ThreeDimensionalConfigurations/DescriptionOfRiemannTypeI - Revision history

Jet53man: /* Examination of Lagrangian Flow in One Specific Model */

2022-03-13T23:21:04Z

Examination of Lagrangian Flow in One Specific Model

← Older revision		Revision as of 19:21, 13 March 2022
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	<td align="right">		<td align="right">
	<math>~		<math>~
	\dot\varphi		\|\dot\varphi\|
	</math>		</math>
	</td>		</td>
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	<td align="center">       </td>		<td align="center">       </td>
	<td align="right">		<td align="right">
	<math>~\dot\varphi =</math>		<math>\|\dot\varphi\| =</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~+1.299300</math>   <font color="green">Correct!</font>		<math>1.299300</math>   <font color="green">Correct!</font>
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Examination of Lagrangian Flow in One Specific Model */

2022-03-06T00:10:12Z

Examination of Lagrangian Flow in One Specific Model

← Older revision		Revision as of 20:10, 5 March 2022
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	<math>~		<math>~
	\biggl\{ \Lambda \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_3 }{\cos\theta} \biggr\}^{1 / 2}		\biggl\{ \Lambda \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_3 }{\cos\theta} \biggr\}^{1 / 2}
	</math>   <font color="~~red~~">~~Wrong~~!</font>		</math>   <font color="green">Correct!</font>
	</td>		</td>
	<td align="center">       </td>		<td align="center">       </td>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~+1.299300</math>   <font color="~~red~~">~~Wrong~~!</font>		<math>~+1.299300</math>   <font color="green">Correct!</font>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~-2.~~21016~~</math>		<math>~-1.299300</math>
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Extracted from XXVIII */

2022-02-28T21:44:05Z

Extracted from XXVIII

← Older revision		Revision as of 17:44, 28 February 2022
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	Given the values of <math>\Omega_2</math> and <math>\Omega_3</math> from this table, we have reverse-engineered this problem and determined what numerical values of <math>A_1</math>, <math>A_2</math>, and <math>A_3</math> were used by {{ Chandrasekhar66_XXVIII }} for various models. ~~The~~ expressions that have been derived via ~~our~~ reverse-engineering effort ~~have been recorded in the raw text of this subsection of our MediaWiki chapter, but this text has been "commented out" for brevity; our determination of the~~ values of <math>A_1</math>, <math>A_2</math>, and <math>A_3</math> have been recorded in the last three columns of the table.		<span id="ReverseEngineered">Given the values</span> of <math>\Omega_2</math> and <math>\Omega_3</math> from this table, we have reverse-engineered this problem and determined what numerical values of <math>A_1</math>, <math>A_2</math>, and <math>A_3</math> were used by {{ Chandrasekhar66_XXVIII }} for various models. What follows are the expressions that have been derived via this reverse-engineering effort. The values of <math>A_1</math>, <math>A_2</math>, and <math>A_3</math> that we have derived in this manner have been recorded in the last three columns of the table.

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	First …		First …

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	===Examination of Lagrangian Flow in One Specific Model===		===Examination of Lagrangian Flow in One Specific Model===

