User:Tohline/ThreeDimensionalConfigurations/JacobiEllipsoids - Revision history

Tohline: /* Constraint on Axis-Ratio Relationship */

2020-06-13T19:47:09Z

Constraint on Axis-Ratio Relationship

← Older revision		Revision as of 19:47, 13 June 2020
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	</table>		</table>

	<b>With regard to our Table 1 (immediately above):</b> Given each pair of axis ratios, <math>~(\tfrac{b}{a},\tfrac{c}{a})</math> — copied from Table IV of EFE (see columns 1 and 2 of our Table 1) — and the corresponding coefficient values, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math>, as tabulated in [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#Table2\|Table 2 of our accompanying discussion]], we calculated corresponding values of <math>~\Omega^2</math> (column 3) and total angular momentum (column 4) in the units used in EFE's Table IV, as well as (column 5) the total angular momentum in units used by [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Christodoulou, ''et al.'' (1995, ApJ, 446, 472)] — see [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#Example_Evaluations\|our related discussion of these physical quantities]]. We also have tabulated associated values of the function, <math>~f_J</math>, (column 6) and its first derivative, <math>~f_J^'</math>, (column 7) as defined immediately above. Notice that <math>~f_J</math> is very nearly zero in all cases, which indicates that each axis-ratio pair indeed identifies a configuration that lies along the Jacobi sequence.		<span id="Table2"><b>With regard to our Table 1 (immediately above):</b></span> Given each pair of axis ratios, <math>~(\tfrac{b}{a},\tfrac{c}{a})</math> — copied from Table IV of EFE (see columns 1 and 2 of our Table 1) — and the corresponding coefficient values, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math>, as tabulated in [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#Table2\|Table 2 of our accompanying discussion]], we calculated corresponding values of <math>~\Omega^2</math> (column 3) and total angular momentum (column 4) in the units used in EFE's Table IV, as well as (column 5) the total angular momentum in units used by [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Christodoulou, ''et al.'' (1995, ApJ, 446, 472)] — see [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#Example_Evaluations\|our related discussion of these physical quantities]]. We also have tabulated associated values of the function, <math>~f_J</math>, (column 6) and its first derivative, <math>~f_J^'</math>, (column 7) as defined immediately above. Notice that <math>~f_J</math> is very nearly zero in all cases, which indicates that each axis-ratio pair indeed identifies a configuration that lies along the Jacobi sequence.

	<table border="1" cellpadding="5" align="center">		<table border="1" cellpadding="5" align="center">

Tohline: /* Constraint on Axis-Ratio Relationship */

2020-06-11T17:16:40Z

Constraint on Axis-Ratio Relationship

← Older revision		Revision as of 17:16, 11 June 2020
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	</div>		</div>

	Our plan is to employ the Newton-Raphson method to find the root(s) of the <math>~f_J = 0</math> relation, typically holding <math>~\upsilon</math> fixed and using the Newton-Raphson technique to identify the corresponding "root" value of <math>~\chi</math>. Using this approach, the Newton-Raphson technique requires specification of, not only the function, <math>~f_J</math>, but also its first derivative,		Our plan is to employ the [https://brilliant.org/wiki/newton-raphson-method/ Newton Raphson method] to find the root(s) of the <math>~f_J = 0</math> relation, typically holding <math>~\upsilon</math> fixed and using the Newton-Raphson technique to identify the corresponding "root" value of <math>~\chi</math>. Using this approach, the [https://brilliant.org/wiki/newton-raphson-method/ Newton Raphson technique] requires specification of, not only the function, <math>~f_J</math>, but also its first derivative,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</div>		</div>

	Let's determine the requisite expression, using a prime superscript to indicate differentiation with respect to <math>~x</math>.		Let's determine the requisite expression, using a prime superscript to indicate differentiation with respect to <math>~\chi</math>.
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline at 01:58, 30 June 2019

2019-06-30T01:58:38Z

← Older revision		Revision as of 01:58, 30 June 2019
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	<!-- __FORCETOC__ will force the creation of a Table of Contents -->		<!-- __FORCETOC__ will force the creation of a Table of Contents -->
	<!-- __NOTOC__ will force TOC off -->		<!-- __NOTOC__ will force TOC off -->

			=Jacobi Ellipsoids=
	{{LSU_HBook_header}}		{{LSU_HBook_header}}

	~~=Jacobi Ellipsoids=~~
	==General Coefficient Expressions==		==General Coefficient Expressions==

Tohline: /* Angular Momentum Constraint */

2016-07-06T01:44:42Z

Angular Momentum Constraint

← Older revision		Revision as of 01:44, 6 July 2016
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	</div>		</div>

