ThreeDimensionalConfigurations/RiemannStype - Revision history

Jet53man: /* Summary */

2023-05-10T22:43:58Z

Summary

← Older revision		Revision as of 18:43, 10 May 2023
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	{{ LL96 }}, p. 700, §2, Eq. (1)<br />		{{ LL96 }}, p. 700, §2, Eq. (1)<br />
			{{ LL96b }}, p. 930, §3, Eqs. (3.1) & (3.2)<br />
	{{ Ou2006 }}, p. 550, §2, Eqs. (3) & (17)		{{ Ou2006 }}, p. 550, §2, Eqs. (3) & (17)
	</div>		</div>

Jet53man: /* Determination and Plot */

2022-04-02T21:25:14Z

Determination and Plot

← Older revision		Revision as of 17:25, 2 April 2022
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	</td></tr></table>		</td></tr></table>



			<table border="1" align="center" cellpadding="8">
			<tr>
			<td align="center" colspan="13"><b>Table:   Key Values Along the Limiting Maclaurin Spheroid Sequence</b></td>
			</tr>

			<tr>
			<td align="center" rowspan="2">Model</td>
			<td align="center" rowspan="2">e</td>
			<td align="center" rowspan="2"><math>\Omega^2_\mathrm{Mc}</math></td>
			<td align="center" rowspan="2"><math>B_{11}</math></td>
			<td align="center" rowspan="2"><math>\mathcal{H} \equiv \frac{4B_{11}}{\Omega^2_\mathrm{Mc}}</math></td>
			<td align="center" colspan="4">Direct</td>
			<td align="center" colspan="4">Adjoint</td>
			</tr>

			<tr>
			<td align="center"><math>\Omega_3</math></td>
			<td align="center"><math>\zeta_3</math></td>
			<td align="center"><math>\frac{\Omega_3}{\Omega_\mathrm{Mc}}</math></td>
			<td align="center"><math>f = \frac{\zeta_3}{\Omega_3}</math></td>
			<td align="center"><math>\Omega_3</math></td>
			<td align="center"><math>\zeta_3</math></td>
			<td align="center"><math>\frac{\Omega_3}{\Omega_\mathrm{Mc}}</math></td>
			<td align="center"><math>f = \frac{\zeta_3}{\Omega_3}</math></td>
			</tr>

			<tr>
			<td align="center" rowspan="1">Nonrotating Sphere & <br />"<math>x = -1</math>" Bifurcation</td>
			<td align="right">0.00000</td>
			<td align="right">0.00000</td>
			<td align="center"><math>\frac{4}{15}</math></td>
			<td align="center"><math>\infty</math></td>
			<td align="center"><math>-~\biggl(\frac{4}{15}\biggr)^{1 / 2}</math><br />Note: (a)</td>
			<td align="right"><math>+~\biggl(\frac{16}{15}\biggr)^{1 / 2}</math></td>
			<td align="center"><math>\infty</math></td>
			<td align="center"><math>-~2</math><br />Note: (a)</td>
			<td align="center"><math>+~\biggl(\frac{4}{15}\biggr)^{1 / 2}</math><br />Note: (a)</td>
			<td align="right"><math>-~\biggl(\frac{16}{15}\biggr)^{1 / 2}</math></td>
			<td align="center"><math>\infty</math></td>
			<td align="center"><math>-~2</math><br />Note: (a)</td>
			</tr>

			<tr>
			<td align="center" rowspan="1">Jacobi/Dedekind Bifurcation</td>
			<td align="center">0.81267<br />Note: (b)</td>
			<td align="center">0.37423<br />Note: (b)</td>
			<td align="center"><math>\frac{\Omega^2_\mathrm{Mc}}{2}</math><br />Note: (c)</td>
			<td align="center"><math>2</math></td>
			<td align="center"><math>\Omega_\mathrm{Mc}</math><br />Note: (c)</td>
			<td align="center"><math>0</math><br />Note: (c)</td>
			<td align="center"><math>1</math></td>
			<td align="center"><math>0</math></td>
			<td align="center"><math>0</math><br />Note: (c)</td>
			<td align="center"><math>2\Omega_\mathrm{Mc}</math><br />Note: (c)</td>
			<td align="center"><math>0</math><br />Note: (c)</td>
			<td align="center"><math>\pm \infty</math><br />Note: (c)</td>
			</tr>

