ThreeDimensionalConfigurations/JacobiEllipsoids - Revision history

Jet53man: /* Bifurcation of Dumbbell sequence from Jacobian Sequence */

2023-05-18T19:12:01Z

Bifurcation of Dumbbell sequence from Jacobian Sequence

← Older revision		Revision as of 15:12, 18 May 2023
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	This value of <math>j^2</math> is very close to the value obtained by {{ EHS82 }}.		This value of <math>j^2</math> is very close to the value obtained by {{ EHS82 }}.

	In the paragraph at the top of the right-hand column of p. 467 of { {Hachisu86bfull }}, we find …		In the paragraph at the top of the right-hand column of p. 467 of {{Hachisu86bfull }}, we find …
	<ul>		<ul>
	<li><math>\Omega^2/(4\pi G\rho) = 0.0535</math></li>		<li><math>\Omega^2/(4\pi G\rho) = 0.0535</math></li>

Jet53man: /* Bifurcation of Poincaré's Sequence of Pear-Shaped Configurations from the Jacobian Sequence */

2023-05-18T19:09:08Z

Bifurcation of Poincaré's Sequence of Pear-Shaped Configurations from the Jacobian Sequence

← Older revision		Revision as of 15:09, 18 May 2023
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	<ul>		<ul>
	<li><math>\Omega^2/(4\pi G \rho) = 0.07101 </math></li>		<li><math>\Omega^2/(4\pi G \rho) = 0.07101 </math></li>
	<li><math>j^2 = \biggl( \frac{3}{2^8 \pi^4} \biggr)^{1/3} \frac{L^2}{(GM^3\bar{a})} = 0.~~07821~~ </math></li>		<li><math>j^2 = \biggl( \frac{3}{2^8 \pi^4} \biggr)^{1/3} \frac{L^2}{(GM^3\bar{a})} = 0.007821 </math></li>
	<li><math>\tau \equiv T/\|W\| = 0.1628 </math></li>		<li><math>\tau \equiv T/\|W\| = 0.1628 </math></li>
	</ul>		</ul>

Jet53man: /* Bifurcation of Dumbbell sequence from Jacobian Sequence */

2023-05-17T23:06:47Z

Bifurcation of Dumbbell sequence from Jacobian Sequence

← Older revision		Revision as of 19:06, 17 May 2023
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	</ul>		</ul>

	{{ CKST95bfull }} grab parameter values from a variety of sources. In the first paragraph of their §3.2 (p. 492), they state …		{{ 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 …
	<ul>		<ul>
	<li><math>(b/a, c/a) = (0.29720, 0.25746)</math></li>		<li><math>(b/a, c/a) = (0.29720, 0.25746)</math></li>

Jet53man: /* Bifurcation of Dumbbell sequence from Jacobian Sequence */

2023-05-17T23:00:48Z

Bifurcation of Dumbbell sequence from Jacobian Sequence

← Older revision		Revision as of 19:00, 17 May 2023
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	</ul>		</ul>
	This value of <math>j^2</math> is very close to the value obtained by {{ EHS82 }}.		This value of <math>j^2</math> is very close to the value obtained by {{ EHS82 }}.

			In the paragraph at the top of the right-hand column of p. 467 of { {Hachisu86bfull }}, we find …
			<ul>
			<li><math>\Omega^2/(4\pi G\rho) = 0.0535</math></li>
			<li><math>j^2 = 0.01157</math></li>
			</ul>

	{{ CKST95bfull }} grab parameter values from a variety of sources. In the first paragraph of their §3.2 (p. 492), they state …		{{ CKST95bfull }} grab parameter values from a variety of sources. In the first paragraph of their §3.2 (p. 492), they state …

Jet53man: /* Bifurcation from Maclaurin to Jacobi Sequence */

2023-05-17T22:58:09Z

Bifurcation from Maclaurin to Jacobi Sequence

← Older revision		Revision as of 18:58, 17 May 2023
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	<li><math>(a/\bar{a}, b/\bar{a}, c/\bar{a}) = (1.197234, 1.197234, 0.697657) ~~~~ \Rightarrow ~~~ (b/a, c/a) = (1.000000, 0.582724)</math></li>		<li><math>(a/\bar{a}, b/\bar{a}, c/\bar{a}) = (1.197234, 1.197234, 0.697657) ~~~~ \Rightarrow ~~~ (b/a, c/a) = (1.000000, 0.582724)</math></li>
	<li><math>\Omega^2/(2\pi G \rho) = 0.1871148 ~~~~ \Rightarrow ~~~ \omega_0^2/(4\pi G \rho) = 0.0935574</math></td></li>		<li><math>\Omega^2/(2\pi G \rho) = 0.1871148 ~~~~ \Rightarrow ~~~ \omega_0^2/(4\pi G \rho) = 0.0935574</math></td></li>
	<li><math>J/(G M^3 \bar{a})^{1 / 2} = 0.303751 </math></td></li>		<li><math>J/(G M^3 \bar{a})^{1 / 2} = 0.303751 ~~~~\Rightarrow ~~~~ j^2 = 0.004555</math></td></li>
	<li><math>\tau \equiv T/\|W\| = 0.1375 </math></td></li>		<li><math>\tau \equiv T/\|W\| = 0.1375 </math></td></li>
			</ul>

