The surface geometry and shadow of a Schwarzschild black hole with halo
We have studied the surface geometry and shadows of Schwarzschild black hole with a halo containing the quadrupolar and octopolar terms. We found the quadrupole term makes the Schwarzschild black hole prolate for the quadrupole strength $\mathcal{Q}<0$ and oblate for $\mathcal{Q}>0$, and the octopole term makes shadow stretch upward for the octopolar strength $\mathcal{O}<0$ and stretch downward for $\mathcal{O}>0$. The shadow of Schwarzschild black hole with halo stretches and squeezes along the horizontal direction for $\mathcal{Q}<0$ and $\mathcal{Q}>0$ respectively. Meanwhile, black hole shadow shifts upward for $\mathcal{O}<0$ and shifts downward for $\mathcal{O}>0$. We exhibited the light rays that form the shadow boundary to explain the emergence of the extraordinary patterns of black hole shadow with the quadrupole and octopole terms. From the observable width $W$, height $H$, oblateness $K$ and distortion parameter $δ_{c}$ of black hole shadow, one can determine the quadrupole and octopolar strengths of Schwarzschild black hole with 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 halo.