Editing SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt3 (section)

====Early Thoughts====
Notice that, in each frame of the [[#OverlapPlots|above comparison figure]], the "Virial" curve intersects and crosses the "Stahler" curve at two locations.  In each plot these two crossing points are identified by filled black circles and, in each plot, the crossing point that lies farthest along the curve &#8212; again, starting from near the origin and moving around the curve in a counter-clockwise direction &#8212; is associated with the equilibrium model on the Stahler curve that is defined by the same value of <math>~\tilde\xi</math> that was used to define the coefficients of the virial theorem mass-radius relation.  This is not surprising, as the virial theorem ''should'' be precisely satisfied by every one of the equilibrium models along the Stahler sequence, as long as the value of <math>~\tilde\xi</math> that is used to define the coefficients of the free-energy function and, in turn, the virial theorem mass-radius relation is identical to the value of <math>~\tilde\xi</math> that defines the truncation radius of the detailed force-balance model.  For example, the "Virial" curve that appears in the top-right panel of the comparison figure &#8212; a panel whose title includes the notation, <math>~\xi = 3</math> &#8212; intersects the "Stahler" curve at <math>~\mathcal{Y}_\mathrm{max}</math>, that is, at the location of the detailed force-balance model that, as previously explained, has a truncation radius, <math>~\tilde\xi = 3</math>.  

It is not (yet) clear to us what physical significance should be ascribed to the model along the Stahler sequence that is identified by the ''second'' crossing of the "Virial" curve, given that the value of <math>~\tilde\xi</math> associated with the truncation radius of this ''second'' detailed force-balance model is not the same as the value of <math>~\tilde\xi</math> that was used to define the coefficients of the "Virial" curve.  We note that, at least for the range of values of <math>~\tilde\xi</math> sampled in the above figure, this second crossing point seems to hover around the same limited segment of the Stahler sequence.  

By direct analogy with [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|discussions of Bonnor-Ebert spheres]], the "maximum mass" model associated with <math>~\mathcal{Y}_\mathrm{max}</math> along the Stahler mass-radius relation has important physical significance in astrophysics.  For a given applied external pressure, however, no models exist above some limiting mass &#8212; identified, here, by <math>~\mathcal{Y}_\mathrm{max}</math>.