Editing SSC/Structure/Polytropes (section)

===Boundary Conditions===

Given that it is a <math>2^\mathrm{nd}</math>-order ODE, a solution of the Lane-Emden equation will require specification of two boundary conditions.  Based on our definition of the variable <math>~\Theta_H</math>, one obvious boundary condition is to demand that <math>~\Theta_H = 1</math> at the center <math>~(\xi=0)</math> of the configuration.  In astrophysically interesting structures, we also expect the first derivative of many physical variables to go smoothly to zero at the center of the configuration &#8212; see, for example, the radial behavior that was derived for {{Math/VAR_Pressure01}}, {{Math/VAR_Enthalpy01}}, and {{Math/VAR_NewtonianPotential01}} in a [[SSC/Structure/UniformDensity#Summary|uniform-density sphere]].  Hence, we will seek solutions to the Lane-Emden equation where <math>~d\Theta_H /d\xi = 0</math> at <math>~\xi=0</math> as well.