Determining parameters of Kerr black holes at finite distance by shadow observation

We propose a new method to determine the physical parameters of Kerr black holes (namely, specific angular momentum $a$, inclination angle $i$, and distance $D$ from an observer) only from the shadow’s information such as the size and shape. Key points in our method are (i) to treat the distance as a free parameter, (ii) to expand the shadow’s outline as a Fourier series, and (iii) to construct principal components from the Fourier coefficients. These points enable us to obtain a one-to-one mapping between three principal components, being observables characterizing the size and deviation of shadow’s shape from a circular disk, and the values of three parameters ($D\mathrm{/}M,a\mathrm{/}M,i$), where $M$ is the mass of black hole. Our method is applicable to various type of black holes and even to ones with accretion disks.