On Self-Gravitating Polytropic Cylinders and Slabs

In this review paper the 2-D Lane-Emden equation (LEEq) model of a self-gravitating gas distribution in the form of an infinitely long cylinder shaped polytrope of finite radius is obtained and its basic radial properties are outlined. Similarly reviewed is the derivation of the 1-D LEEq model of an infinitely wide planar polytrope of finite thickness and its basic properties across thickness are discussed. These two polytropes are solved numerically along with the 3-D models for comparison using the 2 nd order Euler-Richardson method (ERM) and their index based parameters are determined. The Python script used in these computations has been shown to be not only fast but is capable of matching fourth order performance. However, these models are found to have finite radii for all polytropic indices unlike the restricted spherical analogs and have astrophysical applications. Distortion due to rotation in polytropic rings has also been computed using ERM.