Unified $f(R)$ gravity at local scales

We explore the shifted $f(R) (\propto R^{1+\delta })$ model with ${\delta }$ as a distinguishing physical parameter for the study of constraints at local scales. The corresponding dynamics confronted with different geodesics (null and non-null) along with their conformal analog are investigated. For null geodesics, we discuss the light deflection angle, whereas, for non-null geodesics under the weak field limit, we investigate the perihelion advance of the Mercury orbit in f(R) Schwarzschild background, respectively. The extent of an additional force, appearing for non-null geodesics, depends on $\delta $. Such phenomenological investigations allow us to strictly constrain $\delta $ to be approximately ${\mathcal {O}}(10^{-6})$ with a difference of unity in orders at galactic and planetary scales and seem to provide a unique f(R) at local scales. Our results suggest that the present form of the model is suitable for the alternative explanation of dark matter-like effects at local scales.