Mapping of initial conditions for libration point orbits

Highlights

• Structure and properties of LPO families are studied by mapping approach.

• Periodic families, which bifurcates from vertical and halo orbits are constructed.

• A modified bounding planes method is used for the generation of LPOs.

• The applicability area of the bounding planes method is presented.

Abstract

In the framework of circular restricted three-body problem the libration point orbits form the families of periodic and quasi-periodic solutions. In the paper, the mapping of initial conditions is utilized to describe and study the structure and properties of these families. A new numerical method for orbit generation, which is applicable both for the periodic and quasi-periodic orbits, is provided and explored. The applicability area of the proposed method is constructed and analyzed for the vicinity of Sun-Earth L1. Based on the comprehensive numerical investigation of this area, the $\mathit{xz}$-maps aligning the initial conditions of the orbits crossing this plane perpendicularly with their properties were constructed and analyzed. The quasiperiodic Lissajous and quasi-halo orbit families appears on these maps as the domains surrounding the curves corresponding to periodic halo and vertical families. Accurate analysis of these domains made in possible to study the structure of resonant $k$-periodic families, which bifurcate from halo and vertical orbits. The initial conditions of these families are presented by the curves, which thread the quasi-halo and Lissajous domains. These k-periodic families are computed up to $k=10$ for the resonant orbits bifurcating from the halo family, and up to $k=25$ for the resonant orbits bifurcating from the vertical family. Some examples of such orbits different provided. The maps of initial conditions illustrate several important properties of the libration point orbit families, and can be useful for mission design as a tool to select the orbit fitting the mission requirements.

Introduction

The libration point orbits (LPOs) provide a variety of potential applications, which have been extensively used in mission design in the last decades. Sun-Earth L1 LPOs were successfully utilized in ISEE-3 3 (Farquhar, 1998), WIND (Sharer et al., 1995), SOHO (Dunham et al., 1992), ACE (Sharer and Harrington, 1996) and Genesis (Burnett et al., 2003) and are considered as a destination of future missions implementing a concept of hazardous NEO detection and impact warning system (Dunham et al., 2013, Ikenaga et al., 2019, Kovalenko et al., 2018). The Circular Restricted Three-Body Problem (CR3BP) approximating the actual dynamical system serves as a fundamental basis of trajectory design for libration point missions. Thus, the development of numerical and analytical tools providing a deeper understanding of CR3BP dynamics is increasingly significant.

In the framework of CR3BP, the spacecraft is represented by a point with an infinitesimal mass moving in a gravitational field of two primary point masses following circular orbits around their common barycenter. The spacecraft location is specified by the coordinates $\mathit{xyz}$ in the rotating coordinate frame. The $x$-axis is directed from the heavier primary point to the lighter one, the $z-$axis is perpendicular to the primary masses motion plane, and the $y-$axis completes the right-handed orthonormal triad.

LPOs are represented by periodic families (such as planar Lyapunov orbits, halo orbits, vertical orbits, and axial orbits), and quasi-periodic ones (Lissajous, quasi-halo orbits, quasi-periodic Lyapunov orbits). Goudas (1963) showed that in the CR3BP there are only three types of periodic orbits: (A) Orbits symmetric with respect to $\mathit{xz}$-plane; (B) Orbits symmetric to $x$-axis; (C) Double symmetry orbits with respect to the $\mathit{xz}$-plane and the $x$-axis. Halo orbits, firstly defined by Farquhar (1970), are symmetric with respect to $\mathit{xz}$-plane and have equal frequencies of in $\mathit{xy}$-plane and out of$\mathit{xy}$-plane oscillations. Vertical orbits are three-dimensional orbits with the period of in-plane component of motion two times less as the period of out-of-plane component of motion. Axial orbits are symmetric with respect to $x$-axis. A detailed overview of the periodic orbits in CR3BP is provided in the study of Doedel et al. (2007), which, however, doesn’t include the resonant $k$-period families bifurcating from halo and vertical families. Such $k$-period families bifurcating from the halo family were described and studied by Howell and Campbell (1999) up to $k=5$. Nevertheless, there is a lack of papers describing the $k$-period families bifurcating from the vertical family. If the ratio of periods of in-plane and out-of-plane components of motions is irrational, quasi-periodic motion occurs.

