Poisson type relativistic perfect fluid spheres

Static spherically symmetric solutions of the Einstein's field equations in isotropic coordinates representing perfect fluid matter distributions from Newtonian potential-density pairs are investigated. The approach is illustrated with three simple examples based on the potential-density pairs corresponding to a harmonic oscillator (homogeneous sphere), the well-known Plummer model and a massive spherical dark matter halo model with a logarithmic potential. Moreover, the geodesic circular motion of test particles around such structures is studied. The stability of the orbits against radial perturbations is also analyzed using an extension of the Rayleigh criteria of stability of a fluid in rest in a gravitational field. The models considered satisfy all the energy conditions.