			<tr>
			<td align="center" rowspan="1">"<math>x = +1</math>" Bifurcation</td>
			<td align="right">0.952886702</td>
			<td align="right">0.440219895</td>
			<td align="center"><math>\frac{\Omega^2_\mathrm{Mc}}{4}</math><br />Note: (d)</td>
			<td align="center"><math>1</math></td>
			<td align="center"><math>\frac{\Omega_\mathrm{Mc}}{2}</math><br />Note: (d)</td>
			<td align="center"><math>\Omega_\mathrm{Mc}</math><br />Note: (d)</td>
			<td align="center"><math>\frac{1}{2}</math></td>
			<td align="center"><math>+~2</math></td>
			<td align="center"><math>\frac{\Omega_\mathrm{Mc}}{2}</math><br />Note: (d)</td>
			<td align="center"><math>\Omega_\mathrm{Mc}</math><br />Note: (d)</td>
			<td align="center"><math>\frac{1}{2}</math></td>
			<td align="center"><math>+~2</math></td>
			</tr>

			<tr>
			<td align="left" colspan="13">
			Notes:
			<ol type="a">
			<li>[<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], p. 142, last line of Table VI includes … <math>\Omega^2 = 4/15 \approx 0.26667</math> and <math>f = -2</math>.</li>
			<li>[<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], p. 78, Table I.</li>
			<li>[<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], p. 138, §48b(iv).</li>
			<li>[<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], p. 138, §48b(v).</li>
			</ol>
			</td>
			</tr>
			</table>

	==Models Examined by Ou (2006)==		==Models Examined by Ou (2006)==

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 </tr>
 <tr><td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 3, &sect;21, Eq. (105)</font></td></tr>
 </table>

Jet53man: /* Determination and Plot */

2022-04-01T23:02:28Z

Determination and Plot

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	<td align="left" width="50%">Analogous to Figure 5 from §32, p. 79 of [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>]; shows how the square of the normalized rotation frequency varies with eccentricity, <math>e = (1 - a_3/a_1)^{1 / 2},</math> along the (black-dotted) Maclaurin sequence and along the Jacobi sequence (series of purple circular markers).</td>		<td align="left" width="50%">Analogous to Figure 5 from §32, p. 79 of [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>]; shows how the square of the normalized rotation frequency varies with eccentricity, <math>e = (1 - a_3/a_1)^{1 / 2},</math> along the (black-dotted) Maclaurin sequence and along the Jacobi sequence (series of purple circular markers).</td>
	<td align="left">		<td align="left">
	Each Riemann S-type ellipsoid sequence — uniquely identified by its frequency ratio, <math>f = \zeta_3/\Omega_3</math> — intersects the Maclaurin spheroid sequence at a specific value of the meridional-plane eccentricity, <math>e = (1 - a_3/a_1)^{1 / 2}.</math> Analogous to Figure 12 from §48, p. 139 of [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], this plot displays how <math>f_\mathrm{Mc}</math> varies with <math>e</math> along the Maclaurin spheroid sequence; for a given <math>e</math>, the solid black curve is associated with the ''direct'' configuration while the dashed orange curve is associated with the ''adjoint'' configuration. The circular, solid-purple marker identifies the point where the Jacobi sequence <math>(f_\mathrm{Mc} = 0)</math> bifurcates from the Maclaurin spheroid sequence; bifurcation to the Dedekind sequence is not displayed here because its frequency ratio <math>(f_\mathrm{Mc} = \pm \infty)</math> is off scale.		Each Riemann S-type ellipsoid sequence — uniquely identified by its frequency ratio, <math>f = \zeta_3/\Omega_3</math> — intersects the Maclaurin spheroid sequence at a specific value of the meridional-plane eccentricity, <math>e = (1 - a_3/a_1)^{1 / 2}.</math> Analogous to Figure 12 from §48, p. 139 of [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], this plot displays how <math>f_\mathrm{Mc}</math> varies with <math>e</math> along the Maclaurin spheroid sequence; for a given <math>e</math>, the solid black curve is associated with the ''direct'' configuration while the dashed orange curve is associated with the ''adjoint'' configuration. The circular, solid-purple marker identifies the point where the Jacobi sequence <math>(f_\mathrm{Mc} = 0)</math> bifurcates from the Maclaurin spheroid sequence; bifurcation to the Dedekind sequence is not displayed here because its frequency ratio <math>(f_\mathrm{Mc} = \pm \infty)</math> is off scale. The square, solid-green markers identify the location of the pair of models (direct/adjoint) for which <math>\Omega^2_\mathrm{Mc}</math> is maximum (see the square, solid-green marker in the left-hand panel). The circular, solid-orange marker identifies the (degenerate) pair of models <math>(f_\mathrm{Mc} = +2)</math> where the "<math>x = +1</math>" sequence bifurcates from the Maclaurin spheroid sequence.
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Determination and Plot */

