Communications in Nonlinear Science and Numerical Simulation
High-order solutions around triangular libration points in the elliptic restricted three-body problem and applications to low energy transfers
Introduction
There are five equilibrium points for the equations of motion in circular restricted three-body problem (CRTBP), three of them, called collinear libration points ( and ), are unstable, and the remaining two, named triangular libration points ( and ), are stable for all values of mass parameter less than (Routh’s critical value) except for a set of values, which can be satisfied for most cases of three-body system in our Solar system, such as Sun–Earth and Earth–Moon systems [1]. Due to the special configuration relative to the primaries and the complicated dynamics of equilibrium points, the region of space in the vicinity of these points has long captured the interest of scientists and engineers. The applicative importance of equilibrium points in the Sun–Earth and Earth–Moon systems has been recognized by Farquhar [2]. Hence, it is of great practical significance to analytically study the dynamics around equilibrium points.
Usually, the numerical methods are used to study the dynamics around equilibrium points, such as numerical integration, differential correction, optimization technique, and so on. Trajectory design by numerical methods relies on the experience of the designers, and is a trial and error process. While, analytical solutions can explicitly describe the dynamics around equilibrium points by several related parameters, and provide deep insights for understanding the dynamics. In addition, analytical solutions around equilibrium points perform efficiently in the process of preliminary trajectory design. Lindstedt–Poincaré (L–P) method is commonly used to construct the analytical solutions of the motion around equilibrium points. Richardson [3] analytically derived the third-order analytical solution of halo orbits around the collinear libration point in CRTBP. The third-order analytical solution provides the initial guess halo orbits, and then the accurate halo orbits in CRTBP can be calculated by differential correction method. Jorba and Masdemont [4] expressed Lissajous and halo orbits around the collinear libration points in CRTBP as formal series of two amplitudes: the in-plane and out-of-plane amplitudes, and the series expansions truncated at arbitrary order are constructed by means of L–P method. As it is known, the linearized equations of motion near collinear libration points act center × center × saddle behaviors, such that there are center manifolds (such as Lissajous and halo orbits) as well as hyperbolic manifolds (stable and unstable manifolds). Taking into account the center and hyperbolic dynamics of collinear libration points in CRTBP, Masdemont [5] expanded the invariant manifolds as power series of four amplitudes: the amplitude of unstable manifold , the amplitude of stable manifold , the in-plane amplitude and the out-of-plane amplitude , and then high-order solutions of invariant manifolds up to arbitrary order are constructed by using L–P method. By using the series expansions of invariant manifolds, the general dynamics around the collinear libration points, such as Lissajous and halo orbits, stable and unstable manifolds, transit and non-transit trajectories, can all be parameterized. The analytical solutions of invariant manifolds in CRTBP have been applied to look for the rescue trajectories that leave the lunar surface [6], and to study two-impulse transfers between the low Earth parking orbit and Lissajous orbits in the Earth–Moon system [7]. Gómez and Marcote [8] utilized the L–P method to construct the analytical solutions of the bounded orbits of Hill’s equation, which corresponds to the reduced case of CRTBP when the mass parameter satisfies . For triangular libration points, Zagouras [9] analytically derived a fourth-order solution for the three-dimensional periodic orbits. Considering the stable dynamics of triangular points in CRTBP, Lei and Xu [10] expanded the motion around them as formal series of three amplitudes: the long periodic amplitude , short periodic amplitude and vertical periodic amplitude , and the series expansions up to arbitrary order have been constructed, then the practical convergence of the series expansions constructed has been computed for Sun-Jupiter, Sun–Earth and Earth–Moon systems. Taking into account the perturbation of solar gravity and the orbital eccentricity of the Moon, Farquhar et al. [11] analytically constructed the third order solution of the quasi-periodic motion around collinear libration points of Earth–Moon system. In the real Earth–Moon system, periodic orbits around equilibrium points no longer exist and are replaced by quasi-periodic orbits; Hou and Liu [12], [13] analytically studied the quasi-periodic motion around triangular and collinear libration points.
