User:Tohline/ThreeDimensionalConfigurations/EFE Energies - Revision history

Tohline at 19:42, 25 July 2020

2020-07-25T19:42:19Z

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	~~{{LSU_HBook_header}}~~

	=Properties of Homogeneous Ellipsoids (2)=		=Properties of Homogeneous Ellipsoids (2)=

	In addition to pulling from §53 of [[User:Tohline/Appendix/References#EFE\|Chandrasekhar's EFE]], here, we lean heavily on the papers by [https://ui.adsabs.harvard.edu/abs/1983ApJ...271..586W/abstract M. D. Weinberg & S. Tremaine (1983, ApJ, 271, 586)] (hereafter, WT83) and by [https://ui.adsabs.harvard.edu/abs/1995ApJ...446..472C/abstract D. M. Christodoulou, D. Kazanas, I. Shlosman, & J. E. Tohline (1995a, ApJ, 446, 472)] (hereafter, Paper I).		In addition to pulling from §53 of [[User:Tohline/Appendix/References#EFE\|Chandrasekhar's EFE]], here, we lean heavily on the papers by [https://ui.adsabs.harvard.edu/abs/1983ApJ...271..586W/abstract M. D. Weinberg & S. Tremaine (1983, ApJ, 271, 586)] (hereafter, WT83) and by [https://ui.adsabs.harvard.edu/abs/1995ApJ...446..472C/abstract D. M. Christodoulou, D. Kazanas, I. Shlosman, & J. E. Tohline (1995a, ApJ, 446, 472)] (hereafter, Paper I).

			{{LSU_HBook_header}}

	==Sequence-Defining Dimensionless Parameters==		==Sequence-Defining Dimensionless Parameters==

Tohline: /* Conserve Only L */

2019-06-22T17:35:48Z

Conserve Only L

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	</div>		</div>

	Figure 3 of [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I] presents a black-and-white contour plot of this <math>~E_L</math> function for the specific case of <math>~L = 4.71488</math>, which, for reference, is the total angular momentum of an equilibrium [[User:Tohline/Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29\|Maclaurin spheroid]] having an eccentricity, <math>~e = 0.85</math> (see [[#Table1\|Table 1, below]]). We have digitally extracted this black-and-white contour plot from p. 477 of the (PDF-formatted) [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I] publication and have reprinted it as the left-hand panel of our Figure 1. Note that we have flipped the plot horizontally and rotated it by 90° so that the orientation of the axis pair, <math>~(\tfrac{b}{a},\tfrac{c}{a})</math>, conforms with the orientation of a related, information-rich diagram presented by [~~http~~://adsabs.harvard.edu/abs/1965ApJ...142..890C Chandrasekhar (1965)] — see also our [[User:Tohline/ThreeDimensionalConfigurations/JacobiEllipsoids#Sequence_Plots\|accompanying discussion of equilibrium sequence plots]].		Figure 3 of [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I] presents a black-and-white contour plot of this <math>~E_L</math> function for the specific case of <math>~L = 4.71488</math>, which, for reference, is the total angular momentum of an equilibrium [[User:Tohline/Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29\|Maclaurin spheroid]] having an eccentricity, <math>~e = 0.85</math> (see [[#Table1\|Table 1, below]]). We have digitally extracted this black-and-white contour plot from p. 477 of the (PDF-formatted) [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I] publication and have reprinted it as the left-hand panel of our Figure 1. Note that we have flipped the plot horizontally and rotated it by 90° so that the orientation of the axis pair, <math>~(\tfrac{b}{a},\tfrac{c}{a})</math>, conforms with the orientation of a related, information-rich diagram presented by [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract Chandrasekhar (1965)] — see also our [[User:Tohline/ThreeDimensionalConfigurations/JacobiEllipsoids#Sequence_Plots\|accompanying discussion of equilibrium sequence plots]].

