Compactness of supermassive dark objects at galactic centers

We define compactness of a gravitational lens as the scaled closest distance of approach (i.e., $r_0/M$) of the null geodesic giving rise to an image. We model forty supermassive dark objects as Schwarzschild lenses and compute compactness of lenses (determined by the formation of the first order relativistic image). We then obtain a novel formula for the compactness of a lens as a function of mass to the distance ratio ($M/D_d$) and the ratio of lens-source to the observer-source distances ($D_{ds}/D_s$). This formula yields a very important result: Just an observation of a relativistic image would give an incredibly accurate upper bound to the physical compactness (the ratio of the radius to mass) of the lens without having any knowledge of mass of the lens, angular source position, and observer-source and lens-source distances. Similarly, we show that the observation of the second order relativistic image would give a lower value of upper bound to the physical compactness. These results, though obtained for supermassive dark objects at galactic centers, are valid for any object compact enough to give rise to relativistic images.