Jet53man at 22:34, 27 February 2022

2022-02-27T22:34:17Z

← Older revision		Revision as of 18:34, 27 February 2022
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	! style="height: 125px; width: 125px; background-color:#ffeeee;" \|[[H_BookTiledMenu#Three-Dimensional_Configurations\|<b>Type I<br />Riemann<br />Ellipsoids</b>]]		! style="height: 125px; width: 125px; background-color:#ffeeee;" \|[[H_BookTiledMenu#Three-Dimensional_Configurations\|<b>Type I<br />Riemann<br />Ellipsoids</b>]]
	\|}		\|}
			A more succinct version of this chapter is titled, [[3Dconfigurations/DescriptionOfRiemannTypeI\|3Dconfigurations/DescriptionOfRiemannTypeI]].
	==Analytic Determination of Equilibrium Model Parameters==		==Analytic Determination of Equilibrium Model Parameters==
	Drawing heavily from §47 (pp. 129 - 132) of [[Appendix/References#EFE\|[<font color="red">EFE</font>] ]], in a separate chapter we show how the steady-state 2<sup>nd</sup>-order tensor virial equations can be used to derive the equilibrium structure of [[VE/RiemannEllipsoids#Riemann_Ellipsoids_of_Types_I,_II,_&_III\|Riemann Ellipsoids of Type I, II, & III]]. By definition, for these types of Riemann Ellipsoids, the two vectors <math>\vec{\Omega}</math> and <math>~\vec\zeta</math> are not parallel to any of the principal axes of the ellipsoid, and they are not aligned with each other, but they both lie in the <math>y-z</math> plane — that is to say, <math>~(\Omega_1, \zeta_1) = (0, 0)</math>. For a given specified density <math>(\rho)</math> and choice of the three semi-axes <math>(a_1, a_2, a_3) \leftrightarrow (a, b, c)</math>, all five of the expressions displayed in that chapter's [[VE/RiemannEllipsoids#SummaryTable\|''Summary Table'']] must be used in order to determine the equilibrium configuration's associated values of the five unknowns: <math>\Pi, (\Omega_2, \zeta_2), (\Omega_3, \zeta_3)</math>.		Drawing heavily from §47 (pp. 129 - 132) of [[Appendix/References#EFE\|[<font color="red">EFE</font>] ]], in a separate chapter we show how the steady-state 2<sup>nd</sup>-order tensor virial equations can be used to derive the equilibrium structure of [[VE/RiemannEllipsoids#Riemann_Ellipsoids_of_Types_I,_II,_&_III\|Riemann Ellipsoids of Type I, II, & III]]. By definition, for these types of Riemann Ellipsoids, the two vectors <math>\vec{\Omega}</math> and <math>~\vec\zeta</math> are not parallel to any of the principal axes of the ellipsoid, and they are not aligned with each other, but they both lie in the <math>y-z</math> plane — that is to say, <math>~(\Omega_1, \zeta_1) = (0, 0)</math>. For a given specified density <math>(\rho)</math> and choice of the three semi-axes <math>(a_1, a_2, a_3) \leftrightarrow (a, b, c)</math>, all five of the expressions displayed in that chapter's [[VE/RiemannEllipsoids#SummaryTable\|''Summary Table'']] must be used in order to determine the equilibrium configuration's associated values of the five unknowns: <math>\Pi, (\Omega_2, \zeta_2), (\Omega_3, \zeta_3)</math>.

Jet53man: /* See Also */

2022-02-27T22:27:20Z

Jet53man: /* Extracted from XXVIII */

2022-02-27T17:43:43Z

Extracted from XXVIII

← Older revision		Revision as of 13:43, 27 February 2022
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	Given the values of <math>\Omega_2</math> and <math>\Omega_3</math> from this table, ~~let's~~ reverse-~~engineer the~~ problem and ~~determine~~ what numerical values of <math>A_1</math>, <math>A_2</math>, and <math>A_3</math> were used for various models. First …		Given the values of <math>\Omega_2</math> and <math>\Omega_3</math> from this table, we have reverse-engineered this problem and determined what numerical values of <math>A_1</math>, <math>A_2</math>, and <math>A_3</math> were used by {{ Chandrasekhar66_XXVIII }} for various models. The expressions that have been derived via our reverse-engineering effort have been recorded in the raw text of this subsection of our MediaWiki chapter, but this text has been "commented out" for brevity; our determination of the values of <math>A_1</math>, <math>A_2</math>, and <math>A_3</math> have been recorded in the last three columns of the table.

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	===Examination of Lagrangian Flow in One Specific Model===		===Examination of Lagrangian Flow in One Specific Model===

Jet53man: /* Extracted from XXVIII */

2022-02-27T17:27:48Z

Extracted from XXVIII

← Older revision		Revision as of 13:27, 27 February 2022
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	<td align="center" rowspan="10" bgcolor="lightgrey" width="3%"> </td>		<td align="center" rowspan="10" bgcolor="lightgrey" width="3%"> </td>
	<td align="center" colspan="6">Adjoint</td>		<td align="center" colspan="6">Adjoint</td>
	<td align="center" rowspan="2"><~~math~~>~~A_1~~</~~math~~></td>		<td align="center" rowspan="2" bgcolor="red"><font color="white">A<sub>1</sub></font></td>
	<td align="center" rowspan="2"><~~math~~>~~A_2~~</~~math~~></td>		<td align="center" rowspan="2" bgcolor="red"><font color="white">A<sub>2</sub></font></td>
	<td align="center" rowspan="2"><~~math~~>~~A_3~~</~~math~~></td>		<td align="center" rowspan="2" bgcolor="red"><font color="white">A<sub>3</sub></font></td>
	</tr>		</tr>
	<tr>		<tr>
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	<td align="right">-0.74657</td>		<td align="right">-0.74657</td>

	<td align="right">00</td>		<td align="right">-102.706</td>
	<td align="right">00</td>		<td align="right">-2.61422</td>