	<div align="center" id="Figure1">		<div align="center" id="Figure1">

Tohline: /* Animation */

2019-06-22T17:32:40Z

Animation

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	===Animation===		===Animation===

	The animation sequence presented, below, as Figure 3 displays the warped free-energy surface (right) in conjunction with its projection onto the <math>~(\tfrac{b}{a},\tfrac{c}{a})</math> plane (left) for configurations having nineteen different total angular momentum values, <math>~L</math>, as detailed in column 5 of Table 1. The four-digit number that tags each frame of this animation sequence identifies the eccentricity (column 1 of Table 1) of the Maclaurin spheroid that is associated with each selected value of <math>~L</math>. In each frame of the animation, the equilibrium configuration associated with that Maclaurin spheroid is identified by the extremum of the free energy that appears along the right-hand edge <math>~(\tfrac{b}{a} = 1)</math> of the warped surface. For values of <math>~e < 0.81267</math> — corresponding to <math>~L < 4.23296</math> — the Maclaurin spheroid (marked by a small white circle/sphere) sits at the location of the absolute minimum of the free-energy surface and the configuration is stable. But for all larger values of the eccentricity/angular momentum, the Maclaurin spheroid (marked by a small dark-blue circle/sphere) is associated with a ''saddle point'' of the free-energy surface — that is, the configuration is in equilibrium, but it is (secularly) unstable — and the absolute energy minimum shifts off-axis to the location of a Jacobi ellipsoid (marked by a small white circle/sphere) having the same total angular momentum. As [http://adsabs.harvard.edu/abs/1995ApJ...446..472C ~~Christodoulou et al. (1995)~~] ~~point~~ out, evolution from the unstable axisymmetric equilibrium configuration to the stable triaxial configuration occurs along the narrow valley/canyon connecting the two extrema of the free energy.		The animation sequence presented, below, as Figure 3 displays the warped free-energy surface (right) in conjunction with its projection onto the <math>~(\tfrac{b}{a},\tfrac{c}{a})</math> plane (left) for configurations having nineteen different total angular momentum values, <math>~L</math>, as detailed in column 5 of Table 1. The four-digit number that tags each frame of this animation sequence identifies the eccentricity (column 1 of Table 1) of the Maclaurin spheroid that is associated with each selected value of <math>~L</math>. In each frame of the animation, the equilibrium configuration associated with that Maclaurin spheroid is identified by the extremum of the free energy that appears along the right-hand edge <math>~(\tfrac{b}{a} = 1)</math> of the warped surface. For values of <math>~e < 0.81267</math> — corresponding to <math>~L < 4.23296</math> — the Maclaurin spheroid (marked by a small white circle/sphere) sits at the location of the absolute minimum of the free-energy surface and the configuration is stable. But for all larger values of the eccentricity/angular momentum, the Maclaurin spheroid (marked by a small dark-blue circle/sphere) is associated with a ''saddle point'' of the free-energy surface — that is, the configuration is in equilibrium, but it is (secularly) unstable — and the absolute energy minimum shifts off-axis to the location of a Jacobi ellipsoid (marked by a small white circle/sphere) having the same total angular momentum. As [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I] points out, evolution from the unstable axisymmetric equilibrium configuration to the stable triaxial configuration occurs along the narrow valley/canyon connecting the two extrema of the free energy.


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	</tr>		</tr>
	<tr>		<tr>
	<td align="left" colspan="8"><sup>&dagger;</sup>Here, the units of angular momentum are as used in [http://adsabs.harvard.edu/abs/1995ApJ...446..472C ~~Christodoulou et al. (1995)~~]. In order to convert to units of <math>~L</math> as used in EFE (see, for example, Table I, in Chapter 5, §32), multiply by <math>~[2^2/(3\cdot 5^{10})]^{1/6} = 0.0717585</math>.		<td align="left" colspan="8"><sup>&dagger;</sup>Here, the units of angular momentum are as used in [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I]. In order to convert to units of <math>~L</math> as used in EFE (see, for example, Table I, in Chapter 5, §32), multiply by <math>~[2^2/(3\cdot 5^{10})]^{1/6} = 0.0717585</math>.
	</tr>		</tr>
	<tr>		<tr>

Tohline: /* Conserve Only L */

2019-06-22T17:31:07Z

Conserve Only L

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	Acknowledging that we are primarily interested in identifying extrema of this free-energy function, the discussion presented in §3.2 of [http://adsabs.harvard.edu/abs/1995ApJ...446..472C ~~Christodoulou et al. (1995)~~] shows us how to reduce the dimensionality of this problem by one. There, it is shown that, as long as <math>~\tfrac{b}{a} \ne 1</math>, extrema exist in the <math>~x</math>-coordinate direction — that is, <math>~\partial E_L/\partial x = 0</math> — only if <math>~x = 0.</math> For a given choice of <math>~L</math>, therefore, the relevant ''two-dimensional'' free-energy surface is defined by the expression,		Acknowledging that we are primarily interested in identifying extrema of this free-energy function, the discussion presented in §3.2 of [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I] shows us how to reduce the dimensionality of this problem by one. There, it is shown that, as long as <math>~\tfrac{b}{a} \ne 1</math>, extrema exist in the <math>~x</math>-coordinate direction — that is, <math>~\partial E_L/\partial x = 0</math> — only if <math>~x = 0.</math> For a given choice of <math>~L</math>, therefore, the relevant ''two-dimensional'' free-energy surface is defined by the expression,

