Mathematical Problems in Engineering

Volume 2018, Article ID 7489120, 10 pages

https://doi.org/10.1155/2018/7489120

## Numerical Study of the Zero Velocity Surface and Transfer Trajectory of a Circular Restricted Five-Body Problem

School of Mathematical Science, Yangzhou University, Yangzhou 225002, China

Correspondence should be addressed to F. B. Gao; moc.anis@oabafoag

Received 14 May 2018; Revised 26 July 2018; Accepted 5 September 2018; Published 15 October 2018

Academic Editor: Chris Goodrich

Copyright © 2018 R. F. Wang and F. B. Gao. 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

We focus on a type of circular restricted five-body problem in which four primaries with equal masses form a regular tetrahedron configuration and circulate uniformly around the center of mass of the system. The fifth particle, which can be regarded as a small celestial body or probe, obeys the law of gravity determined by the four primaries. The geometric configuration of zero-velocity surfaces of the fifth particle in the three-dimensional space is numerically simulated and addressed. Furthermore, a transfer trajectory of the fifth particle skimming over four primaries then is designed.

#### 1. Introduction

The restricted N-body problem has attracted the attention of many researchers from the fields of mathematics, astronomy, and mechanics because of its wide application in deep space exploration. Here we list some recent or interesting research results. Gao and Zhang [1] studied the existence of periodic orbits of the circular restricted three-body problem. According to the existing literature, the first type of Poincaré periodic orbit generally requires that the mass parameter of the system is sufficiently small, and the periodic orbit studied in this paper is applied to any between , solving the problem that the first type of Poincaré’s periodic orbit has always been considered to occur only when the masses of primaries are quite different. Baltagiannis and Papadakis [2] obtained the zero-velocity surfaces and corresponding equipotential curves in the planar restricted four-body problem where the primaries were always at the vertices of an equilateral triangle. Álvarez-Ramírez and Vidal [3] analyzed the zero-velocity surfaces and zero-velocity curves of the spatial equilateral restricted four-body problem. Singh [4] investigated the permissible regions of motion and the zero-velocity surfaces under the influence of small perturbations in the Coriolis and centrifugal forces in the restricted four-body problem. Mittal et al. [5] found the zero-velocity surfaces and regions of motion in the restricted four-body problem with variable mass where the three primaries formed an equilateral triangle. Asique et al. [6] drew the zero-velocity surfaces to determine the possible permissible boundary regions in the photogravitational restricted four-body problem.

However, limited research studies have been performed on circular restricted five-body problem while it is compared with related three-body and four-body problems. A spatial circular restricted five-body problem wherein the fifth particle (the small celestial body or probe) with negligible mass is moving under the gravity of the four primaries, which move in circular periodic orbits around their centers of mass fixed at the origin of the coordinate system. Because the mass of the fifth particle is small, it does not affect the motion of four primaries.

Kulesza et al. [7] observed the region of motion of the restricted rhomboidal five-body problem whose configuration is a rhombus using the Hamiltonian structure and proved the existence of periodic solutions. Albouy and Kaloshin [8] confirmed there were a finite number of isometry classes of planar central configurations, also called relative equilibria, in the Newtonian five-body problem. Marchesin and Vidal [9] determined the regions of possible motion in the spatial restricted rhomboidal five-body problem by using the Hamiltonian structure. Llibre and Valls [10] found that the unique cocircular central configuration is the regular 5-gon with equal masses for the five-body problem. Bengochea et al. [11] studied the necessary and sufficient conditions for periodicity of some doubly symmetric orbits in the planar -body problem and studied numerically these types of orbits for the case n=2. Shoaib et al. [12] considered the central configuration of different types of symmetric five-body problems that have two pairs of equal masses; the fifth mass can be both inside the trapezoid and outside the trapezoid, but the triangular configuration is impossible. Xu et al. [13] discussed the prohibited areas of the Sun-Jupiter-Trojans-Greeks-Spacecraft system and designed a transfer trajectory from Jupiter to Trojans. Han et al. [14] obtained many new periodic orbits in the planar equal-mass five-body problem by using the variational method. Gao et al. [15] investigated the zero-velocity surfaces and regions of motion for specific configurations in the axisymmetric restricted five-body problem. Saari and Xia [16] verified that, without collisions, the Newtonian N-body problem of point masses could eject a particle to infinity in finite time and that three-dimensional examples exist for all .

In the present paper, it is assumed that the four primaries with equal masses constitute a regular tetrahedron configuration. A dynamic equation of the circular restricted five-body problem is established, and the relationship between energy surface structure of the fifth particle and the corresponding Jacobi constant is discussed. Moreover, the critical position of the fifth particle’s permissible and forbidden regions of motion is also addressed. In addition, based on Matlab software, a transfer trajectory of the fifth particle skimming over four primaries is designed numerically. Because of the gravity of the four primaries, the transfer trajectory will reduce the consumption of fuel for the fifth particle effectively.

#### 2. Equations of Motion

For a circular restricted five-body system, assume each of the four primaries, namely, , , , and , lie at one of the vertices of a regular tetrahedron. Furthermore, their motions are opposite to the center of mass. The small orbiter is only subjected to the gravity of the four primaries, and the origin of the inertial coordinate system is located at the center of mass of the four primaries, with one of them, say , located on the -axis. The plane defined by the remaining three primaries is parallel to plane. The origin of the rotational coordinate system coincides with . The -axis and -axis of the rotational coordinate system are rotated counterclockwise around the centroid of the unit angular velocity relative to the -axis and -axis of the inertial coordinate system , respectively.

Suppose the masses of four primaries are , the mass of the fifth particle is , the angle that the system rotates around its center of mass is , and the two coordinate systems coincide with each other when the time is 0. Hence, we obtain is equal to . The dimensionless distance between each two primaries is 1, and the distances between four primaries and the center of mass are . Thus, the coordinates of four primaries are In the coordinate system , we hypothesize the coordinate of is ; thus, the coordinates of four primaries are

Suppose that the coordinate of the fifth particle is in the inertial coordinate system; thus, the Lagrange function satisfies the following relationship: where the gravitational potential energy is defined as and denotes the distance between the fifth particle and one primary: By substituting (3) into the following Euler-Lagrange equation, we obtain the dimensionless equations of motion of the fifth particle in the inertial coordinate system where , , and denote the derivative of with respect to , , and , respectively.

Suppose that the coordinates of the orbiter in the rotating coordinate system are . The relationship between the two coordinate systems is where is

Thus, the dimensionless equations of motion of the fifth particle in the rotational coordinate system are where the generalized potential energy is defined as and Thus, a first Jacobi type integral is where is the motion velocity of the fifth particle and is the Jacobi constant. The permissible motion region and prohibited region are defined by and , respectively. Therefore, when the velocity of the fifth particle is zero, the curve shown by the above equation is called zero-velocity curve on the plane and is called zero-velocity surface in space.

#### 3. Zero-Velocity Surfaces

The diagrams of relationship between the zero-velocity surfaces and the Jacobi constant are shown in the following.

For a given value of , we can obtain the zero-velocity surfaces and the zero-velocity curves of the circular restricted five-body problem on the , , and planes, as shown in Figures 1(a)–1(c), respectively.