Spherically Symmetric Configurations (Stability — Part III)

Suppose we now want to study the stability of one of the spherically symmetric, equilibrium structures that have been derived elsewhere. The identified set of simplified, time-dependent governing equations will tell us how the configuration will respond to an applied radial (i.e., spherically symmetric) perturbation that pushes the configuration slightly away from its initial equilibrium state.

The Eigenvalue Problem

As has been derived in an accompanying discussion, the second-order ODE that defines the Eigenvalue problem is,

<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>

where, <math>P_0(r_0)</math> and <math>\rho_0(r_0)</math> are the pressure and density distributions in the unperturbed initial equilibrium model and the gravitational acceleration at each radial location in the unperturbed model is,

<math> g_0(r_0) \equiv \frac{GM_r(r_0)}{r_0^2} = - \frac{1}{\rho_0} \frac{dP_0}{dr_0} . </math>

Let's write the governing ODE and the key physical variables as dimensionless expressions. First, multiply through by <math>R^2</math> and define the dimensionless radius as,

<math> \chi_0 \equiv \frac{r_0}{R} </math>

to obtain,

<math> \frac{d^2x}{d\chi_0^2} + \biggl[\frac{4}{\chi_0} - \biggl(\frac{R g_0 \rho_0}{P_0}\biggr) \biggr] \frac{dx}{d\chi_0} + \biggl(\frac{R^2 \rho_0}{\gamma_\mathrm{g} P_0} \biggr)\biggl[\omega^2 + (4 - 3\gamma_\mathrm{g})\frac{g_0}{R \chi_0} \biggr] x = 0 . </math>

Now normalize <math>P_0</math> to <math>P_c</math> and <math>\rho_0</math> to <math>\rho_c</math> to obtain,

<math> \frac{d^2x}{d\chi_0^2} + \biggl[\frac{4}{\chi_0} - \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{P_c}{P_0}\biggr) \biggl(\frac{R g_0 \rho_c}{P_c}\biggr) \biggr] \frac{dx}{d\chi_0} + \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{P_c}{P_0}\biggr) \biggl(\frac{R^2 \rho_c}{\gamma_\mathrm{g} P_c} \biggr)\biggl[\omega^2 + (4 - 3\gamma_\mathrm{g})\frac{g_0}{R \chi_0} \biggr] x = 0 . </math>

The characteristic time for dynamical oscillations in spherically symmetric configurations (SSC) appears to be,

<math> \tau_\mathrm{SSC} \equiv \biggl[ \frac{R^2 \rho_c}{P_c} \biggr]^{1/2} , </math>

and the characteristic gravitational acceleration appears to be,

<math> g_\mathrm{SSC} \equiv \frac{P_c}{R \rho_c} . </math>

So we can write,

<math> \frac{d^2x}{d\chi_0^2} + \biggl[\frac{4}{\chi_0} - \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{P_c}{P_0}\biggr) \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \biggr] \frac{dx}{d\chi_0} + \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{P_c}{P_0}\biggr) \biggl(\frac{1}{\gamma_\mathrm{g}} \biggr)\biggl[\tau_\mathrm{SSC}^2 \omega^2 + (4 - 3\gamma_\mathrm{g})\biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \frac{1}{\chi_0} \biggr] x = 0 . </math>

Uniform-Density Configuration

Setup

From our derived Structure of a uniform-density sphere, in terms of the configuration's radius <math>R</math> and mass <math>M</math>, the central pressure and density are, respectively,

<math>P_c = \frac{3G}{8\pi}\biggl( \frac{M^2}{R^4} \biggr) </math> ,

and

<math>\rho_c = \frac{3M}{4\pi R^3} </math> .

Hence the characteristic time and acceleration are, respectively,

<math> \tau_\mathrm{SSC} = \biggl[ \frac{R^2 \rho_c}{P_c} \biggr]^{1/2} = \biggl[ \frac{2R^3 }{GM} \biggr]^{1/2} = \biggl[ \frac{3}{2\pi G\rho_c} \biggr]^{1/2}, </math>

and,

<math> g_\mathrm{SSC} = \frac{P_c}{R \rho_c} = \biggl( \frac{GM}{2R^2} \biggr) . </math>

The required functions are,

Density:

<math>\frac{\rho_0(r_0)}{\rho_c} = 1 </math> ;

Pressure:

<math>\frac{P_0(r_0)}{P_c} = 1 - \chi_0^2 </math> ;

Gravitational acceleration:

<math> \frac{g_0(r_0)}{g_\mathrm{SSC}} = 2\chi_0 . </math>

So our desired Eigenvalues and Eigenvectors will be solutions to the following ODE:

<math> \frac{1}{(1 - \chi_0^2)} \biggl\{ (1 - \chi_0^2) \frac{d^2x}{d\chi_0^2} + \frac{4}{\chi_0}\biggl[1 - \frac{3}{2}\chi_0^2 \biggr] \frac{dx}{d\chi_0} + \frac{1}{\gamma_\mathrm{g}} \biggl[\tau_\mathrm{SSC}^2 \omega^2 + 2 (4 - 3\gamma_\mathrm{g}) \biggr] x \biggr\} = 0 . </math>

First few lowest-order modes

Mode 0:

<math>x_0 = \mathrm{constant}</math>, in which case,

<math> \omega_0^2 = - 2(4 - 3\gamma_\mathrm{g})\biggl[ \frac{2\pi G\rho_c}{3} \biggr] = 4\pi G \rho_c \biggl[ \gamma_\mathrm{g}- \frac{4}{3} \biggr] </math>

Mode 1:

<math>x_1 = a + b\chi_0^2</math>, in which case,

<math> \frac{1}{(1 - \chi_0^2)} \biggl\{ 2b (1 - \chi_0^2) + 8b \biggl[1 - \frac{3}{2}\chi_0^2 \biggr] + A_1 \biggl(1 - \frac{b}{a}\chi_0^2 \biggr) \biggr\} = 0 , </math>

where,

<math> A_1 \equiv \frac{a}{\gamma_\mathrm{g}}\biggl[ \biggl( \frac{3}{2\pi G\rho_c} \biggr) \omega_1^2+ 2(4 - 3\gamma_\mathrm{g}) \biggr] . </math>

Therefore,

<math> (A_1 + 10b) - \biggl[ 14b + \biggl(\frac{b}{a}\biggr) A_1 \biggr] \chi_0^2 = 0 , </math>

<math> \Rightarrow ~~~~~ A_1 = - 10b ~~~~~\mathrm{and} ~~~~~ A_1 = -14a </math>

<math> \Rightarrow ~~~~~ \frac{b}{a} = \frac{7}{5} ~~~~~\mathrm{and} ~~~~~ \frac{A_1}{a} = -14 = \frac{1}{\gamma_\mathrm{g}}\biggl[ \biggl( \frac{3}{2\pi G\rho_c} \biggr) \omega_1^2+ 2(4 - 3\gamma_\mathrm{g}) \biggr] .

</math>

Hence,

<math> \biggl( \frac{3}{2\pi G\rho_c} \biggr) \omega_1^2 = -8 (\gamma_\mathrm{g} +1) </math>

User:Tohline/SSC/UniformDensity

Contents

Spherically Symmetric Configurations (Stability — Part III)

The Eigenvalue Problem

Uniform-Density Configuration

Setup

First few lowest-order modes

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