Symplectic map description of Halley's comet dynamics

The main features of 1P/Halley chaotic dynamics can be described by a two dimensional symplectic map. Using Mel'nikov integral we semi-analytically determine such a map for 1P/Halley taking into account gravitational interactions from the Sun and the eight planets. We determine the Solar system kick function ie the energy transfer to 1P/Halley along one passage through the Solar system. Our procedure allows to compute for each planet its contribution to the Solar system kick function which appears to be the sum of the Keplerian potential of the planet and of a rotating circular gravitational dipole potential due to the Sun movement around Solar system barycenter. We test the robustness of the symplectic Halley map by directly integrating Newton's equations over $\sim 2.4\cdot 10^4$ yr around Y2K and by reconstructing the Solar system kick function. Our results show that the Halley map with fixed parameters gives a reliable description of comet dynamics on time scales of $10^4$ yr while on a larger scales the parameters of the map are slowly changing due to slow oscillations of orbital momentum.