  The behavior of the angular velocity of a test particle moving in a stable
circular orbit in the vacuum on the brane is considered. In the brane world
scenario, the four dimensional effective Einstein equation acquire extra terms,
called dark radiation and dark pressure, respectively, which arise from the
embedding of the 3-brane in the bulk. A large number of independent
observations have shown that the rotational velocities of test particles
gravitating around galaxies tend, as a function of the distance from the
galactic center, toward constant values. By assuming a constant tangential
velocity, the general solution of the vacuum gravitational field equations on
the brane can be obtained in an exact analytic form. This allows us to obtain
the explicit form of the projections of the bulk Weyl tensor on the brane, and
the equation of state of the dark pressure as a function of the dark radiation.
The physical and geometrical quantities are expressed in terms of
observable/measurable parameters, like the tangential velocity, the baryonic
mass and the radius of the galaxy. We also analyze the dynamics of test
particles by using methods from the qualitative analysis of dynamical systems,
by assuming a simple linear equation of state for the dark pressure. The
obtained results provide a theoretical framework for the observational testing
at the extra-galactic scale of the predictions of the brane world models.
