Appendix/Ramblings/ForCohlHoward - Revision history

Jet53man: /* Dumbbell-Shaped Distortion */

2023-05-23T23:10:33Z

Dumbbell-Shaped Distortion

← Older revision		Revision as of 19:10, 23 May 2023
Line 1,706:		Line 1,706:
	<tr>		<tr>
	<td align="center" colspan="1" rowspan="2">		<td align="center" colspan="1" rowspan="2">
	[[File:~~PearAndDumbbellModelE3~~.png\|600px\|Pear and Dumbbell Sequences]]		[[File:PearAndDumbbellModelE4.png\|600px\|Pear and Dumbbell Sequences]]
	</td>		</td>
	<td align="center" colspan="1" rowspan="1">		<td align="center" colspan="1" rowspan="1">
Line 1,718:		Line 1,718:
	<tr>		<tr>
	<td align="left" colspan="2">		<td align="left" colspan="2">
	''Left panel (primary plot):'' Same as the ''primary plot'' displayed in [[#RRSTEMfigure4\|RRSTEM Figure 4, immediately above]]. A red cross labeled <font color="red">E</font> identifies the position along the Jacobi ellipsoid sequence that is a neutral point against a 4<sup>th</sup>-order harmonic perturbation. A so-called dumbbell-shaped sequence branches off at this point; ~~here~~, ~~we have displayed the behavior of this~~ sequence ~~by drawing the properties of equilibrium models from Table II (p. 1073)~~ of ~~{{ EHS82 }}~~.		''Left panel (primary plot):'' Same as the ''primary plot'' displayed in [[#RRSTEMfigure4\|RRSTEM Figure 4, immediately above]]. A red cross labeled <font color="red">E</font> identifies the position along the Jacobi ellipsoid sequence that is a neutral point against a 4<sup>th</sup>-order harmonic perturbation. A so-called dumbbell-shaped sequence branches off at this point; it, in turn, transitions to a sequence of equal-mass binaries.

	''Left panel (inset box):'' An ''inset box'' ~~is used to magnify~~ the ~~segment~~ of ~~the Maclaurin spheroid sequence where bifurcation to the so-called one-ring (Dyson-Wong toroid) sequence occurs. The neutral point~~ that ~~is believed to be associated with the bifurcation point, itself, carries~~ the ~~geometric distortion label, <math>~~(~~n, m~~) = (~~4, 0~~)~~</math>, and is identified as our <font color="red">Model C</font>~~. ~~As has been detailed in [[Appendix/Ramblings/MacSphCriticalPoints#Model-Sequence_Details\|our separate chapter discussion]]~~, the ~~smooth (pink) curve that connects the spheroid~~ sequence ~~to the one-ring sequence has been~~ defined by ~~the~~ set of 18 equilibrium models ~~presented by~~ {{ ~~ES81~~ }}; the set of ~~small green square markers identify nine~~ equilibrium models ~~obtained~~ from ~~Table I~~ of ~~the separate study by {{ HES82 }}; and the small purple triangular markers identify eight equilibrium models obtained from~~ Table ~~Ia of~~ {{ ~~Hachisu86a~~ }}.		''Left panel (inset box):'' An ''inset box'' shows more clearly the sequence of equilibrium models that make up the dumbbell (green markers and curve) and binary (blue markers and curve) sequences. Here, the dumbbell sequence is defined by a set of equilibrium models drawn from Table II (p. 1073) of {{ EHS82 }} while the binary sequence is defined by a set of equilibrium models drawn from the subsection (p. 243) of Table 1 labeled "N = 0" in {{ HE84b }}.

