The Lane–Emden equation is , where dimensionless variables and relate to the density and radius (see Details). The equation is used in the study of polytropic gaseous spheres and the modeling of stars. Solutions of this second-order differential equation use polytropes to relate pressure and density as a function of , the radial coordinate measured from the center of the polytropic sphere (the star); and have the polytropic relation , where is the polytropic constant and is the polytropic index that you can choose with the control.

Here the equation is solved for the polytropic index value for the dimensionless function . Click the checkbox to see all solutions from 0 to the currently selected value. The first root of each solution, derivatives at that root value, and the critical/mean density ratio are also displayed.