His theories also suggested that stars have a boundary at a finite radius. This work, in conjunction with Jonathan Homer Lane, became known as the Lane-Emden equation. The Lane-Emden equation can be described as "a second-order ordinary differential equation that applies to polytropic profiles in density". The Lane-Emden equations were later studied by Ralph H. Fowler who developed a new set of solutions for different values of n and for all types of boundary conditions. These became known as Emden-Fowler-type differential equations. Emden's convective stellar models have been supplanted by radiative theory.