  We consider the perturbative, fully explicit, analytical behaviour of photon
escape cones in the Kerr spacetime. When one conducts the fully general
non-perturbative Kerr analysis, one quickly finds that one must at some point
appeal to numerical and/or graphical methods. Herein we find that we are able
to say much more if we look at the slow rotation limit (i.e., $a \ll m$).
Indeed we give explicit and tractable expressions for the first and second
order (in $a$) contributions to both the shape of the escape cone, and the
solid angle subtended by the escape cone. We then look at a few special cases
at each order, thereby leading to explicit expressions for the black hole
silhouette, expressions which are of great interest to the observational
community when studying images of black hole silhouettes ("shadows").
