  The standard cosmological model predicts the existence of a Cosmic Neutrino
Background, which has not yet been observed directly. Some experiments aiming
at its detection are currently under development, despite the tiny kinetic
energy of the cosmological relic neutrinos, which makes this task incredibly
challenging. Since massive neutrinos are attracted by the gravitational
potential of our Galaxy, they can cluster locally. Neutrinos should be more
abundant at the Earth position than at an average point in the Universe. This
fact may enhance the expected event rate in any future experiment. Past
calculations of the local neutrino clustering factor only considered a
spherical distribution of matter in the Milky Way and neglected the influence
of other nearby objects like the Virgo cluster, although recent $N$-body
simulations suggest that the latter may actually be important. In this paper,
we adopt a back-tracking technique, well established in the calculation of
cosmic rays fluxes, to perform the first three-dimensional calculation of the
number density of relic neutrinos at the Solar System, taking into account not
only the matter composition of the Milky Way, but also the contribution of the
Andromeda galaxy and the Virgo cluster. The effect of Virgo is indeed found to
be relevant and to depend non-trivially on the value of the neutrino mass. Our
results show that the local neutrino density is enhanced by 0.53% for a
neutrino mass of 10 meV, 12% for 50 meV, 50% for 100 meV or 500% for 300 meV.
