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 relevant Eigenvalue problem is,
<math>
\frac{d^2x}{d\chi_0^2} + \biggl[\frac{4}{\chi_0} - \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{P_0}{P_c}\biggr)^{-1} \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \biggr] \frac{dx}{d\chi_0} + \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{P_0}{P_c}\biggr)^{-1} \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>
where the dimensionless radius,
<math>
\chi_0 \equiv \frac{r_0}{R} ,
</math>
the characteristic time for dynamical oscillations in spherically symmetric configurations (SSC) is,
<math>
\tau_\mathrm{SSC} \equiv \biggl[ \frac{R^2 \rho_c}{P_c} \biggr]^{1/2} ,
</math>
and the characteristic gravitational acceleration is,
<math>
g_\mathrm{SSC} \equiv \frac{P_c}{R \rho_c} .
</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{dx}{d\chi_0} = 2b\chi_0; ~~~~ \frac{d^2 x}{d\chi_0^2} = 2b; </math>
<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[ \biggl(\frac{b}{a}\biggr) A_1 - 14b \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 = 20\gamma_\mathrm{g} -8
</math>
<math>
\Rightarrow ~~~~~ \omega_1^2 = \frac{2}{3}\biggl( 4\pi G\rho_c \biggr) (5\gamma_\mathrm{g} -2)
</math>
and, to within an arbitrary normalization factor,
<math> x_1 = 1 - \frac{7}{5}\chi_0^2 . </math>
n = 1 Polytrope
Setup
From our derived Structure of an n = 1 polytrope, 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{\pi G}{4}\biggl( \frac{M^2}{R^4} \biggr) </math> ,
and
<math>\rho_c = \frac{\pi M}{4 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{R^3 }{GM} \biggr]^{1/2} =
\biggl[ \frac{\pi}{4 G\rho_c} \biggr]^{1/2},
</math>
and,
<math>
g_\mathrm{SSC} = \frac{P_c}{R \rho_c} = \biggl( \frac{GM}{R^2} \biggr) .
</math>
The required functions are,
- Density:
<math>\frac{\rho_0(\chi_0)}{\rho_c} = \frac{\sin(\pi\chi_0)}{\pi\chi_0} </math> ;
- Pressure:
<math>\frac{P_0(\chi_0)}{P_c} = \biggl[ \frac{\sin(\pi\chi_0)}{\pi\chi_0} \biggr]^2 </math> ;
- Gravitational acceleration:
<math>
\frac{g_0(r_0)}{g_\mathrm{SSC}} = \frac{1}{\chi_0^2} \biggl[ \frac{M_r(\chi_0)}{M}\biggr] =
\frac{1}{\pi \chi_0^2} [ \sin (\pi\chi_0 ) - \pi\chi_0 \cos (\pi\chi_0 ) ].
</math>
So our desired Eigenvalues and Eigenvectors will be solutions to the following ODE:
<math>
\frac{d^2x}{d\chi_0^2} + \biggl[\frac{4}{\chi_0} - \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{P_0}{P_c}\biggr)^{-1} \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \biggr] \frac{dx}{d\chi_0} + \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{P_0}{P_c}\biggr)^{-1} \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>
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