	~~<!-- TEMP DELETE~~

	What values of <math>~L</math> should we choose? In association with our [[User:Tohline/ThreeDimensionalConfigurations/EFE_Energies#Conserve_Only_L\|discussion of warped free-energy surfaces]], we'd like to specify the eccentricity, <math>~e</math>, of a Maclaurin spheroid and adopt the angular momentum of ''that'' configuration. According to our [[User:Tohline/Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29\|accompanying discussion of the properties of Maclaurin spheroids]],		What values of <math>~L</math> should we choose? In association with our [[User:Tohline/ThreeDimensionalConfigurations/EFE_Energies#Conserve_Only_L\|discussion of warped free-energy surfaces]], we'd like to specify the eccentricity, <math>~e</math>, of a Maclaurin spheroid and adopt the angular momentum of ''that'' configuration. According to our [[User:Tohline/Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29\|accompanying discussion of the properties of Maclaurin spheroids]],
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	Note, for example, that if <math>~e = 0.85</math>, the square-root of this expression gives, <math>~L_\mathrm{Mac} = 4.7148806</math>, which matches the angular momentum that was used by [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Christodoulou ''et al'' (1995)] to generate their Figure 3.		Note, for example, that if <math>~e = 0.85</math>, the square-root of this expression gives, <math>~L_\mathrm{Mac} = 4.7148806</math>, which matches the angular momentum that was used by [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Christodoulou ''et al'' (1995)] to generate their Figure 3.

	~~-->~~

	==Sequence Plots==		==Sequence Plots==

Tohline: /* Angular Momentum Constraint */ Finished deriving f_L^' (needs to be checked)

2016-07-06T01:05:11Z

Angular Momentum Constraint: Finished deriving f_L^' (needs to be checked)

← Older revision		Revision as of 01:05, 6 July 2016
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	</td>		</td>
	</tr>		</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \biggl[ \frac{3\cdot 5^4}{2^5} \biggr]^{1/3}\upsilon^{-4/3} \chi^{-7/3} (1 + \chi^2)\biggl\{
			3\chi (1 + \chi^2)(1-\chi^2)^{-1} [ (1+\chi^2)A_1^' + \chi^2A_3^' ]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~+[8\chi^2-4]A_1
			+ 2(1-\chi^2)^{-2} [ 2-2A_1-A_3] [ - \chi^2 - 9\chi^4 + 4\chi^6 ] \biggr\}
			</math>
			</td>
			</tr>
			<!--
			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			+ 4\chi^2 - 8\chi^4 + 4\chi^6 -2+4\chi^2 - 2\chi^4 + 2 - 2\chi^4 - 6\chi^{2} + 6\chi^4 - 3 \chi^{2} - 3 \chi^4
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~-2 + 2
			+ 4\chi^2 +4\chi^2 - 6\chi^{2} - 3 \chi^{2} - 8\chi^4 - 2\chi^4 - 2\chi^4 + 6\chi^4 - 3 \chi^4 + 4\chi^6
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~- \chi^2 - 9\chi^4 + 4\chi^6
			</math>
			</td>
			</tr>
			-->
	</table>		</table>
	</div>		</div>

Tohline: /* Angular Momentum Constraint */ Continue derivation of f_L^'

2016-07-05T22:26:59Z

Angular Momentum Constraint: Continue derivation of f_L^'

← Older revision		Revision as of 22:26, 5 July 2016
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	- 6 \chi^{2} (1 + \chi^2)[ 2(1-A_1)-A_3] \}		- 6 \chi^{2} (1 + \chi^2)[ 2(1-A_1)-A_3] \}
	\biggr\}		\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \biggl[ \frac{3\cdot 5^4}{2^5} \biggr]^{1/3}\upsilon^{-4/3} \chi^{-7/3} (1 + \chi^2)\biggl\{
			[8\chi^2-4]A_1
			+ 3\chi (1 + \chi^2)(1-\chi^2)^{-1} [ (1+\chi^2)A_1^' + \chi^2A_3^' ]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			+ 2(1-\chi^2)^{-2} [ 2-2A_1-A_3] [ (4\chi^2-2)(1-\chi^2)^{2} +
			2 (1 + \chi^2)(1-\chi^2) - 6\chi^{2} (1-\chi^2) - 3 \chi^{2} (1 + \chi^2)
			] \biggr\}
	</math>		</math>
	</td>		</td>

Tohline: /* Roots of the Governing Relation */ In the middle of deriving f_L^'

2016-07-05T21:46:56Z

Roots of the Governing Relation: In the middle of deriving f_L^'

← Older revision		Revision as of 21:46, 5 July 2016
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	</table>		</table>
	</div>		</div>

