Editing SSC/Stability/BiPolytropes/RedGiantToPN (section)

==Abstract==

The analysis presented by {{ EFC98 }} is essentially an analysis of the <math>q - v</math> diagram.  We can determine analytically at what value of <math>\xi_i</math> the core-to-total mass ratio reaches a maximum (<math>\nu_\mathrm{max}</math>) for various values of <math>\mu_e/\mu_c \le 1/3</math>.  For example, <math>(\mu_e/\mu_c, \xi_i, \nu_\mathrm{max}) = (\tfrac{1}{4}, 4.9379256, 0.139270157)</math>.  Our LAWE analysis shows that '''none''' of these turning points is associated with the onset of a dynamical instability.

On the other hand, our LAWE analysis '''does''' identify a marginally unstable equilibrium configuration along every sequence; even sequences with <math>\tfrac{1}{3} \le \mu_e/\mu_c \le 1</math>.  [[SSC/Stability/BiPolytropes/Pt3#Eigenvectors_for_Marginally_Unstable_Models_with_(γc,_γe)_=_(6/5,_2)|For example]], <math>(\mu_e/\mu_c, \xi_i, \nu) = (1, 1.6686460157, 0.497747626)</math>.