Difference between revisions of "User:Tohline/Appendix/Ramblings/Azimuthal Distortions"
(Begin describing individual eigenvector properties) 
(→Adopted Notation: Rewrite using Kojima's notation) 

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</div>  </div>  
Adopting Kojima's (1986) notation, that is, defining,  
<div align="center">  
<table border="0" cellpadding="5" align="center">  
<tr>  
<td align="right">  
<math>~y_1 \equiv \frac{\omega_R}{\Omega_0}  m</math>  
</td>  
<td align="center">  
and  
</td>  
<td align="left">  
<math>~y_2 \equiv \frac{\omega_I}{\Omega_0} \, ,</math>  
</td>  
</tr>  
</table>  
</div>  
the eigenvector's behavior can furthermore be described by the expression,  
<div align="center">  
<table border="0" cellpadding="5" align="center">  
<tr>  
<td align="right">  
<math>~\biggl[ \frac{\rho}{\rho_0}  1 \biggr]</math>  
</td>  
<td align="center">  
<math>~=</math>  
</td>  
<td align="left">  
<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>  
</td>  
</tr>  
<tr>  
<td align="right">  
 
</td>  
<td align="center">  
<math>~=</math>  
</td>  
<td align="left">  
<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>  
</td>  
</tr>  
</table>  
</div>  
Note that, as viewed from a frame of reference that is rotating with the mode pattern speed,  
<div align="center">  
<math>\Omega_p \equiv \frac{\omega_R}{m} = \Omega_0\biggl(\frac{y_1}{m}+1\biggr) \, ,</math>  
</div>  
we should find an eigenvector of the form,  
<div align="center">  
<table border="0" cellpadding="5" align="center">  
<tr>  
<td align="right">  
<math>~\biggl[ \frac{\rho}{\rho_0}  1 \biggr]_\mathrm{rot} \equiv \biggl[ \frac{\rho}{\rho_0}  1 \biggr]e^{im\Omega_p t}</math>  
</td>  
<td align="center">  
<math>~=</math>  
</td>  
<td align="left">  
<math>~\biggl\{ f_m(\varpi)e^{im[\phi_m(\varpi)]} \biggr\} e^{y_2 (\Omega_0 t)} \, ,</math>  
</td>  
</tr>  
</table>  
</div>  
which is unchanging (inside the curly braces) except for a uniform exponential amplitude growth.  
=See Also=  =See Also= 
Revision as of 23:57, 3 January 2016
Analyzing Azimuthal Distortions
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A standard technique that is used throughout astrophysics to test the stability of selfgravitating fluids involves perturbing physical variables away from their initial (usually equilibrium) values then linearizing each of the principal governing equations before seeking solutions describing the timedependent behavior of the variables that simultaneously satisfy all of the equations. When the effects of the fluid's self gravity are ignored and this analysis technique is applied to an initially homogeneous medium, the combined set of linearized governing equations generates a wave equation — whose general properties are well documented throughout the mathematics and physical sciences literature — that, specifically in our case, governs the propagation of sound waves. It is quite advantageous, therefore, to examine how the wave equation is derived in the context of an analysis of sound waves before applying the standard perturbation & linearization technique to inhomogeneous and selfgravitating fluids.
In what follows, we borrow heavily from Chapter VIII of Landau & Lifshitz (1975), as it provides an excellent introductory discussion of sound waves.
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 speed,
<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> 
which is unchanging (inside the curly braces) except for a uniform exponential amplitude growth.
See Also
© 2014  2021 by Joel E. Tohline 