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

Can we determine the spin parameter of a black hole by observing the black hole's shadow on an accretion disk? To answer this question, we make qualitative and quantitative analyses of the shape and position of the shadow cast by a rotating black hole on an optically thick accretion disk and its dependence on the angular momentum of the black hole. We consider two types of inner edges of accretion disks, an event horizon and a marginally stable orbit. We have found black hole shadows of quite similar sizes and shapes for very different black hole spin parameters and the same black hole mass. Thus, in practice it is difficult to determine the spin parameter of a black hole from the size and shape of its shadow on an accretion disk. On the other hand, a frame-dragging effect in the vicinity of a black hole causes a displacement of the shadow from the rotation axis of the black hole. The extent of the displacement largely depends on the black hole's spin parameter. However, it is difficult to determine the position of the rotation axis of a black hole observationally. So, we cannot practically use the displacement of the rotation axis of a black hole shadow to measure the spin parameter. We here introduce a bisector axis of black hole shadows that we call the "shadow axis." We define this as the bisector perpendicular to a line segment of the maximum width of a black hole shadow. We can determine the position of a shadow axis through observation of a black hole shadow. For a nonrotating black hole, the minimum interval between the mass center of a black hole and the shadow axis is null. On the other hand, for a rotating black hole the shape and position of the shadow are not symmetric with respect to its rotation axis. So, in this case the minimum interval between the mass center of the black hole and the shadow axis is finite. The extent of this minimum interval is roughly proportional to the spin parameter of a black hole for a fixed inclination angle between a rotation axis and the direction of an observer. The maximum extent of these minimum intervals is about 1.5rg. Here rg is a gravitational radius. This is realized in the case of a maximum inclination angle and maximally rotating black hole in an accretion disk in which the inner edge is an event horizon. In order to measure the spin parameter of a black hole, if the shadow axis is determined observationally, it is crucially important to determine the position of the mass center in a region of the shadow. We also discuss how to determine a mass center of a black hole by observation of the shadow on an accretion disk.