Editing SSC/Structure/BiPolytropes (section)

===Polytropic Envelope===
Again following the notation of Table 1, we will use <math>\eta</math> to identify the dimensionless radial coordinate through the envelope, and the relevant solution of the Lane-Emden equation will be the function, <math>\phi(\eta)</math>.  The ''particular'' solution that we seek for the envelope will differ from the solution obtained in the core not only because the governing polytropic index is different but also because the envelope solution will be constrained by different boundary conditions:  The value of the function <math>\phi(\eta)</math> as well as its first derivative, <math>d\phi/d\eta</math>, must be specified at some radial location within the envelope.  Because the envelope does not extend all the way to the center of the structure, it makes more sense to choose a solution in which <math>\phi</math> is set to unity at the inner edge of the envelope &#8212; that is, at the radial location <math>\eta_i</math> that corresponds to <math>r_i</math> &#8212; rather than at <math>\eta = 0</math>.  By setting <math>\phi = 1</math> at the base of the envelope, it should be clear from the expression, 
<div align="center">
<math>
\rho = \rho_e \phi^{n_e} \, ,
</math>
</div>
(taken from the <math>3^\mathrm{rd}</math> column of Table 1) that <math>\rho_e</math> represents the value of the gas density at the base of the envelope. The value of <math>\rho_e</math> can be obtained from knowledge of the gas density at the outer edge of the core (''i.e.'', at <math>r_i</math>) combined with the specified molecular-weight jump condition at the interface. Looking ahead at the first interface condition for a polytropic core catalogued in Table 2, we see more specifically that, after setting <math>\phi_i = 1</math>,
<div align="center">
<math>
\rho_e = \rho_0 \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{n_c}_i \, .
</math>
</div> 

As we will show in connection with Table 3, the value of <math>d\phi/d\eta</math> at the base of the envelope (''i.e.'', at <math>r_i</math>) also will be set by the properties of the core  at the interface &#8212; specifically, by the values of <math>(d\theta/d\xi)_i</math> and <math>\theta_i</math>.