			In the paragraph on p. 467 of { {Hachisu86bfull }} that immediately follows Eq. (47), we find …
			<ul>
			<li><math>\Omega^2/(4\pi G\rho) = 0.09356</math></li>
			<li><math>j^2 = 0.004555</math></li>
	</ul>		</ul>

Jet53man: /* Bifurcation of Dumbbell sequence from Jacobian Sequence */

2023-05-17T22:42:32Z

Bifurcation of Dumbbell sequence from Jacobian Sequence

← Older revision		Revision as of 18:42, 17 May 2023
Line 1,200:		Line 1,200:
	</ul>		</ul>
	This value of <math>j^2</math> is very close to the value obtained by {{ EHS82 }}.		This value of <math>j^2</math> is very close to the value obtained by {{ EHS82 }}.

			{{ CKST95bfull }} grab parameter values from a variety of sources. In the first paragraph of their §3.2 (p. 492), they state …
			<ul>
			<li><math>(b/a, c/a) = (0.29720, 0.25746)</math></li>
			<li><math>\Omega^2/(4\pi G \rho) = 0.0532790</math></li>
			<li><math>j^2 = 0.0115082</math></li>
			</ul>

	=See Also=		=See Also=

Jet53man: /* Bifurcation of Dumbbell sequence from Jacobian Sequence */

2023-05-17T22:31:18Z

Bifurcation of Dumbbell sequence from Jacobian Sequence

← Older revision		Revision as of 18:31, 17 May 2023
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	</ul>		</ul>

	If we combine the value of <math>\Omega^2</math> obtained from {{ EHS82 }} with the values of <math>(b/a, c/a)</math> obtained from [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], we find …		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 {{ EHS82 }} with the values of <math>(b/a, c/a)</math> obtained from [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], we find …
	<ul>		<ul>
	<li><math>\Omega^2/(\pi G \rho) = </math></li>		<li><math>\Omega^2/(\pi G \rho) = 0.2128</math></li>
	<li>From the [[#Angular_Momentum_Constraint\|above expression]], <math>\frac{L}{(GM^3\bar{a})^{1 / 2}} = </math> and <math>j^2 = \biggl( \frac{3}{2^8 \pi^4} \biggr)^{1/3} \frac{L^2}{(GM^3\bar{a})} = </math></li>		<li>From the [[#Angular_Momentum_Constraint\|above expression]], <math>\frac{L}{(GM^3\bar{a})^{1 / 2}} = 0.48242</math>   and   <math>j^2 = \biggl( \frac{3}{2^8 \pi^4} \biggr)^{1/3} \frac{L^2}{(GM^3\bar{a})} = 0.01149</math></li>
	</ul>		</ul>
			This value of <math>j^2</math> is very close to the value obtained by {{ EHS82 }}.

	=See Also=		=See Also=

Jet53man: /* Bifurcation of Dumbbell sequence from Jacobian Sequence */

2023-05-17T22:23:03Z

Bifurcation of Dumbbell sequence from Jacobian Sequence

← Older revision		Revision as of 18:23, 17 May 2023
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	</ul>		</ul>
	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 }} — we find <math>\cos^{-1}(c/a) = 75.068~\mathrm{degrees}</math>, along with a footnote [5] which states, <font color="darkgreen">"The value <math>\cos^{-1}(c/a) = 75.081~\mathrm{degrees}</math> found earlier differs slightly; but the difference is not outside the limits of accuracy of the numerical evaluation."</font>		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 }} — we find <math>\cos^{-1}(c/a) = 75.068~\mathrm{degrees}</math>, along with a footnote [5] which states, <font color="darkgreen">"The value <math>\cos^{-1}(c/a) = 75.081~\mathrm{degrees}</math> found earlier differs slightly; but the difference is not outside the limits of accuracy of the numerical evaluation."</font>
	~~<ul>~~
	~~<li><math>\Omega^2/(\pi G \rho) = </math></li>~~
	<li>From the [[#Angular_Momentum_Constraint\|above expression]], <math>\frac{L}{(GM^3\bar{a})^{1 / 2}} = </math> and <math>j^2 = \biggl( \frac{3}{2^8 \pi^4} \biggr)^{1/3} \frac{L^2}{(GM^3\bar{a})} = </math></li>
	~~</ul>~~
	According to the first row of properties in Table II of {{ EHS82 }} …		According to the first row of properties in Table II of {{ EHS82 }} …
	<ul>		<ul>
Line 1,195:		Line 1,192:
	<li><math>j^2 = \biggl( \frac{3}{2^8 \pi^4} \biggr)^{1/3} \frac{L^2}{(GM^3\bar{a})} = 0.01157 </math></li>		<li><math>j^2 = \biggl( \frac{3}{2^8 \pi^4} \biggr)^{1/3} \frac{L^2}{(GM^3\bar{a})} = 0.01157 </math></li>
	<li><math>\tau \equiv T/\|W\| = 0.1863 </math></li>		<li><math>\tau \equiv T/\|W\| = 0.1863 </math></li>
			</ul>