	''Right panel:'' Figure 1 (plus caption) from {{ CKST95b }} has been reprinted here to emphasize its similarity to, and overlap with our ''inset box''. According to the caption of this reprinted figure, the filled circular marker labeled "A" identifies the bifurcation point on the ~~Maclaurin spheroid~~ sequence, where <math>e = 0.~~985226~~</math>. Accordingly, we have annotated the reprinted figure to indicate that the ~~axisymmetric equilibrium~~ model associated with point "A" is exactly our <font color="red">Model C</font>. As is stated in the caption of this reprinted figure, the dotted line XBC denotes the (hypothesized) onset of a secular instability that — in the nonlinear regime and conserving total angular momentum (vertical dotted line) — should ~~deform~~ the ~~spheroid~~ into ~~a ring~~-~~like configuration~~.		''Right panel:'' Figure 4 (plus caption) from {{ CKST95b }} has been reprinted here to emphasize its similarity to, and overlap with our ''inset box''. According to the caption of this reprinted figure, the filled circular marker labeled "A" identifies the bifurcation point on the Jacobi ellipsoid sequence, where <math>(b/a, c/a) = (0.2972, 0.2575)</math>. Accordingly, we have annotated the reprinted figure to indicate that the Jacobi ellipsoid model associated with point "A" is exactly our <font color="red">Model E</font>. As is stated in the caption of this reprinted figure, the dotted line XBC denotes the (hypothesized) onset of a secular instability that — in the nonlinear regime and conserving total angular momentum (vertical dotted line) — should lead to fission of the ellipsoid into an equal-mass binary system.
	</td>		</td>
	</tr>		</tr>
	</table>		</table>


	~~<font color="red"><b>Model E</b></font>:~~

	==Details==		==Details==

Jet53man: /* Dumbbell-Shaped Distortion */

2023-05-23T19:12:26Z

Dumbbell-Shaped Distortion

← Older revision		Revision as of 15:12, 23 May 2023
Line 1,697:		Line 1,697:
	<li>		<li>
	{{ CKST95bfull }} grab parameter values from a variety of sources. In subsection "B" (''Jacobi Ellipsoid to Binary'') of their Table 1 (p. 494) and in the first paragraph of their §3.2 (p. 492), they state … <math>(b/a, c/a) = (0.29720, 0.25746)</math>;  <math>\Omega^2/(4\pi G \rho) = 0.0532790</math>;   and, <math>j^2 = 0.0115082</math>.</li>		{{ CKST95bfull }} grab parameter values from a variety of sources. In subsection "B" (''Jacobi Ellipsoid to Binary'') of their Table 1 (p. 494) and in the first paragraph of their §3.2 (p. 492), they state … <math>(b/a, c/a) = (0.29720, 0.25746)</math>;  <math>\Omega^2/(4\pi G \rho) = 0.0532790</math>;   and, <math>j^2 = 0.0115082</math>.</li>
	<li><font color="red">NOTE:</font>   Plugging the axis values from p. 128 of [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>] — that is, <math>(b/a, c/a) = (0.2972, 0.2575)</math> — into our "Riemann01.for" application, we find <math>(A_1, A_2, A_3) = 0.171772322973, 0.844742744895, 0.983484932132)</math>, and <math>\Omega^2/(4\pi G \rho) = 0.053286</math>, and <math>j^2 = 0.011507</math>.</li>		<li><font color="red">NOTE (23 May 2023):</font>   Plugging the axis values from p. 128 of [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>] — that is, <math>(b/a, c/a) = (0.2972, 0.2575)</math> — into our "Riemann01.for" application, we find <math>(A_1, A_2, A_3) = 0.171772322973, 0.844742744895, 0.983484932132)</math>, and <math>\Omega^2/(4\pi G \rho) = 0.053286</math>, and <math>j^2 = 0.011507</math>.</li>
	</ul>		</ul>

Jet53man: /* Jacobi Ellipsoids */

2023-05-23T19:08:42Z

Jacobi Ellipsoids

← Older revision		Revision as of 15:08, 23 May 2023
Line 1,619:		Line 1,619:
	<td align="center"><math>0.2972</math></td>		<td align="center"><math>0.2972</math></td>
	<td align="center"><math>0.2575</math></td>		<td align="center"><math>0.2575</math></td>
	<td align="center"><math>0.~~0532~~</math></td>		<td align="center"><math>0.053286</math></td>
	<td align="center"><math>0.1863</math></td>		<td align="center"><math>0.1863</math></td>
	<td align="center"><math>1.~~157~~\times 10^{-2}</math></td>		<td align="center"><math>1.1507\times 10^{-2}</math></td>
	<td align="center">Dumbbell-shaped</td>		<td align="center">Dumbbell-shaped</td>
	<td align="center">Uniform <math>\omega_0</math></td>		<td align="center">Uniform <math>\omega_0</math></td>

