  We consider the behavior of the tangential velocity of test particles moving
in stable circular orbits in f(R) modified theories of gravity. A large number
of observations at the galactic scale have shown that the rotational velocities
of massive test particles (hydrogen clouds) tend towards constant values at
large distances from the galactic center. We analyze the vacuum gravitational
field equations in f(R) models in the constant velocity region, and the general
form of the metric tensor is derived in a closed form. The resulting
modification of the Einstein-Hilbert Lagrangian is of the form R^{1+n}, with
the parameter n expressed in terms of the tangential velocity. Therefore we
find that to explain the motion of test particles around galaxies requires only
very mild deviations from classical general relativity, and that modified
gravity can explain the galactic dynamics without the need of introducing dark
matter.