			If we combine the value of <math>\Omega^2</math> obtained from {{ EHS82 }} with the values of <math>(b/a, c/a)</math> obtained from [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], we find …
			<ul>
			<li><math>\Omega^2/(\pi G \rho) = </math></li>
			<li>From the [[#Angular_Momentum_Constraint\|above expression]], <math>\frac{L}{(GM^3\bar{a})^{1 / 2}} = </math> and <math>j^2 = \biggl( \frac{3}{2^8 \pi^4} \biggr)^{1/3} \frac{L^2}{(GM^3\bar{a})} = </math></li>
	</ul>		</ul>

Jet53man: /* Bifurcation of Dumbbell sequence from Jacobian Sequence */

2023-05-17T22:11:29Z

Bifurcation of Dumbbell sequence from Jacobian Sequence

← Older revision		Revision as of 18:11, 17 May 2023
Line 1,183:		Line 1,183:
	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, §45</font>, a dumbbell-shaped configuration (neutral point belonging to the fourth harmonic distortion) bifurcates from the Jacobi Sequence at …		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, §45</font>, a dumbbell-shaped configuration (neutral point belonging to the fourth harmonic distortion) bifurcates from the Jacobi Sequence at …
	<ul>		<ul>
	<li><math>(b/a, c/a) = (0.~~2972~~, 0.~~2575~~)</math></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>
			</ul>
			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 }} — we find <math>\cos^{-1}(c/a) = 75.068~\mathrm{degrees}</math>, along with a footnote [5] which states, <font color="darkgreen">"The value <math>\cos^{-1}(c/a) = 75.081~\mathrm{degrees}</math> found earlier differs slightly; but the difference is not outside the limits of accuracy of the numerical evaluation."</font>
			<ul>
	<li><math>\Omega^2/(\pi G \rho) = </math></li>		<li><math>\Omega^2/(\pi G \rho) = </math></li>
	<li>From the [[#Angular_Momentum_Constraint\|above expression]], <math>\frac{L}{(GM^3\bar{a})^{1 / 2}} = </math> and <math>j^2 = \biggl( \frac{3}{2^8 \pi^4} \biggr)^{1/3} \frac{L^2}{(GM^3\bar{a})} = </math></li>		<li>From the [[#Angular_Momentum_Constraint\|above expression]], <math>\frac{L}{(GM^3\bar{a})^{1 / 2}} = </math> and <math>j^2 = \biggl( \frac{3}{2^8 \pi^4} \biggr)^{1/3} \frac{L^2}{(GM^3\bar{a})} = </math></li>

Jet53man: /* Bifurcation of Poincaré's Sequence of Pear-Shaped Configurations from the Jacobian Sequence */

2023-05-17T19:35:03Z

Bifurcation of Poincaré's Sequence of Pear-Shaped Configurations from the Jacobian Sequence

← Older revision		Revision as of 15:35, 17 May 2023
Line 1,173:		Line 1,173:
	<li>From the [[#Angular_Momentum_Constraint\|above expression]], <math>\frac{L}{(GM^3\bar{a})^{1 / 2}} = 0.389536</math> and <math>j^2 = \biggl( \frac{3}{2^8 \pi^4} \biggr)^{1/3} \frac{L^2}{(GM^3\bar{a})} = 0.0074907 </math></li>		<li>From the [[#Angular_Momentum_Constraint\|above expression]], <math>\frac{L}{(GM^3\bar{a})^{1 / 2}} = 0.389536</math> and <math>j^2 = \biggl( \frac{3}{2^8 \pi^4} \biggr)^{1/3} \frac{L^2}{(GM^3\bar{a})} = 0.0074907 </math></li>
	</ul>		</ul>
	According to the first row of properties in Table 1 of {{ EHS82 }} …		According to the first row of properties in Table I of {{ EHS82 }} …
	<ul>		<ul>
	<li><math>\Omega^2/(4\pi G \rho) = 0.07101 </math></li>		<li><math>\Omega^2/(4\pi G \rho) = 0.07101 </math></li>