Optimising the H$α$ index for the identification of activity signals in FGK stars. Improvement of the correlation between H$α$ and CaII H&K

In this work we investigate the effect of the H$α$ bandpass width in the correlation between the CaII H&K and H$α$ indices with the aim of improving the H$α$ index to better identify and model the signals coming from activity variability. We used a sample of 152 FGK dwarfs observed with HARPS for more than 13 years with enough cadence to be able to detect rotational modulations and cycles in activity proxies. We calculated the CaII H&K and H$α$ activity indices using a range of bandwidths for H$α$ between 0.1 and 2.0 Ang. We studied the correlation between the indices time series at long and short timescales and analysed the impact of stellar parameters, activity level and variability on the correlations. The correlation between CaII H&K and H$α$ both at short and long timespans is maximised when using narrow H$α$ bandwidths, with a maximum at 0.6 Ang. For some inactive stars, as the activity level increases, the flux in the H$α$ line core increases while the flux in the line wings decreases as the line becomes shallower and broader. The balance between these fluxes can cause stars to show the negative correlations observed in the literature when using a wide bandwidth on H$α$. These anti-correlations may become positive correlations if using the 0.6 Ang bandwidth. Calculating the H$α$ index using a bandpass of 0.6 Ang maximises the correlation between CaII H&K and H$α$ both at short and long timescales. On the other hand, the use of the broader 1.6 Ang, generally used in exoplanet detection to identify stellar activity signals, degrades the signal by including the flux in the line wings. In face to these results we strongly recommend the use of a 0.6 Ang bandwidth when computing the H$α$ index for the identification of activity rotational modulation and magnetic cycle signals in solar-type stars. (Abridged)