Constraints on the black-hole charges of M87* and Sagittarius A* by changing rates of photon spheres can be relaxed

The Event Horizon Telescope (EHT) Collaboration observed ring images called the shadows of M87* and Sagittarius~A* (Sgr~A*), which are supermassive objects in M87 and our galaxy, respectively, and their general relativistic magnetohydrodynamic simulations of black holes imply that the observed rings are formed by the gravitational lensing of synchrotron radiations from a hot plasma near outside of supermassive black holes. The EHT Collaboration gave constrains on the electrical or alternative charges of M87* and Sgr A* under an assumption that the radius of the observed ring should be proportional to the changing rates of photon spheres by the charges. Since the validness of this assumption is not sure, it is worth to checking the same constraints under another assumption. In this paper, we consider the changing rates of not only the photon spheres but also lensing rings in a simple model and we test whether aforementioned constraint is robust. We conclude that EHT Collaboration's constraints based on the changing rates of the photon spheres can be relaxed compared to that based on the changing rate of the lensing rings while we do not claim that the observed rings are formed by the photon spheres and the lensing rings in our simple model. We concentrate on Reissner-Nordstr\"{o}m black hole spacetimes in this paper, but our result implies the relaxation of the bound of the charge parameters on other black hole spacetimes.