  Nonzero neutrino masses are required by the existence of flavour
oscillations, with values of the order of at least 50 meV. We consider the
gravitational clustering of relic neutrinos within the Milky Way, and used the
$N$-one-body simulation technique to compute their density enhancement factor
in the neighbourhood of the Earth with respect to the average cosmic density.
Compared to previous similar studies, we pushed the simulation down to smaller
neutrino masses, and included an improved treatment of the baryonic and dark
matter distributions in the Milky Way. Our results are important for future
experiments aiming at detecting the cosmic neutrino background, such as the
Princeton Tritium Observatory for Light, Early-universe, Massive-neutrino Yield
(PTOLEMY) proposal. We calculate the impact of neutrino clustering in the Milky
Way on the expected event rate for a PTOLEMY-like experiment. We find that the
effect of clustering remains negligible for the minimal normal hierarchy
scenario, while it enhances the event rate by 10 to 20% (resp. a factor 1.7 to
2.5) for the minimal inverted hierarchy scenario (resp. a degenerate scenario
with 150 meV masses). Finally we compute the impact on the event rate of a
possible fourth sterile neutrino with a mass of 1.3 eV.
