Advances in Astronomy

Volume 2019, Article ID 6324901, 17 pages

https://doi.org/10.1155/2019/6324901

## Initial Parameter Analysis about Resonant Orbits in Earth-Moon System

Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, China

Correspondence should be addressed to Ying-Jing Qian; moc.361@jyqecidnac

Received 17 September 2018; Revised 27 November 2018; Accepted 26 December 2018; Published 23 January 2019

Academic Editor: Geza Kovacs

Copyright © 2019 Li-Bo Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The initial parameters about resonant orbits in the Earth-Moon system were investigated in this study. Resonant orbits with different ratios are obtained in the two-body problem and planar circular restricted three-body problem (i.e., PCRTBP). It is found that the eccentricity and initial phase are two important initial parameters of resonant orbits that affect the closest distance between the spacecraft and the Moon. Potential resonant transition or resonant flyby may occur depending on the possibility of the spacecraft approaching the Moon. Based on an analysis of ballistic capture and flyby, the Kepler energy and the planet’s perturbed gravitational sphere are used as criteria to establish connections between the initial parameters and the possible “steady” resonant orbits. The initial parameter intervals that can cause instability of the resonant orbits in the CRTBP are obtained. Examples of resonant orbits in 1:2 and 2:1 resonances are provided to verify the proposed criteria.

#### 1. Introduction

Mean motion resonance is a common phenomenon that exists when there is a simple integer relationship between frequencies or periods [1, 2]. For example, Earth is in 8:13 resonance with Venus about the Sun; Io is in 2:1 resonance with Europa about Jupiter. The mean motion resonance of the Jupiter-Saturn is approximately 5:2 [3]. Research on resonant orbits for spacecraft can provide an alternative option for trajectory design and have potential to save propellant in real missions.

The two-body theory can be applied to many-body systems with only one major primary, where other primaries’ gravitational influences are negligible. Many resonant orbits can be found with a resonant ratio that is equal to the ratio of the orbital periods corresponding to the bodies in resonance, such as the 3:1 mean motion resonance found in the 55 Cancri planetary system [4, 5] and the 2:1 motion resonance in the extrasolar planetary systems HD 82943 [6] and Gliese 876 [7, 8]. Interestingly, most planets locked in resonance exhibit maximum separation in that they avoid a close approach to the other primary. Man-made missions also can be designed to have a resonant orbit. The Interstellar Boundary Explorer (IBEX) spacecraft is currently in a highly elliptical orbit around Earth in a 3:1 resonance with the Moon [9]. It is important to note that IBEX’s orbital apogee is also oriented to avoid the Moon. Voyatzis [10] carried out extensive numerical studies of planets trapped in 3:1 mean motion resonance. Antoniadou and Voyatzis [11] also found that the temporal evolution of the orbital elements depends on the mean motion resonances, which affects the dynamics and not vice versa. Murray and Dermott [2] summarized the perturbation equation for orbital elements in their monograph and showed significant periodic and quasi-periodic variations in the orbital elements. Therefore, the solution of the secular problem by Laplace for resonant orbits allows us to make assertions regarding the long-term stability of the solar system [12]. Laplace-Lagrange secular theory is valid for small eccentricities; i.e., without significant perturbations, resonant orbits for small eccentricities can exhibit constant resonant ratios after long-time evolution.

In the restricted three-body problem (i.e., CRTBP), some resonant orbits may also exist with resonant ratio that is* approximately* equal to the ratio of the orbital periods due to the secondary primary’s significant gravitational influence [13]. To be more specific, the time required to complete a revolution is not constant. Instead, for a* m*:*n* resonance in the circular restricted three-body problem, the spacecraft completes* m* orbits around the primary in approximately the same time required for the Moon to complete* n* revolutions. In this study, we call this type of resonant ratio a “*steady”* resonant ratio. Although the resonant ratio is not precise, those resonant orbits in the CRTBP are still closed and periodic [14]. Previous studies showed the existence of two and three dimensional resonant orbits in the Saturn system and KOI-730 system [15]. The targeting scheme, similar to that used to compute periodic orbits in the vicinity of the libration points, is widely used to calculate periodic resonant orbits. The initial vector seeds a correction scheme to target a perpendicular crossing of the* x*-axis in a nonlinear propagation [13]. Vaquero [16] numerically determined the orbit families for different resonant ratios and discussed the geometry feature of resonant orbits with different eccentricities. Similarly, Antoniadou [17] presented five types of resonant orbits with different resonant ratios and found that the resonant orbits with steady ratios generally had large eccentricities expect for a few special cases.

