Quasi-singular methods for determining the limiting properties of the spacecraft entry trajectories into the planet's atmosphere

The problem of the limiting dynamic properties determining of the spacecraft (SC) trajectories with non-zero aerodynamic quality entered the atmosphere of a planet (Earth, Venus, Mars etc.) often has singularities. First of all, the singularity of the optimal control and of the SC' dynamic characteristics is appeared after the interplanetary flight or after the SC maneuver in deep space. The target landing point is either the regular landing site on the Earth, or is the selected region with the specified geological characteristics on the surface of the planet (for example, Venus). The bulk and very laborious numerical experiments have shown that the optimal in terms of maximum coverage of the planet's surface by virtual trajectories of the descending spacecraft often presented as the trajectory with non-monotonic altitude changes (up to secondary spacecraft departure from the atmosphere and secondary entrance) and with the compact atmospheric regions of the spacecraft velocity break points. As it turned out, at the same time, the reachability areas on the surface of the planet obtained by optimization methods, have non-trivial geometric details. One of the most interesting nuances is the non-smooth beake-like border of the reverse side of the reachable area. In the certain sense, we can talk about generalized SC ricochets on the planet's atmosphere, which allow us to expand the reachability of the SC (both in terms of range and lateral range). The paper presents a phenomenological semi-analytical model for the phase beams constructing of the quasi-singular SC descent in the planet's atmosphere, which allows us to synthesize the main nuances of the reachability areas, instead the long-time big data optimization. As the result, the time of the preliminary mission ballistic analysis can be significantly reduced (paper)