Difference between revisions of "User:Tohline/SSC/Stability/InstabilityOnsetOverview"

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Yabushita ([http://adsabs.harvard.edu/abs/1974MNRAS.167…95Y 1974], [http://adsabs.harvard.edu/abs/1975MNRAS.172..441Y 1975]) showed that one valid, analytically specifiable eigenvector is, <math>~\sigma_c^2 = 0</math>, and,
Yabushita ([http://adsabs.harvard.edu/abs/1974MNRAS.167…95Y 1974], [http://adsabs.harvard.edu/abs/1975MNRAS.172..441Y 1975]) showed that one valid,  
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  <td align="center" colspan="3"><font color="maroon"><b>Precise Solution to the Isothermal LAWE</b></font></td>
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<math>~x</math>
<math>~\sigma_c^2 = 0</math>
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<math>~=</math>
&nbsp;and &nbsp;
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<math>~1 - \biggl( \frac{1}{\xi e^{-\psi}}\biggr) \frac{d\psi}{d\xi} \, .</math>
<math>~x = 1 - \biggl( \frac{1}{\xi e^{-\psi}}\biggr) \frac{d\psi}{d\xi} \, .</math>
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the relevant configuration is precisely defined by the surface condition, xxx, which is identical to the configuration at the turning point.
the relevant configuration is precisely defined by the surface condition, xxx, which is identical to the configuration at the turning point.


==Polytropic==
==Polytropic==

Revision as of 16:24, 19 March 2017

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,

Precise Solution to the Isothermal LAWE

<math>~\sigma_c^2 = 0</math>

 and  

<math>~x = 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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