  Radial-velocity (RV) planet searches are often polluted by signals caused by
gas motion at the star's surface. Stellar activity can mimic or mask changes in
the RVs caused by orbiting planets, resulting in false positives or missed
detections. Here we use Gaussian process (GP) regression to disentangle the
contradictory reports of planets vs. rotation artifacts in Kapteyn's star
(Anglada-Escude et al. 2014, Robertson et al. 2015, Anglada-Escude et al.
2016). To model rotation, we use joint quasi-periodic kernels for the RV and
H-alpha signals, requiring that their periods and correlation timescales be the
same. We find that the rotation period of Kapteyn's star is 125 days, while the
characteristic active-region lifetime is 694 days. Adding a planet to the RV
model produces a best-fit orbital period of 100~years, or 10 times the
observing time baseline, indicating that the observed RVs are best explained by
star rotation only. We also find no significant periodic signals in residual RV
data sets constructed by subtracting off realizations of the best-fit rotation
model and conclude that both previously reported "planets" are artifacts of the
star's rotation and activity. Our results highlight the pitfalls of using
sinusoids to model quasi-periodic rotation signals.
