Detection of co-orbital planets by combining transit and radial-velocity measurements

Co-orbital planets have not yet been discovered, although they constitute a frequent by-product of planetary formation and evolution models. This lack may be due to observational biases, since the main detection methods are unable to spot co-orbital companions when they are small or near the Lagrangian equilibrium points. However, for a system with one known transiting planet (with mass $m_1$), we can detect a co-orbital companion (with mass $m_2$) by combining the time of mid-transit with the radial-velocity data of the star. Here, we propose a simple method that allows the detection of co-orbital companions, valid for eccentric orbits, that relies on a single parameter $\alpha$, which is proportional to the mass ratio $m_2/m_1$. Therefore, when $\alpha$ is statistically different from zero, we have a strong candidate to harbour a co-orbital companion. We also discuss the relevance of false positives generated by different planetary configurations.