Numerous semi-analytical and numerical methods were developed for the computation of LPOs. Farquhar and Kamel (1973) used the method of Lindstedt-Poincaré and computerized algebraic manipulations to obtain a third-order analytic solution for quasi-periodic orbits about the translunar libration point. A similar approach was implemented by Richardson (1980), who provided an analytical solution for halo-type periodic motion about the collinear libration points in the Sun-Earth system. The method of Lindstedt-Poincaré was also utilized for computation of other periodic and quasi-periodic solutions (Archambeau et al., 2011, Gómez et al., 2003, Gómez et al., 1997, Masdemont, 2005). Another analytical approach known as a reduction to the center manifold is based on the normalization of Hamiltonian and separation of the hyperbolic and elliptic directions (Gómez et al., 1997, Jorba and Masdemont, 1999, Jorba and Villanueva, 1998, Zhang and Li, 2018). The numerical method utilizing differential corrections was used by Breakwell and Brown (1979) for a comprehensive study of L1 and L2 halo families in the CR3BP corresponding to the Earth-Moon system. Howell (1984) extended these results to systems with other mass ratios. The multiple-shooting numerical approach for the calculation of quasi-periodic LPOs was proposed by Howell and Pernicka (1987). Kolemen et al. (2012) utilized truncated Fourier series for parametrization of the Poincaré sections employed to numerical computation of quasi-periodic orbits in CR3BP.

Direct computation of LPOs, using classical numerical methods for integration of ordinary differential equations, is not feasible due to their strong instability. A small error in the state vector provides the appearance of an unstable component of motion which is rising with time, and results in escaping the vicinity of the libration point. Thus, even if the initial conditions providing an unstable LPO are known with numerical precision, the application of the trajectory extension technique is required to prevent the escape.

Hechler and Cobos (2003) proposed the bisection approach which is able to prevent the rising of the unstable component by periodic corrections of the velocity vector. The approach is based on the separation of the escape scenarios on two types: escaping towards the Earth and escaping away from the Earth. The value of velocity correction is then calculated iteratively by the bisection method, aiming to remain between these two scenarios. The method is successfully implemented to construct several quasi-periodic orbits in the ephemeric force model, but the separation criteria are not formalized, which makes it complicated to apply the method for other orbits and evaluate the area of its applicability. The corrections are performed in the direction of 28.6° from the Sun-Earth axis, defined by the linear dynamics around the libration point. The matter of adjustment of this direction for the orbits of large amplitude, which do not follow the linear dynamics, is not considered.

Ren and Shan (2014) implemented the bisection approach for the calculation of LPOs around Earth-Moon L2. As the separation criteria, they used the spherical surfaces whose centers are located in the heavier point mass. The escape direction is evaluated by the first crossing of one of the surfaces by the trajectory. The initial conditions are taken on the parking orbit around the Moon. The first velocity correction is made in the direction, parallel to the spacecraft velocity, to generate an LPO together with the transfer orbit. The trajectory extension is performed by periodical corrections of the state vector (both the coordinates and the velocity), maximizing the time between the bounding spherical surfaces. Five components of the state vector are varied while the sixth one is computed in order to preserve the Hamiltonian. Compared to the method used by Hechler and Cobos (2003) this technique ensures the optimal direction of the corrections, but the number of varied parameters increases the computation time. The applicability area of the proposed algorithm is not discussed.

In order to extend the applicability of the bisection approach to the orbits with higher energy, Zhang and Li (2016) proposed a more complicated approach, utilizing the surfaces produced by rotation of a planar Lyapunov orbit around the $x$-axis. The calculation of velocity corrections required for the trajectory extension is performed using a multistep procedure involving the Brent’s root-bracketing method, minimization of a two-variable function using the Newton iteration method, and computation of the intersections between the trajectory and nonlinear surface. The method is used to calculate large-amplitude orbits, but the computation is time-consuming. As reported, the computation of each correction takes about 5 s to compute on a PC with 2.93 GHz Intel i3-530 processor. The applicability of the algorithm is demonstrated on a number of high energy orbits and their Poincaré sections.

