Using the Sun to estimate Earth-like planets detection capabilities IV. Correcting for the convective component

Context. Radial velocity (RV) time series are strongly impacted by the presence of stellar activity. In a series of papers, we have reconstructed solar RV variations over a full solar cycle from observed solar structures (spots and plages) and studied their impact on the detectability of an Earth-mass planet in the habitable zone of the Sun as seen edge-on from a neighbour star in several typical cases. We found that the convective contribution dominates the RV times series. Aims. The objective of this paper is twofold: to determine detection limits on a Sun-like star seen edge-on with different levels of convection and to estimate the performance of the activity correction using a Ca index. Methods. We apply two methods to compute the detection limits: a correlation-based method and a local power analysis method, which both take into account the temporal structure of the observations. Furthermore, we test two methods using a Ca index to correct for the convective contribution to the RV: a sinusoidal fit to the Ca variations and a linear fit to the RV-Ca relation. In both cases, we use observed Ca and reconstructed Ca to study the various effects and limitations of our estimations. Results. We confirm that an excellent sampling is necessary to have detection limits below 1 MEarth (e.g. 0.2−0.3 MEarth) when there is no convection and a low RV noise. With convection, the detection limit is always above 7 MEarth. The two correction methods perform similarly when the Ca time series are noisy, leading to a significant improvement (down to a few MEarth), which is above the 1 MEarth limit. With a very good Ca noise (signal to noise ratio, S/N, around 130), the sinusoidal method does not get significantly better because it is dominated by the fact that the solar cycle is not sinusoidal, but the RV-Ca method can reach the 1 MEarth for an excellent Ca noise level. Conclusions. For Sun-like conditions and under the simplifying assumptions considered, we first conclude that the detection limit of a few MEarth planet can be reached providing good sampling and Ca noise. The detection of a 1 MEarth may be possible, but only with an excellent temporal sampling and an excellent Ca index noise level: we estimate that a probability larger than 50% to detect a1 MEarth at 1.2 AU requires more than 1000 well-sampled observations and a Ca S/N larger than 130.