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

	Figure 3 of [http://adsabs.harvard.edu/abs/1995ApJ...446..472C ~~Christodoulou et al. (1995)~~] presents a black-and-white contour plot of this <math>~E_L</math> function for the specific case of <math>~L = 4.71488</math>, which, for reference, is the total angular momentum of an equilibrium [[User:Tohline/Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29\|Maclaurin spheroid]] having an eccentricity, <math>~e = 0.85</math> (see [[#Table1\|Table 1, below]]). We have digitally extracted this black-and-white contour plot from p. 477 of the (PDF-formatted) [http://adsabs.harvard.edu/abs/1995ApJ...446..472C ~~Christodoulou et al. (1995)~~] publication and have reprinted it as the left-hand panel of our Figure 1. Note that we have flipped the plot horizontally and rotated it by 90° so that the orientation of the axis pair, <math>~(\tfrac{b}{a},\tfrac{c}{a})</math>, conforms with the orientation of a related, information-rich diagram presented by [http://adsabs.harvard.edu/abs/1965ApJ...142..890C Chandrasekhar (1965)] — see also our [[User:Tohline/ThreeDimensionalConfigurations/JacobiEllipsoids#Sequence_Plots\|accompanying discussion of equilibrium sequence plots]].		Figure 3 of [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I] presents a black-and-white contour plot of this <math>~E_L</math> function for the specific case of <math>~L = 4.71488</math>, which, for reference, is the total angular momentum of an equilibrium [[User:Tohline/Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29\|Maclaurin spheroid]] having an eccentricity, <math>~e = 0.85</math> (see [[#Table1\|Table 1, below]]). We have digitally extracted this black-and-white contour plot from p. 477 of the (PDF-formatted) [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I] publication and have reprinted it as the left-hand panel of our Figure 1. Note that we have flipped the plot horizontally and rotated it by 90° so that the orientation of the axis pair, <math>~(\tfrac{b}{a},\tfrac{c}{a})</math>, conforms with the orientation of a related, information-rich diagram presented by [http://adsabs.harvard.edu/abs/1965ApJ...142..890C Chandrasekhar (1965)] — see also our [[User:Tohline/ThreeDimensionalConfigurations/JacobiEllipsoids#Sequence_Plots\|accompanying discussion of equilibrium sequence plots]].

	<div align="center" id="Figure1">		<div align="center" id="Figure1">
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	<tr>		<tr>
	<td align="center" width="240px"><b>Left-hand Panel:</b><br />Black-and-white contour plot<br />		<td align="center" width="240px"><b>Left-hand Panel:</b><br />Black-and-white contour plot<br />
	extracted from p. 477 of [http://adsabs.harvard.edu/abs/1995ApJ...446..472C ~~Christodoulou et al. (1995)~~]<br />		extracted from p. 477 of [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I]<br />
	"''Phase-Transition Theory of Instabilities. I. Second-Harmonic Instability and Bifurcation Points''"<p></p>		"''Phase-Transition Theory of Instabilities. I. Second-Harmonic Instability and Bifurcation Points''"<p></p>
	ApJ, vol. 446, pp. 472-484 © [http://aas.org/ AAS]		ApJ, vol. 446, pp. 472-484 © [http://aas.org/ AAS]
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	</div>		</div>

