User:Tohline/SSC/Stability/Isothermal
Radial Oscillations of Pressure-Truncated Isothermal Spheres
Here we draw primarily from the following three sources:
- §5.3.8 (p. 372) of Horedt's (2004) treatise on Polytropes: Applications in Astrophysics and Related Fields
- S. Yabushita (1968, MNRAS, 140, 109) — Jeans's Type Gravitational Instability of Finite Isothermal Gas Spheres
- L. G. Taff & H. M. Horn (1974, MNRAS, 168, 427-432) — Radial Pulsations of Finite Isothermal Gas Spheres
See also:
- S. Yabushita (1975, MNRAS, 171, 85) — Jeans's Type Gravitational Instability of Finite Isothermal Gas Spheres — II
- S. Yabushita (1992, Astrophys. & Space Sciences, 193, 173 - 183) — Similarity between the structure and stability of isothermal and polytropic gas spheres
- J. P. Chieze (1987, A&A, 171, 225 - 232) — The fragmentation of molecular clouds. I - The mass-radius-velocity dispersion relations
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Groundwork
Equilibrium Model
In an accompanying discussion, while reviewing the original derivations of Ebert (1955) and Bonnor (1956), we have detailed the equilibrium properties of pressure-truncated isothermal spheres. A parallel presentation of these details can be found in §2 — specifically, equations (2.4) through (2.10) — of Yabushita (1968). Each of Yabushita's key mathematical expressions can be mapped to ours via the variable substitutions presented here in Table 1.
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Table 1: Mapping from Yabushita's (1968) Notation to Ours |
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| Yabushita's (1968) Notation: | <math>~x</math> | <math>~\psi</math> | <math>~\mu</math> | <math>~M</math> | <math>~x_0</math> | <math>~p_0</math> |
| Our Notation: | <math>~\xi</math> | <math>~-\psi</math> | <math>~\bar\mu</math> | <math>~M_{\xi_e}</math> | <math>~\xi_e</math> | <math>~P_e</math> |
For example, given the system's sound speed, <math>~c_s</math>, and total mass, <math>~M_{\xi_e}</math>, the expression from our presentation that shows how the bounding external pressure, <math>~P_e</math>, depends on the dimensionless Lane-Emden function, <math>~\psi</math>, is,
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<math>~P_e</math> |
<math>~=</math> |
<math>~\biggl( \frac{c_s^8}{4\pi G^3 M_{\xi_e}^2} \biggr) ~\xi_e^4 \biggl(\frac{d\psi}{d\xi}\biggr)^2_e e^{-\psi_e}</math> |
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<math>~\Rightarrow ~~~ \xi_e^2 \biggl(\frac{d\psi}{d\xi}\biggr)_e e^{-(1/2)\psi_e}</math> |
<math>~=</math> |
<math>~\frac{1}{c_s^4}\biggl[ G^3 M_{\xi_e}^2 ~(4\pi P_e)\biggr]^{1 / 2} \, ,</math> |
which exactly matches Yabushita's (1968) equation (2.9), after recalling that the system's sound speed is related to its temperature via the relation,
<math>c_s^2 = \frac{\Re T}{\bar{\mu}} \, .</math>
And, our expression for the truncated configuration's equilibrium radius is,
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<math>~R</math> |
<math>~=</math> |
<math>~\frac{GM_{\xi_e}}{c_s^2} \biggl[ - \xi \biggl(\frac{d\psi}{d\xi}\biggr) \biggr]_e^{-1}</math> |
which exactly matches Yabushita's (1968) equation (2.10).
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Equations extracted† from S. Yabushita (1968, MNRAS, 140, 109)
"Jeans's Type Gravitational Instability of Finite Isothermal Gas Spheres"
MNRAS, vol. 140, pp. 109-120 © Royal Astronomical Society |
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†Mathematical expressions displayed here with layout modified from the original publication. |
Linearized Wave Equation
In an accompanying discussion, we derived the so-called,
Adiabatic Wave (or Radial Pulsation) Equation
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<math>~ \frac{d^2x}{dr_0^2} + \biggl[\frac{4}{r_0} - \biggl(\frac{g_0 \rho_0}{P_0}\biggr) \biggr] \frac{dx}{dr_0} + \biggl(\frac{\rho_0}{\gamma_\mathrm{g} P_0} \biggr)\biggl[\omega^2 + (4 - 3\gamma_\mathrm{g})\frac{g_0}{r_0} \biggr] x = 0 </math> |
whose solution gives eigenfunctions that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations.
See Also
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© 2014 - 2021 by Joel E. Tohline |