Jet53man: /* Dumbbell-Shaped Distortion */

2023-05-23T19:06:02Z

Dumbbell-Shaped Distortion

← Older revision		Revision as of 15:06, 23 May 2023
Line 1,697:		Line 1,697:
	<li>		<li>
	{{ CKST95bfull }} grab parameter values from a variety of sources. In subsection "B" (''Jacobi Ellipsoid to Binary'') of their Table 1 (p. 494) and in the first paragraph of their §3.2 (p. 492), they state … <math>(b/a, c/a) = (0.29720, 0.25746)</math>;  <math>\Omega^2/(4\pi G \rho) = 0.0532790</math>;   and, <math>j^2 = 0.0115082</math>.</li>		{{ CKST95bfull }} grab parameter values from a variety of sources. In subsection "B" (''Jacobi Ellipsoid to Binary'') of their Table 1 (p. 494) and in the first paragraph of their §3.2 (p. 492), they state … <math>(b/a, c/a) = (0.29720, 0.25746)</math>;  <math>\Omega^2/(4\pi G \rho) = 0.0532790</math>;   and, <math>j^2 = 0.0115082</math>.</li>
	<li><font color="red">NOTE:</font>   Plugging the axis values from p. 128 of [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>] — that is, <math>(b/a, c/a) = (0.2972, 0.2575)</math> — into our "Riemann01.for" application, we find <math>(A_1, A_2, A_3) = 0.171772322973, 0.844742744895, 0.983484932132)</math> and <math>\Omega^2/(4\pi G \rho) = 0.053286</math>.</li>		<li><font color="red">NOTE:</font>   Plugging the axis values from p. 128 of [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>] — that is, <math>(b/a, c/a) = (0.2972, 0.2575)</math> — into our "Riemann01.for" application, we find <math>(A_1, A_2, A_3) = 0.171772322973, 0.844742744895, 0.983484932132)</math>, and <math>\Omega^2/(4\pi G \rho) = 0.053286</math>, and <math>j^2 = 0.011507</math>.</li>
	</ul>		</ul>

Jet53man: /* Dumbbell-Shaped Distortion */

2023-05-23T18:58:45Z

Dumbbell-Shaped Distortion

← Older revision		Revision as of 14:58, 23 May 2023
Line 1,697:		Line 1,697:
	<li>		<li>
	{{ CKST95bfull }} grab parameter values from a variety of sources. In subsection "B" (''Jacobi Ellipsoid to Binary'') of their Table 1 (p. 494) and in the first paragraph of their §3.2 (p. 492), they state … <math>(b/a, c/a) = (0.29720, 0.25746)</math>;  <math>\Omega^2/(4\pi G \rho) = 0.0532790</math>;   and, <math>j^2 = 0.0115082</math>.</li>		{{ CKST95bfull }} grab parameter values from a variety of sources. In subsection "B" (''Jacobi Ellipsoid to Binary'') of their Table 1 (p. 494) and in the first paragraph of their §3.2 (p. 492), they state … <math>(b/a, c/a) = (0.29720, 0.25746)</math>;  <math>\Omega^2/(4\pi G \rho) = 0.0532790</math>;   and, <math>j^2 = 0.0115082</math>.</li>
			<li><font color="red">NOTE:</font>   Plugging the axis values from p. 128 of [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>] — that is, <math>(b/a, c/a) = (0.2972, 0.2575)</math> — into our "Riemann01.for" application, we find <math>(A_1, A_2, A_3) = 0.171772322973, 0.844742744895, 0.983484932132)</math> and <math>\Omega^2/(4\pi G \rho) = 0.053286</math>.</li>
	</ul>		</ul>

Jet53man: /* Dumbbell-Shaped Distortion */

2023-05-22T23:23:23Z

Dumbbell-Shaped Distortion

← Older revision		Revision as of 19:23, 22 May 2023
Line 1,717:		Line 1,717:
	<tr>		<tr>
	<td align="left" colspan="2">		<td align="left" colspan="2">
	''Left panel (primary plot):'' Same as the ''primary plot'' displayed in [[#RRSTEMfigure4\|RRSTEM Figure 4, immediately above]].		''Left panel (primary plot):'' Same as the ''primary plot'' displayed in [[#RRSTEMfigure4\|RRSTEM Figure 4, immediately above]]. A red cross labeled <font color="red">E</font> identifies the position along the Jacobi ellipsoid sequence that is a neutral point against a 4<sup>th</sup>-order harmonic perturbation. A so-called dumbbell-shaped sequence branches off at this point; here, we have displayed the behavior of this sequence by drawing the properties of equilibrium models from Table II (p. 1073) of {{ EHS82 }}.