	In our Figure 2, this same <math>~E_L</math> function has been displayed as a warped, two-dimensional free-energy surface draped across the three-dimensional <math>~(\tfrac{b}{a},\tfrac{c}{a},E)</math> domain, where depth as well as color has been used to tag energy values. The two-dimensional, colored contour plot presented in the right-hand panel of our Figure 1 results from the projection of this free-energy surface onto the <math>~(\tfrac{b}{a},\tfrac{c}{a})</math> plane; it reproduces in quantitative detail the black-and-white contour plot that we have extracted from [http://adsabs.harvard.edu/abs/1995ApJ...446..472C ~~Christodoulou et al. (1995)~~]. In an effort to (qualitatively) illustrate this agreement, we have digitally "pasted" the black-and-white contour plot from [http://adsabs.harvard.edu/abs/1995ApJ...446..472C ~~Christodoulou et al. (1995)~~] onto our colored contour plot and presented the combined image in the middle panel of our Figure 1.		In our Figure 2, this same <math>~E_L</math> function has been displayed as a warped, two-dimensional free-energy surface draped across the three-dimensional <math>~(\tfrac{b}{a},\tfrac{c}{a},E)</math> domain, where depth as well as color has been used to tag energy values. The two-dimensional, colored contour plot presented in the right-hand panel of our Figure 1 results from the projection of this free-energy surface onto the <math>~(\tfrac{b}{a},\tfrac{c}{a})</math> plane; it reproduces in quantitative detail the black-and-white contour plot that we have extracted from [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I]. In an effort to (qualitatively) illustrate this agreement, we have digitally "pasted" the black-and-white contour plot from [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I] onto our colored contour plot and presented the combined image in the middle panel of our Figure 1.


	Our Figure 2 image of the free-energy surface helps illuminate the description of this surface that appears in the caption of Fig. 3 from [http://adsabs.harvard.edu/abs/1995ApJ...446..472C ~~Christodoulou et al. (1995)~~]. Quoting from that figure caption:  "The [equilibrium] Maclaurin spheroid sits on a saddle point <math>~[(\tfrac{b}{a},\tfrac{c}{a}) = (1.0,0.52678); E_0 = -7.81842]</math>, while a global minimum with <math>~E_0 = -7.83300</math> exists at <math>~(\tfrac{b}{a},\tfrac{c}{a}) = (0.588,0.428)</math>."		Our Figure 2 image of the free-energy surface helps illuminate the description of this surface that appears in the caption of Fig. 3 from [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I]. Quoting from that figure caption:  "The [equilibrium] Maclaurin spheroid sits on a saddle point <math>~[(\tfrac{b}{a},\tfrac{c}{a}) = (1.0,0.52678); E_0 = -7.81842]</math>, while a global minimum with <math>~E_0 = -7.83300</math> exists at <math>~(\tfrac{b}{a},\tfrac{c}{a}) = (0.588,0.428)</math>."

	<div align="center" id="Figure2">		<div align="center" id="Figure2">

Tohline: /* Conserved Quantities */

2019-06-22T17:26:01Z

Conserved Quantities

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	[ 1<sup>st</sup> expression — [[User:Tohline/Appendix/References#EFE\|EFE]], <font color="#00CC00">§53, Eq. (239)</font> ]<br />		[ 1<sup>st</sup> expression — [[User:Tohline/Appendix/References#EFE\|EFE]], <font color="#00CC00">§53, Eq. (239)</font> ]<br />
	[ 2<sup>nd</sup> expression — [~~http~~://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I], <font color="#00CC00">Eq. (2.7)</font> ]<br />		[ 2<sup>nd</sup> expression — [https://ui.adsabs.harvard.edu/abs/1995ApJ...446..472C/abstract Paper I], <font color="#00CC00">Eq. (2.7)</font> ]<br />
	</div>		</div>
	where — see an [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#Triaxial_Configurations\|accompanying discussion]] for the definitions of <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math>,		where — see an [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#Triaxial_Configurations\|accompanying discussion]] for the definitions of <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math>,
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	[ 1<sup>st</sup> expression — [[User:Tohline/Appendix/References#EFE\|EFE]], <font color="#00CC00">§53, Eq. (239)</font> ]<br />		[ 1<sup>st</sup> expression — [[User:Tohline/Appendix/References#EFE\|EFE]], <font color="#00CC00">§53, Eq. (239)</font> ]<br />
	[ 2<sup>nd</sup> expression — [~~http~~://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I], <font color="#00CC00">Eq. (2.8)</font> ]<br />		[ 2<sup>nd</sup> expression — [https://ui.adsabs.harvard.edu/abs/1995ApJ...446..472C/abstract Paper I], <font color="#00CC00">Eq. (2.8)</font> ]<br />
	</div>		</div>

