/Appendix/References#T78\|<font color="red">T78</font>]]</b>];		Fig. 4.2 (p. 88) & Fig. 10.12 (p. 237) of [<b>[[User:Tohline/Appendix/References#T78\|<font color="red">T78</font>]]</b>]
	</div>		</div>
	</td>		</td>

Tohline: /* Angular Velocity vs. Angular Momentum */

2020-07-30T22:57:13Z

Angular Velocity vs. Angular Momentum

← Older revision		Revision as of 22:57, 30 July 2020
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	Following Tassoul, our Figure 3 shows how the square of the angular velocity varies with <math>~\tau</math>, and our Figure 4 shows how the system angular momentum varies with <math>~\tau</math>. In these plots, respectively, the square of the angular velocity has been normalized by <math>~2\pi G \rho</math> — that is, by a quantity that is a factor of two larger than the normalization adopted in EFE — while the angular momentum has been normalized to the same quantity used in EFE. As above, the small solid-green square marker identifies the location along the sequence where the system with the maximum angular velocity resides.		Following Tassoul, our Figure 3 shows how the square of the angular velocity varies with <math>~\tau</math>, and our Figure 4 shows how the system angular momentum varies with <math>~\tau</math>. In these plots, respectively, the square of the angular velocity has been normalized by <math>~2\pi G \rho</math> — that is, by a quantity that is a factor of two larger than the normalization adopted in EFE — while the angular momentum has been normalized to the same quantity used in EFE. As above, the small solid-green square marker identifies the location along the sequence where the system with the maximum angular velocity resides.

	===Angular Velocity ''vs.'' Angular Momentum===		===Angular Velocity or T/\|W\| ''vs.'' Angular Momentum===

	Figure 5 shows how the square of the angular velocity varies with the square of the angular momentum for models along the Maclaurin spheroid sequence. In generating this plot, following the lead of [https://ui.adsabs.harvard.edu/abs/1983PThPh..69.1131E/abstract Eriguchi & Hachisu (1983)], we have normalized the square of the angular velocity by <math>~4\pi G \rho</math> — a factor of four larger than the normalization used in EFE — and we have adopted a slightly different angular-momentum normalization, namely,		Figure 5 shows how the square of the angular velocity varies with the square of the angular momentum for models along the Maclaurin spheroid sequence. In generating this plot, following the lead of [https://ui.adsabs.harvard.edu/abs/1983PThPh..69.1131E/abstract Eriguchi & Hachisu (1983)], we have normalized the square of the angular velocity by <math>~4\pi G \rho</math> — a factor of four larger than the normalization used in EFE — and we have adopted a slightly different angular-momentum normalization, namely,
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	<tr>		<tr>
	<td align="center">'''Figure 5'''</td>		<td align="center">'''Figure 5'''</td>
			<td align="center">'''Figure 6'''</td>
	</tr>		</tr>
	<tr>		<tr>
	<td align="center">		<td align="center">
	[[File:EH83Fig3.png\|center\|350px\|Maclaurin Spheroid Sequence]]		[[File:EH83Fig3.png\|center\|350px\|Maclaurin Spheroid Sequence]]
			</td>
			<td align="center">
			[[File:EH83Fig4.png\|center\|350px\|Maclaurin Spheroid Sequence]]
	</td>		</td>
	</tr>		</tr>
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	<div align="center">		<div align="center">
	Fig. 3 (p. 1134) of [https://ui.adsabs.harvard.edu/abs/1983PThPh..69.1131E/abstract Eriguchi & Hachisu (1983)]<br />		Fig. 3 (p. 1134) of [https://ui.adsabs.harvard.edu/abs/1983PThPh..69.1131E/abstract Eriguchi & Hachisu (1983)]<br />
			Fig. 3 (p. 487) of [https://ui.adsabs.harvard.edu/abs/1986ApJS...61..479H/abstract Hachisu (1986)]<br />
	Fig. 4 (p. 4507) of [https://ui.adsabs.harvard.edu/abs/2019MNRAS.487.4504B/abstract Basillais & Huré (2019)]		Fig. 4 (p. 4507) of [https://ui.adsabs.harvard.edu/abs/2019MNRAS.487.4504B/abstract Basillais & Huré (2019)]
			</div>
			</td>
			<td align="center">
			This solid black curve also appears in:
			<div align="center">
			Fig. 4 (p. 487) of [https://ui.adsabs.harvard.edu/abs/1986ApJS...61..479H/abstract Hachisu (1986)]
	</div>		</div>
	</td>		</td>

