Application to the longitude of Titan

International audienceContext. The ephemerides of natural satellites resulting from numerical integration have a very good precision on the fitting to recent observations, in a limited interval. Meanwhile, synthetic ephemerides like the Théorie Analytique des Satellites de Saturne (TASS) by Vienne and Duriez describe in detail the dynamical system by a representation based on the combinations of the proper frequencies. Some theoretical studies need to have both advantages. For example, to study the rotation of Titan, one needs to know the representation of its longitude.Aims. We aim to use these two types of ephemerides in order to rebuild a long-lasting and high-precision ephemeris with proper frequencies based on the numerical integration ephemeris. The aim is to describe the numerical ephemerides with formulas similar to analytical ones.Methods. We used the representation of the orbital elements from the TASS ephemeris analysed over 10 000 years as a reference template. We obtained the proper frequencies with both numerical and the TASS ephemeris over 1000 years only. A least-square procedure allowed us to get the analytical representation of an orbital element in this limited interval.Results. We acquire the representation of the mean longitude of Titan from JPL ephemeris over 1000 years. For almost all components, the corresponding amplitudes and phases are similar to the relative terms from TASS. The biggest difference between our representation and the mean longitude of Titan of JPL is less than 100 km over 1000 years, and the standard deviation is about 26 km