Qualitative Properties of the Solutions to the Lane–Emden Equation in the Cylindrical Setup

We analyze the Lane–Emden equations in the cylindrical framework. Although the explicit forms of the solutions (which are also called polytropes) are not known, we identify some of their qualitative properties. In particular, possible critical points and zeros of the polytropes are investigated and discussed, leading to possible improvements in the approximation methods which are currently employed. The cases when the critical parameter is odd and even are separately analyzed. Furthermore, we propose a technique to evaluate the distance between a pair of polytropes in small intervals.