Compared to CRTBP, the elliptic restricted three-body problem (ERTBP) could approximate the real three-body system in our Solar system more accurately. The equations of motion of ERTBP in the pulsating synodic system are non-autonomous, but there are still five equilibrium points behaving similar dynamics to the ones in CRTBP. Hou and Liu [14] expressed Lissajous and halo orbits around the collinear libration points in ERTBP as formal series of three amplitudes: the orbital eccentricity of primary, the in-plane and out-of-plane amplitudes. Based on the works of [5], [14], Lei et al. [15] analytically investigated the invariant manifolds associated with Lissajous and halo orbits around collinear libration points in ERTBP. Ren et al. [16] derived the third-order solution of the elliptic equations of relative motion, which correspond to the reduced case of ERTBP when the mass parameter satisfies , and conceptually presented the general procedure of constructing high-order analytical solutions by means of L–P method. About the analytical studies for Earth–Moon libration point satellites, such as flight mechanics, utilization and station-keeping problems, the readers are suggested to refer to literature [2], [17]. The systematic investigations about the dynamics and applications to mission design near collinear and triangular libration points have been done by the Barcelona research group [18], [19], [20], [21].
The first goal of this paper is to construct the high-order solutions of the bounded orbits around triangular libration points in ERTBP. Considering the stable dynamics of triangular points, the general solutions around them in ERTBP are expressed as formal series of four amplitudes: the orbital eccentricity, the long, short and vertical periodic amplitudes, and the series expansions up to arbitrary order are constructed by means of L–P method. The series expansions constructed can explicitly describe the long, short and vertical quasi-periodic orbits around triangular libration points in ERTBP. Since CRTBP is a particular case of ERTBP when , the analytical solutions constructed in ERTBP can be reduced to describe the motion around triangular libration points in CRTBP. In order to check the validity of the series expansions, numerical integration with the initial state provided by the analytical solutions is carried out to study the practical convergence.
Taking advantage of the specific configuration and dynamics of triangular points, some applications have been proposed. For example, Schutz [22] proposed that the triangular point orbits of Earth–Moon system could be ideal locations to place astronomical telescopes or a space station; O’Neill [23] presented an idea about building space colonies at triangular libration points. Due to the stable dynamics of triangular libration points, the advantage for a spacecraft orbiting around these points lies in that less fuel is required for station-keeping. Usually, the bounded orbits (long, short, vertical or quasi-periodic orbits) around these points are considered as the target orbits of these missions. With the aid of the analytical solutions constructed, the bounded orbits around triangular points can be determined by several related parameters.
Due to the promising applications of these points in Earth–Moon system, some researchers have investigated the problem about transferring a spacecraft from the Earth to the vicinity of triangular libration points. Traditionally, Hohmann-like transfer is a simple way, and the flight time is about 5 days. Salazar et al. [24] proposed a novel transfer strategy to send a spacecraft to the vicinity of triangular points, combing trajectories derived from the periodic orbits of Family G in three-body system with swing-by maneuvers. Vaquero and Howell [25] have recently studied three-maneuver transfers to the short periodic orbit around of the Earth–Moon system, based on the invariant manifolds associated with the orbits in resonance with the Moon. In [26], [27], the traveling between the equilibrium points of Earth–Moon system and the Moon as well as the Earth are studied in CRTBP. In general, the transfers between the Earth and triangular points of Earth–Moon system in previous research are fuel consuming. For a long-term unmanned mission, it is essential to reduce the fuel cost in order to increase the scientific load, which is the reason that low energy transfers are of great interest in deep space exploration. Generally, there are two kinds of low energy transfers. The first kind of low energy transfer corresponding to the interior capture (called interior WSB transfer) takes advantage of the gravity assist of the Moon to reduce fuel cost [28], [29], [30]. The other kind of low energy transfer corresponding to the exterior capture is based on the Weak Stability Boundary (WSB) theory [31], [32], and called exterior WSB transfer. Exterior WSB transfer takes the gravity assist of the Sun to raise the altitude of periapsis of the spacecraft to the height of lunar orbit, and then less fuel cost is required to insert the spacecraft into the target orbit. In [33], the ballistic capture mechanism of exterior WSB transfer are explained by couple CRTBP approximation, that is, exterior WSB transfer can be constructed with the aid of the intersection of invariant manifolds of two three-body systems (Sun–Earth + Moon and Earth–Moon systems). The same concept of exterior WSB transfer is applied to design an impulsive trajectory transferring a spacecraft from LEO to a lunar halo orbit [34]. The Japanese mission MUSES-A, renamed HITEN, is the first mission that adopted the exterior WSB transfer to realize its objectives, including ten lunar swingbys, insertion of a sub-satellite into an orbit around the Moon, an excursion to the triangular points in the Earth–Moon system, and so on [35]. HITEN is also the first spacecraft that flies to the vicinity of triangular libration points of Earth–Moon system. Recently, the low energy transfers are adopted by the Gravity Recovery and Interior Laboratory (GRAIL) mission [36].
