Mutsuko Y. Morimoto

Artificial Equilibrium Points in the Low-Thrust Restricted Three-Body Problem

T HE present Note describes the concept of the artificial equilibrium point (AEP) assisted by continuous thrust in the restricted three-body problem (RTBP). In the RTBP, there are five libration points called Lagrange points. Each of the Lagrange points is in equilibrium between the gravitational forces of the two primary bodies and the centrifugal force in the rotating frame. The Lagrange points have been investigated in a number of studies in the field of celestial mechanics. Since the 1950s, space engineers have been interested in these Lagrange points and have investigated the applicability of these points to space missions. Farquhar [1] introduced the concept of telecommunication systems using Lagrange points in the Earth–moon system, and in subsequent studies, he investigated ballistic periodic orbits about equilibrium points not only in the Earth–moon system [2–5], but also in the sun– Earth system [6,7]. Currently, low-thrust propulsion systems such as electric propulsion and the solar sail are being developed not only for controlling satellite orbits, but also asmain engines for interplanetary transfer. These low-thrust propulsion systems are able to provide continuous control acceleration to the spacecraft and thus to increase mission design flexibility. When we attempt to use Lagrange points, the positions are normally restricted to only five points. In terms of mission design, however, Lagrange points are not always the best positions. For example, we must operate the satellite at midnight every day when a spacecraft is placed at L2 points, unless we can use a deep space network. L4 and L5 are stable points, but these points are far from the Earth. Therefore, the transfer time required to reach L4/L5 and to telecommunicate is longer than that for points closer to the Earth. To achieve various mission objectives, positions other than the Lagrange points might be suitable in some cases. These points are in nonequilibrium, but it is possible to keep the spacecraft at these positions by using continuous thrust. Libration points with continuous control acceleration have also been studied (Duseck [8], Simmons et al. [9], McInnes et al. [10,11], and Broschart and Sheeres [12]). In [11], McInnes investigated the magnitude of control acceleration and stability for a two-body problem. These studies reported specific libration points with a certain mass ratio or those with certain low-thrust accelerations. However, for more flexible and generic mission designs, we must analyze arbitral points for the general mass-ratio range with continuous control acceleration by an idealized continuous thrust. In a previous paper [13], we focused on resonant periodic orbits existing on the line connecting two primary bodies with a continuous-low-thrust propulsion system. On the other hand, in the present Note, we investigate the magnitude and direction of the required acceleration creating the AEP in three-dimensional space. In addition, we discuss its stability by linearizing the equations of motion and carrying out a linear stability analysis.

Periodic Orbits with Low-Thrust Propulsion in the Restricted Three-body Problem

Periodic orbits around nonequilibrium points are generated systematically by using continuous low-thrust propulsion in the restricted three-body problem, with a mass ratio varying from 0 to 1/2. A continuous constant acceleration is applied to cancel the gravitational forces of two primary bodies and the centrifugal force at a nonequilibrium point, which is changed into an artificial equilibrium point. The equations of motion are linearized to analytically generate periodic orbits with constant acceleration. Then, periodic orbits around artificial equilibrium points which exist on the line connecting two primary bodies are investigated. The frequencies of these periodic motions are expressed by a parameter that is a function of the mass ratio and the position of the orbits around artificial equilibrium points. By choosing the frequencies of motions that are small-integer resonant, we have found the existence of points at which in-plane and out-of-plane motions are synchronized.

Periodic orbits with constant control acceleration in the restricted three body problem

In the restricted three body problem, each of the Lagrange points is in equilibrium between gravitational forces of the two primary bodies and centrifugal force in the rotating frame. A non-equilibrium point, which is not at Lagrange points, can be turned into an artificial equilibrium point (AEP) by canceling any residual acceleration at the non-equilibrium points with continuous control acceleration. This study describes an investigation of periodic orbits around the AEPs in the restricted three body problem. The non-linearized equations of motion in this problem are linearized around the AEPs, and periodic orbits are generated by simple constant control acceleration in three-dimensional space.