Editing SSC/Stability/InstabilityOnsetOverview (section)

===Isolated Polytropes===
Given a value of the polytropic index, <math>n</math>, the internal structure of an isolated polytrope is provided via the function, <math>\Theta_H(\xi) \equiv (\rho/\rho_c)^{1/n}</math>, which is a solution of the,
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<font color="maroon"><b>Polytropic Lane-Emden Equation</b></font> <p></p>
{{ Math/EQ_SSLaneEmden01 }}
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subject to the boundary conditions, <math>\Theta_H = 1</math> and <math>d\Theta_H/d\xi = 0</math> at <math>\xi = 0</math>.  In an [[SSC/Structure/Polytropes#Polytropic_Spheres|accompanying chapter]], we have reviewed what the structural properties are of polytropes that have a range of polytropic indexes.  Our emphasis has been on systems (n = 0, 1, and 5) for which the Lane-Emden equation can be solved analytically, but we also have discussed systems of astrophysical interest (n = 2.5, 3.0, 3.5, and 6) whose structural properties can only be described in terms of numerical solutions of this governing equation.  In isolation, systems having <math>n \ge 5</math> extend to infinity &#8212; as does the isothermal sphere discussed above.  But systems with <math>0 \le n < 5</math> have finite radii, that is, the function, <math>\Theta_H(\xi)</math>, naturally drops to zero at a radial-coordinate location, <math>\xi = \xi_\mathrm{surf} < \infty</math>.