Trajectories Derived from Periodic Orbits around the Lagrangian Point L1 and Lunar Swing-Bys: Application in Transfers to Near-Earth Asteroids

Abstract

To present a set of trajectories derived from the retrograde periodic orbits around the Lagrangian equilibrium point L1, this paper considers the Circular Restricted Three-body Problem with Earth-Moon masses (CR3BP), the Restricted Bicircular, and Full Four-Body Sun-Earth-Moon-spacecraft Problems (BCR4BP and FR4BP, respectively). These periodic orbits are predicted by the dynamics of the CR3BP. To generate the trajectories of this set, first, slightly different increments of velocity (∆Vs) from those needed to generate periodic orbits around L1 are applied to a spacecraft in circular low Earth orbits in the same direction of their motion when the Earth, the spacecraft, and the Moon are aligned in this order. Thus, translunar trajectories derived from the periodic orbits are obtained and they will lead the spacecraft to the vicinity of the Moon. Depending on the values of the |∆Vs|, which are also functions of the relative positioning between the Sun, the Earth, and the Moon, three types of trajectories of interest are found: Collision with the Moon, escape, and geocentric orbits with large semi-major axes. For a well-defined interval of the |∆Vs|, the trajectories accomplish swing-bys with the Moon and obtain energy to escape from the Earth–Moon system and reach Near-Earth Asteroids (NEAs) between the orbits of Venus and Mars. This procedure reduces the costs of inserting spacecraft into transfer trajectories to a set of NEAs in terms of the required |∆V| by up to 5% when compared to Lambert’s problem, for example. This work also presents analyses of examples of transfers to the NEAs 3361 Orpheus, 99942 Apophis, and 65803 Didymos, from 2025 on.

Alternative approach to the DART mission by the use of gravity assist maneuvers with the Moon and solar sails

“…Low-cost escape trajectories are found in this system for a variety of initial conditions of position and velocity 20 . Starting from a circular LEO and the initial conditions for a periodic orbits of family G 16 , as the one showed in Fig.…”

“…Starting from a circular LEO and the initial conditions for a periodic orbits of family G 16 , as the one showed in Fig. 1 a in an inertial geocentric reference frame ( ), as defined by Ribeiro, de Melo e Prado 20 , small increments of velocity are introduced in order to change the trajectory of the vehicle and to provoke a swing-by with the Moon, in what was called “trajectories G of escape” (TGE). The required injection velocity to produce trajectories G similar to the one shown in Fig.…”

“…1 b is given by Eq. ( 2 ) 20 as a function of the altitude of the initial parking LEO ( ), the longitude of the ascending node of the Moon ( ), and the true anomaly of the Moon ( ). A scheme showing these variables is presented in Fig.…”

“…Complementary to swing-by maneuvers, periodic orbits of family G 16 have been used to produce low-cost escape trajectories in the Earth-Moon system. The periodic orbits of family G were first introduced by Broucke 16 for the Restricted Three Body Problem (R3BP), but its existence in more complex dynamics have already been documented in several works 17 – 20 . For example, in de Melo, Macau and Winter 18 , as these orbits pass naturally close to the surface of both the Earth and the Moon, minimal variations in the velocity were introduced in the trajectory of the vehicle inside a periodic orbit of family G to cause a swing-by with the Moon and its escape from the Earth-Moon system.…”

“…In more recent works 20 , different types of trajectories derived from periodic orbits of family G, called trajectories G (TGs), were mapped as function of the launch velocity, starting from a Low-Earth Orbit (LEO), and the position of all the bodies involved in the dynamics of the system. Ribeiro, de Melo and Prado 20 demonstrated that the use of these TGs could lead to a decrease of 2.5–5% in the necessary s to perform transfers to NEAs, when compared to reference methods as the method of Lambert. In " Restricted four-body problem and trajectories G " Section, more details about these trajectories are presented.…”