A brief analysis of self-gravitating polytropic models with a non-zero cosmological constant

Context. We investigate the equilibrium and stability of polytropic spheres in the presence of a non-zero cosmological constant. Aims. We solve the Newtonian gravitational equilibrium equation for a system with a polytropic equation of state of the matter P = Kρ^γ introducing a non-zero cosmological constant Λ. Methods. We consider the cases of n = 1, 1.5, 3 and construct series of solutions with a fixed value of Λ. For each value of n, the non-dimensional equilibrium equation has a family of solutions, instead of the unique solution of the Lane-Emden equation at Λ = 0. Results. The equilibrium state exists only for central densities ρ0 higher than the critical value ρc. There are no static solutions at ρ0 < ρc. We investigate the stability of equilibrium solutions in the presence of a non-zero Λ and show that dark energy reduces the dynamic stability of the configuration. We apply our results to the analysis of the properties of the equilibrium states of clusters of galaxies in the present universe with non-zero Λ.