User:Tohline/SSC/Stability/InstabilityOnsetOverview

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Overview: Marginally Unstable Pressure-Truncated Configurations

Additional details may be found here.

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

The internal structure of a detailed force-balance model is provided via the function, <math>~\psi(\xi)</math>, which is a solution to the,

Isothermal Lane-Emden Equation

LSU Key.png

<math>~\frac{1}{\xi^2} \frac{d}{d\xi}\biggl( \xi^2 \frac{d\psi}{d\xi} \biggr) = e^{-\psi}</math>

Equilibrium sequence for pressure-truncated configurations is displayed in three ways.

Figure 1:   Bonnor's P-V Diagram
(see related discussion)

Bonnor (1956, MNRAS, 116, 351)
Pressure-Truncated Isothermal Equilibrium Sequence

This equation — in the following, slightly rewritten form — can be found among our selected set of key equations associated with the study of radial pulsation, and will henceforth be referred to as the,

Isothermal LAWE

LSU Key.png

<math>~0 = \frac{d^2x}{d\xi^2} + \biggl[4 - \xi \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} + \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_\mathrm{g}}\biggr)\xi^2 - \alpha \xi \biggl( \frac{d\psi}{d\xi} \biggr) \biggr] \frac{x}{\xi^2} </math>

where:    <math>~\sigma_c^2 \equiv \frac{3\omega^2}{2\pi G\rho_c}</math>     and,     <math>~\alpha \equiv \biggl(3 - \frac{4}{\gamma_\mathrm{g}}\biggr)</math>

Yabushita (1974, 1975) showed that one valid, analytically specifiable eigenvector is, <math>~\sigma_c^2 = 0</math>, and,

<math>~x</math>

<math>~=</math>

<math>~1 - \biggl( \frac{1}{\xi e^{-\psi}}\biggr) \frac{d\psi}{d\xi} \, .</math>

When viewed in concert with the surface boundary condition,

<math>~\frac{d\ln x}{d\ln\xi}</math>

<math>~=</math>

<math>~- 3 \, ,</math>

the relevant configuration is precisely defined by the surface condition, xxx, which is identical to the configuration at the turning point.


Polytropic

References

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