Appendix/Ramblings/MacSphCriticalPoints - Revision history

Jet53man: /* Model-Sequence Details */

2023-04-24T22:06:50Z

Model-Sequence Details

Jet53man: /* Model-Sequence Details */

2023-04-21T19:26:03Z

Model-Sequence Details

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	</table>		</table>

			<span id="PhaseTransition"> </span>
	<table border="1" align="center" cellpadding="5">		<table border="1" align="center" cellpadding="5">
	<tr>		<tr>

Jet53man: /* Axisymmetric Equilibrium Sequences that Display a Topological Change */

2023-04-14T19:14:48Z

Axisymmetric Equilibrium Sequences that Display a Topological Change

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	<li>A so-called Maclaurin toroid sequence — the toroidal segment was first constructed by {{ MPT77 }}.</li>		<li>A so-called Maclaurin toroid sequence — the toroidal segment was first constructed by {{ MPT77 }}.</li>
	</ol>		</ol>
	Along the first of these sequences, every equilibrium configuration has a surface that is precisely a spheroid. The other two bifurcate from the Maclaurin spheroid sequence; contain a segment of concave, hamburger-shaped (spheroidal-like) configurations; and, upon further extension, blend into ~~extend~~		Along the first of these sequences, every equilibrium configuration has a surface that is precisely a spheroid. The other two bifurcate from the Maclaurin spheroid sequence; contain a segment of concave, hamburger-shaped (spheroidal-like) configurations; and, upon further extension, blend into the (separately identified) sequence of toroidal-shaped configurations. In other words, moving along the respective model sequences, we encounter a spheroidal-like to ring-like topological change. The overriding question that requires nonlinear dynamical modeling to answer is: "Is there a model (or a range of models) along the Maclaurin spheroid sequence that is unstable toward evolution away from that sequence and toward one of the ring-like sequences?"

			'''TWO KEY CONCEPTS:'''
			<ol type="A">
			<li>The model parameterization that distinguishes the two ring-like sequences from one another is the specified ''distribution of angular momentum.'' Dyson-Wong tori are uniformly rotating, whereas, each Maclaurin toroid has an angular momentum distribution that is the same as is present in all Maclaurin spheroids. It is this single structural feature that drives the bifurcation points for these sequences to two quite different locations along the Maclaurin spheroid sequence.</li>
			<li>The bifurcation points of both ring-like sequences arise at positions on the Maclaurin spheroid sequence where the models are expected to be violently unstable toward the development of nonaxisymmetric distortions. Hence, in order to study how unstable spheroids undergo a transition to a ring-like configuration (the topological change), a numerical code must be developed with the capability to suppress all nonaxisymmetric distortions. This can be accomplished, for example, by adopting a cylindrical coordinate representation of the fluid equations then performing two-dimensional rather than three-dimensional simulations.</li>
			</ol>

	~~that bifurcates from the Maclaurin spheroid sequence and, upon extension, blends into the sequence of Dyson-Wong tori~~

	===One-Ring (Dyson-Wong) Sequence===		===One-Ring (Dyson-Wong) Sequence===

			It is important to remember — as emphasized above — that ''all'' equilibrium models along the one-ring (Dyson-Wong) sequence are uniformly rotating.

	====Background Storyline====		====Background Storyline====

Jet53man: /* Axisymmetric Equilibrium Sequences that Display a Topological Change */

2023-04-14T18:29:00Z

Axisymmetric Equilibrium Sequences that Display a Topological Change

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	In the context of discussions of self-gravitating configurations that are rotating and have a uniform density, there are three well-known ''axisymmetric'' equilibrium sequences:		In the context of discussions of self-gravitating configurations that are rotating and have a uniform density, there are three well-known ''axisymmetric'' equilibrium sequences:
	<ul>		<ol>
	<li>The Maclaurin spheroid sequence.</li>		<li>The Maclaurin spheroid sequence — first constructed in 1742 by Maclaurin.</li>
	<li>A so-called "one-ring" sequence ~~that bifurcates from~~ the ~~Maclaurin spheroid sequence and, upon extension, blends into the sequence of~~ Dyson~~-Wong tori.~~</li>		<li>A so-called one-ring (Dyson-Wong toroid) sequence — the ring-like segment was first mapped out in 1893 by Dyson</li>
	<li>~~The~~ Maclaurin ~~spheroid~~ sequence.</li>		<li>A so-called Maclaurin toroid sequence — the toroidal segment was first constructed by {{ MPT77 }}.</li>
	</ul>		</ol>
			Along the first of these sequences, every equilibrium configuration has a surface that is precisely a spheroid. The other two bifurcate from the Maclaurin spheroid sequence; contain a segment of concave, hamburger-shaped (spheroidal-like) configurations; and, upon further extension, blend into extend


