Stellar magnetic activity produces time-varying distortions in the
photospheric line profiles of solar-type stars. These lead to systematic errors
in high-precision radial-velocity measurements, which limit efforts to discover
and measure the masses of low-mass exoplanets with orbital periods of more than
a few tens of days. We present a new data-driven method for separating Doppler
shifts of dynamical origin from apparent velocity variations arising from
variability-induced changes in the stellar spectrum. We show that the
autocorrelation function (ACF) of the cross-correlation function used to
measure radial velocities is effectively invariant to translation. By
projecting the radial velocities on to a subspace labelled by the observation
identifiers and spanned by the amplitude coefficients of the ACF's principal
components, we can isolate and subtract velocity perturbations caused by
stellar magnetic activity. We test the method on a 5-year time sequence of 853
daily 15-minute observations of the solar spectrum from the HARPS-N instrument
and solar-telescope feed on the 3.58-m Telescopio Nazionale Galileo. After
removal of the activity signals, the heliocentric solar velocity residuals are
found to be Gaussian and nearly uncorrelated. We inject synthetic low-mass
planet signals with amplitude K=40 cm s−1 into the solar observations at
a wide range of orbital periods. Projection into the orthogonal complement of
the ACF subspace isolates these signals effectively from solar activity
signals. Their semi-amplitudes are recovered with a precision of ∼6.6 cm
s−1, opening the door to Doppler detection and characterization of
terrestrial-mass planets around well-observed, bright main-sequence stars
across a wide range of orbital periods