Difference between revisions of "User:Tohline/Appendix/Ramblings/Azimuthal Distortions"

From VistrailsWiki
Jump to navigation Jump to search
(→‎Adopted Notation: Insert Figure 2 from Woodward, Tohline, & Hachisu (1994) to illustrate four factors in perturbation equation)
(→‎Adopted Notation: Insert figure that compares with Imamura)
Line 166: Line 166:
<tr>
<tr>
   <td align="center">
   <td align="center">
[[File:Diagram01.png|500px|Rearranged Figure 2 from Woodward, Tohline, and Hachisu (1994)]]
[[File:Diagram01.png|650px|Rearranged Figure 2 from Woodward, Tohline, and Hachisu (1994)]]
   </td>
   </td>
</tr>
</tr>
Line 172: Line 172:
</table>
</table>
</div>
</div>
<div align="center">
<div align="center">
 
<table border="1" cellpadding="8" align="center">
<tr><td align="center">
[[File:ImamuraMontage.png|Comparison with Immamura]]
</td></tr>
</table>
</div>
</div>



Revision as of 23:49, 4 January 2016

Analyzing Azimuthal Distortions

Whitworth's (1981) Isothermal Free-Energy Surface
|   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 time-invariant coefficients and functions that characterize the intrinsic eigenvector of each azimuthal eigen-mode, 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, Self-gravitating Disks in Protostellar Systems"

ApJ, vol. 420, pp. 247-267 © American Astronomical Society

Rearranged Figure 2 from Woodward, Tohline, and Hachisu (1994)

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.


Comparison with Immamura

See Also

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
|   H_Book Home   |   YouTube   |
Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS |
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