			that bifurcates from the Maclaurin spheroid sequence and, upon extension, blends into the sequence of Dyson-Wong tori

	===One-Ring (Dyson-Wong) Sequence===		===One-Ring (Dyson-Wong) Sequence===

Jet53man: /* One-Ring Sequence */

2023-04-14T17:51:29Z

One-Ring Sequence

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	</table>		</table>

	===One-Ring Sequence===		==Axisymmetric Equilibrium Sequences that Display a Topological Change==

			In the context of discussions of self-gravitating configurations that are rotating and have a uniform density, there are three well-known ''axisymmetric'' equilibrium sequences:
			<ul>
			<li>The Maclaurin spheroid sequence.</li>
			<li>A so-called "one-ring" sequence that bifurcates from the Maclaurin spheroid sequence and, upon extension, blends into the sequence of Dyson-Wong tori.</li>
			<li>The Maclaurin spheroid sequence.</li>
			</ul>

			===One-Ring (Dyson-Wong) Sequence===

	====Background Storyline====		====Background Storyline====

Jet53man: /* Background Storyline */

2023-04-14T00:09:25Z

Background Storyline

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	</ul>		</ul>
	</li>		</li>
	<li>{{ ES81 }} claim that the one-ring sequence bifurcates from the Maclaurin sequence precisely at the point where the spheroid has an eccentricity, <math>e = e_\mathrm{cr} = 0.98523</math> — in which case, also, <math>\Omega^2/(4\pi G \rho) = 0.08726</math> and <math>j^2 = 0.02174</math>. In support of this conjecture, they point out that, Chandrasekhar (1967; publication XXX) and {{ Bardeen71 }} have shown that this is <font color="darkgreen">… a neutral point on the Maclaurin sequence against the perturbation of <math>P_4(\eta)</math> displacement at the surface where <math>\eta</math> is one of the spheroidal coordinates."</font> This is also the "bifurcation point along the Maclaurin sequence" that is labeled by the quantum numbers, <math>(n, m) = (4, 0)</math> in Table 1 of {{ HE84 }}.</li>		<li>{{ ES81 }} claim that the one-ring sequence bifurcates from the Maclaurin sequence precisely at the point where the spheroid has an eccentricity, <math>e = e_\mathrm{cr} = 0.98523</math> — in which case, also, <math>\Omega^2/(4\pi G \rho) = 0.08726</math> and <math>j^2 = 0.02174</math>. In support of this conjecture, they point out that, Chandrasekhar (1967; publication XXX) and {{ Bardeen71 }} have shown that this is <font color="darkgreen">… a neutral point on the Maclaurin sequence against the perturbation of <math>P_4(\eta)</math> displacement at the surface where <math>\eta</math> is one of the spheroidal coordinates."</font> This is also the "neutral point" on the Maclaurin sequence labeled "F" in Table I of {{ HE82 }}; and the "bifurcation point" along the Maclaurin sequence that is labeled by the quantum numbers, <math>(n, m) = (4, 0)</math> in Table 1 of {{ HE84 }}.</li>
	</ol>		</ol>

