Stability of Spherically Symmetric Configurations (Eulerian Perspective)

A standard technique that is used throughout astrophysics to test the stability of self-gravitating fluids involves perturbing physical variables away from their initial (usually equilibrium) values then linearizing each of the principal governing equations before seeking solutions describing the time-dependent behavior of the variables that simultaneously satisfy all of the equations. When the effects of the fluid's self gravity are ignored and this analysis technique is applied to an initially homogeneous medium, the combined set of linearized governing equations generates a wave equation that governs the propagation of sound waves. Here we build on our separate, introductory discussion of sound waves and apply standard perturbation & linearization techniques to spherically symmetric, inhomogeneous and self-gravitating fluids. We will assume that the reader has read this separate introductory discussion and, in particular, understands how the linear wave equation that governs the propagation of sound waves is derived from the set of nonlinear, principal governing equations.

Assembling the Key Relations

Governing Equations and Supplemental Relations

We begin with the set of principal governing equations that provides the foundation for all of our discussions in this H_Book, namely, the

Eulerian Representation
of the Continuity Equation,

<math>~\frac{\partial\rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0</math>

Eulerian Representation
of the Euler Equation,

<math>~\frac{\partial\vec{v}}{\partial t} + (\vec{v}\cdot \nabla) \vec{v}= - \frac{1}{\rho} \nabla P - \nabla \Phi</math>

Adiabatic Form of the
First Law of Thermodynamics

<math>~\frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\biggr) = 0</math> .

Poisson Equation

<math>\nabla^2 \Phi = 4\pi G \rho</math>

As was done in our separate, introductory discussion of sound waves, we will assume that we are dealing with an ideal gas and supplement this set of equations with a barotropic (polytropic) equation of state,

<math>~P = K\rho^{\gamma_\mathrm{g}}</math> … with … <math>\gamma_\mathrm{g} \equiv \frac{d\ln P_0}{d\ln \rho_0} = \frac{\rho_0}{P_0} \biggl( \frac{dP}{d\rho} \biggr)_0 \, ,</math>

which will ensure that the adiabatic form of the first law of thermodynamics is satisfied.

Perturbation then Linearization of Equations

In this Eulerian analysis, we are investigating how conditions vary with time at a fixed point in space, <math>~\vec{r}</math>. By analogy with our separate introductory analysis of sound waves, we will write the four primary variables in the form,

<math>~\rho</math>	<math>~=</math>	<math>~\rho_0(\vec{r}) + \rho_1(\vec{r},t) \, ,</math>
<math>~\vec{v}</math>	<math>~=</math>	<math>~\cancelto{0}{\vec{v}_0} + \vec{v}_1(\vec{r},t) = \vec{v}(\vec{r},t) \, ,</math>
<math>~P</math>	<math>~=</math>	<math>~P_0(\vec{r}) + P_1(\vec{r},t) \, ,</math>
<math>~\Phi</math>	<math>~=</math>	<math>~\Phi_0(\vec{r}) + \Phi_1(\vec{r},t) \, ,</math>

where quantities with subscript "0" are initial values — independent of time, but not necessarily spatially uniform, and usually specified via some choice of an initial equilibrium configuration — and quantities with subscript "1" denote variations away from the initial state, which are assumed to be small in amplitued — for example, <math>~|\rho_1/\rho_0 | \ll 1</math> and <math>~| P_1/P_0 | \ll 1</math>. As indicated, we will assume that the fluid configuration is initially stationary <math>~(\mathrm{i.e.,}~\vec{v}_0 = 0)</math> and, for simplicity, will not append the subscript "1" to the velocity perturbation. It is to be understood, however, that the velocity, <math>~\vec{v}</math>, is small also. In what follows, by definition, <math>~P_1</math>, <math>~\rho_1</math>, <math>~\Phi_1</math>, and <math>~\vec{v}</math> are considered to be of first order in smallness, while products of these quantities are of second (or even higher) order in smallness.

