Neutron monitors and muon detectors for solar modulation studies: 2. $\phi$ time series

The level of solar modulation at different times (related to the solar activity) is a central question of solar and galactic cosmic-ray physics. In the first paper of this series, we have established a correspondence between the uncertainties on ground-based detectors count rates and the parameter $\phi$ (modulation level in the force-field approximation) reconstructed from these count rates. In this second paper, we detail a procedure to obtain a reference $\phi$ time series from neutron monitor data. We show that we can have an unbiased and accurate $\phi$ reconstruction ($\Delta\phi/\phi\simeq 10\%$). We also discuss the potential of Bonner spheres spectrometers and muon detectors to provide $\phi$ time series. Two by-products of this calculation are updated $\phi$ values for the cosmic-ray database and a web interface to retrieve and plot $\phi$ from the 50's to today (\url{http://lpsc.in2p3.fr/crdb}).