Co-orbital period: an analytical prescription

The transit detection of planets interacting in a 1:1 mean motion resonance (MMR) could provide with both the size and masses of the planets, due to the nature of the interaction. Regardless of the observability of each planet or the co-orbital configuration (horseshoe or tadpole). However, to properly identify a co-orbital configuration, the TTV signal needs to be as long as the 1:1 libration period (co-orbital period). For practical purposes, an analytical function that describes this period is desirable. The tadpole period has previously been described analytically, while the horseshoe period counts with a semi-analytical formulation. In this work, we seek to find a functional form for the horseshoe period to then constrain the length of observations required to identify 1:1 MMR. Using numerical simulations of co-orbital systems within a range of planetary masses, we obtain the libration period of the stable configurations. The co-orbital period is obtained through the periodogram of the planets’ mutual separation angle. Subsequently, the correlation, if any, of the co-orbital period with planetary mass is analyzed to find the best analytical fit. Our analysis revealed that the horseshoe period depends both on the planetary mass and the separation angle, showing that for a given total planetary mass there is a range of possible horseshoe periods. In particular, the mean horseshoe period is a power law of the planetary mass with exponent -0.347, constrained by the minimum and maximum separation angle of a given mass. Moreover, the shortest horseshoe period is ~20 times the orbital period, corresponding to a total planetary mass of 1 Jovian mass. Suggesting that the TTV of a transiting hot gas giant should span at least 60 days to properly identify a co-orbital companion. Co-orbital configurations with lower total planetary mass would require as long as 2000 orbits to be identified as such. A detailed description of the methodology and results will be provided at the conference.