Editing SSC/Structure/BiPolytropes (section)

===Polytropic Core===
Using the notation established in Table 1, if the core obeys a polytropic equation of state, the variable <math>\xi</math> will denote the dimensionless radial coordinate through the core and the relevant solution is a function, <math>\theta(\xi)</math>, whose value goes to <math>1</math> and whose first derivative, <math>d\theta/d\xi</math>, goes to <math>0</math> at <math>\xi = 0</math>.  Then, given a value of the central density, <math>\rho_0</math>, the density throughout the core is,
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<math>\rho(\xi) = \rho_0 \theta(\xi)^{n_c}</math> ;
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and, given a value of the polytropic constant in the core, <math>K_c</math>, the pressure throughout the core is,
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<math>P(\xi) = K_c \rho_0^{1+1/n_c} \theta(\xi)^{n_c + 1}</math>.
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Likewise, given <math>\rho_0</math> and <math>K_c</math>, the radial coordinate <math>r</math> (in dimensional rather than dimensionless units) and the mass enclosed within this radius, <math>M_r</math>, are given by the bottom two expressions shown in the <math>2^\mathrm{nd}</math> column of Table 1.  

The structure of an [[SSC/Structure/Polytropes#Lane-Emden_Equation|isolated polytrope]] would be described by following the function <math>\theta(\xi)</math> all the way out to the surface, that is, to the radial location <math>\xi_\mathrm{surf}</math> where <math>\theta(\xi)</math> first drops to zero.  (Analytic solutions of this type are presented elsewhere for [[SSC/Structure/Polytropes#n_.3D_0_Polytrope|<math>n = 0</math>]], [[SSC/Structure/Polytropes#n_.3D_1_Polytrope|<math>n = 1</math>]], and [[SSC/Structure/Polytropes#n_.3D_5_Polytrope|<math>n = 5</math>]].)  In constructing a bipolytrope, we will instead follow <math>\theta(\xi)</math> out to a radius <math>\xi_i < \xi_\mathrm{surf}</math>, then build an envelope whose inner radius &#8212; or ''base'' &#8212; is at the ''interface'' radius, <math>r_i</math>, that corresponds to <math>\xi_i</math>.  For any choice of the pair of polytropic indexes, <math>n_c</math> and <math>n_e</math>, a series of bipolytropes can then be constructed by choosing a variety of different interface radii.