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	[ 1<sup>st</sup> expression — [[User:Tohline/Appendix/References#EFE\|EFE]], <font color="#00CC00">§53, Eq. (240)</font> ]<br />		[ 1<sup>st</sup> expression — [[User:Tohline/Appendix/References#EFE\|EFE]], <font color="#00CC00">§53, Eq. (240)</font> ]<br />
	[ 2<sup>nd</sup> expression — [~~http~~://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I], <font color="#00CC00">Eq. (2.5)</font> ]<br />		[ 2<sup>nd</sup> expression — [https://ui.adsabs.harvard.edu/abs/1995ApJ...446..472C/abstract Paper I], <font color="#00CC00">Eq. (2.5)</font> ]<br />
	</div>		</div>

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	[ 1<sup>st</sup> expression — [[User:Tohline/Appendix/References#EFE\|EFE]], <font color="#00CC00">§53, Eq. (241)</font> ]<br />		[ 1<sup>st</sup> expression — [[User:Tohline/Appendix/References#EFE\|EFE]], <font color="#00CC00">§53, Eq. (241)</font> ]<br />
	[ 2<sup>nd</sup> expression — [~~http~~://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I], <font color="#00CC00">Eq. (2.6)</font> ]<br />		[ 2<sup>nd</sup> expression — [https://ui.adsabs.harvard.edu/abs/1995ApJ...446..472C/abstract Paper I], <font color="#00CC00">Eq. (2.6)</font> ]<br />
	</div>		</div>

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	</table>		</table>

	[ [~~http~~://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I], <font color="#00CC00">Eq. (3.4)</font> ]<br />		[ [https://ui.adsabs.harvard.edu/abs/1995ApJ...446..472C/abstract Paper I], <font color="#00CC00">Eq. (3.4)</font> ]<br />
	</div>		</div>


	Note that, based on the units chosen in [~~http~~://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I], <math>~M = 5</math>, and <math>~abc = 15/4</math>.		Note that, based on the units chosen in [https://ui.adsabs.harvard.edu/abs/1995ApJ...446..472C/abstract Paper I], <math>~M = 5</math>, and <math>~abc = 15/4</math>.

	=Aside: Chandra's Notation=		=Aside: Chandra's Notation=

Tohline: /* Sequence-Defining Dimensionless Parameters */

2019-06-22T17:24:26Z

Sequence-Defining Dimensionless Parameters

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	[ [[User:Tohline/Appendix/References#EFE\|EFE]], <font color="#00CC00">§48, Eq. (31)</font> ]<br />		[ [[User:Tohline/Appendix/References#EFE\|EFE]], <font color="#00CC00">§48, Eq. (31)</font> ]<br />
	[ [https://ui.adsabs.harvard.edu/abs/1983ApJ...271..586W/abstract WT83], <font color="#00CC00">Eq. (5)</font> ]<br />		[ [https://ui.adsabs.harvard.edu/abs/1983ApJ...271..586W/abstract WT83], <font color="#00CC00">Eq. (5)</font> ]<br />
	[ [~~http~~://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I], <font color="#00CC00">Eq. (2.1)</font> ]<br />		[ [https://ui.adsabs.harvard.edu/abs/1995ApJ...446..472C/abstract Paper I], <font color="#00CC00">Eq. (2.1)</font> ]<br />
	</div>		</div>
	where, <math>~\zeta</math> is the system's vorticity as measured in a frame rotating with angular velocity, <math>~\Omega</math>. Alternatively, we can use the dimensionless parameter,		where, <math>~\zeta</math> is the system's vorticity as measured in a frame rotating with angular velocity, <math>~\Omega</math>. Alternatively, we can use the dimensionless parameter,
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	</table>		</table>
	[ [[User:Tohline/Appendix/References#EFE\|EFE]], <font color="#00CC00">§48, Eq. (40)</font> ]<br />		[ [[User:Tohline/Appendix/References#EFE\|EFE]], <font color="#00CC00">§48, Eq. (40)</font> ]<br />
	[ [~~http~~://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I], <font color="#00CC00">Eq. (2.2)</font> ]<br />		[ [https://ui.adsabs.harvard.edu/abs/1995ApJ...446..472C/abstract Paper I], <font color="#00CC00">Eq. (2.2)</font> ]<br />
	</div>		</div>
	or,		or,

Tohline: /* Properties of Homogeneous Ellipsoids (2) */

2019-06-22T17:23:54Z

Properties of Homogeneous Ellipsoids (2)

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	=Properties of Homogeneous Ellipsoids (2)=		=Properties of Homogeneous Ellipsoids (2)=

