Gravitational lensing and energy emission of a black hole immersed in dark matter and plasma

In this study, we investigate the influence of plasma on the deflection angle and energy emission rate of a rotating black hole immersed in a dark matter medium. Using several homogeneous and inhomogeneous plasma profiles, we find that increasing the plasma frequency enhances the deflection angle in homogeneous plasma, but reduces it in inhomogeneous plasma. For rapidly rotating black holes, the overall deflection angle remains greater in homogeneous plasma versus inhomogeneous, whereas for slower spins, the opposite trend occurs. We also compare the shadow observable. By comparing our theoretical results with the observed properties of M87* and Sgr A*, we obtain constraints on key black hole parameters. Regarding the energy emission rate, increasing plasma frequency leads to a decrease in emission, with the highest rate occurring in homogeneous plasma and the lowest in inhomogeneous plasma. This ordering reflects how the plasma distribution modifies the shadow radius, which determines the effective emission area. Our findings demonstrate that the combined effects of plasma and dark matter significantly alter photon trajectories, thereby changing both the bending angle of light and the black hole’s energy emission behavior.