Inhibition of the convective blueshift in active regions is a major contribution to the radial velocity variations, at least for solar-like stars. A common technique to correct for this component is to model the RV as a linear function of chromospheric emission, because both are strongly correlated with the coverage by plages. This correction is not perfect: the aim of the present study is to understand the limits of this correction and to improve it. We investigate these questions by analysing a large set of synthetic time series corresponding to old main sequence F6-K4 stars modelled using a consistent set of parameters. We focus here on the analysis of the correlation between time series, in particular between RV and chromospheric emission on different timescales. We also study the temporal variation for each time series. Inclination strongly impacts these correlations, as well as additional signals (granulation and supergranulation). Although RV and LogRHK are often well correlated, a combination of geometrical effects (butterfly diagrams related to dynamo processes and inclination) and activity level variations over time create an hysteresis pattern during the cycle, which produces a departure from an excellent correlation: for a given activity level, the RV is higher or lower during the ascending phase compared to the descending phase of the cycle depending on inclination, with a reversal for inclinations about 60 deg from pole-on. This hysteresis is also observed for the Sun and other stars. This property is due to the spatio-temporal distribution of the activity pattern and to the difference in projection effects of the RV and chromospheric emission. These results allow us to propose a new method which significantly improves the correction for long timescales, and could be crucial to improving detection rates of planets in the habitable zone around F6-K4 stars.
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