ThreeDimensionalConfigurations/ChallengesPt4 - Revision history

Jet53man: /* See Also */

2022-04-15T17:45:39Z

Jet53man: /* See Also */

2022-02-17T23:41:21Z

Jet53man at 02:30, 6 February 2022

2022-02-06T02:30:05Z

← Older revision		Revision as of 22:30, 5 February 2022
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	=See Also=		=See Also=
	* [[~~User:Tohline/~~ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids\|Riemann Type 1 Ellipsoids]]		* [[ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids\|Riemann Type 1 Ellipsoids]]
	* [[~~User:Tohline/~~ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations\|Construction Challenges (Pt. 1)]]		* [[ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations\|Construction Challenges (Pt. 1)]]
	* [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt2\|Construction Challenges (Pt. 2)]]		* [[ThreeDimensionalConfigurations/ChallengesPt2\|Construction Challenges (Pt. 2)]]
	* [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt3\|Construction Challenges (Pt. 3)]]		* [[ThreeDimensionalConfigurations/ChallengesPt3\|Construction Challenges (Pt. 3)]]
	* [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt4\|Construction Challenges (Pt. 4)]]		* [[ThreeDimensionalConfigurations/ChallengesPt4\|Construction Challenges (Pt. 4)]]
	* [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt5\|Construction Challenges (Pt. 5)]]		* [[ThreeDimensionalConfigurations/ChallengesPt5\|Construction Challenges (Pt. 5)]]
			* [[ThreeDimensionalConfigurations/ChallengesPt6\|Construction Challenges (Pt. 6)]]
	* Related discussions of models viewed from a rotating reference frame:		* Related discussions of models viewed from a rotating reference frame:
	** [[~~User:Tohline/~~PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]		** [[PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]
	** <font color="red"><b>NOTE to Eric Hirschmann & David Neilsen... </b></font>I have moved the earlier contents of this page to a new Wiki location called [[~~User:Tohline/~~Apps/RiemannEllipsoids_Compressible\|Compressible Riemann Ellipsoids]].		** <font color="red"><b>NOTE to Eric Hirschmann & David Neilsen... </b></font>I have moved the earlier contents of this page to a new Wiki location called [[Apps/RiemannEllipsoids_Compressible\|Compressible Riemann Ellipsoids]].


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Jet53man: /* Challenges Constructing Ellipsoidal-Like Configurations (Pt. 4) */

2022-02-06T02:20:09Z

Challenges Constructing Ellipsoidal-Like Configurations (Pt. 4)

← Older revision		Revision as of 22:20, 5 February 2022
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	=Challenges Constructing Ellipsoidal-Like Configurations (Pt. 4)=		=Challenges Constructing Ellipsoidal-Like Configurations (Pt. 4)=

	This chapter extends the accompanying chapters titled, [[~~User:Tohline/~~ThreeDimensionalConfigurations/Challenges\|''Construction Challenges (Pt. 1)'']], [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt2\|''(Pt. 2)'']], and [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt3\|''(Pt. 3)'']]. The focus here is on firming up our understanding of the relationships between various "tilted" Cartesian coordinate frames.		This chapter extends the accompanying chapters titled, [[ThreeDimensionalConfigurations/Challenges\|''Construction Challenges (Pt. 1)'']], [[ThreeDimensionalConfigurations/ChallengesPt2\|''(Pt. 2)'']], and [[ThreeDimensionalConfigurations/ChallengesPt3\|''(Pt. 3)'']]. The focus here is on firming up our understanding of the relationships between various "tilted" Cartesian coordinate frames.

