User:Tohline/Appendix/Ramblings/ForPaulFisher - Revision history

Tohline: /* For Paul Fisher */

2021-04-02T16:54:23Z

For Paul Fisher

← Older revision		Revision as of 16:54, 2 April 2021
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	</tr>		</tr>

			</table>

			When viewed from this primed frame, the potential associated with a Maclaurin spheroid becomes,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~(\pi G \rho)^{-1} \Phi(x', y', z') + I_\mathrm{BT} a_1^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			A_1 \biggl[ (x')^2 + \biggl(y'\cos i_0 - z' \sin i_0\biggr)^2 \biggr]
			+
			A_3 \biggl[ z_0 + z' \cos i_0 + y' \sin i_0 \biggr]^2
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			A_1 \biggl[ (x')^2 + (y')^2 \cos^2 i_0 + (z')^2 \sin^2 i_0 - 2(y' z')\sin i_0 \cos i_0\biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

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

			</td>
			<td align="left">
			<math>~
			+
			A_3 \biggl[ z_0^2 + 2 z' z_0 \cos i_0 + 2z_0 y' \sin i_0 + (z')^2 \cos^2 i_0 + 2y' z' \sin i_0 \cos i_0 + (y')^2 \sin^2 i_0 \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			A_1 (x')^2 + (y')^2 \biggl[A_1 \cos^2 i_0 + A_3 \sin^2 i_0 \biggr] + (z')^2 \biggl[ A_1 \sin^2 i_0 + A_3\cos^2 i_0 \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

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

			</td>
			<td align="left">
			<math>~
			+
			z_0 A_3 \biggl[ z_0 + 2 z' \cos i_0 + 2 y' \sin i_0 \biggr]
			+
			2(A_3 - A_1 )y' z' \sin i_0 \cos i_0 \, .
			</math>
			</td>
			</tr>
	</table>		</table>

Tohline: /* Thoughts Moving Forward */

2021-04-02T16:08:50Z

Thoughts Moving Forward

Show changes

Tohline: /* Overview of Dissertation */

2021-04-02T03:01:02Z

Overview of Dissertation

← Older revision		Revision as of 03:01, 2 April 2021
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	</td></tr></table>		</td></tr></table>

	All three-dimensional hydrodynamic simulations employ Richstone's (1980) time-independent "axisymmetric logarithmic potential" that is prescribed by the expression,		All three-dimensional hydrodynamic simulations employ [https://ui.adsabs.harvard.edu/abs/1980ApJ...238..103R/abstract Richstone's (1980)] time-independent "axisymmetric logarithmic potential" that is prescribed by the expression,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Thoughts Moving Forward */

2021-04-02T02:49:33Z

Thoughts Moving Forward

← Older revision		Revision as of 02:49, 2 April 2021
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	==Thoughts Moving Forward==		==Thoughts Moving Forward==

	Let's continue to examine a collection of Lagrangian fluid elements that are orbiting in an (axisymmetric) oblate-spheroidal potential with flattening "q." But rather than adopting the Richstone potential, we will consider the [[User:Tohline/Apps/MaclaurinSpheroids#Gravitational_Potential\|potential generated inside an homogeneous (''i.e.,'' Maclaurin) spheroid]] whose eccentricity is, <math>~e = (1 - q^2)^{1 / 2}</math>. What does the potential field look like from the perspective of a particle/fluid-element whose orbital angular momentum vector is tipped at an angle, <math>~i_0</math>, to the symmetry axis of the oblate-spheroidal potential? Presumably the potential is "observed" to vary with position around the orbit as though the underlying potential is non-axisymmetric. Does it appear to be the potential inside a Riemann S-Type ellipsoid? If so, what values of <math>~(b/a, c/a)</math> correspond to the chosen parameter pair, <math>~(q, i_0</math>?		Let's continue to examine a collection of Lagrangian fluid elements that are orbiting in an (axisymmetric) oblate-spheroidal potential with flattening "q." But rather than adopting the Richstone potential, we will consider the [[User:Tohline/Apps/MaclaurinSpheroids#Gravitational_Potential\|potential generated inside an homogeneous (''i.e.,'' Maclaurin) spheroid]] whose eccentricity is, <math>~e = (1 - q^2)^{1 / 2}</math>. What does the potential field look like from the perspective of a particle/fluid-element whose orbital angular momentum vector is tipped at an angle, <math>~i_0</math>, to the symmetry axis of the oblate-spheroidal potential? Presumably the potential is "observed" to vary with position around the orbit as though the underlying potential is non-axisymmetric. Does it appear to be the potential inside a Riemann S-Type ellipsoid? If so, what values of <math>~(b/a, c/a)</math> correspond to the chosen parameter pair, <math>~(q, i_0)</math>?