	<td align="right">+0.57083</td>		<td align="right">+0.57083</td>
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	<td align="right">-0.66992</td>		<td align="right">-0.66992</td>

	<td align="right">00</td>		<td align="right">-2.52580</td>
	<td align="right">00</td>		<td align="right">-2.10501</td>

	<td align="right">~~---~~</td>		<td align="right">0.10258739</td>
	<td align="right">~~---~~</td>		<td align="right">0.04358267</td>
	<td align="right">~~---~~</td>		<td align="right">1.85382994</td>
	</tr>		</tr>
	<tr>		<tr>

Jet53man: /* Extracted from XXVIII */

2022-02-27T17:19:51Z

Extracted from XXVIII

← Older revision		Revision as of 13:19, 27 February 2022
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	<td align="right">-0.84580</td>		<td align="right">-0.84580</td>

	<td align="right">00</td>		<td align="right">-297.565</td>
	<td align="right">00</td>		<td align="right">-2.85743</td>

	<td align="right">+0.52221</td>		<td align="right">+0.52221</td>
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	<td align="right">-0.64518</td>		<td align="right">-0.64518</td>

	<td align="right">00</td>		<td align="right">-1.19634</td>
	<td align="right">00</td>		<td align="right">-1.66262</td>

	<td align="right">~~---~~</td>		<td align="right">0.08593434</td>
	<td align="right">~~---~~</td>		<td align="right">0.04618515</td>
	<td align="right">~~---~~</td>		<td align="right">1.86788051</td>
	</tr>		</tr>
	<tr>		<tr>

Jet53man: /* Extracted from XXVIII */

2022-02-27T17:12:11Z

Extracted from XXVIII

← Older revision		Revision as of 13:12, 27 February 2022
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	<td align="right">-1.02536</td>		<td align="right">-1.02536</td>

	<td align="right">00</td>		<td align="right">-39.7800</td>
	<td align="right">00</td>		<td align="right">-2.66300</td>

	<td align="right">+0.73061</td>		<td align="right">+0.73061</td>
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	<td align="right">-0.87944</td>		<td align="right">-0.87944</td>

	<td align="right">00</td>		<td align="right">-1.85200</td>
	<td align="right">00</td>		<td align="right">-1.95897</td>

	<td align="right">~~---~~</td>		<td align="right">0.18178501</td>
	<td align="right">~~---~~</td>		<td align="right">0.08464699</td>
	<td align="right">~~---~~</td>		<td align="right">1.73356799</td>
	</tr>		</tr>
	<tr>		<tr>

Jet53man: /* Extracted from XXVIII */

2022-02-27T17:05:49Z

Extracted from XXVIII

← Older revision		Revision as of 13:05, 27 February 2022
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	<td align="right">-1.24612</td>		<td align="right">-1.24612</td>

	<td align="right">00</td>		<td align="right">-118.657</td>
	<td align="right">00</td>		<td align="right">-3.06119</td>

	<td align="right">+0.63035</td>		<td align="right">+0.63035</td>
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	<td align="right">-0.85376</td>		<td align="right">-0.85376</td>

	<td align="right">00</td>		<td align="right">-0.94731</td>
	<td align="right">00</td>		<td align="right">-1.43697</td>

	<td align="right">~~---~~</td>		<td align="right">0.14374587</td>
	<td align="right">~~---~~</td>		<td align="right">0.09152713</td>
	<td align="right">~~---~~</td>		<td align="right">1.76472699</td>
	</tr>		</tr>
	<tr>		<tr>

← Older revision		Revision as of 18:27, 27 February 2022
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	=See Also=		=See Also=

	* [[ThreeDimensionalConfigurations/DescriptionOfRiemannTypeI\|Description of Riemann Type I Ellipsoids]]		* [[3Dconfigurations/DescriptionOfRiemannTypeI\|Description of Riemann Type I Ellipsoids]] (best)
	* [[ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids\|Riemann Type 1 Ellipsoids]] (~~old~~ introduction)		* [[ThreeDimensionalConfigurations/DescriptionOfRiemannTypeI\|Description of Riemann Type I Ellipsoids]] (older introduction)
			* [[ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids\|Riemann Type 1 Ellipsoids]] (oldest introduction)
	* [[ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations\|Construction Challenges (Pt. 1)]]		* [[ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations\|Construction Challenges (Pt. 1)]]
	* [[ThreeDimensionalConfigurations/ChallengesPt2\|Construction Challenges (Pt. 2)]]		* [[ThreeDimensionalConfigurations/ChallengesPt2\|Construction Challenges (Pt. 2)]]