Jet53man: /* One-Ring Sequence */

2023-04-14T00:00:05Z

One-Ring Sequence

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	<li>Table I of {{ ES81 }} provides quantitative data describing the properties of eighteen models that lie along this combined "one-ring" sequence, such as: <math>\Omega^2/(4\pi G \rho), j^2, T_\mathrm{rot}/\|W_\mathrm{grav}\|,</math> and <math>(T_\mathrm{rot} + W_\mathrm{grav})/E_0</math>. We have copied the values of two of these parameters from their Table I into the first two (pink) columns of our table, immediately below.</li>		<li>Table I of {{ ES81 }} provides quantitative data describing the properties of eighteen models that lie along this combined "one-ring" sequence, such as: <math>\Omega^2/(4\pi G \rho), j^2, T_\mathrm{rot}/\|W_\mathrm{grav}\|,</math> and <math>(T_\mathrm{rot} + W_\mathrm{grav})/E_0</math>. We have copied the values of two of these parameters from their Table I into the first two (pink) columns of our table, immediately below.</li>
	<li>This data has been used to generate the pink-colored one-ring sequence shown in our plot, below; see especially the plot inset. Over the years, the same set of data has been used to display the behavior of the one-ring sequence in numerous publications; see, for example, the reproduction of Figure 1 (p. 488) from {{ CKST95b }} that we have presented, below.</li>		<li>This data has been used to generate the pink-colored one-ring sequence shown in our plot, below; see especially the plot inset. Over the years, the same set of data has been used to display the behavior of the one-ring sequence in numerous publications; see, for example, the reproduction of Figure 1 (p. 488) from {{ CKST95b }} that we have presented, below.</li>
			<li>Figure 2 of {{ ES81 }} displays meridional-plane cross-sections through five of their eighteen models in an effort to illustrate how the surface geometry smoothly changes along the complete sequence: from spheroid, to "hamburger" shape, to torus.</li>
	</ul>		</ul>
	</li>		</li>
			<li>{{ ES81 }} claim that the one-ring sequence bifurcates from the Maclaurin sequence precisely at the point where the spheroid has an eccentricity, <math>e = e_\mathrm{cr} = 0.98523</math> — in which case, also, <math>\Omega^2/(4\pi G \rho) = 0.08726</math> and <math>j^2 = 0.02174</math>. In support of this conjecture, they point out that, Chandrasekhar (1967; publication XXX) and {{ Bardeen71 }} have shown that this is <font color="darkgreen">… a neutral point on the Maclaurin sequence against the perturbation of <math>P_4(\eta)</math> displacement at the surface where <math>\eta</math> is one of the spheroidal coordinates."</font> This is also the "bifurcation point along the Maclaurin sequence" that is labeled by the quantum numbers, <math>(n, m) = (4, 0)</math> in Table 1 of {{ HE84 }}.</li>
	</ol>		</ol>

Jet53man: /* One-Ring Sequence */

2023-04-13T23:14:59Z

One-Ring Sequence

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	<li>Over 125 years ago via a pair of detailed publications — {{ Dyson1893full }} and {{ Dyson1893Part2full }} — Dyson demonstrated that a sequence of rapidly rotating, self-gravitating equilibrium models could be constructed that had a uniform density, were uniformly rotating, and had a toroidal (ring) shape. [Configurations that, in every respect except their ''shape'', were like Maclaurin spheroids.] See [[Apps/DysonPotential#Dyson_(1893)\|our review and discussion]] of this work.</li>		<li>Over 125 years ago via a pair of detailed publications — {{ Dyson1893full }} and {{ Dyson1893Part2full }} — Dyson demonstrated that a sequence of rapidly rotating, self-gravitating equilibrium models could be constructed that had a uniform density, were uniformly rotating, and had a toroidal (ring) shape. [Configurations that, in every respect except their ''shape'', were like Maclaurin spheroids.] See [[Apps/DysonPotential#Dyson_(1893)\|our review and discussion]] of this work.</li>
	<li>{{ Wong74full }} tackled this same problem, improving on, and extending Dyson's work. As a consequence, this sequence of models is often referred to as "Dyson-Wong tori." See [[Apps/DysonWongTori#Self-Gravitating,_Incompressible_(Dyson-Wong)_Tori\|our detailed review and discussion]].</li>		<li>{{ Wong74full }} tackled this same problem, improving on, and extending Dyson's work. As a consequence, this sequence of models is often referred to as "Dyson-Wong tori." See [[Apps/DysonWongTori#Self-Gravitating,_Incompressible_(Dyson-Wong)_Tori\|our detailed review and discussion]].</li>
			<li>Motivated by the work of {{ FESB-K80full }}, {{ ES81full }} demonstrated that the Dyson-Wong toroidal sequence can be "smoothly connected" to the Maclaurin spheroid sequence via an intermediate branch of models having a <font color="darkgreen">… concave hamburger-like shape of equilibrium …"</font>
			<ul>
			<li>Table I of {{ ES81 }} provides quantitative data describing the properties of eighteen models that lie along this combined "one-ring" sequence, such as: <math>\Omega^2/(4\pi G \rho), j^2, T_\mathrm{rot}/\|W_\mathrm{grav}\|,</math> and <math>(T_\mathrm{rot} + W_\mathrm{grav})/E_0</math>. We have copied the values of two of these parameters from their Table I into the first two (pink) columns of our table, immediately below.</li>
			<li>This data has been used to generate the pink-colored one-ring sequence shown in our plot, below; see especially the plot inset. Over the years, the same set of data has been used to display the behavior of the one-ring sequence in numerous publications; see, for example, the reproduction of Figure 1 (p. 488) from {{ CKST95b }} that we have presented, below.</li>
			</ul>
			</li>
	</ol>		</ol>

