The GAPS Programme with HARPS-N at TNG XVII. Line profile indicators and kernel regression as diagnostics of radial-velocity variations due to stellar activity in solar-like stars

Stellar activity is the ultimate source of radial-velocity (RV) noise in the search for Earth-mass planets orbiting late-type main-sequence stars. We analyse the performance of four different indicators and the chromospheric index $\log R'_{\rm HK}$ in detecting RV variations induced by stellar activity in 15 slowly rotating ($v\sin i \leq 5$ km/s), weakly active ($\log R'_{\rm HK} \leq -4.95$) solar-like stars observed with the high-resolution spectrograph HARPS-N. We consider indicators of the asymmetry of the cross-correlation function (CCF) between the stellar spectrum and the binary weighted line mask used to compute the RV, that is the bisector inverse span (BIS), $\Delta V$, and a new indicator $V_{\rm asy(mod)}$ together with the full width at half maximum (FWHM) of the CCF. We present methods to evaluate the uncertainties of the CCF indicators and apply a kernel regression (KR) between the RV, the time, and each of the indicators to study their capability of reproducing the RV variations induced by stellar activity. The considered indicators together with the KR prove to be useful to detect activity-induced RV variations in $47 \pm 18$ percent of the stars over a two-year time span when a significance (two-sided p-value) threshold of one percent is adopted. In those cases, KR reduces the standard deviation of the RV time series by a factor of approximately two. The BIS, the FWHM, and the newly introduced $V_{\rm asy(mod)}$ are the best indicators, being useful in $27 \pm 13$, $13 \pm 9$, and $13 \pm 9$ percent of the cases, respectively. The relatively limited performances of the activity indicators are related to the very low activity level and $v\sin i$ of the considered stars. For the application of our approach to sun-like stars, a spectral resolution of at least $10^5$ and highly stabilized spectrographs are recommended.