	In addition to pulling from §53 of [[User:Tohline/Appendix/References#EFE\|Chandrasekhar's EFE]], here, we lean heavily on the papers by [https://ui.adsabs.harvard.edu/abs/1983ApJ...271..586W/abstract M. D. Weinberg & S. Tremaine (1983, ApJ, 271, 586)] (hereafter, WT83) and by [~~http~~://adsabs.harvard.edu/abs/1995ApJ...446..472C D. M. Christodoulou, D. Kazanas, I. Shlosman, & J. E. Tohline (~~1995~~, ApJ, 446, 472)] (hereafter, Paper I).		In addition to pulling from §53 of [[User:Tohline/Appendix/References#EFE\|Chandrasekhar's EFE]], here, we lean heavily on the papers by [https://ui.adsabs.harvard.edu/abs/1983ApJ...271..586W/abstract M. D. Weinberg & S. Tremaine (1983, ApJ, 271, 586)] (hereafter, WT83) and by [https://ui.adsabs.harvard.edu/abs/1995ApJ...446..472C/abstract D. M. Christodoulou, D. Kazanas, I. Shlosman, & J. E. Tohline (1995a, ApJ, 446, 472)] (hereafter, Paper I).

	==Sequence-Defining Dimensionless Parameters==		==Sequence-Defining Dimensionless Parameters==

Tohline: /* Properties of Homogeneous Ellipsoids (2) */

2019-06-22T17:19:26Z

Properties of Homogeneous Ellipsoids (2)

← Older revision		Revision as of 17:19, 22 June 2019
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	=Properties of Homogeneous Ellipsoids (2)=		=Properties of Homogeneous Ellipsoids (2)=

	In addition to pulling from §53 of [[User:Tohline/Appendix/References#EFE\|Chandrasekhar's EFE]], here, we lean heavily on the papers by [~~http~~://adsabs.harvard.edu/abs/1983ApJ...271..586W M. D. Weinberg & S. Tremaine (1983, ApJ, 271, 586)] (hereafter, WT83) and by [http://adsabs.harvard.edu/abs/1995ApJ...446..472C D. M. Christodoulou, D. Kazanas, I. Shlosman, & J. E. Tohline (1995, ApJ, 446, 472)] (hereafter, Paper I).		In addition to pulling from §53 of [[User:Tohline/Appendix/References#EFE\|Chandrasekhar's EFE]], here, we lean heavily on the papers by [https://ui.adsabs.harvard.edu/abs/1983ApJ...271..586W/abstract M. D. Weinberg & S. Tremaine (1983, ApJ, 271, 586)] (hereafter, WT83) and by [http://adsabs.harvard.edu/abs/1995ApJ...446..472C D. M. Christodoulou, D. Kazanas, I. Shlosman, & J. E. Tohline (1995, ApJ, 446, 472)] (hereafter, Paper I).

	==Sequence-Defining Dimensionless Parameters==		==Sequence-Defining Dimensionless Parameters==
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	[ [[User:Tohline/Appendix/References#EFE\|EFE]], <font color="#00CC00">§48, Eq. (31)</font> ]<br />		[ [[User:Tohline/Appendix/References#EFE\|EFE]], <font color="#00CC00">§48, Eq. (31)</font> ]<br />
	[ [~~http~~://adsabs.harvard.edu/abs/1983ApJ...271..586W WT83], <font color="#00CC00">Eq. (5)</font> ]<br />		[ [https://ui.adsabs.harvard.edu/abs/1983ApJ...271..586W/abstract WT83], <font color="#00CC00">Eq. (5)</font> ]<br />
	[ [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I], <font color="#00CC00">Eq. (2.1)</font> ]<br />		[ [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I], <font color="#00CC00">Eq. (2.1)</font> ]<br />
	</div>		</div>
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	</tr>		</tr>
	</table>		</table>
	[ [~~http~~://adsabs.harvard.edu/abs/1983ApJ...271..586W WT83], <font color="#00CC00">Eq. (4)</font> ]<br />		[ [https://ui.adsabs.harvard.edu/abs/1983ApJ...271..586W/abstract WT83], <font color="#00CC00">Eq. (4)</font> ]<br />
	</div>		</div>

Tohline: /* Animation */ Insert new, longer animation, and improve description in text

2016-07-09T19:21:26Z

Animation: Insert new, longer animation, and improve description in text

← Older revision		Revision as of 19:21, 9 July 2016
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	===Animation===		===Animation===