Jet53man: /* One-Ring Sequence */

2023-04-13T22:10:31Z

One-Ring Sequence

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	===One-Ring Sequence===		===One-Ring Sequence===

	~~<table border~~=~~"1" align~~=~~"center" cellpadding~~=~~"5">~~		====Background Storyline====
	~~<tr>~~
	~~<td align~~=~~"center" colspan~~=~~"1" rowspan~~=~~"2">~~		<ol>
	~~[[File:ES81OneRingWithInsetBox.png\|800px\|One-Ring Sequence]]~~		<li>Over 125 years ago via a pair of detailed publications — {{ Dyson1893full }} and {{ Dyson1893Part2full }} — Dyson demonstrated that a sequence of rapidly rotating, self-gravitating equilibrium models could be constructed that had a uniform density, were uniformly rotating, and had a toroidal (ring) shape. [Configurations that, in every respect except their ''shape'', were like Maclaurin spheroids.] See [[Apps/DysonPotential#Dyson_(1893)\|our review and discussion]] of this work.</li>
	<~~/td~~>		<li>{{ Wong74full }} tackled this same problem, improving on, and extending Dyson's work. As a consequence, this sequence of models is often referred to as "Dyson-Wong tori." See [[Apps/DysonWongTori#Self-Gravitating,_Incompressible_(Dyson-Wong)_Tori\|our detailed review and discussion]].</li>
	<~~td align="center" colspan="1" rowspan="1"~~>		</ol>
	~~Figure 1 extracted from §2.2, p. 488~~ of &~~hellip~~;~~<br />~~{{ ~~CKST95bfigure~~ }}
	</~~td>~~
	</tr>
	<tr>
	~~<td align=~~"~~center~~" ~~colspan="1">~~[[~~File:CKST95bFig1annotated.png~~\|~~400px\|CKST95b Figure 1~~]]</~~td>~~
	~~</tr~~>
	</~~table~~>

			====Model-Sequence Details====

	<table border="1" align="center" cellpadding="5">		<table border="1" align="center" cellpadding="5">
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	</ul>		</ul>
	</td>		</td>
			</tr>
			</table>


			<table border="1" align="center" cellpadding="5">
			<tr>
			<td align="center" colspan="1" rowspan="2">
			[[File:ES81OneRingWithInsetBox.png\|800px\|One-Ring Sequence]]
			</td>
			<td align="center" colspan="1" rowspan="1">
			Figure 1 extracted from §2.2, p. 488 of …<br />{{ CKST95bfigure }}
			</td>
			</tr>
			<tr>
			<td align="center" colspan="1">[[File:CKST95bFig1annotated.png\|400px\|CKST95b Figure 1]]</td>
	</tr>		</tr>
	</table>		</table>

Jet53man: /* One-Ring Sequence */

2023-04-13T17:22:15Z

One-Ring Sequence

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	</tr>		</tr>
	<tr>		<tr>
	<td align="center" colspan="1">[[File:~~CKST95bFig1~~.png\|400px\|CKST95b Figure 1]]</td>		<td align="center" colspan="1">[[File:CKST95bFig1annotated.png\|400px\|CKST95b Figure 1]]</td>
	</tr>		</tr>
	</table>		</table>