Editing Appendix/Ramblings/OriginOfPlanetaryNebulae (section)

===Commentary===

====Evolutionary Tracks====
When we embarked on this investigation, we thought that it would be difficult to quantitatively compare the evolution of our <math>(n_c, n_e) = (5, 1)</math> bipolytrope with more realistic stellar evolutionary tracks &#8212; as published, for example, by {{ Iben67full }} &#8212; because our toy model does not provide a mechanism for assessing the variation in time of the configuration's surface temperature.  In the preceding subsection of this discussion, we have demonstrated that a comparison of evolutions is possible if we focus on an examination of how the star's radius varies as the mass of its core steadily increases.  

Not surprisingly, we find that the ''trend'' is the same in our toy model as it is in {{ Iben67hereafter }}'s <math>1 M_\odot</math> model: &nbsp; as the core's mass monotonically increases, the star's radius monotonically increases as well.  But there appear to be two significant mismatches.  (1) In the earliest stage of its evolution, {{ Iben67hereafter }}'s model exhibits a steeper rise in the mass of the core for a given radius.  (2) If we set <math>\mu_e/\mu_c = 1/2</math> &#8212; which is the ''expected'' value when considering the fusion of hydrogen into helium &#8212; overall, our toy model dumps significantly more mass into the core than does {{ Iben67hereafter }}'s evolutionary model. Dropping the mean-molecular-weight ratio from <math>1 / 2</math> to <math>1 / 4</math> provides a better match, but it is difficult to come up with an astrophysical argument that would justify such a low ratio. 

Given that the ''trend'' matches, we will consider this a win!  Hopefully a more quantitative match will be obtained when we switch to a bipolytrope that has an isothermal <math>(n_c = \infty)</math> core, instead of an <math>n_c = 5</math> core.

====Dynamical Instability====

We have placed one solid-red circular marker on each of the evolutionary tracks that have been drawn from our toy model.  In both cases, this marks the point along the track &#8212; at a radius less than <math>4 R_\odot</math> &#8212; where the <math>(n_c, n_e) = (5, 1)</math> model becomes dynamically unstable.  It is tempting to suggest that this indicates that {{ Iben67hereafter }}'s model should also become dynamically unstable before expanding to a radius greater than <math>4 R_\odot</math>.  Let's pursue building a bipolytrope that has an isothermal core and see if that system strengthens this argument.