User:Tohline/Appendix/Ramblings/ForPaulFisher
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For Paul Fisher
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Overview of Dissertation
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:
Threedimensional hydrodynamic simulations show that, in the absence of selfgravity, an axisymmetric, gaseous galaxy disk whose angular momentum vector is initially tipped at an angle, , 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, , to the equatorial plane of the halo, where 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. . 
All threedimensional hydrodynamic simulations employ Richstone's (1980) timeindependent "axisymmetric logarithmic potential" that is prescribed by the expression,



Thoughts Moving Forward
Let's continue to examine a collection of Lagrangian fluid elements that are orbiting in an (axisymmetric) oblatespheroidal potential with flattening "q." But rather than adopting the Richstone potential, we will consider the potential generated inside an homogeneous (i.e., Maclaurin) spheroid whose eccentricity is, , namely,
[ST83], §7.3, p. 169, Eq. (7.3.1)
where, the coefficients , , and are functions only of the spheroid's eccentricity. What does the potential field look like from the perspective of a particle/fluidelement whose orbital angular momentum vector is tipped at an angle, , to the symmetry axis of the oblatespheroidal potential? Presumably the potential is "observed" to vary with position around the orbit as though the underlying potential is nonaxisymmetric. Does it appear to be the potential inside a Riemann SType ellipsoid? If so, what values of correspond to the chosen parameter pair, ?
Well, let's define a primed (Cartesian) coordinate system whose z'axis is tipped at this angle, , with respect to the symmetry axis of the oblatespheroidal potential. Drawing from a discussion in which we have presented a closely analogous methodical derivation of orbital parameters, we have,





When viewed from this primed frame, the potential associated with a Maclaurin spheroid becomes,















See Also
 Type I Riemann Ellipsoids.
 Dimitris M. Christodoulou's (1989) doctoral dissertation (accessible via the LSU Digital Commons) titled, Using TiltedRing Models and Numerical Hydrodynamics to Study the Structure, Kinematics and Dynamics of HI Disks in Galaxies.
© 2014  2021 by Joel E. Tohline 