	=See Also=		=See Also=

Tohline: /* For Paul Fisher */

2021-04-02T02:48:14Z

For Paul Fisher

← Older revision		Revision as of 02:48, 2 April 2021
Line 35:		Line 35:
	==Thoughts Moving Forward==		==Thoughts Moving Forward==

	Let's continue to examine a collection of Lagrangian fluid elements that are orbiting in an (axisymmetric) oblate-spheroidal potential with flattening "q." But rather than adopting the Richstone potential, we will consider the [[User:Tohline/Apps/MaclaurinSpheroids#Gravitational_Potential\|potential generated inside an homogeneous (''i.e.,'' Maclaurin) spheroid]] whose eccentricity is, <math>~e = (1 - 1/q^2)^{1 / 2}</math>. What does the potential field look like from the perspective of a particle/fluid-element whose orbital angular momentum vector is tipped at an angle, <math>~i_0</math>, to the symmetry axis of the oblate-spheroidal potential? Presumably the potential is "observed" to vary with position around the orbit as though the underlying potential is non-axisymmetric. Does it appear to be the potential inside a Riemann S-Type ellipsoid? If so, what values of <math>~(b/a, c/a)</math> correspond to the chosen parameter pair, <math>~(q, i_0</math>?		Let's continue to examine a collection of Lagrangian fluid elements that are orbiting in an (axisymmetric) oblate-spheroidal potential with flattening "q." But rather than adopting the Richstone potential, we will consider the [[User:Tohline/Apps/MaclaurinSpheroids#Gravitational_Potential\|potential generated inside an homogeneous (''i.e.,'' Maclaurin) spheroid]] whose eccentricity is, <math>~e = (1 - q^2)^{1 / 2}</math>. What does the potential field look like from the perspective of a particle/fluid-element whose orbital angular momentum vector is tipped at an angle, <math>~i_0</math>, to the symmetry axis of the oblate-spheroidal potential? Presumably the potential is "observed" to vary with position around the orbit as though the underlying potential is non-axisymmetric. Does it appear to be the potential inside a Riemann S-Type ellipsoid? If so, what values of <math>~(b/a, c/a)</math> correspond to the chosen parameter pair, <math>~(q, i_0</math>?

	=See Also=		=See Also=

Tohline: /* For Paul Fisher */

2021-04-02T02:46:59Z

For Paul Fisher

← Older revision		Revision as of 02:46, 2 April 2021
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	=For Paul Fisher=		=For Paul Fisher=

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	==Overview of Dissertation==		==Overview of Dissertation==
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	</table>		</table>

			==Thoughts Moving Forward==

			Let's continue to examine a collection of Lagrangian fluid elements that are orbiting in an (axisymmetric) oblate-spheroidal potential with flattening "q." But rather than adopting the Richstone potential, we will consider the [[User:Tohline/Apps/MaclaurinSpheroids#Gravitational_Potential\|potential generated inside an homogeneous (''i.e.,'' Maclaurin) spheroid]] whose eccentricity is, <math>~e = (1 - 1/q^2)^{1 / 2}</math>. What does the potential field look like from the perspective of a particle/fluid-element whose orbital angular momentum vector is tipped at an angle, <math>~i_0</math>, to the symmetry axis of the oblate-spheroidal potential? Presumably the potential is "observed" to vary with position around the orbit as though the underlying potential is non-axisymmetric. Does it appear to be the potential inside a Riemann S-Type ellipsoid? If so, what values of <math>~(b/a, c/a)</math> correspond to the chosen parameter pair, <math>~(q, i_0</math>?

	=See Also=		=See Also=

Tohline: /* For Paul Fisher */

2021-03-29T22:31:59Z

For Paul Fisher

← Older revision		Revision as of 22:31, 29 March 2021
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			==Overview of Dissertation==
	[https://digitalcommons.lsu.edu/gradschool_disstheses/6940/ Paul Fisher's (1999) doctoral dissertation] (accessible via the LSU Digital Commons) is titled, ''Nonaxisymmetric Equilibrium Models for Gaseous Galaxy Disks.'' Its abstract reads, in part:		[https://digitalcommons.lsu.edu/gradschool_disstheses/6940/ Paul Fisher's (1999) doctoral dissertation] (accessible via the LSU Digital Commons) is titled, ''Nonaxisymmetric Equilibrium Models for Gaseous Galaxy Disks.'' Its abstract reads, in part:

