  We provide evidence for a correspondence between the formation of black holes
and the stability of circular null geodesics around the collapsing
perturbation. We first show that the critical threshold of the compaction
function to form a black hole in radiation is well approximated by the critical
threshold for the appearance of the first unstable circular orbit in a
spherically symmetric background. We also show that the critical exponent in
the scaling law of the primordial black hole mass close to the threshold is set
by the inverse of the Lyapunov coefficient of the unstable orbits when a
self-similar stage is developed close to criticality.
