P. Bretagnon's research while affiliated with French National Centre for Scientific Research and other places

... where the spatial gradient operator is given by ∂ i ≡ ∂/∂ x i , the Newtonian gravitational potential is given by U and the subscripts j, k denote the j-th gravitational and k-th nongravitational perturbation, respectively. Owing to the increased accuracy requirements in fields such as astrometry (Soffel and Han 2019), geodesy (Müller et al. 2008;Soffel and Langhans 2012), the rapid improvements in time and frequency stability (Exertier et al. 2019), and further, the development and utilisation of advanced telecommunication systems for radio tracking of interplanetary probes, we require that Einstein's general theory of relativity (d'Inverno 1992;Weinberg 1972) be taken into account for any mission requiring highly accurate orbit information and practically all astronomical and geodynamical observations (Brumberg et al. 1998). Within the Solar System, the so-called effects of general relativity are accounted for using the first post-Newtonian (PN) approximation (Soffel and Han 2019) and we refer the reader to the seminal papers Damour et al. (1991Damour et al. ( , 1992bDamour et al. ( , 1993Damour et al. ( , 1994 for extensive technical details. ...

... To construct a planetary theory two major ways can be considered: the use of a numerical integrator [8], [6] or the use of analytical methods to integrate the problem [2], [22], [23], [26]. ...

... In the present case, the same effect should eventually be observed by performing a much longer integration. It might be interesting to remark that a similar calculation, namely approximating the dynamics of the SJS system with a quasi–periodic motion, has been performed by Bretagnon and Simon ([3]). Their method was based on the technique introduced by Krylov and Bogoliubov (which is reminiscent of the Lindstedt's series method). ...

... The mean orbital elements of the inner planets are given in the form of segments of the power series to the second power of time (23). Bretagnon (1982c) complemented the theory of motion of the inner planets by the third approximation that includes the relativistic perturbations from the Sun (Lestrade and Bretagnon, 1982). As a result, the ephemerides were computed with an accuracy of 0.0005′′ for Mercury, 0.0030′′ for Venus and the Earth, and 0.0047′′ for Mars. ...

... The Lagrange equations were presented in compact form (Chapront et al., 1975), which makes it possible to obtain the perturbations using an iterative process. The mutual perturbations of all pairs of outer planets are given in Simon and Bretagnon (1975b). ...

... Expressions for the non-rigid Earth nutations in the longitude and the obliquity, presented in (2), in the paper of Bretagnon et al. (1999) contain only the real parts of the complex values. The exact expressions for the non-rigid Earth nutations in the longitude and the obliquity, in the complex values, are obtained in this investigation: ...

... The angular arguments, ζ ( j) (t), are linear combinations of the Delaunay arguments, whose linear change rates then specify the respective frequencies (Simon et al. 1994;Petit and Luzum 2010). The real constants C 2m( j) , S 2m( j) measure the frequency-dependent contribution of the potential raised by Earth and Sun with respect to the lunar reference sphere of radius R, more specifically, through ...

... Comparison of the frequencies obtained analytically and numerically by Applegate et al. (1986), Bretagnon and Francou (1992), Laskar (1990), andNobili et al. (1989) showed good agreement between the results (Table 3). Bretagnon (1996) considered the relationship between the general planetary theory constructed by an iterative method and the classical theory containing secular terms in the inclinations and eccentricities. The interval of applicability of the general planetary theory reaches several million years at a low ephemeris accuracy. ...

... This equation can be solved only by numerical integration and the results are given in polynomial form. To date, the best approximation is given by Williams (1994) and Simon et al. (1994); however, these solutions are applied only in the best professional computer codes, while the large majority of commercial programs use the older Lieske et al. (1977) solution (the so called "IAU formula", since it is based upon the International Astronomical Union IAU/1976/ system of astronomical constants) or its �rst order approximation or even a simple proportional correction with the average value of -0.024 arcsec per century. ...

... In order to get the time transformation between t TX and t TB , the differential (54) must be integrated for the whole time span we consider for the radio-science experiment (nominally, the duration of the space mission). In literature, there are also some approximate analytical solutions of (54) (see [22]), which can be used to perform the time transformations. ...