  The groundbreaking image of the black hole at the center of the M87 galaxy
has raised questions at the intersection of observational astronomy and black
hole physics. How well can the radius of a black hole shadow can be measured,
and can this measurement be used to distinguish general relativity from other
theories of gravity? We explore these questions using a simple spherical flow
model in general relativity, scalar Gauss--Bonnet gravity, and the Rezzolla and
Zhidenko parameterized metric. We assume an optically thin plasma with
power-law emissivity in radius. Along the way we present a generalized Bondi
flow as well as a piecewise-analytic model for the brightness profile of a cold
inflow. We use the second moment of a synthetic image as a proxy for EHT
observables and compute the ratio of the second moment to the radius of the
black hole shadow. We show that corrections to this ratio from modifications to
general relativity are subdominant compared to corrections to the critical
impact parameter, and argue that this is generally true. We find that
astrophysical model parameters are the dominant source of uncertainty in this
calculation, emphasizing the importance of understanding the astrophysical
model. Given a sufficiently accurate astrophysical model, however, it is
possible using measurements of the black hole shadow to distinguish between
general relativity and other theories of gravity.
