Determining parameters of a spherical black hole with a thin accretion disk by observing its shadow

We revisit the classic system of a spherically symmetric black hole in general relativity (i.e., a Schwarzschild black hole) surrounded by a geometrically thin accretion disk. Our purpose is to examine whether one can determine three parameters of this system (i.e., black hole mass $M$, distance between the black hole and an observer $r_o$, inclination angle $i$) solely by observing the accretion disk and the black hole shadow. A point in our analysis is to allow $r_o$ to be finite, which is set to be infinite in most relevant studies. First, it is shown that one can determine the values of $(r_o/M, i)$, where $M/r_o$ is the so-called angular gravitational radius, from the size and shape of shadow. Then, it is shown that if one additionally knows the accretion rate $\dot{M}$ (respectively, mass $M$) by any independent theoretical or observational approach, one can determine the values of $(M, r_o, i)$ [respectively, $(\dot{M}, r_o, i)$] without degeneracy, in principle, from the value of flux at any point on the accretion disk.