  Gravitational lensing of luminous matter that surrounds a black hole or some
other sufficiently compact object produces an infinite sequence of images.
Besides the direct (or primary) image, it comprises demagnified and deformed
replicas of the original known as photon rings which are progressively nearing
the boundary of the socalled shadow. In the present paper, we present
analytical approximation formulas for higher-order photon rings for an
asymptotically flat, static, spherically symmetric spacetime that admits a
photon sphere. We consider a geometrically thin disk of light sources in the
equatorial plane and an observer at arbitrary inclination far away from the
center. Fixing the emission radius and leveraging the strong deflection limit,
which provides an analytical logarithmic approximation for the deflection
angle, we find the deformed shape of higher-order photon rings in the form of a
polar equation on the observer's screen. It has been suggested by other authors
to use the relative size of photon rings for characterizing the underlying
spacetime. In particular, the relative separation between two neighboring
photon rings, which we call "gap parameter", was considered. We analytically
calculate the gap parameter of higher-order photon rings for metrics of the
considered class that may depend on multiple parameters. The advantage of using
this quantity is in the fact that, to within the assumed approximations, it is
independent of the mass of the central object (or of some other characteristic
parameter if the mass is zero) and of the distance of the observer.
Measurements of the gap parameter, which may become possible in the near
future, will restrict the spacetime models that are in agreement with the
observations. Even without knowing the inner and outer radii of the shining
disk, it will conclusively rule out some metrics. Some examples are provided.
