User:Tohline/ThreeDimensionalConfigurations/RiemannTypeI - Revision history

Tohline: /* Chandrasekhar's Detailed Analysis */

2021-04-03T00:11:26Z

Chandrasekhar's Detailed Analysis

← Older revision		Revision as of 00:11, 3 April 2021
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	* [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract S. Chandrasekhar (1965)], ApJ, 142, 890 - 961. ''The Equilibrum and the Stability of the Riemann Ellipsoids. I.'' — This work is referenced as [[User:Tohline/Appendix/References#Appendix_of_EFE\|Paper XXV in EFE]] and focuses on S-type Riemann ellipsoids.		* [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract S. Chandrasekhar (1965)], ApJ, 142, 890 - 961. ''The Equilibrum and the Stability of the Riemann Ellipsoids. I.'' — This work is referenced as [[User:Tohline/Appendix/References#Appendix_of_EFE\|Paper XXV in EFE]] and focuses on S-type Riemann ellipsoids.
	* [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..842C/abstract S. Chandrasekhar (1966)], ApJ, 145, 842 - 877. ''The Equilibrum and the Stability of the Riemann Ellipsoids. II.'' — This work is referenced as [[User:Tohline/Appendix/References#Appendix_of_EFE\|Paper XXVIII in EFE]] and focuses on Riemann ellipsoids of Types I, II and III.		* [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..842C/abstract S. Chandrasekhar (1966)], ApJ, 145, 842 - 877. ''The Equilibrum and the Stability of the Riemann Ellipsoids. II.'' — This work is referenced as [[User:Tohline/Appendix/References#Appendix_of_EFE\|Paper XXVIII in EFE]] and focuses on Riemann ellipsoids of Types I, II and III.

			Potentially relevant citations:

			* [https://ui.adsabs.harvard.edu/abs/1997Ap%26SS.254..269C/abstract T. T. Chia & S. Y. Pung (1997)] Astrophysics and Space Science, 254 (issue 2), pp. 269 - 294, "Dynamical Behaviour of Compressible Homogeneous Uniformly Rotating Ellipsoids with Nonparallel Angular Velocity and Vorticity"
			* [https://ui.adsabs.harvard.edu/abs/1989PhLB..218..386R/abstract D. J. Rowe, P. Rochford, & J. Carvalho (1989)], Physics Letters B, 218 (issue 4), pp. 386 - 390, "Vorticity as intrinsic spin in the nuclear collective model.

	==Finite-Amplitude Oscillations==		==Finite-Amplitude Oscillations==

Tohline: /* Other */

2020-08-09T21:06:13Z

Other

← Older revision		Revision as of 21:06, 9 August 2020
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	* [[User:Tohline/Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29\|Properties of Maclaurin Spheroids]]		* [[User:Tohline/Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29\|Properties of Maclaurin Spheroids]]
	* [[User:Tohline/Apps/MaclaurinSpheroids/GoogleBooks#Excerpts_from_A_Treatise_of_Fluxions\|Excerpts from Maclaurin's (1742) ''A Treatise of Fluxions'']]		* [[User:Tohline/Apps/MaclaurinSpheroids/GoogleBooks#Excerpts_from_A_Treatise_of_Fluxions\|Excerpts from Maclaurin's (1742) ''A Treatise of Fluxions'']]
			* [https://ui.adsabs.harvard.edu/abs/2020PhRvL.124e2501S/abstract N. Sensharma, ''et al.'' (2020)], Physical Review Letters, Volume 124, Issue 5, article id.052501. ''Longitudinal Wobbling Motion in <sup>187</sup>Au.''



