User:Tohline/Appendix/Ramblings/Azimuthal Distortions
Analyzing Azimuthal Distortions
 Tiled Menu  Tables of Content  Banner Video  Tohline Home Page  
Adopted Notation
We will adopt the notation of J. E. Tohline & I. Hachisu (1988, ApJ, 361, 394). Specifically, drawing on their equation (2) but ignoring variations in the vertical coordinate, the mass density is given by the expression,
<math>~\rho</math> 
<math>~=</math> 
<math>~\rho_0 \biggl[ 1 + f(\varpi)e^{i(\omega t  m\phi)} \biggr] \, ,</math> 
where it is understood that <math>~\rho_0</math>, which defines the structure of the initial axisymmetric equilibrium configuration, is generally a function of the cylindrical radial coordinate, <math>~\varpi</math>.
Using the subscript, <math>~m</math>, to identify the timeinvariant coefficients and functions that characterize the intrinsic eigenvector of each azimuthal eigenmode, and acknowledging that the associated eigenfrequency will in general be imaginary, that is,
<math>~\omega_m</math> 
<math>~=</math> 
<math>~\omega_R + i\omega_I \, ,</math> 
we expect each unstable mode to display the following behavior:
<math>~\biggl[ \frac{\rho}{\rho_0}  1 \biggr]</math> 
<math>~=</math> 
<math>~f_m(\varpi)e^{i[\omega_R t + i \omega_I t  m\phi_m(\varpi)]} </math> 

<math>~=</math> 
<math>~\biggl\{ f_m(\varpi)e^{im\phi_m(\varpi)}\biggr\} e^{i\omega_R t } \cdot e^{\omega_I t} </math> 

<math>~=</math> 
<math>~\biggl\{ f_m(\varpi)e^{i[\omega_R t + m\phi_m(\varpi)]} \biggr\} e^{\omega_I t} \, .</math> 
Adopting Kojima's (1986) notation, that is, defining,
<math>~y_1 \equiv \frac{\omega_R}{\Omega_0}  m</math> 
and 
<math>~y_2 \equiv \frac{\omega_I}{\Omega_0} \, ,</math> 
the eigenvector's behavior can furthermore be described by the expression,
<math>~\biggl[ \frac{\rho}{\rho_0}  1 \biggr]</math> 
<math>~=</math> 
<math>~\biggl\{ f_m(\varpi)e^{i[(y_1+m) (\Omega_0 t) + m\phi_m(\varpi)]} \biggr\} e^{y_2 (\Omega_0 t)} </math> 

<math>~=</math> 
<math>~\biggl\{ f_m(\varpi)e^{im[(y_1/m+1) (\Omega_0 t) + \phi_m(\varpi)]} \biggr\} e^{y_2 (\Omega_0 t)} \, .</math> 
Note that, as viewed from a frame of reference that is rotating with the mode pattern frequency,
<math>\Omega_p \equiv \frac{\omega_R}{m} = \Omega_0\biggl(\frac{y_1}{m}+1\biggr) \, ,</math>
we should find an eigenvector of the form,
<math>~\biggl[ \frac{\rho}{\rho_0}  1 \biggr]_\mathrm{rot} \equiv \biggl[ \frac{\rho}{\rho_0}  1 \biggr]e^{im\Omega_p t}</math> 
<math>~=</math> 
<math>~\biggl\{ f_m(\varpi)e^{im[\phi_m(\varpi)]} \biggr\} e^{y_2 (\Omega_0 t)} \, ,</math> 
whose relative amplitude — with a radial structure as specified inside the curly braces — is undergoing a uniform exponential growth but is otherwise unchanging.
Four panels from figure 2 extracted^{†} from p. 252 of J. W. Woodward, J. E. Tohline & I. Hachisu (1994)
"The Stability of Thick, Selfgravitating Disks in Protostellar Systems"
ApJ, vol. 420, pp. 247267 © American Astronomical Society 
^{†}As displayed here, the layout of figure panels (a, b, c, d) has been modified from the original publication layout; otherwise, each panel is unmodified. 
See Also
© 2014  2021 by Joel E. Tohline 