2022-04-01T22:52:14Z

Determination and Plot

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	</td>		</td>
	<td align="left">		<td align="left">
	<math>1 \pm (\mathcal{H} + 1)^{1 / 2} \, ,</math>		<math>1 \pm (\mathcal{H} - 1)^{1 / 2} \, ,</math>
	</td>		</td>
	<td align="center">      and,		<td align="center">      and,
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	<td align="left" width="50%">Analogous to Figure 5 from §32, p. 79 of [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>]; shows how the square of the normalized rotation frequency varies with eccentricity, <math>e = (1 - a_3/a_1)^{1 / 2},</math> along the (black-dotted) Maclaurin sequence and along the Jacobi sequence (series of purple circular markers).</td>		<td align="left" width="50%">Analogous to Figure 5 from §32, p. 79 of [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>]; shows how the square of the normalized rotation frequency varies with eccentricity, <math>e = (1 - a_3/a_1)^{1 / 2},</math> along the (black-dotted) Maclaurin sequence and along the Jacobi sequence (series of purple circular markers).</td>
	<td align="left">		<td align="left">
	Analogous to Figure 12 from §48, p. 139 of [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>]~~; shows~~ how ~~the frequency ratio,~~ <math>f_\mathrm{Mc} ~~= \zeta_3~~/~~\Omega_3)~~</math>, ~~varies~~ with ~~eccentricity~~, <math>~~e =~~ (~~1 - a_3/a_1)^~~{~~1 / 2~~},</math> ~~along~~ the ~~(black-dotted)~~ Maclaurin sequence ~~and along~~ the ~~Jacobi~~ sequence (~~series of purple circular markers~~).		Each Riemann S-type ellipsoid sequence — uniquely identified by its frequency ratio, <math>f = \zeta_3/\Omega_3</math> — intersects the Maclaurin spheroid sequence at a specific value of the meridional-plane eccentricity, <math>e = (1 - a_3/a_1)^{1 / 2}.</math> Analogous to Figure 12 from §48, p. 139 of [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], this plot displays how <math>f_\mathrm{Mc}</math> varies with <math>e</math> along the Maclaurin spheroid sequence; for a given <math>e</math>, the solid black curve is associated with the ''direct'' configuration while the dashed orange curve is associated with the ''adjoint'' configuration. The circular, solid-purple marker identifies the point where the Jacobi sequence <math>(f_\mathrm{Mc} = 0)</math> bifurcates from the Maclaurin spheroid sequence; bifurcation to the Dedekind sequence is not displayed here because its frequency ratio <math>(f_\mathrm{Mc} = \pm \infty)</math> is off scale.
	</td>		</td>
	</tr>		</tr>

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	</td>		</td>
	<td align="left">		<td align="left">
	<math>\Omega_\mathrm{Mc} \pm \biggl[4 B_{11} + \Omega^2_\mathrm{Mc}\biggr]^{1 / 2} \, ,</math>		<math>\Omega_\mathrm{Mc} \pm \biggl[4 B_{11} - \Omega^2_\mathrm{Mc}\biggr]^{1 / 2} \, ,</math>
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Maclaurin Spheroid Limit */

