Shapes and Positions of Black Hole Shadows in Accretion Disks and Spin Parameters of Black Holes

Can we determine a spin parameter of a black hole by observation of a black hole shadow in an accretion disk? In order to answer this question, we make a qualitative analysis and a quantitative analysis of a shape and a position of a black hole shadow casted by a rotating black hole on an optically thick accretion disk and its dependence on an angular momentum of a black hole. We consider two types of inner edges, an event horizon and a marginally stable orbit, of accretion disks. We have found black hole shadows with a quite similar size and a shape for largely different black hole spin parameters and a same black hole mass. Thus, it is practically difficult to determine a spin parameter of a black hole from a size and a shape of a black hole shadow in an accretion disk. On the other hand, a displacement of a black hole shadow from a rotation axis of a black hole is caused by a frame dragging effect in the vicinity of a black hole. The extent of the displacement largely depend on a black hole spin parameter. However, it is difficult to determine a position of a rotation axis of a black hole observationally. So, we can not practically measure a spin parameter of a black hole from the displacement from a rotation axis of a black hole shadow. We newly introduce a bisector axis of a black hole shadow named a shadow axis. We define this shadow axis as a bisector perpendicular to a line segment of the maximum width of a black hole shadow. We can determine a position of a shadow axis of by observation of a black hole shadows. For a non-rotating black hole, the minimum interval between a mass center of a black hole and a shadow axis is null. On the other hand, for a rotating black hole a shape and a position of a black hole shadow are not symmetric with respect to a rotation axis of a black hole shadow. So, in this case the minimum interval between a mass center of a black hole and a shadow axis is finite. An extent of this minimum interval is roughly proportional to a spin parameter of a black hole for a fixed inclination angle between a rotation axis of a black hole and a direction of an observer. The maximum extent of these minimum intervals is about 1.5rg. Here, rg is a gravitational radius. This is realized when the maximum inclination angle and the maximally rotating black hole in an accretion disk with its inner edge of an event horizon. In order to measure a spin parameter of a black hole, if a shadow axis is determined observationally, it is crucially important to determine a position of a mass center of a black hole in a region of a black hole shadow. We also discuss how to determine a mass center of a black hole by observation of a black hole shadow in an accretion disk. Subject headings: accretion, accretion disks—black hole physics— galaxies: nuclei