Incorporating the low-thrust propulsion into the multi-body environment, the transfers perform more fuel-efficiently compared to the impulsive transfers due to the high specific impulse of low-thrust propulsion, but the problem of low-thrust trajectory design becomes more complicated. Recently, some new techniques for the trajectory design, combining the low-thrust propulsion and low energy transfer, have been proposed. Dellnitz et al. [37] used the invariant manifold technique and the low-thrust with set oriented methods to construct the Earth–Venus low-thrust transfer. Pergola et al. [38] combined the low thrust propulsion with ballistic trajectories on invariant manifolds associated with multiple CRTBPs to design the transfers between halo orbits of Sun–Earth and Sun–Mars systems. The concept of attainable set proposed and summarized in [39], playing the same role as invariant manifold in trajectory design, has been applied to the design of interplanetary low-thrust transfers [40], [41], and the design of low energy, low-thrust transfers to the Moon [39], [42], [43], [44].
The other goal of this paper is to investigate how to transfer a spacecraft from the low Earth orbit to triangular libration point orbits of Earth–Moon system with less fuel cost, based on the analytical solutions constructed. In order to save fuel consumption, the gravitational attraction of the Sun is incorporated into the Earth–Moon CRTBP to design two-impulse, low energy transfers to triangular libration point orbits of Earth–Moon system. In addition, due to the high specific impulse of the low-thrust propulsion, the low-thrust, low energy transfers are also constructed in order to further reduce the fuel cost. The reasons why low energy transfers (both two-impulse and low-thrust) are fuel-efficient are analyzed from the viewpoint of Jacobi energy.
The remainder of this paper is organized as follows. Section 2 presents the process of constructing high-order solutions around triangular libration points in ERTBP. Based on the analytical solutions, two missions are briefly described in Section 3. The optimization problems of two-impulse, low energy transfer and low-thrust, low energy transfer are established and solved in Sections 4 Two-impulse, low energy transfers to triangular libration point orbits, 5 Low-thrust, low energy transfers to triangular libration point orbits, respectively. At last, the conclusions together with discussions are drawn in Section 6.
Section snippets
Elliptic restricted three-body problem
The motion of an infinitesimal particle, such as a spacecraft , is studied in the gravitational field generated by two primaries and , moving around their barycenter in an elliptic orbit. Such a system is called ERTBP, in which the motion of the spacecraft in the barycentric synodic coordinate system is governed by [1]where is the orbital eccentricity of the secondary, and is the pseudo potential function of the three-body
Mission description
Based on the analytical solutions constructed in Section 2, two missions are planned. The first one is about transferring a spacecraft from LEO to the vicinity of point, and the second one is about transferring to the vicinity of point in the Earth–Moon system. In order to save fuel consumption, low energy transfers (impulsive and low-thrust) are explored to complete these missions in Sections 4 Two-impulse, low energy transfers to triangular libration point orbits, 5 Low-thrust, low
Two-impulse, low energy transfers to triangular libration point orbits
In this section, two-impulse, low energy transfers to triangular libration point orbits are designed in combination with the gravitational attraction of the Sun. The first velocity impulse (accelerating velocity impulse) is employed by the launch vehicle to force the spacecraft to depart from the low Earth orbit (LEO), and the second impulse (braking velocity impulse) is applied to insert the spacecraft into the target orbit. Such an impulsive low energy transfer problem is solved by
Low-thrust, low energy transfers to triangular libration point orbits
Due to the high specific impulse of low-thrust propulsion, this kind of transfer may be more fuel-efficient compared to the impulsive transfers. In this section, the low-thrust, low energy transfers to triangular point orbits are to be investigated in the multi-body environment in combination with low-thrust propulsion. Such a transfer can be described like this: a spacecraft is initially located on LEO, an accelerating velocity impulse, , is carried out by the launcher to force the
Conclusions and discussion
In this paper, firstly, high-order analytical solutions of the motion around triangular libration points in ERTBP are studied; secondly, taking short periodic orbits around and points of Earth–Moon system as target orbits, two-impulsive low energy transfers as well as low-thrust, low energy transfers are investigated by means of numerical optimization methods (including global and local optimization methods).
For triangular libration points in ERTBP, the solutions around them are expanded
Acknowledgments
Hanlun Lei wishes to thank Dr. Xiyun Hou for many help discussions and the authors are grateful to three anonymous reviewers for their constructive comments that substantially improved the quality of this paper. This work was carried out with financial support from the National Basic Research Program 973 of China (2013CB834103), the National High Technology Research and Development Program 863 of China (2012AA121602), the National Natural Science Foundation of China (Grant No. 11078001) and the
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