	The animation sequence presented, below, as Figure 3 displays the warped free-energy surface (right) in conjunction with its projection onto the <math>~(\tfrac{b}{a},\tfrac{c}{a})</math> plane (left) for configurations having ~~thirteen~~ different total angular momentum values, <math>~L</math>, as detailed in column 3 of Table 1. The four-digit number that tags each frame of this animation sequence identifies the eccentricity (column 1 of Table 1) of the Maclaurin spheroid that is associated with each selected value of <math>~L</math>. In each frame of the animation, the equilibrium configuration associated with that Maclaurin spheroid is identified by the extremum of the free energy that appears along the right-hand edge <math>~(\tfrac{b}{a} = 1)</math> of the warped surface. For values of <math>~e < 0.81267</math> — corresponding to <math>~L < 4.23296</math> — the Maclaurin spheroid sits at the location of the absolute minimum of the free-energy surface and the configuration is stable. But for all larger values of the eccentricity/angular momentum, the Maclaurin spheroid is associated with a ''saddle point'' of the free-energy surface — that is, the configuration is in equilibrium, but it is (secularly) unstable — and the absolute energy minimum shifts off-axis to the location of a Jacobi ellipsoid having the same total angular momentum, As [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Christodoulou et al. (1995)] point out, evolution from the unstable axisymmetric equilibrium configuration to the stable triaxial configuration occurs along the narrow valley/canyon connecting the two extrema of the free energy.		The animation sequence presented, below, as Figure 3 displays the warped free-energy surface (right) in conjunction with its projection onto the <math>~(\tfrac{b}{a},\tfrac{c}{a})</math> plane (left) for configurations having nineteen different total angular momentum values, <math>~L</math>, as detailed in column 5 of Table 1. The four-digit number that tags each frame of this animation sequence identifies the eccentricity (column 1 of Table 1) of the Maclaurin spheroid that is associated with each selected value of <math>~L</math>. In each frame of the animation, the equilibrium configuration associated with that Maclaurin spheroid is identified by the extremum of the free energy that appears along the right-hand edge <math>~(\tfrac{b}{a} = 1)</math> of the warped surface. For values of <math>~e < 0.81267</math> — corresponding to <math>~L < 4.23296</math> — the Maclaurin spheroid (marked by a small white circle/sphere) sits at the location of the absolute minimum of the free-energy surface and the configuration is stable. But for all larger values of the eccentricity/angular momentum, the Maclaurin spheroid (marked by a small dark-blue circle/sphere) is associated with a ''saddle point'' of the free-energy surface — that is, the configuration is in equilibrium, but it is (secularly) unstable — and the absolute energy minimum shifts off-axis to the location of a Jacobi ellipsoid (marked by a small white circle/sphere) having the same total angular momentum. As [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Christodoulou et al. (1995)] point out, evolution from the unstable axisymmetric equilibrium configuration to the stable triaxial configuration occurs along the narrow valley/canyon connecting the two extrema of the free energy.


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	<tr>		<tr>
	<td align="center" colspan="1" bgcolor="#CCFFFF">		<td align="center" colspan="1" bgcolor="#CCFFFF">
	[[File:~~JacobiMaclaurin1~~.gif\|640px\|~~Christodoulou1995Fig3 Flipped~~]]		[[File:JacobiMaclaurin2.gif\|640px\|Animation related to Fig. 3 from Christodoulou1995]]
	</td>		</td>
	</tr>		</tr>

Tohline: /* Adopted Evolutionary Constraints */ Added 6 more Maclaurin spheroid models (6 more rows) to Table 1

2016-07-09T15:48:21Z

Adopted Evolutionary Constraints: Added 6 more Maclaurin spheroid models (6 more rows) to Table 1

← Older revision		Revision as of 15:48, 9 July 2016
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	</div>		</div>

