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The accomplishments of space flights and introduction of new astrometric methods (radar ranging, lunar laser-ranging, VLBI measurements) in the 1960s required considerably more precise planetary ephemerides than it was possible with classical analytical theories by Leverrier, Hill, Newcomb and Clemence. On the other hand, these modern data made possible the creation of such ephemerides. Two series of numerical ephemerides of planets most complete up to now, and of the same level of accuracy, are considered in this paper. There are the well-known numerical DE ephemerides of JPL as well as the EPM (Ephemerides of Planets and the Moon) ephemerides produced at the Institute of Applied Astronomy. The description of the dynamical models, the brief characteristics of DE118, DE200, DE403, DE405, DE410, EPM87, EPM98, EPM2000, EPM2004 ephemerides, and the comparison between DE410 and EPM2004 are given. The latest DE410 and EPM2004 ephemerides have resulted from a least squares adjustment to observational data totaling about 300000 position observations (1911–2003) of different types. The accurate radar observations of planets and spacecraft have made it possible not only to improve the orbital elements of planets but to determine a broad set of astronomical constants from the value of the astronomical unit (AU) to parameters of PPN formalism. Recent estimates of different astronomical constant are presented, and progress is shown in the improvement of the AU value, the parameters $\beta, \gamma $, as well as possible variability of the gravitational constant $G$.