2022-04-01T22:16:24Z

Maclaurin Spheroid Limit

Show changes

Jet53man: /* Expressions Supplied by EFE */

2022-04-01T17:24:20Z

Expressions Supplied by EFE

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	<td align="center" rowspan="2"><math>\Omega^2_\mathrm{Mc}</math></td>		<td align="center" rowspan="2"><math>\Omega^2_\mathrm{Mc}</math></td>
	<td align="center" rowspan="2"><math>B_{11}</math></td>		<td align="center" rowspan="2"><math>B_{11}</math></td>
	<td align="center" rowspan="2"><math>\mathcal{H}</math></td>		<td align="center" rowspan="2"><math>\mathcal{H} \equiv \frac{4B_{11}}{\Omega^2_\mathrm{Mc}}</math></td>
	<td align="center" colspan="4">Direct</td>		<td align="center" colspan="4">Direct</td>
	<td align="center" colspan="4">Adjoint</td>		<td align="center" colspan="4">Adjoint</td>
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	<tr>		<tr>
	<td align="center" rowspan="1">Jacobi/Dedekind Bifurcation</td>		<td align="center" rowspan="1">Jacobi/Dedekind Bifurcation</td>
	<td align="~~right~~">0.81267<br />Note: (b)</td>		<td align="center">0.81267<br />Note: (b)</td>
	<td align="~~right~~">0.37423<br />Note: (b)</td>		<td align="center">0.37423<br />Note: (b)</td>
	<td align="~~right~~">~~---~~</td>		<td align="center"><math>\frac{\Omega^2_\mathrm{Mc}}{2}</math><br />Note: (c)</td>
	<td align="~~right~~">~~1.00000~~</td>		<td align="center"><math>2</math></td>
	<td align="~~right~~">~~---~~</td>		<td align="center"><math>\Omega_\mathrm{Mc}</math><br />Note: (c)</td>
	<td align="~~right~~">~~---~~</td>		<td align="center"><math>0</math><br />Note: (c)</td>
	<td align="~~right~~"><math>~~\tfrac{1}{2}(~~1 ~~- \sqrt{2})~~</math></td>		<td align="center"><math>1</math></td>
	<td align="~~right~~"><math>~~\frac{2}{(1 - \sqrt{2})}~~</math></td>		<td align="center"><math>0</math></td>
	<td align="~~right~~">~~---~~</td>		<td align="center"><math>0</math><br />Note: (c)</td>
	<td align="~~right~~">~~---~~</td>		<td align="center"><math>2\Omega_\mathrm{Mc}</math><br />Note: (c)</td>
	<td align="~~right~~"><math>~~\tfrac{1}{2}(1 + \sqrt{2})~~</math></td>		<td align="center"><math>0</math><br />Note: (c)</td>
	<td align="~~right~~"><math>\~~frac{2}{(1 +~~ \~~sqrt{2})}~~</math></td>		<td align="center"><math>\pm \infty</math><br />Note: (c)</td>
	</tr>		</tr>

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	<td align="right">0.952886702</td>		<td align="right">0.952886702</td>
	<td align="right">0.440219895</td>		<td align="right">0.440219895</td>
	<td align="~~right~~">~~---~~</td>		<td align="center"><math>\frac{\Omega^2_\mathrm{Mc}}{4}</math><br />Note: (d)</td>
	<td align="~~right~~">1~~.00000~~</td>		<td align="center"><math>1</math></td>
	<td align="~~right~~">~~---~~</td>		<td align="center"><math>\frac{\Omega_\mathrm{Mc}}{2}</math><br />Note: (d)</td>
	<td align="~~right~~">~~---~~</td>		<td align="center"><math>\Omega_\mathrm{Mc}</math><br />Note: (d)</td>
	<td align="right"><math>\tfrac{1}{2}(1 - \sqrt{2})</math></td>		<td align="right"><math>\tfrac{1}{2}(1 - \sqrt{2})</math></td>
	<td align="right"><math>\frac{2}{(1 - \sqrt{2})}</math></td>		<td align="right"><math>\frac{2}{(1 - \sqrt{2})}</math></td>
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	<ol type="a">		<ol type="a">
	<li>[<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], p. 142, last line of Table VI includes … <math>\Omega^2 = 4/15 \approx 0.26667</math> and <math>f = -2</math>.</li>		<li>[<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], p. 142, last line of Table VI includes … <math>\Omega^2 = 4/15 \approx 0.26667</math> and <math>f = -2</math>.</li>
	<li>~~See~~ [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], p. 78, Table I.</li>		<li>[<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], p. 78, Table I.</li>
			<li>[<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], p. 138, §48b(iv).</li>
			<li>[<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], p. 138, §48b(v).</li>
	</ol>		</ol>
	</td>		</td>

