Stars are often modeled as self-gravitating, spherically symmetric polytropic gas spheres. The Lane-Emden equation describes the density profile of such a system, with different systems being represen..
Stars are often modeled as self-gravitating, spherically symmetric polytropic gas spheres. The Lane-Emden equation describes the density profile of such a system, with different systems being represented by their polytropic index. Although analytical solutions exist for polytropic indices 0, 1 and 5, it is still open to find the analytical solution for other polytropic indices. However, numerical methods yield approximate solutions that offer valuable insights into the nature of stars. In this project, we begin by exploring known analytic solutions to the equation. Then we implement numerical methods to find approximate solutions for various polytropic indices using Python. To this end we utilize Euler’s Method and a 4th order Runge-Kutta Method (RKM) which are commonly used. In our computation, the RKM produced a more accurate solution at a higher step size than Euler’s Method, while being more computationally expensive. Euler’s method provided a less complex way to estimate solutions, trading a higher iteration requirement and lower accuracy with ease-of-use.
