Editing Appendix/Ramblings/ForPaulFisher

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=For Paul Fisher=

==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:

<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>

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">

<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>

==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 [[Apps/MaclaurinSpheroids#Gravitational_Potential|potential generated inside an homogeneous (''i.e.,'' Maclaurin) spheroid]] whose eccentricity is, <math>~e = (1 - q^2)^{1 / 2}</math>, namely, 

<div align="center">
<p><math>
 \Phi(\varpi,z) = -\pi G \rho \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 \varpi^2 + A_3 z^2 \biggr) \biggr],
</math>
</p>
[<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>], &sect;7.3, p. 169, Eq. (7.3.1)
</div>

where, the coefficients <math>~A_1</math>, <math>~A_3</math>, and <math>~I_\mathrm{BT}</math> are functions only of the spheroid's eccentricity.  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>?

Well, let's define a primed (Cartesian) coordinate system whose z'-axis is tipped at this angle, <math>~i_0</math>, with respect to the symmetry axis of the oblate-spheroidal potential.  Drawing from a discussion in which we have presented a closely analogous [[ThreeDimensionalConfigurations/RiemannTypeI#Methodical_Derivation_of_Orbital_Parameters|methodical derivation of orbital parameters]], we have,
<table border="1" align="center" cellpadding="10">
<tr>
<td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~y'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y \cos i_0 + (z-z_0)\sin i_0 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(z-z_0)\cos i_0 - y\sin i_0 \, .</math>
  </td>
</tr>
</table>

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

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x' \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y' \cos i_0 - z'\sin i_0 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z-z_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~z'\cos i_0 + y'\sin i_0 \, .</math>
  </td>
</tr>
</table>

</td>
</tr>

<tr>
<td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\dot{x}'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\dot{x} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\dot{y}'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\dot{y} \cos i_0 + \dot{z}\sin i_0 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\cancelto{0}{\dot{z}'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\dot{z} \cos i_0 - \dot{y}\sin i_0 \, .</math>
  </td>
</tr>
</table>

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

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\dot{x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\dot{x}' \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\dot{y}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\dot{y}' \cos i_0 - \cancelto{0}{\dot{z}'}\sin i_0 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\dot{z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\cancelto{0}{\dot{z}'}\cos i_0 + \dot{y}'\sin i_0 \, .</math>
  </td>
</tr>
</table>

</td>
</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">
&nbsp;
  </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">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </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">
&nbsp;
  </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">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </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>

=See Also=

* [[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) titled, ''Using Tilted-Ring Models and Numerical Hydrodynamics to Study the Structure, Kinematics and Dynamics of HI Disks in Galaxies.''

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