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Tohline: /* Riemann Type 1 Ellipsoids */

2020-06-13T17:20:36Z

Riemann Type 1 Ellipsoids

← Older revision		Revision as of 17:20, 13 June 2020
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	</ol>		</ol>

	==Regarding thought "A"==		===Regarding thought "A"===

	For the example model depicted in Figure 2, we have,		For the example model depicted in Figure 2, we have,
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	==Regarding thought "B"==		===Regarding thought "B"===
	When a fluid element crosses the x-axis we should expect z = 0. <b><font color="red">This needs to be rethought because the center of the closed elliptical trajectory of most fluid elements will be offset from the body's inherent z-axis.</font></b>		When a fluid element crosses the x-axis we should expect z = 0. <b><font color="red">This needs to be rethought because the center of the closed elliptical trajectory of most fluid elements will be offset from the body's inherent z-axis.</font></b>

	==Regarding thought "D"==		===Regarding thought "D"===
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	===First Trial Value of θ===		====First Trial Value of θ====
	Now, from above, we determined that the "tip" angle must obey the expression,		Now, from above, we determined that the "tip" angle must obey the expression,

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	===Second Trial Value of θ===		====Second Trial Value of θ====

	Instead, what if we obtain the tip angle straight from the tabulated expression,		Instead, what if we obtain the tip angle straight from the tabulated expression,
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	</table>		</table>

	==Try Again==		===Try Again===

	===Methodical Derivation of Orbital Parameters===		====Methodical Derivation of Orbital Parameters====
	Let the unprimed coordinates (x, y, z) represent the body frame of the ellipsoid, and the primed coordinates (x', y', z') represent a coordinate system in which the z'-axis is "tipped" — about the x' = x axis — away from the z-axis by an angle, θ. We assume that the motion of each Lagrangian fluid element will be restricted to an x'-y' equatorial plane — that is, we assume that <math>~\dot{z}' \equiv dz'/dt = 0</math> — but in general its velocity vector in the unprimed "body" frame will have the three nonzero components as specified in EFE and above. Here are some relevant transformations between these two coordinate systems.		Let the unprimed coordinates (x, y, z) represent the body frame of the ellipsoid, and the primed coordinates (x', y', z') represent a coordinate system in which the z'-axis is "tipped" — about the x' = x axis — away from the z-axis by an angle, θ. We assume that the motion of each Lagrangian fluid element will be restricted to an x'-y' equatorial plane — that is, we assume that <math>~\dot{z}' \equiv dz'/dt = 0</math> — but in general its velocity vector in the unprimed "body" frame will have the three nonzero components as specified in EFE and above. Here are some relevant transformations between these two coordinate systems.
	<table border="1" align="center" cellpadding="10">		<table border="1" align="center" cellpadding="10">
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	</table>		</table>

	===Example Implementation===		====Example Implementation====

	<span id="Tipped">Drawing from</span> the set of [[#Case_I\|''Case I'' parameters listed above]], we will set a = 1, b = 1.25, c = 0.4703, ζ<sub>2</sub> = -2.2794, and ζ<sub>3</sub> = -1.9637. This means that,		<span id="Tipped">Drawing from</span> the set of [[#Case_I\|''Case I'' parameters listed above]], we will set a = 1, b = 1.25, c = 0.4703, ζ<sub>2</sub> = -2.2794, and ζ<sub>3</sub> = -1.9637. This means that,

Tohline: /* Riemann Type 1 Ellipsoids */

2020-06-13T17:14:22Z

Riemann Type 1 Ellipsoids

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2020-06-13T17:11:07Z

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=Riemann Type 1 Ellipsoids=
{{LSU_HBook_header}}

==General Coefficient Expressions==

=See Also=

* Discussion of [[User:Tohline/ThreeDimensionalConfigurations/RiemannStype|Ou's Riemann-Like Ellipsoids]]

* [[User:Tohline/ThreeDimensionalConfigurations/MeetsCOLLADAandOculusRiftS|Riemann Meets COLLADA & Oculus Rift S]]
** [[User:Tohline/Appendix/Ramblings/VirtualReality#Virtual_Reality_and_3D_Printing|Virtual Reality and 3D Printing]]
** [[User:Tohline/Appendix/Ramblings/OculusRift_S|Success Importing Animated Scene into Oculus Rift S]]
** [[User:Tohline/Appendix/Ramblings/RiemannMeetsOculus|Carefully (Re)Build Riemann Type S Ellipsoids Inside Oculus Rift Environment]]: Example (b/a, c/a) = (0.41, 0.385)
** Other Example S-type Riemann Ellipsoids:
*** [[User:Tohline/Appendix/Ramblings/RiemannB90C333|(b/a, c/a) = (0.90, 0.333)]]
*** [[User:Tohline/Appendix/Ramblings/RiemannB74C692|(b/a, c/a) = (0.74, 0.692)]]
*** [[User:Tohline/Appendix/Ramblings/RiemannB28C256|(b/a, c/a) = (0.28, 0.256)]]

