Weak deflection angle of a dirty black hole

In this paper, we present the weak deflection angle in a Schwarzschild black hole of mass $m$ surrounded by the dark matter of mass $M$ and thickness $\Delta r_{s}$. The Gauss-Bonnet theorem, formulated for asymptotic spacetimes, is found to be ill-behaved in the third-order of $1/\Delta r_{s}$ for very large $\Delta r_{s}$. Using the finite-distance for the radial locations of the source and the receiver, we derived the expression for the weak deflection angle up to the third-order of $1/\Delta r_{s}$ using Ishihara (\textit{et al.}) method. The result showed that the required dark matter thickness is $\sim2\sqrt{3mM}$ for the deviations in the weak deflection angle to occur. Such thickness requirement is better by a factor of 2 as compared to the deviations in the shadow radius ($\sim\sqrt{3mM}$). It implies that the use of the weak deflection angle in detecting dark matter effects in one's galaxy is better than using any deviations in the shadow radius.