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

We study the weak-field deflection of light by mass distributions described by two-power-law densities $\rho(R)=\rho_0 R^{-\alpha}(R+1)^{\beta-\alpha}$, 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 agrees 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.