==Chandrasekhar's Detailed Analysis==
* [https://books.google.com/books?id=vo9VAAAAYAAJ&pg=PR6&dq=Gesammelte+Mathematische+Werke+B.+Riemann&source=gbs_selected_pages&cad=3#v=onepage&q=Gesammelte%20Mathematische%20Werke%20B.%20Riemann&f=false Bernhard Riemann (1876)] ''Gesammelte Mathematische Werke und Wissenschaftlicher'', especially Chapter X (p. 168) titled (something along the following line), "A Contribution to Research on Rotating Ellipsoidal Fluids"
* [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract S. Chandrasekhar (1965)], ApJ, 142, 890 - 961. ''The Equilibrum and the Stability of the Riemann Ellipsoids. I.'' — This work is referenced as [[User:Tohline/Appendix/References#Appendix_of_EFE|Paper XXV in EFE]] and focuses on S-type Riemann ellipsoids.
* [https://ui.adsabs.harvard.edu/abs/1966ApJ...145..842C/abstract S. Chandrasekhar (1966)], ApJ, 145, 842 - 877. ''The Equilibrum and the Stability of the Riemann Ellipsoids. II.'' — This work is referenced as [[User:Tohline/Appendix/References#Appendix_of_EFE|Paper XXVIII in EFE]] and focuses on Riemann ellipsoids of Types I, II and III.

==Finite-Amplitude Oscillations==

* [https://ui.adsabs.harvard.edu/abs/1967ApJ...149..145R/abstract L. F. Rossner (1967)], ApJ, 149, 145. ''The Finite-Amplitude Oscillations of the Maclaurin Spheroids'' — This work is referenced as [[User:Tohline/Appendix/References#Appendix_of_EFE|Paper XXXVIII in EFE]].
* [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..523F/abstract M. Fujimoto (1968)], ApJ, 152, 523. ''Gravitational Collapse of Rotating Gaseous Ellipsoids''
* [https://ui.adsabs.harvard.edu/abs/1995Ap%26SS.229..215C/abstract T. T. Chia & S. Y. Pung (1995)], Astrophysics and Space Science, 229, issue 2, 215 - 233. ''Effects of Variations of Parallel Angular Velocity and Vorticity on the Oscillations of Compressible Homogeneous Rotating Ellipsoids''
* [https://ui.adsabs.harvard.edu/abs/1997Ap%26SS.254..269C/abstract T. T. Chia & S. Y. Pung (1997)], Astrophysics and Space Science, 254, 269 - 294. ''Dynamical Behaviour of Compressible Homogeneous Uniformly Rotating Ellipsoids with Nonparallel Angular Velocity and Vorticity''

==In the Context of Galaxy Disks==
* [https://ui.adsabs.harvard.edu/abs/1970ApJ...162...97H/abstract C. Hunter (1970)], ApJ, 162, 97 - 103. ''The Disklike Riemann Ellipsoids.''

==Other==

* [[User:Tohline/Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29|Properties of Maclaurin Spheroids]]
* [[User:Tohline/Apps/MaclaurinSpheroids/GoogleBooks#Excerpts_from_A_Treatise_of_Fluxions|Excerpts from Maclaurin's (1742) ''A Treatise of Fluxions'']]

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User:Tohline/ThreeDimensionalConfigurations/RiemannTypeI - Revision history

Tohline: /* Chandrasekhar's Detailed Analysis */

Tohline: /* Other */

Tohline: /* Riemann Type 1 Ellipsoids */

Tohline: /* Riemann Type 1 Ellipsoids */

Tohline: Created page with '__FORCETOC__ =Riemann Type 1 Ellipsoids= {{LSU_HBook_header}} ==General Coefficien…'

Tohline: Created page with 'FORCETOC =Riemann Type 1 Ellipsoids= {{LSU_HBook_header}} ==General Coefficien…'