Tohline: /* Equilibrium Angular Velocity */

2020-07-29T21:22:12Z

Equilibrium Angular Velocity

← Older revision		Revision as of 21:22, 29 July 2020
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	<tr>		<tr>
	<td align="center">		<td align="center">
	The dark blue circular markers locate the 18 individual models identified in Table 1. The solid black curve derives from our evaluation of the function, <math>~\omega_0^2(e)</math>; this curve also may be found in:		The dark blue circular markers locate 15 of the 18 individual models identified in Table 1. The solid black curve derives from our evaluation of the function, <math>~\omega_0^2(e)</math>; this curve also may be found in:
	<div align="center">		<div align="center">
	Fig. 5 (p. 79) of [<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>];<br />		Fig. 5 (p. 79) of [<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>];<br />

Tohline: /* Equilibrium Angular Velocity */

2020-07-29T21:18:09Z

Equilibrium Angular Velocity

← Older revision		Revision as of 21:18, 29 July 2020
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	</tr>		</tr>
	<tr><td align="center">		<tr><td align="center">
	[[File:EFE ~~Omega2vsECCwithThomsonTait~~.png\|center\|350px\|Maclaurin Spheroid Sequence]]		[[File:EFE Omega2vsECCwithThomsonTait2.png\|center\|350px\|Maclaurin Spheroid Sequence]]
	</td></tr>		</td></tr>
	<tr>		<tr>

Tohline: /* Equilibrium Angular Velocity */

2020-07-29T21:11:53Z

Equilibrium Angular Velocity

← Older revision		Revision as of 21:11, 29 July 2020
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	<tr>		<tr>
	<td align="center">		<td align="center">
	The solid black curve derives from our evaluation of the function, <math>~\omega_0^2(e)</math>; it also may be found in:		The dark blue circular markers locate the 18 individual models identified in Table 1. The solid black curve derives from our evaluation of the function, <math>~\omega_0^2(e)</math>; this curve also may be found in:
	<div align="center">		<div align="center">
	Fig. 5 (p. 79) of [<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>];<br />		Fig. 5 (p. 79) of [<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>];<br />
	Fig. 7.2 (p. 173) of [<b>[[User:Tohline/Appendix/References#ST83\|<font color="red">ST83</font>]]</b>]<br />		Fig. 7.2 (p. 173) of [<b>[[User:Tohline/Appendix/References#ST83\|<font color="red">ST83</font>]]</b>]<br />
	~~The dark blue circular markers locate the 18 individual models identified in Table 1.~~
	</div>		</div>
	</td>		</td>

Tohline: /* Equilibrium Angular Velocity */

2020-07-29T21:08:14Z

Equilibrium Angular Velocity

← Older revision		Revision as of 21:08, 29 July 2020
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	</tr>		</tr>
	<tr><td align="center">		<tr><td align="center">
	[[File:~~EFE_Omega2vsEcc~~.png\|center\|350px\|Maclaurin Spheroid Sequence]]		[[File:EFE Omega2vsECCwithThomsonTait.png\|center\|350px\|Maclaurin Spheroid Sequence]]
	</td></tr>		</td></tr>
	<tr>		<tr>

Tohline: /* Energy Ratio, T/|W| */

2020-07-29T20:53:05Z

Energy Ratio, T/|W|

← Older revision		Revision as of 20:53, 29 July 2020
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	<tr>		<tr>
	<td colspan="3" align="center">		<td colspan="3" align="center">
	'''Table 1:'''  Limiting Values		'''Table 2:'''  Limiting Values
	</td>		</td>
	</tr>		</tr>
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	</table>		</table>

	Building on an [[User:Tohline/Apps/MaclaurinSpheroids#Gravitational_Potential\|accompanying discussion of the structure of Maclaurin spheroids]], Table 1 — shown just above, on the right — lists the limiting values of several key functions. Note, in particular, that as the eccentricity varies smoothly from zero (spherical configuration) to unity (infinitesimally thin disk), the energy ratio, <math>~\tau</math>, varies smoothly from zero to one-half. In his examination of the Maclaurin spheroid sequence, [[User:Tohline/Appendix/References#T78\|Tassoul (1978)]] chose to use this energy ratio as the ''order parameter'', rather than the eccentricity.		Building on an [[User:Tohline/Apps/MaclaurinSpheroids#Gravitational_Potential\|accompanying discussion of the structure of Maclaurin spheroids]], Table 2 — shown just above, on the right — lists the limiting values of several key functions. Note, in particular, that as the eccentricity varies smoothly from zero (spherical configuration) to unity (infinitesimally thin disk), the energy ratio, <math>~\tau</math>, varies smoothly from zero to one-half. In his examination of the Maclaurin spheroid sequence, [[User:Tohline/Appendix/References#T78\|Tassoul (1978)]] chose to use this energy ratio as the ''order parameter'', rather than the eccentricity.

	<table border="0" align="center"><tr><td align="center">		<table border="0" align="center"><tr><td align="center">