			As [[#Constraint_on_Axis-Ratio_Relationship\|above]], we will hold <math>~\upsilon</math> fixed and use the Newton-Raphson technique to identify the corresponding "root" value of <math>~\chi</math>. Hence, we need to specify, not only the function, <math>~f_L</math>, but also its first derivative,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~f_L^'</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~\frac{\partial f_L}{\partial \chi} \, .</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \biggl[ \frac{3^4\cdot 5^4}{2^5} \biggr]^{1/3}\upsilon^{-4/3} \frac{\partial}{\partial \chi} \biggl\{
			\chi^{-4/3} (1 + \chi^2)^{2}
			[2 - (A_1+A_3)]
			- \chi^{-4/3} (1 + \chi^2)^{2}(1-\chi^2)^{-1}[ 2(1-A_1)-A_3]
			\biggr\}</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \biggl[ \frac{3^4\cdot 5^4}{2^5} \biggr]^{1/3}\upsilon^{-4/3} \biggl\{
			-\frac{4}{3}\chi^{-7/3} (1 + \chi^2)^{2}[2 - (A_1+A_3)]
			+4\chi^{-1/3} (1 + \chi^2)[2 - (A_1+A_3)]
			-\chi^{-4/3} (1 + \chi^2)^{2}(A_1^'+A_3^')
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			+ \frac{4}{3} \chi^{-7/3} (1 + \chi^2)^{2}(1-\chi^2)^{-1}[ 2(1-A_1)-A_3]
			- 4\chi^{-1/3} (1 + \chi^2)(1-\chi^2)^{-1}[ 2(1-A_1)-A_3]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			- 2 \chi^{-1/3} (1 + \chi^2)^{2}(1-\chi^2)^{-2}[ 2(1-A_1)-A_3]
			- \chi^{-4/3} (1 + \chi^2)^{2}(1-\chi^2)^{-1}[ -2A_1^'-A_3^']
			\biggr\}</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \biggl[ \frac{3\cdot 5^4}{2^5} \biggr]^{1/3}\upsilon^{-4/3} \chi^{-7/3} (1 + \chi^2)\biggl\{
			[12\chi^2-4(1 + \chi^2)][2 - (A_1+A_3)]
			-3\chi (1 + \chi^2)(A_1^'+A_3^')
			+ 3\chi (1 + \chi^2)(1-\chi^2)^{-1}[ 2A_1^' + A_3^']
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			+ (1-\chi^2)^{-2}\{ 4 (1 + \chi^2)(1-\chi^2)[ 2(1-A_1)-A_3]
			- 12\chi^{2} (1-\chi^2)[ 2(1-A_1)-A_3]
			- 6 \chi^{2} (1 + \chi^2)[ 2(1-A_1)-A_3] \}
			\biggr\}
			</math>
			</td>
			</tr>
			</table>
			</div>

			<!-- TEMP DELETE

	What values of <math>~L</math> should we choose? In association with our [[User:Tohline/ThreeDimensionalConfigurations/EFE_Energies#Conserve_Only_L\|discussion of warped free-energy surfaces]], we'd like to specify the eccentricity, <math>~e</math>, of a Maclaurin spheroid and adopt the angular momentum of ''that'' configuration. According to our [[User:Tohline/Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29\|accompanying discussion of the properties of Maclaurin spheroids]],		What values of <math>~L</math> should we choose? In association with our [[User:Tohline/ThreeDimensionalConfigurations/EFE_Energies#Conserve_Only_L\|discussion of warped free-energy surfaces]], we'd like to specify the eccentricity, <math>~e</math>, of a Maclaurin spheroid and adopt the angular momentum of ''that'' configuration. According to our [[User:Tohline/Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29\|accompanying discussion of the properties of Maclaurin spheroids]],
Line 770:		Line 885:

	Note, for example, that if <math>~e = 0.85</math>, the square-root of this expression gives, <math>~L_\mathrm{Mac} = 4.7148806</math>, which matches the angular momentum that was used by [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Christodoulou ''et al'' (1995)] to generate their Figure 3.		Note, for example, that if <math>~e = 0.85</math>, the square-root of this expression gives, <math>~L_\mathrm{Mac} = 4.7148806</math>, which matches the angular momentum that was used by [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Christodoulou ''et al'' (1995)] to generate their Figure 3.