	==The Plan==		==The Plan==
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	As has been shown in [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt2#Intersection_of_Tipped_Plane_With_Ellipsoid_Surface\|our accompanying discussion]], we obtain the following,		As has been shown in [[ThreeDimensionalConfigurations/ChallengesPt2#Intersection_of_Tipped_Plane_With_Ellipsoid_Surface\|our accompanying discussion]], we obtain the following,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
	<tr>		<tr>
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	<table border="1" align="center" cellpadding="10" width="60%" bordercolor="orange">		<table border="1" align="center" cellpadding="10" width="60%" bordercolor="orange">
	<tr><td align="center" bgcolor="lightblue">'''RESULT 3'''<br />(same as [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt2#Result1\|Result 1]], but different from [[#Result2\|Result 2, below]])		<tr><td align="center" bgcolor="lightblue">'''RESULT 3'''<br />(same as [[ThreeDimensionalConfigurations/ChallengesPt2#Result1\|Result 1]], but different from [[#Result2\|Result 2, below]])
	</td></tr>		</td></tr>
	<tr><td align="left">		<tr><td align="left">
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	<tr><td align="left" rowspan="4">		<tr><td align="left" rowspan="4">
	<table border="1" cellpadding="8" align="center">		<table border="1" cellpadding="8" align="center">
	<tr><th align="center" colspan="2">Example Type I<br />Ellipsoid<br />([[~~User:Tohline/~~ThreeDimensionalConfigurations/RiemannTypeI#Example_b1.25c0.470\|see also]])</th></tr>		<tr><th align="center" colspan="2">Example Type I<br />Ellipsoid<br />([[ThreeDimensionalConfigurations/RiemannTypeI#Example_b1.25c0.470\|see also]])</th></tr>
	<tr>		<tr>
	<td align="center"><math>~\frac{b}{a} = \frac{a_2}{a_1}</math></td>		<td align="center"><math>~\frac{b}{a} = \frac{a_2}{a_1}</math></td>
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	</table>		</table>

	As we have summarized in an [[~~User:Tohline/~~ThreeDimensionalConfigurations/RiemannTypeI#EFEvelocities\|accompanying discussion]] of Riemann Type 1 ellipsoids — see also [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt3#Riemann-Derived_Expressions\|our separate discussion]] — [[~~User:Tohline/~~Appendix/References#EFE\|[<font color="red">EFE</font>] ]] provides an expression for the velocity vector of each fluid element, given its instantaneous ''body''-coordinate position (x, y, z) = (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>) — see his Eq. (154), Chapter 7, §51 (p. 156). As viewed from the rotating ''body'' coordinate frame, the three component expressions are,		As we have summarized in an [[ThreeDimensionalConfigurations/RiemannTypeI#EFEvelocities\|accompanying discussion]] of Riemann Type 1 ellipsoids — see also [[ThreeDimensionalConfigurations/ChallengesPt3#Riemann-Derived_Expressions\|our separate discussion]] — [[Appendix/References#EFE\|[<font color="red">EFE</font>] ]] provides an expression for the velocity vector of each fluid element, given its instantaneous ''body''-coordinate position (x, y, z) = (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>) — see his Eq. (154), Chapter 7, §51 (p. 156). As viewed from the rotating ''body'' coordinate frame, the three component expressions are,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</table>		</table>
	<table border="1" align="center" cellpadding="10" width="60%" bordercolor="orange">		<table border="1" align="center" cellpadding="10" width="60%" bordercolor="orange">
	<tr><td align="center" bgcolor="lightblue">'''RESULT 2'''<br />(different from [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt2#Result1\|Result 1]] and [[#Result3\|Result 3, above]])		<tr><td align="center" bgcolor="lightblue">'''RESULT 2'''<br />(different from [[ThreeDimensionalConfigurations/ChallengesPt2#Result1\|Result 1]] and [[#Result3\|Result 3, above]])
	</td></tr>		</td></tr>
	<tr><td align="left">		<tr><td align="left">
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	===Body Frame===		===Body Frame===
	As has been shown in [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt2#Intersection_of_Tipped_Plane_With_Ellipsoid_Surface\|our accompanying discussion]], we obtain the following,		As has been shown in [[ThreeDimensionalConfigurations/ChallengesPt2#Intersection_of_Tipped_Plane_With_Ellipsoid_Surface\|our accompanying discussion]], we obtain the following,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
	<tr>		<tr>
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	* <math>~x_\mathrm{surf}/y'_\mathrm{surf} = 0.92001/0.896819 = 1.0259 \, .</math>		* <math>~x_\mathrm{surf}/y'_\mathrm{surf} = 0.92001/0.896819 = 1.0259 \, .</math>

	At a minimum these numerical values should be compared to the Riemann model parameters [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt2#Example_Equilibrium_Model\|computed in our earlier (Pt. 2) examination]].		At a minimum these numerical values should be compared to the Riemann model parameters [[ThreeDimensionalConfigurations/ChallengesPt2#Example_Equilibrium_Model\|computed in our earlier (Pt. 2) examination]].