However, some resonant orbits in CRTBP are not “*lucky*” enough to complete “*steady”* resonant ratios. Anderson [18] focused on the resonant orbits with multiple loops near the secondary. Flyby is one of those interesting phenomena related to mean motion resonance in the CRTBP that involves a gravity assist maneuver, followed by escape after approaching the secondary primary [18]. They also explored their characteristics using a monodromy matrix and found that multiple flybys dramatically alter the resonant ratio [19]. What is more, observations show that some comets in resonance with Jupiter in the solar system exhibit “resonance transitions” into another resonant orbit when they were close to Jupiter [20]. A resonant transition is a phenomenon when the particle leaves one mean motion resonance and enters another after some time. A number of Jupiter comets such as Oterma exhibit a rapid transition from heliocentric orbits outside the orbit of Jupiter into heliocentric orbits inside the orbit of Jupiter, and vice versa. Koon [20] pointed out that the interior heliocentric orbit is typically close to the 3:2 resonance during resonance transition, while the exterior heliocentric orbit is near the 2:3 resonance. Therefore, we can conclude that resonant orbits might not exhibit “*steady”* resonant ratios mainly due to perturbations caused by the approaching second primary in the CRTBP. Obviously, it is important to discover and analyze the initial orbital geometry factors that affect resonant orbits and will allow resonant orbits involving an approach to the second primary.

Research on dynamical system analysis of resonance related flyby and transition offers an opportunity to focus on the connections between the initial resonant orbit parameters and the possibility of maintaining “steady” resonant ratios. In the analysis of flybys, Anderson [21] defined the flyby range and concluded that the spacecraft’s trajectory is considered a typical two-body orbit when the spacecraft remains some set distance from the perturbing planet. This set distance is often referred to as the sphere of influence or the sphere of action, and its calculation has been the subject of considerable research by Laplace [22] and Tisserand [23]. With the purpose of understanding resonance transitions, Belbruno [24, 25] explained the comet’s temporary capture and provided the necessary energy conditions for resonance transitions. The temporary nonpositive Kepler energy with respect to the secondary primary is called “*definition one*” for ballistic capture. Temporary ballistic capture is analogously referred to as weak capture, and resonant motion with respect to secondary primary was unstable during weak capture.

Based on the definition of ballistic capture [24, 25] and flyby range [19, 21], the Kepler energy and perturbing planet’s sphere of influence are used as indices to establish connections between the initial parameters of resonant orbits and their possibilities of maintaining “steady” resonant ratios.

This study focuses purely on resonant orbits losing steady ratios and attempts to locate the parameter intervals that can cause those orbits to lose their steady ratios as the secondary primary is approached, where the Earth-Moon system is taken as an example.

The remainder of this paper is structured as follows. The basic dynamics of the two-body problem model and the CRTBP model are briefly introduced in Section 2. The basic methodology for constructing representative resonant orbit families in the two-body problem and the CRTBP are presented in Section 3. Two important initial parameters are selected during resonant family analysis. The parameter intervals that can cause those orbits to lose their stable resonant ratio as the secondary primary is approached are determined in Section 4. The connections between the initial parameters and possible weak capture and flyby are clarified. Conclusions are drawn in Section 5.

#### 2. Equation of Motion

The two-body model and the CRTBP model are used to analyze the contributions of the Moon’s gravitational influence and geometry properties on the resonant orbits in this study. Before proceeding further, the definitions of the coordinate systems are defined and illustrated in Figure 1.