User:Tohline/SSC/UniformDensity
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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
From our derived Structure of a uniform-density sphere, in terms of the configuration's radius <math>R</math> and mass <math>R</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> ;
and the required functions are,
- Mass:
- Given the density, <math>\rho_c</math>, and the radius, <math>R</math>, of the configuration, the total mass is,
<math>M = \frac{4\pi}{3} \rho_c R^3 </math> ;
- and, expressed as a function of <math>M</math>, the mass that lies interior to radius <math>r</math> is,
<math>\frac{M_r}{M} = \biggl(\frac{r}{R} \biggr)^3</math> .
- Pressure:
- Given values for the pair of model parameters <math>( \rho_c , R )</math>, or <math>( M , R )</math>, or <math>( \rho_c , M )</math>, the central pressure of the configuration is,
<math>P_c = \frac{2\pi G}{3} \rho_c^2 R^2 = \frac{3G}{8\pi}\biggl( \frac{M^2}{R^4} \biggr) = \biggl[ \frac{\pi}{6} G^3 \rho_c^4 M^2 \biggr]^{1/3}</math> ;
- and, expressed in terms of the central pressure <math>P_c</math>, the variation with radius of the pressure is,
<math>P(r) = P_c \biggl[ 1 -\biggl(\frac{r}{R} \biggr)^2 \biggr]</math> .
- Enthalpy:
- Throughout the configuration, the enthalpy is given by the relation,
<math>H(r) = \frac{P(r)}{ \rho_c} = \frac{GM}{2R} \biggl[ 1 -\biggl(\frac{r}{R} \biggr)^2 \biggr]</math> .
- Gravitational potential:
- Throughout the configuration — that is, for all <math>r \leq R</math> — the gravitational potential is given by the relation,
<math>\Phi_\mathrm{surf} - \Phi(r) = H(r) = \frac{G M}{2R} \biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] </math> .
- Outside of this spherical configuration— that is, for all <math>r \geq R</math> — the potential should behave like a point mass potential, that is,
<math>\Phi(r) = - \frac{GM}{r} </math> .
- Matching these two expressions at the surface of the configuration, that is, setting <math>\Phi_\mathrm{surf} = - GM/R</math>, we have what is generally considered the properly normalized prescription for the gravitational potential inside a uniform-density, spherically symmetric configuration:
<math>\Phi(r) = - \frac{G M}{R} \biggl\{ 1 + \frac{1}{2}\biggl[ 1- \biggl(\frac{r}{R} \biggr)^2 \biggr] \biggr\} = - \frac{3G M}{2R} \biggl[ 1 - \frac{1}{3} \biggl(\frac{r}{R} \biggr)^2 \biggr] </math> .
- Mass-Radius relationship:
- We see that, for a given value of <math>\rho_c</math>, the relationship between the configuration's total mass and radius is,
<math>M \propto R^3 ~~~~~\mathrm{or}~~~~~R \propto M^{1/3} </math> .
- Central- to Mean-Density Ratio:
- Because this is a uniform-density structure, the ratio of its central density to its mean density is unity, that is,
<math>\frac{\rho_c}{\bar{\rho}} = 1 </math> .
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