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

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.

Table 1:  Mapping from Yabushita's (1968) Notation to Ours
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,

<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>
<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 — see the boxed-in excerpt that follows — 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,

<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 — see the boxed-in excerpt that follows — matches Yabushita's (1968) equation (2.10).

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

†Mathematical expressions displayed here with layout modified from the original publication.

Linearized Wave Equation

In an accompanying discussion, we derived the so-called,

Wave Equation for Self-Gravitating Fluids <math>~\frac{\partial^2 s }{\partial t^2} + \frac{\nabla\rho_0}{\rho_0} \cdot \frac{\partial \vec{v}}{\partial t} </math> <math>~=</math> <math>4\pi G \rho_0 s + \nabla^2\biggl[s \biggl( \frac{dP}{d\rho} \biggr)_0 \biggr] </math>

that describes the time-variation at any point in space of the fractional density fluctuation,

<math>s \equiv \frac{\rho_1}{\rho_0} \, .</math>

Multiplying this differential equation through by <math>~\rho_0</math>, and making two substitutions from our accompanying summary of the separately linearized principal governing equations — namely,

<math> ~\frac{\partial \vec{v}}{\partial t} = - \nabla\Phi_1 - \frac{1}{\rho_0} \nabla P_1 + \frac{\rho_1}{\rho_0^2} \nabla P_0 \, , </math>

and,

<math> P_1 = \biggl( \frac{dP}{d\rho} \biggr)_0 \rho_1 ~~~ \Rightarrow ~~~ s \biggl( \frac{dP}{d\rho} \biggr)_0 = \frac{P_1}{\rho_0} \, , </math>

— gives,

<math>~\frac{\partial^2 \rho_1 }{\partial t^2} + \nabla\rho_0 \cdot \biggl[ - \nabla\Phi_1 - \frac{1}{\rho_0} \nabla P_1 + \frac{\rho_1}{\rho_0^2} \nabla P_0\biggr] </math>	<math>~=</math>	<math>4\pi G \rho_0 \rho_1 + \rho_0\nabla^2\biggl[\frac{P_1}{\rho_0} \biggr] </math>
<math>~\Rightarrow ~~~ \frac{\partial^2 \rho_1 }{\partial t^2} + \nabla\rho_0 \cdot \biggl[ - \nabla\Phi_1 - \frac{1}{\rho_0} \nabla P_1 - \frac{\rho_1}{\rho_0} \nabla \Phi_0\biggr] </math>	<math>~=</math>	<math>4\pi G \rho_0 \rho_1 + \nabla^2 P_1 + P_1 \rho_0\nabla^2\biggl[\frac{1}{\rho_0} \biggr] </math>
<math>~\Rightarrow ~~~ \frac{\partial^2 \rho_1 }{\partial t^2} - \nabla^2 P_1 - 4\pi G \rho_0 \rho_1 - \nabla\rho_0 \cdot \nabla\Phi_1 - \nabla \Phi_0 \cdot \nabla\rho_1 </math>	<math>~=</math>	<math> \nabla\rho_0 \cdot \biggl[ \frac{1}{\rho_0} \nabla P_1\biggr] + P_1 \rho_0\nabla^2\biggl[\frac{1}{\rho_0} \biggr] + \nabla \Phi_0 \cdot \biggl[ \frac{\rho_1}{\rho_0} \nabla\rho_0 -\nabla\rho_1 \biggr] </math>
	<math>~=</math>	<math> \nabla\rho_0 \cdot \biggl[ \frac{1}{\rho_0} \nabla P_1\biggr] + P_1 \rho_0\nabla^2\biggl[\frac{1}{\rho_0} \biggr] + \frac{\nabla \Phi_0}{\rho_0} \cdot \biggl[ \rho_1 \nabla\rho_0 - \rho_0 \nabla\rho_1 \biggr] </math>

Alternative Expression

Adiabatic Wave (or Radial Pulsation) Equation

<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

• Radial Oscillations of Polytropic Spheres

© 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