The lunar theory ELP revisited. 
Introduction of new planetary perturbations

On the basis of the semi-analytical theory ELP, a new solution has been built that makes use of the planetary perturbations MPP01 constructed by P. Bidart. This new solution, called ELP/MPP02, is an analytical solution that contains all the perturbations to represent the lunar motion. The level of truncation for the series is the centimeter. This limit induces analytical developments of a reasonable dimension. ELP/MPP02 is compared to the JPL ephemerides DE405 and DE406. After fitting the constants and the reference frame to DE405, over a few centuries, numerical comparisons with DE405 show a significant improvement in precision, in particular for the radius vector. Over the long range of several millennia, comparisons with DE406 show an improvement in the long periodic perturbations due to planets. On the time interval of one century around J2000 we added to ELP/MPP02 numerical complements $\rho_{405}$ based on the differences with DE405 in such a way that ELP/MPP02+$\rho_{405}$ has the precision of the numerical integration. Using this new ephemeris to analyse LLR data provided since 1970, we build the solution ELP/MPP02(LLR) fitted to LLR observations. Various approximations have been also tested on the residuals DE406 - ELP/MPP02. We first determined corrections to the secular variations of the mean longitude, longitudes of the node and perigee under a simple polynomial form; they reduce the differences between ELP/MPP02 and DE406 to less than 3 arcsec in longitude and latitude and 2 km in distance over the whole time interval of DE406 [ -3000; +3000] . Next, the differences themselves between the two solutions are approximated over 2 millennia with Poisson series in a pseudo analytical form similar to ELP/MPP02. We reduce the residuals to less than 0.03 arcsec in longitude and latitude and 30 m in distance