  Among modified theories of gravity, the teleparallel $f(T)$ gravity is an
intensively discussed modelin the literature. The best way to investigate its
viabilityis to derive observable predictions which yield evidence or
constraints for the model, when compared with actual observations. In this
paper we derive the photon sphere and the perihelion shift for weak $f(T)$
perturbations of general relativity. We consistently calculate first order
teleparallel perturbations of Schwarzschild and Minkowski spacetime geometry,
with which we improve and extend existing results in the literature.
