Estimation of the lunar free libration modes based on the recent ephemerides

The accuracy of lunar laser ranging (LLR) experiment has already increased to about 1 cm in recent years. When the LLR data joined to the lunar dynamical model updated from DE430, three independent numerically integrated ephemerides called DE436, EPM2017 and INPOP17a were released in 2017. It prompts us to reanalysis the lunar rotation parameters of the ephemerides, and also, especially estimation of the lunar free libration modes. To this end, we performed a Fourier analysis and least-square fit with the data from these three ephemerides to estimate lunar free librations. The results show that the free librations for each ephemeris are consistent with each other and with the previously published DE421 and DE430 free libration values; minor differences in detail may result from differences in the physical parameters contained in each ephemeris. The Euler angles integrated in each ephemeris however, show a significant difference before 1800, relative to the difference after 1970. This indicates that special attention must be paid to backward integration when building ephemerides.