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A standard technique that is used throughout astrophysics to test the stability of self-gravitating 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 time-dependent 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 [http://en.wikipedia.org/wiki/Wave_equation 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 self-gravitating fluids.
In what follows, we borrow heavily from Chapter VIII of [[User:Tohline/Appendix/References#LL75|Landau & Lifshitz (1975)]], as it provides an excellent introductory discussion of sound waves.


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Adopting Kojima's (1986) notation, that is, defining,
Adopting [http://adsabs.harvard.edu/abs/1986PThPh..75.1464K Kojima's (1986)] notation, that is, defining,
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Revision as of 23:59, 3 January 2016

Analyzing Azimuthal Distortions

Whitworth's (1981) Isothermal Free-Energy Surface
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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 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

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