			-->

	==Sequence Plots==		==Sequence Plots==

Tohline: /* Angular Momentum Constraint */ Give example of L_Mac calculation

2016-07-05T19:22:30Z

Angular Momentum Constraint: Give example of L_Mac calculation

← Older revision		Revision as of 19:22, 5 July 2016
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	</div>		</div>

	What values of <math>~L</math> should we choose? In association with our discussion of warped free-energy surfaces, we'd like to specify the eccentricity, <math>~e</math>, of a Maclaurin spheroid and adopt the angular momentum of ''that'' configuration. According to our [[User:Tohline/Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29\|accompanying discussion of the properties of Maclaurin spheroids]],		What values of <math>~L</math> should we choose? In association with our [[User:Tohline/ThreeDimensionalConfigurations/EFE_Energies#Conserve_Only_L\|discussion of warped free-energy surfaces]], we'd like to specify the eccentricity, <math>~e</math>, of a Maclaurin spheroid and adopt the angular momentum of ''that'' configuration. According to our [[User:Tohline/Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29\|accompanying discussion of the properties of Maclaurin spheroids]],
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<math>~ [2\cdot 3^4\cdot 5^4]^{1/3} \frac{(1-e^2)^{1/3} }{e^2} \biggl\{		<math>~ [2\cdot 3^4\cdot 5^4]^{1/3} \frac{(1-e^2)^{1/3} }{e^2} \biggl\{
	\biggl[\frac{\sin^{-1}e}{e} \biggr](1-e^2)^{-1/2} \biggl[3-2e^2\biggr] - 3		\biggl[\frac{\sin^{-1}e}{e} \biggr](1-e^2)^{-1/2} \biggl[3-2e^2\biggr] - 3
	\biggr\} </math>		\biggr\} \, . </math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	</div>		</div>

			Note, for example, that if <math>~e = 0.85</math>, the square-root of this expression gives, <math>~L_\mathrm{Mac} = 4.7148806</math>, which matches the angular momentum that was used by [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Christodoulou ''et al'' (1995)] to generate their Figure 3.

	==Sequence Plots==		==Sequence Plots==

Tohline: /* Angular Momentum Constraint */ Correct expression for L^2_Mac

2016-07-05T19:15:31Z

Angular Momentum Constraint: Correct expression for L^2_Mac

← Older revision		Revision as of 19:15, 5 July 2016
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~ [2\cdot 3^4\cdot 5^4]^{1/3} (1-e^2)^{-1/6} \biggl[		<math>~ [2\cdot 3^4\cdot 5^4]^{1/3} (1-e^2)^{-2/3} \biggl\{
	\frac{\sin^{-1}e}{e} - (1-e^2)^{1/2}		\frac{1}{e^2} \biggl[\frac{\sin^{-1}e}{e} \biggr](1-e^2)^{1/2}
	\biggr] \biggl[ \frac{3-2e^2}{e^2}\biggr] \~~, .~~</math>		+\frac{2}{e^2} \biggl[\frac{\sin^{-1}e}{e} \biggr](1-e^2)^{3/2}
			-\frac{1}{e^2} \biggl[ (1-e^2)^{1/2} \biggr](1-e^2)^{1/2}
			-\frac{2}{e^2} \biggl[(1-e^2)^{-1/2} \biggr](1-e^2)^{3/2}
			\biggr\} </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ [2\cdot 3^4\cdot 5^4]^{1/3} (1-e^2)^{-2/3} \biggl\{
			\frac{1}{e^2} \biggl[\frac{\sin^{-1}e}{e} \biggr](1-e^2)^{1/2} \biggl[3-2e^2\biggr]
			-\frac{3(1-e^2)}{e^2}
			\biggr\} </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ [2\cdot 3^4\cdot 5^4]^{1/3} \frac{(1-e^2)^{1/3} }{e^2} \biggl\{
			\biggl[\frac{\sin^{-1}e}{e} \biggr](1-e^2)^{-1/2} \biggl[3-2e^2\biggr] - 3
			\biggr\} </math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Angular Momentum Constraint */

2016-07-05T17:59:45Z

Angular Momentum Constraint

← Older revision		Revision as of 17:59, 5 July 2016
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~ [2\cdot 3^4\cdot 5^4]^{1/3} (1-e^2)^{-2/3} ~~\biggl\{ \frac{1}{e^2}~~ \biggl[\frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr]~~(1-e^2)^{1/2}~~		<math>~ [2\cdot 3^4\cdot 5^4]^{1/3} (1-e^2)^{-1/6} \biggl[
	~~-\frac{2}{e^2}~~ \biggl[~~(1-e^2)^{-1/2} -~~\frac{~~\sin~~^~~{-1}e~~}{~~e} \biggr](1-~~e^~~2)^{3/~~2} \biggr\} </math>		\frac{\sin^{-1}e}{e} - (1-e^2)^{1/2}
			\biggr] \biggl[ \frac{3-2e^2}{e^2}\biggr] \, .</math>
	</td>		</td>
	</tr>		</tr>