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	<font color="darkgreen">Three-dimensional hydrodynamic simulations show that, in the absence of self-gravity, an axisymmetric, gaseous galaxy disk whose angular momentum vector is initially tipped at an angle, <math>~i_0</math>, to the symmetry axis of a fixed spheroidal dark matter halo potential does not settle to the equatorial plane of the halo. Instead, the disk settles to a plane that is tipped at an angle, <math>~\alpha = \tan^{-1}[q^2 \tan i_0]</math>, to the equatorial plane of the halo, where <math>~q</math> is the axis ratio of the halo equipotential surfaces. The equilibrium configuration to which the disk settles appears to be flat but it exhibits distinct nonaxisymmetric features. .</font>		<font color="darkgreen">Three-dimensional hydrodynamic simulations show that, in the absence of self-gravity, an axisymmetric, gaseous galaxy disk whose angular momentum vector is initially tipped at an angle, <math>~i_0</math>, to the symmetry axis of a fixed spheroidal dark matter halo potential does not settle to the equatorial plane of the halo. Instead, the disk settles to a plane that is tipped at an angle, <math>~\alpha = \tan^{-1}[q^2 \tan i_0]</math>, to the equatorial plane of the halo, where <math>~q</math> is the axis ratio of the halo equipotential surfaces. The equilibrium configuration to which the disk settles appears to be flat but it exhibits distinct nonaxisymmetric features. .</font>
	</td></tr></table>		</td></tr></table>

			All three-dimensional hydrodynamic simulations employ Richstone's (1980) time-independent "axisymmetric logarithmic potential" that is prescribed by the expression,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\Phi(x, y, z)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{v_0^2}{2}~ \ln\biggl[x^2 + y^2 + \frac{z^2}{q^2} \biggr] \, .
			</math>
			</td>
			</tr>
			</table>

	=See Also=		=See Also=

Tohline: /* For Paul Fisher */

2021-03-29T22:05:03Z

For Paul Fisher

← Older revision		Revision as of 22:05, 29 March 2021
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	{{LSU_HBook_header}}		{{LSU_HBook_header}}


			[https://digitalcommons.lsu.edu/gradschool_disstheses/6940/ Paul Fisher's (1999) doctoral dissertation] (accessible via the LSU Digital Commons) is titled, ''Nonaxisymmetric Equilibrium Models for Gaseous Galaxy Disks.'' Its abstract reads, in part:

			<table border="0" align="center" width="80%" cellpadding="3"><tr><td align="left">
			<font color="darkgreen">Three-dimensional hydrodynamic simulations show that, in the absence of self-gravity, an axisymmetric, gaseous galaxy disk whose angular momentum vector is initially tipped at an angle, <math>~i_0</math>, to the symmetry axis of a fixed spheroidal dark matter halo potential does not settle to the equatorial plane of the halo. Instead, the disk settles to a plane that is tipped at an angle, <math>~\alpha = \tan^{-1}[q^2 \tan i_0]</math>, to the equatorial plane of the halo, where <math>~q</math> is the axis ratio of the halo equipotential surfaces. The equilibrium configuration to which the disk settles appears to be flat but it exhibits distinct nonaxisymmetric features. .</font>
			</td></tr></table>

	=See Also=		=See Also=

Tohline: /* See Also */

2021-03-29T21:52:31Z

Tohline: /* See Also */

2021-03-29T21:47:22Z

← Older revision		Revision as of 21:52, 29 March 2021
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	=See Also=		=See Also=

	* [[User:Tohline/ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids\|Type I Riemann Ellipsoids]]		* [[User:Tohline/ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids\|Type I Riemann Ellipsoids]].
	* [https://digitalcommons.lsu.edu/gradschool_disstheses/4769/ Dimitris M. Christodoulou's (1989) doctoral dissertation] (accessible via the LSU Digital Commons)		* [https://digitalcommons.lsu.edu/gradschool_disstheses/4769/ Dimitris M. Christodoulou's (1989) doctoral dissertation] (accessible via the LSU Digital Commons) titled, ''Using Tilted-Ring Models and Numerical Hydrodynamics to Study the Structure, Kinematics and Dynamics of HI Disks in Galaxies.''


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← Older revision		Revision as of 21:47, 29 March 2021
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	=See Also=		=See Also=

	* ~~Discussion of~~ [[User:Tohline/ThreeDimensionalConfigurations/~~RiemannStype~~\|~~Ou's~~ Riemann~~-Like~~ Ellipsoids]]		* [[User:Tohline/ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids\|Type I Riemann Ellipsoids]]
			* [https://digitalcommons.lsu.edu/gradschool_disstheses/4769/ Dimitris M. Christodoulou's (1989) doctoral dissertation] (accessible via the LSU Digital Commons)


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