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	<tr>		<tr>
	<td align="center" rowspan="1">Nonrotating Sphere</td>		<td align="center" rowspan="1">Nonrotating Sphere & <br />"<math>x = -1</math>" Bifurcation</td>
	<td align="right">0.00000</td>		<td align="right">0.00000</td>
	<td align="right">0.00000</td>		<td align="right">0.00000</td>
	<td align="center"><math>\frac{4}{15}</math></td>		<td align="center"><math>\frac{4}{15}</math></td>
	<td align="center"><math>\infty</math></td>		<td align="center"><math>\infty</math></td>
	<td align="~~right~~"><math>-~\biggl(\frac{4}{15}\biggr)^{1 / 2}</math></td>		<td align="center"><math>-~\biggl(\frac{4}{15}\biggr)^{1 / 2}</math><br />Note: (a)</td>
	<td align="right"><math>\biggl(\frac{16}{15}\biggr)^{1 / 2}</math></td>		<td align="right"><math>+~\biggl(\frac{16}{15}\biggr)^{1 / 2}</math></td>
	<td align="center"><math>\infty</math></td>		<td align="center"><math>\infty</math></td>
	<td align="center">-2</td>		<td align="center"><math>-~2</math><br />Note: (a)</td>
	<td align="~~right~~"><math>\biggl(\frac{4}{15}\biggr)^{1 / 2}</math></td>		<td align="center"><math>+~\biggl(\frac{4}{15}\biggr)^{1 / 2}</math><br />Note: (a)</td>
	<td align="right"><math>-~\biggl(\frac{16}{15}\biggr)^{1 / 2}</math></td>		<td align="right"><math>-~\biggl(\frac{16}{15}\biggr)^{1 / 2}</math></td>
	<td align="center"><math>\infty</math></td>		<td align="center"><math>\infty</math></td>
	<td align="center">-2</td>		<td align="center"><math>-~2</math><br />Note: (a)</td>
			</tr>

			<tr>
			<td align="center" rowspan="1">Jacobi/Dedekind Bifurcation</td>
			<td align="right">0.81267<br />Note: (b)</td>
			<td align="right">0.37423<br />Note: (b)</td>
			<td align="right">---</td>
			<td align="right">1.00000</td>
			<td align="right">---</td>
			<td align="right">---</td>
			<td align="right"><math>\tfrac{1}{2}(1 - \sqrt{2})</math></td>
			<td align="right"><math>\frac{2}{(1 - \sqrt{2})}</math></td>
			<td align="right">---</td>
			<td align="right">---</td>
			<td align="right"><math>\tfrac{1}{2}(1 + \sqrt{2})</math></td>
			<td align="right"><math>\frac{2}{(1 + \sqrt{2})}</math></td>
	</tr>		</tr>

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	<td align="right"><math>\tfrac{1}{2}(1 + \sqrt{2})</math></td>		<td align="right"><math>\tfrac{1}{2}(1 + \sqrt{2})</math></td>
	<td align="right"><math>\frac{2}{(1 + \sqrt{2})}</math></td>		<td align="right"><math>\frac{2}{(1 + \sqrt{2})}</math></td>
			</tr>

			<tr>
			<td align="left" colspan="13">
			Notes:
			<ol type="a">
			<li>[<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], p. 142, last line of Table VI includes … <math>\Omega^2 = 4/15 \approx 0.26667</math> and <math>f = -2</math>.</li>
			<li>See [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], p. 78, Table I.</li>
			</ol>
			</td>
	</tr>		</tr>
	</table>		</table>

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2022-04-01T15:59:19Z

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	<td align="center"><math>\frac{4}{15}</math></td>		<td align="center"><math>\frac{4}{15}</math></td>
	<td align="center"><math>\infty</math></td>		<td align="center"><math>\infty</math></td>
	<td align="right">---</td>		<td align="right"><math>-~\biggl(\frac{4}{15}\biggr)^{1 / 2}</math></td>
	<td align="right">~~---~~</td>		<td align="right"><math>\biggl(\frac{16}{15}\biggr)^{1 / 2}</math></td>
	<td align="center"><math>\infty</math></td>		<td align="center"><math>\infty</math></td>
	<td align="~~right~~">-2~~.00000~~</td>		<td align="center">-2</td>
	<td align="right">~~---~~</td>		<td align="right"><math>\biggl(\frac{4}{15}\biggr)^{1 / 2}</math></td>
	<td align="right">---</td>		<td align="right"><math>-~\biggl(\frac{16}{15}\biggr)^{1 / 2}</math></td>
	<td align="center"><math>\infty</math></td>		<td align="center"><math>\infty</math></td>
	<td align="~~right~~">-2~~.00000~~</td>		<td align="center">-2</td>
	</tr>		</tr>

	<tr>		<tr>
	<td align="center" rowspan="1"><math>x = +1</math> Bifurcation</td>		<td align="center" rowspan="1">"<math>x = +1</math>" Bifurcation</td>
	<td align="right">0.952886702</td>		<td align="right">0.952886702</td>
	<td align="right">0.440219895</td>		<td align="right">0.440219895</td>