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 an interesting result: Just an observation of a relativistic image would
give an incredibly accurate upper bound to the compactness 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 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.
