Black hole surrounded by the pseudo-isothermal dark matter halo

In this paper, we obtain a new spherically symmetric black hole surrounded by the pseudo-isothermal dark matter halo. Furthermore, to explore the effects of the pseudo-isothermal halo profile on a rotating black hole at the M87 galactic center, we derive a rotating black hole solution encompassed by the pseudo-isothermal halo by using the Newman-Janis method. Our investigation focuses on the impact of the pseudo-isothermal halo on the black hole event horizon, time-like and null orbits, as well as the black hole shadow. We find that as the spin parameter a increases, the interval between the inner event horizon and the outer event horizon of the rotating black hole surrounded by the pseudo-isothermal halo in M87 diminishes. This leads to the formation of an extreme black hole. The presence of dark matter, however, has minimal effect on the event horizon. Moreover, in the M87 as the spin parameter a increases, the black hole shadow deviates increasingly from a standard circle, with larger spin parameters causing more pronounced distortion relative to the standard circle. Surprisingly, we observe that the dark matter density has very little influence on the shadow of the black hole surrounded by the pseudo-isothermal halo in the M87. This study contributes to a deeper understanding of black hole structures and the role of dark matter in the universe.