Systematization and analysis of LPO families are usually performed in terms of Poincaré sections representing the coordinates of the intersections with $\mathit{xy}$-plane for the LPOs with the same value of Jacobi constant (Gómez and Mondelo, 2001). In these sections, quasi-periodic orbits form the invariant curves, while the periodic ones are represented by a finite number of points. The quasi-halo and Lissajous zones are clearly identified by concentric invariant curves encircling the fixed points, which correspond to the vertical and halo orbits. The transition zone corresponds to the homoclinic connection of the planar Lyapunov orbit (Gómez et al., 2003). At higher energies, the bifurcations of period multiplication become visible as “island chain” structures on Poincaré sections (Gómez and Mondelo, 2001, Zhang and Li, 2016).

Poincaré section by the $\mathit{xy}$-plane is a useful tool for analysis of the dynamics in the center manifold. Nevertheless, discussing several important aspects of trajectory design and LPO computation in terms of Poincaré sections is not feasible. For instance, the applicability area of a new LPO generation method cannot be described by a single Poincaré section as it is constructed for a single value of Jacobi constant. Visualization of periodic families and their bifurcations by means of single Poincaré sections is inconvenient. The initial conditions for the orbits cannot be definitely visualized on such Poincaré sections as the velocity direction is different at each point. An invariant curve, associated with a quasi-periodic orbit, reflects the orbit type but it is hard to evaluate the orbit geometry outside of the $\mathit{xy}$-plane.

Aiming to address these issues, this paper proposes an approach based on $\mathit{xz}$-maps of LPOs, and methods for their computation. The position of each point on such a map represents a single orbit, while the color illustrates its characteristics. Thus, it can serve as a tool, providing the initial conditions for an orbit meeting the specified requirements, as the orbit type, amplitude, or avoidance of exclusion zones (zone of eclipse for L2 or solar interference zone for L1). Since $\mathit{xz}$-maps present the orbits with different Jacobi constants, they can also be used for the evaluation of the applicability area of different LPOs computation methods. Each point on the map represents the initial conditions corresponding to the position on the $\mathit{xz}$-plane and the velocity directed along the $y$-axis. The velocity magnitude is calculated to eliminate the unstable component. The orthogonality of the velocity to the $\mathit{xz}$-plane makes it possible to match each point with a single orbit. The orbit families then form the domains on the map that reflect their structure and properties.

The computation of LPOs is performed using the simplest realization of the bisection approach utilizing the planes, orthogonal to the $x$-axis, as the separation criteria (Aksenov and Bober, 2018). In this work, the trajectory extension algorithm is supplemented with the procedure of determination of an optimal direction for each correction and Hamiltonian preservation technique. Exploiting the properties of the state transition matrix made it possible to avoid complex multidimensional minimization procedures implemented in the studies mentioned above (Ren and Shan, 2014, Zhang and Li, 2016) and decrease the computation time. The applicability area of the developed LPO computation algorithm is studied for the Sun-Earth L1.

The analysis of CR3BP dynamics in the vicinity of the Sun-Earth L1 is performed in terms of the proposed mapping approach and compared with the analysis performed in terms of Poincaré sections by the $\mathit{xy}$-plane. The advantages of the mapping approach were used for the analysis of resonant k-periodic orbits of type A and C. The periodic families of type B (as the axial family) were out of the investigation as they do not cross the $\mathit{xz}$-plane perpendicular and therefore can not be mapped. In the considered area, the k-period orbits bifurcate from the halo and vertical families. The initial conditions providing such multi periodic orbits are evaluated using the maps and then adjusted numerically. The k-period families are computed up to $k=10$ for the resonant orbits bifurcating from the halo family, and up to $k=25$ for the resonant orbits bifurcating from the vertical family.