Numerical integration ephemerides are widely used in research and engineering for their high precision. However, subject to their finite available time spans, their use is limited in theoretical research, such as the studies of rotation and evolution. Previously, we successfully experimented on the analytical representation of the mean longitude of Titan of the Jet Propulsion Laboratory (JPL) ephemeris, as a function of combinations of proper frequencies, and related the results with what is given in the synthetic ephemerides obtained by the Theorie Analytique des Satellites de Saturne (TASS)

In this study the analytical representations of the other osculating elements of the JPL Titan ephemeris are accomplished in order to construct the new synthetic representations, which have the advantages of both systems: long-lasting stability, the system details of TASS, and the high precision of JPL.

A frequency analysis process was used to obtain the proper frequencies, amplitudes, and phases of the two ephemerides in the short term and the semi-long terms. For the proper frequency of the ascending node of Titan, which has a very long period, it is challenging to acquire the exact value, and the formula of TASS was used. The amplitude and phase of long terms were further calculated by a least-squares procedure.

Thanks to the accomplishments of the new synthetic representations of the JPL ephemeris, we report the complete combinations of the osculating elements of Titan. These combinations contain important dynamical information such as the proper frequencies. They will be useful in the theoretical research.