Photon regions, shadow observables and constraints from M87* of a Kerr-Newman-like black hole in Bumblebee gravity surrounded by plasma

In this paper, we investigate the photon regions, shadow, and observational constraints of a Kerr-Newman-like black hole in Bumblebee gravity within a plasma medium. By employing a specific non-homogeneous power-law plasma model to ensure the separability of the Hamilton-Jacobi equation, we derive the null geodesic equations, analyze the photon regions, and construct the black hole shadow. Furthermore, we introduce two sets of shadow observables to systematically analyze the distinct effects of each physical parameter (spin $a$, charge $Q_0$, Lorentz-violating parameter $\ell$, and plasma parameter $k$) on the shadow geometry. Specifically, we find that $a$ and $\ell$ mainly enhance the distortion of the shadow, whereas $Q_0$ and $k$ primarily lead to its radial shrinkage. Additionally, a brief evaluation of the energy emission rate shows that an increase in these parameters generally suppresses the emission peak. Finally, by modeling M87* as a charged rotating black hole in Bumblebee gravity surrounded by plasma, we can constrain the physical parameters using observations from the Event Horizon Telescope (EHT). While the angular diameter $\theta_d = 42 \pm 3 \, \mu\text{as}$ narrows the viable parameter space, the circularity deviation $\Delta C \lesssim 0.1$ and axis ratio $1 < D_x \lesssim 4/3$ obey the EHT limits. This suggests that the charged rotating black hole in Bumblebee gravity surrounded by plasma might be a candidate for real astrophysical black holes.