Context. Due to our increasing knowledge on the Galactic and stellar neighborhood of the Solar System, modern long-period comet motion studies have to take into account both stellar perturbations and the overall Galactic potential. Aims. Our aim is t
o propose algorithms and methods to perform numerical integration of a Solar System small body equations of motion much faster and at the same time with greater precision. Methods. We propose a new formulation of the equations of motion formulated in the Solar System barycentric frame but accurately accounting for the differential perturbations caused by the Galactic potential. To use these equations effectively we provide numerical ephemerides of the Galactic positions of the Sun and a set of potential stellar perturbers. Results. The proposed methods offer the precision higher by several orders of magnitude and simultaneously greatly reduce the necessary CPU time. The application of this approach is presented with the example of a detailed dynamical study of the past motion of comet C/2015 XY1.
(Abreviated) Comet C/2017 K2 PANSTARRS drew attention to its activity already at a time of its discovery in May 2017 when it was about 16 au from the Sun. This Oort spike comet will approach its perihelion in December 2022, and the question about its
dynamical past is one of the important issues to explore. To this aim it is necessary to obtain its precise osculating orbit, its original orbit, and propagate its motion backwards in time to the previous perihelion. We study a dynamical evolution of C/2017 K2 to the previous perihelion (backward calculations for about 3-4 Myr) as well as to the future (forward calculations for about 0.033 Myr). Outside the planetary system both Galactic and stellar perturbations were taken into account. We derived that C/2017 K2 is a dynamically old Oort spike comet (1/a$_{prev}$ = (48.7 $pm$ 7,9) x10$^{-6}$ au$^{-1}$) with the previous perihelion distance below 10 au for 97 per cent of VCs (nominal q$_{prev}$ = 3.77 au). It means that C/2017 K2 has already visited our planetary zone during its previous perihelion passage. Thus, it is almost certainly a dynamically old Oort spike comet.
