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For Paul Fisher

Whitworth's (1981) Isothermal Free-Energy Surface
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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:

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,

~\Phi(x, y, z)

~=

~
\frac{v_0^2}{2}~ \ln\biggl[x^2 + y^2 + \frac{z^2}{q^2} \biggr] \, .

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,


 \Phi(\varpi,z) = -\pi G \rho \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 \varpi^2 + A_3 z^2 \biggr) \biggr],

[ST83], §7.3, p. 169, Eq. (7.3.1)

where, the coefficients ~A_1, ~A_3, and ~I_\mathrm{BT} 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'

~=

~x \, ,

~y'

~=

~y \cos i_0 + (z-z_0)\sin i_0 \, ,

~z'

~=

~(z-z_0)\cos i_0 - y\sin i_0 \, .

~x

~=

~x' \, ,

~y

~=

~y' \cos i_0 - z'\sin i_0 \, ,

~z-z_0

~=

~z'\cos i_0 + y'\sin i_0 \, .

~\dot{x}'

~=

~\dot{x} \, ,

~\dot{y}'

~=

~\dot{y} \cos i_0 + \dot{z}\sin i_0 \, ,

~\cancelto{0}{\dot{z}'}

~=

~\dot{z} \cos i_0 - \dot{y}\sin i_0 \, .

~\dot{x}

~=

~\dot{x}' \, ,

~\dot{y}

~=

~\dot{y}' \cos i_0 - \cancelto{0}{\dot{z}'}\sin i_0 \, ,

~\dot{z}

~=

~\cancelto{0}{\dot{z}'}\cos i_0 + \dot{y}'\sin i_0 \, .

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

~(\pi G \rho)^{-1} \Phi(x', y', z') +  I_\mathrm{BT} a_1^2

~=

~
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

 

~=

~
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]

 

 

~
+ 
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]

 

~=

~
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]

 

 

~
+ 
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 \, .

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation

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