  Currently envisioned extensions of the Event Horizon Telescope to space will
soon target the black hole photon ring: a narrow ring-shaped imprint of a black
hole's strong gravity produced in its images by highly bent photon
trajectories. In principle, the shape of the photon ring encodes information
about the geometry of the underlying black hole spacetime. In practice,
however, whether or not this information can be extracted from the ring shape
depends on several factors, ranging from the astrophysical details of the
emitting source (such as the magnitude of its plasma fluctuations) to the
specific configuration of the interferometric array (such as the separation
between its telescopes, or the level of noise in its instruments). Here, we
employ a phenomenological model to assess the impact of astrophysical
fluctuations and instrument noise on the inferred shape of the photon ring. Our
systematic study of several astrophysical profiles suggests that this shape can
be measured even in the presence of instrument noise across a wide range of
baselines. The measurement accuracy and precision appear relatively insensitive
to the noise level, up to a sharp threshold beyond which any measurement
becomes incredibly challenging (at least without recourse to more sophisticated
data analysis methods). Encouragingly, we find that only a few snapshot images
are generally needed to overcome the impact of astrophysical fluctuations and
correctly infer the ring diameter. Inference becomes more challenging when
analyzing the visibility amplitude in a baseline window that is not entirely
dominated by a single photon ring. Nevertheless, in most cases, it is still
possible to fit a ring shape with the correct fractional asymmetry. These
results provide excellent prospects for future precision measurements of black
hole spin and fundamental astrophysics via black hole imaging.