	==Various Coordinate Frames==		==Various Coordinate Frames==
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	====Summary====		====Summary====
	In a [[~~User:Tohline/~~ThreeDimensionalConfigurations/RiemannTypeI#Try_Again\|separate discussion]], we have shown that, as viewed from a frame that "tumbles" with the (purple) body of a Type 1 Riemann ellipsoid, each Lagrangian fluid element moves along an elliptical path in a plane that is tipped by an angle <math>~\theta</math> about the x-axis of the body. As viewed from the (primed) coordinates associated with this tipped plane, by definition, z' = constant and dz'/dt = 0, and the planar orbit is defined by the expression for an,		In a [[ThreeDimensionalConfigurations/RiemannTypeI#Try_Again\|separate discussion]], we have shown that, as viewed from a frame that "tumbles" with the (purple) body of a Type 1 Riemann ellipsoid, each Lagrangian fluid element moves along an elliptical path in a plane that is tipped by an angle <math>~\theta</math> about the x-axis of the body. As viewed from the (primed) coordinates associated with this tipped plane, by definition, z' = constant and dz'/dt = 0, and the planar orbit is defined by the expression for an,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
	<tr>		<tr>
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	</tr>		</tr>
	<tr><td align="center">		<tr><td align="center">
	Given that b/a = 1.25 and c/a = 0.4703 for our chosen [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt2#Example_Equilibrium_Model\|Example Type I Ellipsoid]], we find that, <math>~\theta = - 1.18122 ~\mathrm{rad} = -67.68^\circ</math>.		Given that b/a = 1.25 and c/a = 0.4703 for our chosen [[ThreeDimensionalConfigurations/ChallengesPt2#Example_Equilibrium_Model\|Example Type I Ellipsoid]], we find that, <math>~\theta = - 1.18122 ~\mathrm{rad} = -67.68^\circ</math>.
	</td>		</td>
	</tr>		</tr>
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	</table>		</table>

	We have determined that (numerical value given for our [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt2#Example_Equilibrium_Model\|chosen example Type I ellipsoid]]),		We have determined that (numerical value given for our [[ThreeDimensionalConfigurations/ChallengesPt2#Example_Equilibrium_Model\|chosen example Type I ellipsoid]]),
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	</tr>		</tr>
	</table>		</table>
	where, as has also been specified [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt3#betagamma\|in an accompanying discussion]],		where, as has also been specified [[ThreeDimensionalConfigurations/ChallengesPt3#betagamma\|in an accompanying discussion]],
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	</table>		</table>

	Recognizing, [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt2#Tipped_Orbital_Plane\|as before]], that the relevant coordinate mapping is,		Recognizing, [[ThreeDimensionalConfigurations/ChallengesPt2#Tipped_Orbital_Plane\|as before]], that the relevant coordinate mapping is,

	<table border="1" width="50%" cellpadding="8" align="center">		<table border="1" width="50%" cellpadding="8" align="center">

Jet53man: Created page with "FORCETOC =Challenges Constructing Ellipsoidal-Like Configurations (Pt. 4)= T..."

2021-07-24T22:30:52Z

Created page with "__FORCETOC__  =Challenges Constructing Ellipsoidal-Like Configurations (Pt. 4)= T..."

