Neutron stars accreting the ISM: Are they fast or slow objects ?

Old neutron stars (ONSs) which have radiated away their internal and rotational energy may still shine if accreting the interstellar medium. Rather stringent limits from the analysis of ROSAT surveys indicate that most optimistic predictions on ONSs observability are in excess of a factor as large as $\sim 100$. Here we explore two possible evolutionary scenarios that may account for the paucity of ONSs. In the first it is assumed that the ONS population is not too fast ($V<100 km s^{-1}$) and that magnetic field decay guides the evolution. In the second, NSs move with high speed ($V>100$ km s$^{-1}$) and preserve their magnetic field at birth. We find that according to the former scenario most ONSs are now in the propeller phase, while in the latter nearly all ONSs are silent, dead pulsars.