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Polytropic equations (Lane–Emden [LE] equations) are valuable because they offer a simple explanation for a star’s interior structure, interstellar matter, molecular clouds, and even spiral arms that can be calculated and used to estimate various physical parameters. Many analytical and numerical methods are used to solve the polytropic LE equation. The series expansion method played an essential role in many areas of science and has found application in many branches of physical science. This work uses the series expansion method to examine N-dimensional polytropes (i.e., slab, cylinder, and sphere). To solve LE-type equations, a computational method based on accelerated series expansion (ASE) is applied. We calculate several models for the N-dimensional polytropes. The numerical results show good agreement between the ASE and numerical and analytical models of the N-dimensional polytropes.

This paper presents new relativistic composite polytropic models for compact stars by simultaneously solving Einstein field equations with the polytropic state equation to simulate the spherically symmetric, static matter distribution. Using a non-uniform polytropic index, we get the Tolman–Oppenheimer–Volkoff equation for the relativistic composite polytrope (CTOV). To analyze the star's structure, we numerically solve the CTOV equation and compute the Emden and mass functions for various relativistic parameters and polytropic indices appropriate for neutron stars. The calculation results show that, as the relativistic parameter approaches zero, we recover the well-known Lane-Emden equation from the Newtonian theory of polytropic stars; thus, testing the computational code by comparing composite Newtonian models to those in the literature yields good agreement. We compute composite relativistic models for the neutron star candidates Cen X-3, SAXJ1808.4-3658, and PSR J1614-22304. We compare the findings with various existing models in the literature. Based on the accepted models for PSR J1614-22304 and Cen X-3, the star's core radius is predicted to be between 50 and 60% percent of its total radius, while we found that the radius of the core of star SAXJ1808.4-3658 is around 30% of the total radius. Our findings show that the neutron star structure may be approximated by a composite relativistic polytrope, resulting in masses and radii that are quite consistent with observation. [ABSTRACT FROM AUTHOR]