Paper detail

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 $\hat{C}^2_{xy}(f)$ 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 $\hat{C}^2_{xy}(f)$ indicates that the signal at frequency $f$ has a stellar origin. We use Welch&#39;s method to measure coherence between activity indicators and RVs in archival observations of GJ 581, alpha Cen B, and GJ 3998. High RV-H$α$ coherence at the frequency of GJ 3998 b, and high RV-S index coherence at the frequency of GJ 3998 c, 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 alpha Cen B has activity signals that are not associated with rotation. Welch&#39;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&#39;s method may yield unacceptably large bias for datasets with $N < 75$ observations and works best on datasets with $N > 100$. Tapering the time-domain data can reduce the bias of the Welch&#39;s power spectrum estimator, but observers should not apply tapers to datasets with extremely uneven observing cadence. A software package for calculating magnitude-squared coherence and Welch&#39;s power spectrum estimates is available on github.