Zero impact parameter weak deflection by two-power densities

Abstract

We obtain the trend of the zero impact parameter deflection for two-power densities ρ(R)=ρ₀ R⁻^α(R+1)^α⁻^β beyond α=2 by considering the simpler single-power density ρ=ρ₀ R⁻^α. In a previous study by de Leon and Vega, it was found that the zero impact parameter of two-power densities vanishes when α<2, and approaches a non-zero finite value at α=2 similar to the deflection due to a singular isothermal sphere. Here, we find that the zero impact parameter deflection beyond α=2 diverges. Increasing further the value of α makes this angle diverge faster. We also present here the deflection angle as a function of the impact parameter due to single-power densities with exponent values in 2<α<3. The function is also a power law that decays slower than the density function.