Appendix/Ramblings/ForPaulFisher

For Paul Fisher

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:

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, $i_0$, 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, $\alpha ={\tan }^{-1}[q^2\tan i_0]$, to the equatorial plane of the halo, where $q$ 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,

$\mathrm{\Phi }(x,y,z)$

$=$

$\frac{v_0^2}{2}\ln [x^2+y^2+\frac{z^2}{q^2}].$

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, $e=(1-q^2{)}^{1/2}$, namely,

$\mathrm{\Phi }(\varpi ,z)=-\pi G\rho [I_{\mathrm{B}\mathrm{T}}{a}_1^2-(A_1{\varpi }^2+A_3{z}^2\left)\right],$

[ST83], §7.3, p. 169, Eq. (7.3.1)

where, the coefficients $A_1$, $A_3$, and $I_{\mathrm{B}\mathrm{T}}$ 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, $i_0$, 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 $(b/a,c/a)$ correspond to the chosen parameter pair, $(q,i_0)$?

Well, let's define a primed (Cartesian) coordinate system whose z'-axis is tipped at this angle, $i_0$, with respect to the symmetry axis of the oblate-spheroidal potential. Drawing from a discussion in which we have presented a closely analogous methodical derivation of orbital parameters, we have,

$x^{\prime }$ $=$ $x,$ $y^{\prime }$ $=$ $y\cos i_0+(z-z_0)\sin i_0,$ $z^{\prime }$ $=$ $(z-z_0)\cos i_0-y\sin i_0.$	$x$ $=$ $x^{\prime },$ $y$ $=$ $y^{\prime }\cos i_0-z^{\prime }\sin i_0,$ $z-z_0$ $=$ $z^{\prime }\cos i_0+y^{\prime }\sin i_0.$
${\dot{x}}^{\prime }$ $=$ $\dot{x},$ ${\dot{y}}^{\prime }$ $=$ $\dot{y}\cos i_0+\dot{z}\sin i_0,$ ${\xcancel{{\dot{z}}^{\prime }}}^0$ $=$ $\dot{z}\cos i_0-\dot{y}\sin i_0.$	$\dot{x}$ $=$ ${\dot{x}}^{\prime },$ $\dot{y}$ $=$ ${\dot{y}}^{\prime }\cos i_0-{\xcancel{{\dot{z}}^{\prime }}}^0\sin i_0,$ $\dot{z}$ $=$ ${\xcancel{{\dot{z}}^{\prime }}}^0\cos i_0+{\dot{y}}^{\prime }\sin i_0.$

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

$(\pi G\rho {)}^{-1}\mathrm{\Phi }(x^{\prime },y^{\prime },z^{\prime })+I_{\mathrm{B}\mathrm{T}}{a}_1^2$	$=$	$A_1[(x^{\prime }{)}^2+(y^{\prime }\cos i_0-z^{\prime }\sin i_0{)}^2]+A_3[z_0+z^{\prime }\cos i_0+y^{\prime }\sin i_0{]}^2$
	$=$	$A_1[(x^{\prime }{)}^2+(y^{\prime }{)}^2{\cos }^2{i}_0+(z^{\prime }{)}^2{\sin }^2{i}_0-2(y^{\prime }{z}^{\prime })\sin i_0\cos i_0]$
		$+A_3[z_0^2+2z^{\prime }{z}_0\cos i_0+2z_0{y}^{\prime }\sin i_0+(z^{\prime }{)}^2{\cos }^{}{i}_0+2y^{\prime }{z}^{\prime }\sin i_0\cos i_0+(y^{\prime }{)}^2{\sin }^{}{i}_0]$
	$=$	$A_1(x^{\prime }{)}^2+(y^{\prime }{)}^2[A_1{\cos }^2{i}_0+A_3{\sin }^2{i}_0]+(z^{\prime }{)}^2[A_1{\sin }^2{i}_0+A_3{\cos }^2{i}_0]$
		$+z_0{A}_3[z_0+2z^{\prime }\cos i_0+2y^{\prime }\sin i_0]+2(A_3-A_1)y^{\prime }{z}^{\prime }\sin i_0\cos i_0.$

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.