Optimum trajectories for spacecraft mission to asteroid Apophis with a return to the Earth

A topical task for astronautics is to prepare a mission to the asteroid Apophis that is approaching the Earth. Determination of energetically favorable trajectories for a spacecraft involved in this mission with a return to the Earth has been considered. Two possible variants of engines for this mission are analyzed. The first variant employs electric engines with low jet thrust, while the second variant employs a spacecraft accelerated and controlled by only high-thrust engines. It is shown that both variants can be used in the mission to Apophis, but the use of electric engines with low thrust allows the project characteristics to be significantly improved.