	''Left panel (inset box):'' An ''inset box'' is used to magnify the segment of the Maclaurin spheroid sequence where bifurcation to the so-called one-ring (Dyson-Wong toroid) sequence occurs. The neutral point that is believed to be associated with the bifurcation point, itself, carries the geometric distortion label, <math>(n, m) = (4, 0)</math>, and is identified as our <font color="red">Model C</font>. As has been detailed in [[Appendix/Ramblings/MacSphCriticalPoints#Model-Sequence_Details\|our separate chapter discussion]], the smooth (pink) curve that connects the spheroid sequence to the one-ring sequence has been defined by the set of 18 equilibrium models presented by {{ ES81 }}; the set of small green square markers identify nine equilibrium models obtained from Table I of the separate study by {{ HES82 }}; and the small purple triangular markers identify eight equilibrium models obtained from Table Ia of {{ Hachisu86a }}.		''Left panel (inset box):'' An ''inset box'' is used to magnify the segment of the Maclaurin spheroid sequence where bifurcation to the so-called one-ring (Dyson-Wong toroid) sequence occurs. The neutral point that is believed to be associated with the bifurcation point, itself, carries the geometric distortion label, <math>(n, m) = (4, 0)</math>, and is identified as our <font color="red">Model C</font>. As has been detailed in [[Appendix/Ramblings/MacSphCriticalPoints#Model-Sequence_Details\|our separate chapter discussion]], the smooth (pink) curve that connects the spheroid sequence to the one-ring sequence has been defined by the set of 18 equilibrium models presented by {{ ES81 }}; the set of small green square markers identify nine equilibrium models obtained from Table I of the separate study by {{ HES82 }}; and the small purple triangular markers identify eight equilibrium models obtained from Table Ia of {{ Hachisu86a }}.

Jet53man: /* Dumbbell-Shaped Distortion */

2023-05-22T23:15:42Z

Dumbbell-Shaped Distortion

← Older revision		Revision as of 19:15, 22 May 2023
Line 1,717:		Line 1,717:
	<tr>		<tr>
	<td align="left" colspan="2">		<td align="left" colspan="2">
	''Left panel (primary plot):'' ~~As in~~ the ''~~right panel~~'' of [[#~~RRSTEMfigure1~~\|RRSTEM Figure 1, above]], the solid, multi-colored curve shows how <math>\Omega^2</math> varies with <math>0 \le j^2 \le 0.04</math> along the Maclaurin spheroid sequence. Models A and B are not explicitly labeled, but the plot still shows the small solid circular markers (purple and yellow, respectively) that identify the locations of these models on the spheroid sequence. The point where the Jacobi sequence bifurcates from the Maclaurin spheroid sequence (Model A) is labeled by the quantum numbers, <math>(n, m) = (2, 2)</math>, to indicate what geometric distortion, <math>P_n^m</math>, that is associated with this particular bifurcation. Drawing from Table 1 of {{ HE84 }}, five other neutral points are marked by red crosses; three carry labels in this ''primary plot'' — <math>(3, 3), (4, 4), (3, 1)</math> — and the remaining two are labeled in the ''inset box'' — <math>(4, 2), (4, 0)</math>.		''Left panel (primary plot):'' Same as the ''primary plot'' displayed in [[#RRSTEMfigure4\|RRSTEM Figure 4, immediately above]].