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← Older revision		Revision as of 13:45, 15 April 2022
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	* Related discussions of models viewed from a rotating reference frame:		* Related discussions of models viewed from a rotating reference frame:
	** [[PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]		** [[PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]
	** <font color="red"><b>NOTE to Eric Hirschmann & David Neilsen... </b></font>I have moved the earlier contents of this page to a new Wiki location called [[Apps/~~RiemannEllipsoids_Compressible~~\|Compressible Riemann Ellipsoids]].		** <font color="red"><b>NOTE to Eric Hirschmann & David Neilsen... </b></font>I have moved the earlier contents of this page to a new Wiki location called [[Apps/RiemannEllipsoidsCompressible\|Compressible Riemann Ellipsoids]].


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← Older revision		Revision as of 19:41, 17 February 2022
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	=See Also=		=See Also=
	* [[ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids\|Riemann Type 1 Ellipsoids]]
			* [[ThreeDimensionalConfigurations/DescriptionOfRiemannTypeI\|Description of Riemann Type I Ellipsoids]]
			* [[ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids\|Riemann Type 1 Ellipsoids]] (old 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)]]
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	* [[ThreeDimensionalConfigurations/ChallengesPt5\|Construction Challenges (Pt. 5)]]		* [[ThreeDimensionalConfigurations/ChallengesPt5\|Construction Challenges (Pt. 5)]]
	* [[ThreeDimensionalConfigurations/ChallengesPt6\|Construction Challenges (Pt. 6)]]		* [[ThreeDimensionalConfigurations/ChallengesPt6\|Construction Challenges (Pt. 6)]]

	* Related discussions of models viewed from a rotating reference frame:		* Related discussions of models viewed from a rotating reference frame:
	** [[PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]		** [[PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]

← Older revision		Revision as of 22:30, 5 February 2022
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	=See Also=		=See Also=
	* [[~~User:Tohline/~~ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids\|Riemann Type 1 Ellipsoids]]		* [[ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids\|Riemann Type 1 Ellipsoids]]
	* [[~~User:Tohline/~~ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations\|Construction Challenges (Pt. 1)]]		* [[ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations\|Construction Challenges (Pt. 1)]]
	* [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt2\|Construction Challenges (Pt. 2)]]		* [[ThreeDimensionalConfigurations/ChallengesPt2\|Construction Challenges (Pt. 2)]]
	* [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt3\|Construction Challenges (Pt. 3)]]		* [[ThreeDimensionalConfigurations/ChallengesPt3\|Construction Challenges (Pt. 3)]]
	* [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt4\|Construction Challenges (Pt. 4)]]		* [[ThreeDimensionalConfigurations/ChallengesPt4\|Construction Challenges (Pt. 4)]]
	* [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt5\|Construction Challenges (Pt. 5)]]		* [[ThreeDimensionalConfigurations/ChallengesPt5\|Construction Challenges (Pt. 5)]]
			* [[ThreeDimensionalConfigurations/ChallengesPt6\|Construction Challenges (Pt. 6)]]
	* Related discussions of models viewed from a rotating reference frame:		* Related discussions of models viewed from a rotating reference frame:
	** [[~~User:Tohline/~~PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]		** [[PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]
	** <font color="red"><b>NOTE to Eric Hirschmann & David Neilsen... </b></font>I have moved the earlier contents of this page to a new Wiki location called [[~~User:Tohline/~~Apps/RiemannEllipsoids_Compressible\|Compressible Riemann Ellipsoids]].		** <font color="red"><b>NOTE to Eric Hirschmann & David Neilsen... </b></font>I have moved the earlier contents of this page to a new Wiki location called [[Apps/RiemannEllipsoids_Compressible\|Compressible Riemann Ellipsoids]].


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ThreeDimensionalConfigurations/ChallengesPt4 - Revision history

Jet53man: /* See Also */

Jet53man: /* See Also */

Jet53man at 02:30, 6 February 2022

Jet53man: /* Challenges Constructing Ellipsoidal-Like Configurations (Pt. 4) */

Jet53man: Created page with "__FORCETOC__ =Challenges Constructing Ellipsoidal-Like Configurations (Pt. 4)= T..."

Jet53man: Created page with "FORCETOC =Challenges Constructing Ellipsoidal-Like Configurations (Pt. 4)= T..."