User:Tohline/Appendix/Ramblings/ForPaulFisher
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:
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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. . |
All three-dimensional hydrodynamic simulations employ Richstone's (1980) time-independent "axisymmetric logarithmic potential" that is prescribed by the expression,
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<math>~\Phi(x, y, z)</math> |
<math>~=</math> |
<math>~ \frac{v_0^2}{2}~ \ln\biggl[x^2 + y^2 + \frac{z^2}{q^2} \biggr] \, . </math> |
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 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
- Type I Riemann Ellipsoids.
- 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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