	''Left panel (inset box):'' An ''inset box'' is used to magnify the segment of the Maclaurin spheroid sequence where bifurcation to the so-called one-ring (Dyson-Wong toroid) sequence occurs. The neutral point that is believed to be associated with the bifurcation point, itself, carries the geometric distortion label, <math>(n, m) = (4, 0)</math>, and is identified as our <font color="red">Model C</font>. As has been detailed in [[Appendix/Ramblings/MacSphCriticalPoints#Model-Sequence_Details\|our separate chapter discussion]], the smooth (pink) curve that connects the spheroid sequence to the one-ring sequence has been defined by the set of 18 equilibrium models presented by {{ ES81 }}; the set of small green square markers identify nine equilibrium models obtained from Table I of the separate study by {{ HES82 }}; and the small purple triangular markers identify eight equilibrium models obtained from Table Ia of {{ Hachisu86a }}.		''Left panel (inset box):'' An ''inset box'' is used to magnify the segment of the Maclaurin spheroid sequence where bifurcation to the so-called one-ring (Dyson-Wong toroid) sequence occurs. The neutral point that is believed to be associated with the bifurcation point, itself, carries the geometric distortion label, <math>(n, m) = (4, 0)</math>, and is identified as our <font color="red">Model C</font>. As has been detailed in [[Appendix/Ramblings/MacSphCriticalPoints#Model-Sequence_Details\|our separate chapter discussion]], the smooth (pink) curve that connects the spheroid sequence to the one-ring sequence has been defined by the set of 18 equilibrium models presented by {{ ES81 }}; the set of small green square markers identify nine equilibrium models obtained from Table I of the separate study by {{ HES82 }}; and the small purple triangular markers identify eight equilibrium models obtained from Table Ia of {{ Hachisu86a }}.

Jet53man: /* Dumbbell-Shaped Distortion */

2023-05-22T23:12:43Z

Dumbbell-Shaped Distortion

@@ Line 1,688: / Line 1,688: @@
 <font color="red"><b>Model E</b></font>:
-&#176;
+<ul><li>
+According to the last pair of equations on p. 128 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 6, &sect;45</font>, a neutral point belonging to the fourth harmonic (a dumbbell-shaped) distortion arises on the Jacobi Sequence at <math>(b/a) = 0.2972</math> and <math>\cos^{-1}(c/a) = 75\overset{\circ}{.}081~~~~\Rightarrow ~~~~ (c/a) = 0.2575</math>.  Chronologically, this result for <math>(b/a, c/a)</math> appears first in Eq. (93) on p. 635 of {{ Chandrasekhar67_XXXIIfull }}.  Then, in Eq. (66) on p. 302 of {{ Chandrasekhar68_XXXVfull }} &#8212; we find <math>\cos^{-1}(c/a) = 75\overset{\circ}{.}068</math>, along with a footnote [5] which states, <font color="darkgreen">"The value <math>\cos^{-1}(c/a) = 75\overset{\circ}{.}081</math> found earlier differs slightly; but the difference is not outside the limits of accuracy of the numerical evaluation."</font>
-According the last pair of equations on p. 128 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 6, &sect;45</font>, a neutral point belonging to the fourth harmonic (a dumbbell-shaped) distortion arises on the Jacobi Sequence at <math>(b/a) = (0.2972)</math> and <math>\cos^{-1}(c/a) = 75^{\arcmin}081~\mathrm{degrees}~~~~\Rightarrow ~~~~ (c/a) = 0.2575</math>.
+</li>
+<li>
+According to the first row of properties in Table II of {{ EHS82 }}, we find that <font color="red">Model E</font> is characterized by the properties &hellip;  <math>\Omega^2/(4\pi G \rho) = 0.0532</math>; <math>j^2 = ( 3\cdot 2^{-8} \pi^{-4} )^{1/3}  L^2/(GM^3\bar{a}) = 0.01157</math>; and <math>\tau \equiv T/|W| = 0.1863 </math>.  &nbsp;I have not (yet) found the corresponding value of <math>\Omega^2</math> in any of Chandrasekhar's publications, but if we combine the value of <math>\Omega^2</math> obtained from {{ EHS82hereafter }} with the values of <math>(b/a, c/a)</math> obtained from [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], we find &hellip; <math>\Omega^2/(\pi G \rho) = 0.2128</math>; from the [[#Angular_Momentum_Constraint|above expression]], <math>L_* = 0.48242</math>; and &nbsp; <math>j^2 = ( 3\cdot 2^{-8} \pi^{-4})^{1/3}  L_*^2 = 0.01149</math>.  &nbsp; This value of <math>j^2</math> is very close to the value obtained by {{ EHS82hereafter }}.</li>
-<ul>
+<li>
-<li><math>(b/a) = (0.2972)</math> and <math>\cos^{-1}(c/a) = 75.081~\mathrm{degrees}~~~~\Rightarrow ~~~~ (c/a) = 0.2575</math>.</li>
+In the paragraph at the top of the right-hand column of p. 467 of {{Hachisu86bfull }}, we find &hellip; <math>\Omega^2/(4\pi G\rho) = 0.0535</math>; <math>j^2 = 0.01157</math>.</li>
 </ul>

