  Radial velocity (RV) searches for Earth-mass exoplanets in the habitable zone
around Sun-like stars are limited by the effects of stellar variability on the
host star. In particular, suppression of convective blueshift and brightness
inhomogeneities due to photospheric faculae/plage and starspots are the
dominant contribution to the variability of such stellar RVs. Gaussian process
(GP) regression is a powerful tool for statistically modeling these
quasi-periodic variations. We investigate the limits of this technique using
800 days of RVs from the solar telescope on the High Accuracy Radial velocity
Planet Searcher for the Northern hemisphere (HARPS-N) spectrograph. These data
provide a well-sampled time series of stellar RV variations. Into this data
set, we inject Keplerian signals with periods between 100 and 500 days and
amplitudes between 0.6 and 2.4 m s$^{-1}$. We use GP regression to fit the
resulting RVs and determine the statistical significance of recovered periods
and amplitudes. We then generate synthetic RVs with the same covariance
properties as the solar data to determine a lower bound on the observational
baseline necessary to detect low-mass planets in Venus-like orbits around a
Sun-like star. Our simulations show that discovering planets with a larger mass
($\sim$ 0.5 m s$^{-1}$) using current-generation spectrographs and GP
regression will require more than 12 yr of densely sampled RV observations.
Furthermore, even with a perfect model of stellar variability, discovering a
true exo-Venus ($\sim$ 0.1 m s$^{-1}$) with current instruments would take over
15 yr. Therefore, next-generation spectrographs and better models of stellar
variability are required for detection of such planets.
