  The images of supermassive black holes surrounded by optically-thin,
radiatively-inefficient accretion flows, like those observed with the Event
Horizon Telescope, are characterized by a bright ring of emission surrounding
the black-hole shadow. In the Kerr spacetime this bright ring, when narrow,
closely traces the boundary of the shadow and can, with appropriate
calibration, serve as its proxy. The present paper expands the validity of this
statement by considering two particular spacetime geometries: a solution to the
field equations of a modified gravity theory and another that parametrically
deviates from Kerr but recovers the Kerr spacetime when its deviation
parameters vanish. A covariant, axisymmetric analytic model of the accretion
flow based on conservation laws and spanning a broad range of plasma conditions
is utilized to calculate synthetic non-Kerr black-hole images, which are then
analysed and characterized. We find that in all spacetimes: (i) it is the
gravitationally-lensed unstable photon orbit that plays the critical role in
establishing the diameter of the rings observed in black-hole images, not the
event horizon or the innermost stable circular orbit, (ii) bright rings in
these images scale in size with, and encompass, the boundaries of the
black-hole shadows, even when deviating significantly from Kerr, and (iii)
uncertainties in the physical properties of the accreting plasma introduce
subdominant corrections to the relation between the diameter of the image and
the diameter of the black-hole shadow. These results provide important new
theoretical justification for using black-hole images to probe and test the
spacetimes of supermassive black holes.