Jet53man: /* Dumbbell-Shaped Distortion */

2023-05-22T21:55:59Z

Dumbbell-Shaped Distortion

@@ Line 1,687: / Line 1,687: @@
 <font color="red"><b>Model E</b></font>:
 <span id="RRSTEMfigure5">&nbsp;</span>
 <table border="1" align="center" cellpadding="5">

Jet53man: /* Pear-Shaped Distortion */

2023-05-22T21:38:09Z

Pear-Shaped Distortion

← Older revision		Revision as of 17:38, 22 May 2023
Line 1,678:		Line 1,678:
	''Primary plot:'' As in the ''right panel'' of [[#RRSTEMfigure1\|RRSTEM Figure 1, above]], the solid, multi-colored curve shows how <math>\Omega^2</math> varies with <math>0 \le j^2 \le 0.04</math> along the Maclaurin spheroid sequence; for reference, <font color="red">Model A</font> and <font color="red">Model B</font> are labeled. Also as in [[#RRSTEMfigure1\|RRSTEM Figure 1]], starting at the Model A bifurcation point, the purple dashed curve identifies the equilibrium sequence of Jacobi ellipsoids. A red cross labeled <font color="red">F</font> identifies the position along the Jacobi ellipsoid sequence that is a "neutral point against the 3<sup>rd</sup>-order harmonic perturbation." A so-called pear-shaped sequence branches off at this point; here, we have displayed the behavior of this sequence by drawing the properties of equilibrium models from Table I (p. 1071) of {{ EHS82 }}.		''Primary plot:'' As in the ''right panel'' of [[#RRSTEMfigure1\|RRSTEM Figure 1, above]], the solid, multi-colored curve shows how <math>\Omega^2</math> varies with <math>0 \le j^2 \le 0.04</math> along the Maclaurin spheroid sequence; for reference, <font color="red">Model A</font> and <font color="red">Model B</font> are labeled. Also as in [[#RRSTEMfigure1\|RRSTEM Figure 1]], starting at the Model A bifurcation point, the purple dashed curve identifies the equilibrium sequence of Jacobi ellipsoids. A red cross labeled <font color="red">F</font> identifies the position along the Jacobi ellipsoid sequence that is a "neutral point against the 3<sup>rd</sup>-order harmonic perturbation." A so-called pear-shaped sequence branches off at this point; here, we have displayed the behavior of this sequence by drawing the properties of equilibrium models from Table I (p. 1071) of {{ EHS82 }}.

	''Inset box:'' Because the pear-shaped sequence is very short, an ''inset box'' is used to magnify the plotted sequence; this was also done by {{ EHS82 }} — see their Figure 1 (p. 1072). A red arrow, that accompanies our <font color="red">Model F</font> label, points to the location along the Jacobi sequence where the relevant neutral point lies. As is suggested by {{ EHS82 }}, the presumption is that this pear-shaped sequence bifurcates from the Jacobi sequence at the neutral point. But this is not the case in our plot. Upon closer inspection, we see that the pear-shaped sequence ''does'' appear to bifurcate from the Jacobi sequence in Figure 1 of {{ EHS82 }} and that this happens because the Jacobi sequence is shifted a bit from ''our'' depiction of the Jacobi sequence location. Curious!		''Inset box:'' Especially Because the pear-shaped sequence is very short, an ''inset box'' is used to magnify the plotted sequence; this was also done by {{ EHS82 }} — see their Figure 1 (p. 1072). A red arrow, that accompanies our <font color="red">Model F</font> label, points to the location along the Jacobi sequence where the relevant neutral point lies. As is suggested by {{ EHS82 }}, the presumption is that this pear-shaped sequence bifurcates from the Jacobi sequence at the neutral point. But this is not the case in our plot. Upon closer inspection, we see that the pear-shaped sequence ''does'' appear to bifurcate from the Jacobi sequence in Figure 1 of {{ EHS82 }} and that this happens because the Jacobi sequence is shifted a bit from ''our'' depiction of the Jacobi sequence location. Curious!
	</td>		</td>
	</tr>		</tr>