High-order solutions around triangular libration points in the elliptic restricted three-body problem and applications to low energy transfers

•The high-order solutions of the motion around triangular points in ERTBP are constructed.•Two-impulse low energy transfers to triangular point orbits of Earth–Moon system are designed.•Low-thrust low energy transfers to triangular point orbits of Earth–Moon system are constructed.•The reason that the low energy transfers are fuel-efficient is analysed from the viewpoint of energy.

High-order series expansions around triangular libration points in the elliptic restricted three-body problem (ERTBP) are constructed first, and then with the aid of the series solutions, two-impulse and low-thrust low energy transfers to the triangular point orbits of the Earth–Moon system are designed in this paper. The equations of motion of ERTBP in the pulsating synodic reference frame have the same symmetries as the ones in circular restricted three-body problem (CRTBP), and also have five equilibrium points. Considering the stable dynamics of triangular libration points, the analytical solutions of the motion around them in ERTBP are expressed as formal series of four amplitudes: the orbital eccentricity of the primary, the long, short and vertical periodic amplitudes. The series expansions truncated at arbitrary order are constructed by means of Lindstedt–Poincaré method, and then the quasi-periodic orbits around triangular libration points in ERTBP can all be parameterized. In particular, when the eccentricity of the primary is zero, the series expansions constructed can be reduced to describe the motion around triangular libration points in CRTBP. In order to check the validity of the series expansions constructed, the domain of convergence corresponding to different orders is studied by using numerical integration. After obtaining the analytical solutions of the bounded orbits around triangular points, the target orbits in practical missions can be expressed by several related parameters. Thanks to the series expansions constructed, two missions are planned to transfer a spacecraft from the Earth to the short periodic orbits around triangular libration points of Earth–Moon system. To complete the missions with less fuel cost, low energy transfers (two-impulse and low-thrust) are investigated by means of numerical optimization methods (both global and local optimization techniques). Simulation results indicate that (a) the low-thrust, low energy transfers outperform the corresponding two-impulse, low energy transfers in terms of propellant fraction, and (b) compared with the traditional Hohmann-like transfers, both the two-impulse, low energy and low-thrust, low energy transfers perform very efficiently, at the cost of flight time.