Circular stable orbits in $f(R)$ realistic static and spherically-symmetric spacetimes

We investigate the geodesic structure of realistic static and spherically symmetric spacetimes embedding neutron stars in metric $f(R)$ gravity, focusing on the quadratic Starobinsky model $f(R)=aR^2$ with $a<0$. Neutron-star solutions are obtained by numerically solving the modified Tolman-Oppenheimer-Volkoff system for several realistic equations of state. Such solutions are then matched consistently to the exterior vacuum geometry by enforcing the full set of junction conditions required in metric $f(R)$ theories. Using an effective potential approach, we show that stable circular orbits appear in discrete radial bands separated by forbidden regions, with a dominant principal band of stability that depends sensitively on the stellar central pressure, the equation of state, and the magnitude of the parameter $|a|$. Outside the stable bands, massive particles can have bound but unstable precessing trajectories as well as unbounded motions. On the other hand, for null geodesics, we find no evidence for photon spheres outside the neutron star within the parameter range studied.