Weak gravitational deflection by two-power-law densities using the Gauss-Bonnet theorem

We study the weak deflection of light by nonrelativistic mass distributions described by two-power-law densities rho(R) = rho R-0(-alpha)(R +1)(beta-alpha), where alpha and beta are non-negative integers. New analytic expressions of deflection angles are obtained via the application of the Gauss-Bonnet theorem to a chosen surface on the optical manifold. Some of the well-known models of this two-power-law form are the Navarro-Frenk-White (NFW) model (alpha, beta) = (1, 3), Hernquist (1,4), Jaffe (2,4), and the singular isothermal sphere (2,2). The calculated deflection angles for Hernquist and NFW agree with that of Keeton and Bartelmann, respectively. The limiting values of these deflection angles (at zero or infinite impact parameter) are either vanishing or similar to the deflection due to a singular isothermal sphere. We show that these behaviors can be attributed to the topological properties of the optical manifold, thus extending the pioneering insight of Werner and Gibbons to a broader class of mass densities.