Extended galactic rotational velocity profiles in $f(R)$ gravity background

An attempt has been made to explore the galactic dynamics via the rotational velocity beyond the Einstein's geometric theory of gravity. It is inspired from the geometric relation obtained in the power law $f(R)$ gravity model in vacuum. We analyse the action with a small positive deviation from the Einstein-Hilbert action (taking $R$ as $f(R)\propto R^{1+\delta}$) at the galactic scales for the explanation of cosmological dark matter problem and obtain the contribution of dynamical $f(R)$ background geometry in accelerating the test mass. In the weak field limits, we obtain the effective acceleration of the test mass due to a massive spherically symmetric source in $f(R)$ background and develop an equation for the rotational velocity. We test the viability of the model by tracing the motion of test mass outside the typical galactic visible boundaries without considering any dark matter halo profile. We obtain a nice agreement in the outer regions (up to few tens of kpc beyond the visible boundary) of the typical galaxy by using the known galaxy data.\\ We further explore the galactic dynamics for a galaxy NGC 1052 of which the dark matter deficient galaxies, i.e., DF2 and DF4 are a part (satellite galaxies) and discuss plots of the dynamical feature of rotation curves in $f(R)$ background for the model parameter $\delta<<1$ and interpret the results for its satellite galaxies.