Continuity Equation

Substituting the expression for <math>~\rho</math> into the lefthand side of the continuity equation and neglecting small quantities of the second order, we have,

<math>~~\frac{\partial}{\partial t} (\rho_0 + \rho_1) + \nabla\cdot [(\rho_0 + \rho_1)\vec{v}]</math>	<math>~=</math>	<math>~ \cancelto{0}{\frac{\partial \rho_0}{\partial t}} + \frac{\partial \rho_1}{\partial t} + \nabla\cdot (\rho_0 \vec{v}) + \nabla\cdot\cancelto{\mathrm{small}}{(\rho_1\vec{v} )} </math>
	<math>~\approx</math>	<math>~ \frac{\partial \rho_1}{\partial t} + \rho_0\nabla\cdot \vec{v} + \vec{v}\cdot \nabla\rho_0 \, , </math>

where, in the first line, the first term on the lefthand side has been set to zero because <math>~\rho_0</math> is independent of time. Hence, we have the desired,

Linearized Continuity Equation

<math>~ \frac{\partial \rho_1}{\partial t} + \rho_0\nabla\cdot \vec{v} + \vec{v}\cdot \nabla\rho_0 </math>

Note that, if we were to assume that the initial configuration is homogeneous, then we could set <math>~\nabla\rho_0 = 0</math> and drop the last term on the righthand side of this expression, retrieving the linearized continuity equation used in our introductory discussion of sound waves.

Euler Equation

Next, we note that the term,

<math>(\vec{v} \cdot \nabla)\vec{v} \, ,</math>

in Euler's equation may be neglected because it is of second order in smallness. Substituting the expressions for <math>~\rho</math> and <math>~P</math> into the righthand side of the Euler equation and neglecting small quantities of the second order, we have,

<math>~\frac{1}{(\rho_0 + \rho_1)} \nabla (P_0 + P_1)</math>	<math>~=</math>	<math>~ \frac{1}{\rho_0} \biggl( 1 + \frac{\rho_1}{\rho_0} \biggr)^{-1} \biggl[ \cancelto{0}{\nabla P_0} + \nabla P_1\biggr] </math>
	<math>~=</math>	<math>~ \frac{1}{\rho_0} \biggl[ 1 - \frac{\rho_1}{\rho_0} + \cancelto{\mathrm{small}}{\biggl(\frac{\rho_1}{\rho_0} \biggr)^2} + \cdots \biggr] \nabla P_1 </math>
	<math>~=</math>	<math>~ \frac{1}{\rho_0} \nabla P_1 - \frac{1}{\rho_0^2} \cancelto{\mathrm{small}}{(\rho_1 \nabla P_1)} \, , </math>

where, in the first line, <math>~\nabla P_0</math> has been set to zero because we have assumed that the initial equilibrium state is homogeneous, and the binomial theorem has been used to obtain the expression on the righthand side of the second line. Combining these simplification steps, we have (see, also, equation 63.3 of LL75) the,

Linearized Euler Equation

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

<math>~ - \frac{1}{\rho_0} \nabla P_1 \, . </math>

Ultimately, as emphasized in LL75, the condition that the linearized governing equations should be applicable to the propagation of sound waves is that the velocity of the fluid particles in the wave should be small compared with the velocity of sound, that is, <math>~|\vec{v}| \ll c_s</math>.

In a similar fashion, perturbing the variables in the barotropic equation of state leads to,

<math>~ P_0 + P_1 </math>	<math>~=</math>	<math>~ K (\rho_0 + \rho_1)^{\gamma_\mathrm{g}} </math>
	<math>~=</math>	<math>~ K\rho_0^{\gamma_\mathrm{g}} \biggl(1 + \frac{\rho_1}{\rho_0} \biggr)^{\gamma_\mathrm{g}} </math>
<math>~\Rightarrow~~~ P_1</math>	<math>~=</math>	<math>~ K\rho_0^{\gamma_\mathrm{g}} \biggl(1 + \frac{\rho_1}{\rho_0} \biggr)^{\gamma_\mathrm{g}} - K\rho_0^{\gamma_\mathrm{g}} </math>
	<math>~=</math>	<math>~ K\rho_0^{\gamma_\mathrm{g}} \biggl[1 + \gamma_\mathrm{g}\biggl(\frac{\rho_1}{\rho_0} \biggr) + \frac{\gamma_\mathrm{g}(\gamma_\mathrm{g}-1)}{2} \cancelto{\mathrm{small}}{\biggl(\frac{\rho_1}{\rho_0} \biggr)^2} + \cdots \biggr] - K\rho_0^{\gamma_\mathrm{g}} </math>
	<math>~\approx</math>	<math>~ \gamma_\mathrm{g} \biggl( \frac{P_0}{\rho_0} \biggr) \rho_1 \, . </math>

Hence, we have (see, also, equation 63.4 of LL75) the,

Linearized Equation of State

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

Summary

In summary, the following three linearized equations govern the time-dependent physical relationship between the three perturbation amplitudes <math>~P_1(\vec{r},t)</math>, <math>~\rho_1(\vec{r},t)</math> and <math>~\vec{v}(\vec{r},t)</math> in the context of sound waves:

