Surface geometry and shadow of a Schwarzschild black hole with a halo

We studied the surface geometry and shadows of a Schwarzschild black hole with a halo containing quadrupolar and octopolar terms. It was found that the quadrupole term causes a Schwarzschild black hole to be prolate at the quadrupole strength Q < 0 and oblateQ at > 0 , and the octopole term causes to the shadow stretch upward at the octopolar strength ngth O < 0 and downward at O > 0 . The shadow of a Schwarzschild black hole with a halo stretches and squeezes along the horizontal directionQ at < 0 and Q > 0 , respectively. Meanwhile, the black hole shadow shifts upwardO at < 0 and downwardO at > 0 . We present the light rays that form the shadow boundary to explain the emergence of the extraordinary patterns of the black hole shadow with quadrupole and octopole terms. From the observable width W, height H, oblateness K, and distortion parameter of the black hole shadow, ?c we can determine the quadrupole and octopolar strengths of a Schwarzschild black hole with a halo. Our results show that the quadrupolar and octopolar terms yield a series of interesting patterns for the shadow of a Schwarzschild black hole with a halo.