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

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,

And, our expression for the truncated configuration's equilibrium radius is,

<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 p. 110 of 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

The linearized wave equation that Yabushita (1968) used to examine the radial pulsation modes of pressure-truncated isothermal spheres is displayed in the following, boxed-in image:

Equation extracted from p. 111 of 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

This equation can be obtained straightforwardly through a strategic combination of three of the following four linearized principal governing equations that we have derived in our accompanying, broad introductory discussion of linear stability analyses, namely,

Linearized
Equation of Continuity
<math> \frac{\partial \rho_1}{\partial t} + \rho_0\nabla\cdot \vec{v} + \vec{v}\cdot \nabla\rho_0 = 0 , </math>

Linearized
Euler Equation
<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>

Linearized
Adiabatic Form of the
First Law of Thermodynamics

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

Linearized
Poisson Equation

<math> \nabla^2 \Phi_1 = 4\pi G \rho_1\, . </math>

Taking the partial time-derivative of the linearized equation of continuity gives,

<math>~- \nabla\cdot \frac{\partial \vec{v}}{\partial t} </math>

<math>~\frac{1}{\rho_0}\frac{\partial^2 \rho_1}{\partial t^2} + \frac{\nabla\rho_0}{\rho_0} \cdot \frac{\partial\vec{v}}{\partial t} \, ;</math>

and, taking the divergence of the linearized Euler equation gives,

<math>~-\nabla\cdot \frac{\partial \vec{v}}{\partial t} </math>

<math>~\nabla^2 \Phi_1 + \nabla\cdot \biggl[\frac{1}{\rho_0} \nabla P_1\biggr] - \nabla \cdot \biggl[ \frac{\rho_1}{\rho_0^2} \nabla P_0 \biggr] \, .</math>

Combining the two, then making two substitutions using (1) the linearized Poisson equation and (2) the linearized Euler equation, we have,

<math>~\frac{\partial^2 \rho_1}{\partial t^2} + \nabla\rho_0 \cdot \frac{\partial\vec{v}}{\partial t} </math>	<math>~=</math>	<math>~\rho_0 \nabla^2 \Phi_1 + \rho_0 \nabla\cdot \biggl[\frac{1}{\rho_0} \nabla P_1\biggr] - \rho_0\nabla \cdot \biggl[ \frac{\rho_1}{\rho_0^2} \nabla P_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_0^2} \nabla P_0 \biggr] </math>	<math>~=</math>	<math>~4\pi G \rho_0 \rho_1 + \nabla^2 P_1 + \rho_0 \nabla P_1 \cdot \nabla \biggl(\frac{1}{\rho_0} \biggr) - \rho_0\nabla \cdot \biggl[ \frac{\rho_1}{\rho_0^2} \nabla P_0 \biggr] \, .</math>

Rearranging terms, and using the replacement equilibrium relation, <math>~\nabla P_0 = - \rho_0\nabla\Phi_0</math>, gives,

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

<math>~\Rightarrow ~~~ \frac{\partial^2 \rho_1}{\partial t^2} - \nabla^2 P_1 - 8\pi G \rho_0 \rho_1 - \nabla\rho_0\cdot\nabla\Phi_1 - \nabla \Phi_0 \cdot \nabla \rho_1 </math>

This is identical to Yabushita's (1968) equation (2.12).

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.

User:Tohline/SSC/Stability/Isothermal

Contents

Radial Oscillations of Pressure-Truncated Isothermal Spheres

Groundwork

Equilibrium Model

Linearized Wave Equation

Alternative Expression

See Also

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