			===Animation===

	The animation sequence presented, below, as Figure 3 displays the warped free-energy surface (right) in conjunction with its projection onto the <math>~(\tfrac{b}{a},\tfrac{c}{a})</math> plane (left) for configurations having thirteen different total angular momentum values, <math>~L</math>, as detailed in column 3 of Table 1. The four-digit number that tags each frame of this animation sequence identifies the eccentricity (column 1 of Table 1) of the Maclaurin spheroid that is associated with each selected value of <math>~L</math>. In each frame of the animation, the equilibrium configuration associated with that Maclaurin spheroid is identified by the extremum of the free energy that appears along the right-hand edge <math>~(\tfrac{b}{a} = 1)</math> of the warped surface. For values of <math>~e < 0.81267</math> — corresponding to <math>~L < 4.23296</math> — the Maclaurin spheroid sits at the location of the absolute minimum of the free-energy surface and the configuration is stable. But for all larger values of the eccentricity/angular momentum, the Maclaurin spheroid is associated with a ''saddle point'' of the free-energy surface — that is, the configuration is in equilibrium, but it is (secularly) unstable — and the absolute energy minimum shifts off-axis to the location of a Jacobi ellipsoid having the same total angular momentum, As [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Christodoulou et al. (1995)] point out, evolution from the unstable axisymmetric equilibrium configuration to the stable triaxial configuration occurs along the narrow valley/canyon connecting the two extrema of the free energy.		The animation sequence presented, below, as Figure 3 displays the warped free-energy surface (right) in conjunction with its projection onto the <math>~(\tfrac{b}{a},\tfrac{c}{a})</math> plane (left) for configurations having thirteen different total angular momentum values, <math>~L</math>, as detailed in column 3 of Table 1. The four-digit number that tags each frame of this animation sequence identifies the eccentricity (column 1 of Table 1) of the Maclaurin spheroid that is associated with each selected value of <math>~L</math>. In each frame of the animation, the equilibrium configuration associated with that Maclaurin spheroid is identified by the extremum of the free energy that appears along the right-hand edge <math>~(\tfrac{b}{a} = 1)</math> of the warped surface. For values of <math>~e < 0.81267</math> — corresponding to <math>~L < 4.23296</math> — the Maclaurin spheroid sits at the location of the absolute minimum of the free-energy surface and the configuration is stable. But for all larger values of the eccentricity/angular momentum, the Maclaurin spheroid is associated with a ''saddle point'' of the free-energy surface — that is, the configuration is in equilibrium, but it is (secularly) unstable — and the absolute energy minimum shifts off-axis to the location of a Jacobi ellipsoid having the same total angular momentum, As [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Christodoulou et al. (1995)] point out, evolution from the unstable axisymmetric equilibrium configuration to the stable triaxial configuration occurs along the narrow valley/canyon connecting the two extrema of the free energy.
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	<td align="center"><math>~\frac{c}{a}</math></td>		<td align="center"><math>~\frac{c}{a}</math></td>
	<td align="center"><math>~E_\mathrm{Jac}</math></td>		<td align="center"><math>~E_\mathrm{Jac}</math></td>
			</tr>
			<tr>
			<td align="center">0.650</td>
			<td align="center">0.7599342</td>
			<td align="center">-8.9018255</td>
			<td align="center">0.0</td>
			<td align="center">2.8270256</td>
			<td align="center">---</td>
			<td align="center">---</td>
			<td align="center">---</td>
			</tr>
			<tr>
			<td align="center">0.675</td>
			<td align="center">0.7378177</td>
			<td align="center">-8.8165100</td>
			<td align="center">0.0</td>
			<td align="center">2.9985043</td>
			<td align="center">---</td>
			<td align="center">---</td>
			<td align="center">---</td>
			</tr>
			<tr>
			<td align="center">0.700</td>
			<td align="center">0.7141428</td>
			<td align="center">-8.7216343</td>
			<td align="center">0.0</td>
			<td align="center">3.1820090</td>
			<td align="center">---</td>
			<td align="center">---</td>
			<td align="center">---</td>
			</tr>
			<tr>
			<td align="center">0.725</td>
			<td align="center">0.6887489</td>
			<td align="center">-8.6155943</td>
			<td align="center">0.0</td>
			<td align="center">3.3796768</td>
			<td align="center">---</td>
			<td align="center">---</td>
			<td align="center">---</td>
			</tr>
			<tr>
			<td align="center">0.750</td>
			<td align="center">0.66143783</td>
			<td align="center">-8.4963506</td>
			<td align="center">0.0</td>
			<td align="center">3.5942337</td>
			<td align="center">---</td>
			<td align="center">---</td>
			<td align="center">---</td>
			</tr>
			<tr>
			<td align="center">0.775</td>
			<td align="center">0.6319612</td>
			<td align="center">-8.3612566</td>
			<td align="center">0.0</td>
			<td align="center">3.8292360</td>
			<td align="center">---</td>
			<td align="center">---</td>
			<td align="center">---</td>
	</tr>		</tr>
	<tr>		<tr>