Rotation Curves of Galaxies by Fourth Order Gravity

We investigate the radial behavior of galactic rotation curves by a Fourth Order Gravity adding also the dark matter component. The Fourth Order Gravity is a theory of gravity described by Lagrangian generalizing the one of Hilbert-Einstein containing a generic function of the Ricci scalar, the Ricci and Riemann tensor. A systematic analysis of rotation curves, in the Newtonian Limit of theory, induced by all galactic substructures of ordinary matter is shown. This analysis is presented for Fourth Order Gravity with and without dark matter. The outcomes are compared with respect to the classical outcomes of General Relativity. The gravitational potential of pointlike mass is the usual potential corrected by two Yukawa terms. The rotation curve is higher or also lower than curve of General Relativity if in the Lagrangian the Ricci scalar square is dominant or not with respect to the contribution of the Ricci tensor square. The theoretical spatial behaviors of rotation curve are compared with the experimental data for the Milky Way and the galaxy NGC 3198. Although the Fourth Order Gravity gives more rotational contributions, in the limit of large distances the Keplerian behavior is ever present, and it is missing only if we add the dark matter component. However by modifying the theory of gravity, consequently, also the spatial description of dark matter could undergo a modification and the free parameters of model can assume different values. After an analytical discussion of theoretical behaviors and the comparing with experimental evidence we can claim that any Fourth Order Gravity is not successful to explain the galactic rotation curves. In the last part of paper we analyze the gravitational potential induced by Lagrangian containing only powers of Ricci scalar. In this case we find an inconsistency in the boundary conditions in the passage from matter to the vacuum.