The models developed so far to account for the Non-Gravitational Perturbations (NGP) acting on the satellites of the Global Navigation Satellite Systems (GNSS) constitute the present limit for further improvements in the Precise Orbit Determination (POD) of these satellites. These models are mainly based on the estimate of empirical parameters (with the goal to absorb unknowns quantities) or on the use of box-wing models. This unsatisfactory modeling has a negative impact both in the use of GNSS data over relatively short arcs, i.e. within the classical use of these satellites for navigation and positioning, as well as over longer arcs, where the contribution of GNSS satellites to geophysical applications is very important. The hardness in the modeling of the subtle effects produced by the NGP is further highlighted in the case of a complex in shape spacecraft, as for the Galileo ones. In fact, solar panels, antennae for microwaves link, thrusters for maneuvers and the mutual shadowing effects among the many surfaces involved, all represent additional complications to be modeled. Indeed, because of these difficulties in the modeling of the NGP - also in the case of the direct solar radiation pressure (SRP), the largest NGP acting on GNSS spacecraft - the European Space Agency (ESA) is currently considering the possibility to equip the Second Generation Galileo spacecraft with an on-board accelerometer in order to directly measure the non-gravitational accelerations. This will allow more refined ephemerides for the orbit of these satellites and, consequently, a more precise and accurate positioning on the Earth as well as more accurate measurements for the parameters of interest in the field of geophysics. We are involved in this study by means of an ESA project named GALileo and ACcelerometry (GALAC), led by the Space Research Centre of the Polish Academy of Sciences (SRC-PAS), Warsaw. The GALAC main objective is to provide the required characteristics and performan...