Editing SSC/Stability/UniformDensity (section)

==Stability==

The [[SSC/Perturbations#2ndOrderODE|Adiabatic Wave Equation]] that defines this [[SSC/Perturbations#The_Eigenvalue_Problem|eigenvalue problem]] has been derived from the fundamental set of nonlinear [[PGE#Principal_Governing_Equations|Principal Governing Equations]] by assuming that, for example, the radial position, <math>r(m,t)</math>, at any time, <math>t</math>, and of each mass shell throughout our spherical configuration can be described by the expression,
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<math>r(m,t) = r_0(m) [ 1 + x(m) e^{i\omega t} ] \, ,</math>
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where, the fractional displacement, <math>|x| \ll 1</math>.  Switching to {{ Sterne37hereafter }}'s variable notation, this should be written as,
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<math>\xi(x,t) = \xi_0(x) [ 1 + A\xi_1(x) e^{i n t} ] \, ,</math>
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with the presumption that the coefficient, <math>|A| \ll 1</math>, and the understanding that, in {{ Sterne37hereafter}}, the variable, <math>x</math>, is used to identify individual mass shells.  Specifically, given <math>R</math> and <math>\bar\rho</math>,
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<math>m \equiv M_r = \frac{4}{3}\pi \xi_0^3 \bar\rho = \frac{4}{3}\pi (R x)^3 \bar\rho </math>
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <math>\Rightarrow</math> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
<math>x = \biggl( \frac{3m}{4\pi R^3 \bar\rho} \biggr)^{1/3} \, .</math>
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The [[#Sterne.27s_General_Solution|general solution of this eigenvalue problem]] presented by {{ Sterne37 }} describes mathematically how a self-gravitating, uniform-density configuration will vibrate if perturbed away from its equilibrium state; the oscillatory behavior associated with each pure radial mode, <math>j</math> &#8212; among an infinite number of possible modes &#8212; is fully defined by the polynomial expression for the eigenvector, <math>\xi_1(x)</math>, and the corresponding value of the square of the eigenfrequency, <math>n^2</math>. 


If, for any mode, <math>j</math>,  the ''square'' of the derived eigenfrequency, <math>n^2</math>, is positive, then the eigenfrequency itself will be a real number &#8212; specifically,
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<math>n = \pm \sqrt{|n^2|} \, .</math>
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As a result, the radial location of every mass shell will vary sinusoidally in time according to the expression,
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<math>\frac{\xi(x,t)}{\xi_0} - 1 \propto e^{\pm i \sqrt{|n^2|} t} \, .</math>
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If, on the other hand, <math>n^2</math>, is negative, then the eigenfrequency will be an imaginary number &#8212; specifically,
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<math>n = \pm i \sqrt{|n^2|} \, .</math>
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As a result, the radial location of every mass shell will grow (or damp) exponentially in time according to the expression,
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<math>\frac{\xi(x,t)}{\xi_0} - 1 \propto e^{\pm \sqrt{|n^2|} t} \, .</math>
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This latter condition is the mark of a dynamically unstable system.  It is in this manner that the solution to an eigenvalue problem can provide critical information regarding the relative stability of equilibrium configurations.

For any given mode number, <math>j</math>, then, the critical configuration separating stable from unstable systems occurs when the dimensionless eigenfrequency is zero.  Therefore, the critical state occurs when,
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<math>0 = \frac{n_\mathrm{crit}^2}{4\pi G \bar\rho}</math>
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<math>=</math>
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<math>\biggl(1 + \frac{\mathfrak{F}}{6} \biggr)\gamma_\mathrm{crit} -\frac{4}{3}  </math>
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<math>\Rightarrow ~~~~~ \gamma_\mathrm{crit}(j)</math>
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<math>=</math>
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<math>\frac{4}{3}\biggl(1 + \frac{\mathfrak{F}}{6} \biggr)^{-1}  </math>
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&nbsp;
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<math>=</math>
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<math>\frac{4}{3}\biggl[1 + \frac{2j(2j+5)}{6} \biggr]^{-1}  </math>
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&nbsp;
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<math>=</math>
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<math>\frac{4}{3 + j(2j+5)}  \, . </math>
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The plot titled, "Critical Adiabatic Index," that is [[#Properties_of_Eigenfunction_Solutions| presented above]] shows graphically how <math>\gamma_\mathrm{crit}</math> varies with mode number over the range of mode numbers, <math>0 \le j \le 11</math>.  All modes are stable as long as <math>\gamma > 4/3</math>.  As the adiabatic index is decreased below this value, the lowest order mode, <math>j = 0</math>, becomes unstable, first; then successively higher order modes become unstable at smaller and smaller values of the index.  A very similar explanation and enunciation of {{ Sterne37hereafter}}'s derived results regarding the stability of uniform-density spheres appears at the bottom of p. 338 of {{ Ledoux46full }}.  The relevant paragraph from {{ Ledoux46 }} follows:

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Paragraph extracted<sup>&dagger;</sup> from the bottom of p. 338 of &hellip;<br />
{{ Ledoux46figure }}
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<!-- [[File:Ledoux1946OnSterne01.png|700px|center|Ledoux (1946)]] -->
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"Another point brought out clearly by Sterne's analysis is that, while the fundamental mode becomes unstable for <math>\Gamma < \tfrac{4}{3}\, ,</math> the higher harmonics continue to be stable.  Indeed, as we may directly verify from equation (19), the first harmonic becomes unstable only if <math>\Gamma < \tfrac{2}{5}</math> and the second one if <math>\Gamma < \tfrac{4}{21} \, .</math>"
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<sup>&dagger;</sup>Our function, <math>~\gamma_\mathrm{crit}(j)</math>, is effectively the expression to which Ledoux is referring when he says, "&hellip; directly verify from equation (19) &hellip;"
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