Magnitude-squared Coherence: A Powerful Tool for Disentangling Doppler Planet Discoveries from Stellar Activity

If Doppler searches for Earth-mass, habitable planets are to succeed, observers must be able to identify and model out stellar activity signals. Here we demonstrate how to diagnose activity signals by calculating the magnitude-squared coherence ${\hat{C}}_{{xy}}^{2}(f)$ between an activity-indicator time series x _t and the radial-velocity (RV) time series y _t . Since planets only cause modulation in RV, not in activity indicators, a high value of ${\hat{C}}_{{xy}}^{2}(f)$ indicates that the signal at frequency f has a stellar origin. We use Welch's method to measure coherence between activity indicators and RVs in archival observations of GJ 581, α Cen B, and GJ 3998. High RV-Hα coherence at the frequency of GJ 3998 b and high RV-S index coherence at the frequency of GJ 3998c, indicate that the planets may actually be stellar signals. We also replicate previous results showing that GJ 581 d and g are rotation harmonics and demonstrate that α Cen B has activity signals that are not associated with rotation. Welch's power spectrum estimates have cleaner spectral windows than Lomb-Scargle periodograms, improving our ability to estimate rotation periods. We find that the rotation period of GJ 581 is 132 days, with no evidence of differential rotation. Welch's method may yield unacceptably large bias for data sets with N < 75 observations and works best on data sets with N > 100. Tapering the time-domain data can reduce the bias of the Welch's power spectrum estimator, but observers should not apply tapers to data sets with extremely uneven observing cadence. A software package for calculating magnitude-squared coherence and Welch's power spectrum estimates is available on github.