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

Linearized
Euler Equation
<math> ~\frac{\partial \vec{v}}{\partial t} = - \frac{1}{\rho_0} \nabla P_1 \, , </math>

Linearized
Adiabatic Form of the
First Law of Thermodynamics

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

Wave Equation Derivation

It is customary to combine these three relations to obtain a single, second-order partial-differential equation in terms of (any) one of the perturbation amplitudes. We begin by using the third equation to replace <math>~P_1</math> in favor of <math>~\rho_1</math> in the second equation. This generates,

<math>~ \rho_0 \frac{\partial \vec{v}}{\partial t} + \biggl( \frac{dP}{d\rho} \biggr)_0 \nabla \rho_1 </math>

Taking the divergence of this equation gives,

<math>~ \rho_0 \frac{\partial}{\partial t}(\nabla\cdot \vec{v}) + \biggl( \frac{dP}{d\rho} \biggr)_0 \nabla^2 \rho_1 </math>

while taking the time derivative of the first (i.e., the linearized continuity) equation gives,

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

(Note that we have freely interchanged the order of the <math>~\nabla</math> and <math>~\partial/\partial t</math> operators because the spatial operator is not a function of time. Also, as before, quantities having a subscript "0" have been pulled outside of both operators because, in this discussion, they have no time- or spatial-dependence.) Finally, taking the difference between these last two relations produces the anticipated,

Wave Equation

<math>~ \frac{\partial^2 \rho_1}{\partial t^2} - c_s^2 \nabla^2 \rho_1 = 0 </math>

exhibiting the wave propagation speed,

<math>~ c_s = \sqrt{\biggl( \frac{dP}{d\rho} \biggr)_0} \, . </math>

As derived, this wave equation describes, from an Eulerian (as opposed to Lagrangian) perspective, how the density perturbation, <math>~\rho_1(\vec{r},t)</math>, varies with time at any coordinate position.

Alternatives

Utilizing the linearized adiabatic form of the first law of thermodynamics, we can trivially replace <math>~\rho_1</math> with <math>~P_1</math> in the above wave equation to obtain a

Wave Equation

<math>~ \frac{\partial^2 P_1}{\partial t^2} - c_s^2 \nabla^2 P_1 = 0 \, , </math>

that exhibits the same wave propagation speed but describes the variation of the pressure, rather than density, fluctuations. Alternatively, adopting the approach preferred by LL75, we can introduce the velocity potential by putting,

<math>\vec{v} = \nabla\phi \, ,</math>

in the linearized governing equations. After also replacing <math>~P_1</math> in favor of <math>~\rho_1</math> in the linearized Euler equation, we have,

<math>~\rho_0 \frac{\partial \nabla\phi}{\partial t} + \biggl( \frac{dP}{d\rho} \biggr)_0 \nabla \rho_1</math>	<math>~=</math>	<math>~0</math>
<math>~ \Rightarrow ~~~ \nabla \biggl[ \rho_0 \frac{\partial \phi}{\partial t} + \biggl( \frac{dP}{d\rho} \biggr)_0 \rho_1 \biggr] </math>	<math>~=</math>	<math>~0</math>
<math>~ \Rightarrow ~~~ \frac{\partial \phi}{\partial t} + \frac{1}{\rho_0}\biggl( \frac{dP}{d\rho} \biggr)_0 \rho_1 </math>	<math>~=</math>	<math>~0 \, .</math>

Taking the time-derivative of this expression gives,

<math>~\frac{1}{\rho_0} \frac{\partial \rho_1}{\partial t} + \biggl( \frac{dP}{d\rho} \biggr)_0^{-1} \frac{\partial^2 \phi}{\partial t^2} </math>

In addition, the linearized continuity equation becomes,

<math> \frac{1}{\rho_0} \frac{\partial \rho_1}{\partial t} + \nabla^2 \phi = 0 . </math>

Taking the difference between these last two expressions generates a (see, also, equation 63.7 of LL75),

Wave Equation

<math>~ \frac{\partial^2 \phi}{\partial t^2} - c_s^2 \nabla^2 \phi = 0 \, , </math>

that exhibits the same wave propagation speed but describes the variation of the velocity potential, rather than density or pressure fluctuations.

User:Tohline/SSC/Stability Eulerian Perspective

Contents

Stability of Spherically Symmetric Configurations (Eulerian Perspective)

Assembling the Key Relations

Governing Equations and Supplemental Relations

Perturbation then Linearization of Equations

Continuity Equation

Euler Equation

Summary

Wave Equation Derivation

Alternatives

See Also

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