Abstract

This work studies the possibility of generating artificial collinear liberation points for the planar circular restricted three-body problem using Lorentz force affecting a charged spacecraft due to the magnetic field of a planet. It is considered to be a magnetic dipole inclined by angle α with the spin axis of the planet. The acceleration components for Lorentz force are first derived in an inertial planet-center coordinate system. Then, they are transformed into the rotating coordinate system of the three-body system, with the planet naturally the smaller primary in a planet-Sun system. The equations for the liberation points are derived including the charge per unit mass as the controlling parameter. Finally, the values of the charge per unit mass required for controlling the collinear liberation point positions are derived. A numerical application for the Sun-Jupiter system is introduced and the relation between the position of the artificial liberation point and the charge per unit mass is presented graphically.

1. Introduction

The history of the restricted problem begins in 1772 by Euler and Lagrange, continues in 1836 by Jacobi and Hill [1], and is followed by Pioncare [2], Levi-Civita, Birkhoff, and then Szebhley [3]. The problem plays an important role in studying the motion of artificial satellites. Eckstien et al. [4] used the method of multiple variable expansion to estimate the motion of a close satellite of the smaller primary in the restricted three-body problem. Not only the orbits of close satellites but also the orbits of distant satellites are determined by Kogan [5]. Zhuravlev et al. [6] investigated the motion of artificial satellites in the restricted problem when the planet has a definite shape. Cordeiro et al. [7] investigate some dynamical properties of the phenomena of satellite capture; moreover, a numerical study of this phenomenon has been performed by Neto [8].

In the frame of studying the stability of Lagrangian points in the RTBP, great work was done. For literature reviewing, Arnol’d [9], Moser [10], Leontovich [11], Arnol’d [12], and Deprit and Deprit-Bartholomé [13] have recently shown that the equilateral triangular solution will indeed be stable for nearly all values of the mass parameter μ in the range 0<μ<0.0385208. Later, Markeev [14] proved that in the plane problem stability will hold for all values of μ in the previous range except the two special values μ₁ = 0.0242938 and μ₂ = 0.0135160, for which the motion will be unstable.

The literature on the RTBP is rich with articles studying equilibrium points in the planar case other than the well-known five points. These can exist when other forces than gravity are included in the problem whether natural or artificial [15–18]. Recently, many articles studied the three-body problem from different aspects. For example, Zotos [19] studied basins of convergence of the equilibrium points in the pseudo-Newtonian circular restricted three-body problem in a plane, and Suraj et al. [20, 21] studied the topology of the basins of convergence in the three-body problem. Zotos [22] studied the fractal basins of attraction with oblateness and radiation pressure. Suraj et al. [23] studied the perturbations effects of the Coriolis and centrifugal forces on the existence of the liberation points. Mittal et al. [24] studied the periodic orbits generated by Lagrangian solution when the primaries are not spherical.

Mostafa et al. [25] studied the artificial triangular points in the planar RTBP using Lorentz Force on a charged satellite due to the planet's magnetic field, neglecting the tilt angle of the magnetic field with the normal to the plane of motion. In this work, the collinear points are studied with the tilt angle considered. The acceleration due to Lorentz force generated by the second primary, in the restricted three-body problem will be driven in the planet-centric coordinate system. Then, the acceleration vector will be transformed into the rotating frame of the RTBP. Finally, the effect of that force on the position of the liberation points using the charge per unit mass as a controller parameter is studied.

2. Equation of Motion

Let the center of the coordinate system be located at the center of mass (C.M) of the primaries in the rotating coordinate system, XYZ, which is rotating with constant unit angular speed where the plane is the plane of motion of the two primaries, and .

The X-axis is the line joining the bigger primary, , and the smaller one , Y-axis is normal to X-axis in their orbital plane while Z-axis is perpendicular to the XY plane (as shown in Figure 1). Let the separation distance between the primaries is l which will be the unit of distances.

Figure 1 

The kinematics of circular planer three body.

Let are the coordinates of the negligible mass, in this coordinate system as shown in Figure 1.

The dimensionless equations of motion of the negligible mass, , in the framework of the circular restricted three-body problem (CRTBP) are as follows (Fitzpatrick [26] and Murray and Dermott [27]):where is the distance of the mass from the center of mass (CM) and is equal to while is the distance of the mass from CM and also equal to , with G is the gravitational constant. and are the position vectors of the mass m from and , respectively.

Let be the unit of distance. Then, , which means that and are dependent parameters, so only one parameter can be chosen to describe the problem. If is chosen to be the dimensionless mass of the smaller primary then, will be the dimensionless mass of the other primary.

Let us change the center of the coordinate system to be in the smaller primary, , in the new coordinate system x, y, and z such that the x-axis is pointing from the bigger primary to the smaller one, the y-axis is normal to it in the primaries orbital plane, and the z-axis is normal to the primaries orbital plan.

Then, the dimensionless equations of motion, in the smaller primary coordinate system, will be

With

The coordinate of the small body in terms of the distances can be given by

To control the motion described by Equations (4)–(6), additional acceleration must add as a controller for the motion.

In our study, we will examine the Lorentz force generated by the magnetic field of the planets. Let this acceleration be denoted by

3. Lorentz Acceleration

If the small body with mass m experienced by a charge q (Coulombs) moving through a planet magnetic field , in planet-center inertial coordinate system , and , as shown in Figure 2.

Figure 2 

The apparent magnetic axis, rotational axis, and SC orbit in the inertial planet-centric coordinate system on the celestial sphere.

Then, the Lorentz force vector, in the inertial coordinate system , will affect the motion of such a body. Lorentz acceleration, , can be given by Streetman [28]:where is the inertial velocity while is the angular velocity of the source of the magnetic field which is rotating with the spin speed , of the planet and can be written as , with is the charge-to-mass ratio of the small body (specific charge) in Coulombs per kilogram (C/kg).

If we assume the source is a magnetic dipole, then, general vector model of a dipole magnetic field is (Rothwell [29])

Note that, the subscript “i” refers to “inertial system.” is unit vector along the north magnetic pole, and is the strength of the field in Weber-meters. For a tilted magnetic dipole, the angle α between and , is the angle between the rotational axis and the magnetic axis, as shown in Figure 2. In the Earth case, the angle α is 11.7^o and is independent of time.

From Figure 2, we can express the direction of the magnetic field in the inertial reference frame, X_iY_iZ_i, as

With,

Finally, we can express the magnetic field described by Equation (13) aswhere , , and are the coordinates of the position vector in the planet-centric inertial coordinate system.

The acceleration components due to the planet magnetic field, in the planet-centric inertial coordinate system, using Equation (12), can give by

Then,

Finally, the Lorenz acceleration components will be:

Equations (19)–(21) represent the acceleration experienced by spacecraft due to an inclined magnetic field by an angle α with the rotational axis of the central planet.

To substitute Lorenz acceleration, Equations (19)–(21), in the equations of motion for the three body, we need to transform Equations (19)–(21) to be in the rotational coordinate system (x, y, z) described by Equations (4)–(6).

Finally, the Lorentz acceleration described in the sidereal planet-centered coordinate system can be derived by substituting the transformation into Equations (19)–(21), assuming the angle ε is constant:where .

4. Lorentz Three-Body Problem

The equation of motion for the three-body problem in existence of Lorenz acceleration, using Equations (22)–(24) into Equations (4)–(6), will be

We note that the units of the quantity are which is after simplifying T⁻¹ L³ where we use C for the dimension of electric charge, M for mass, T for time, and L for length.

Equations (25)–(27) are the dimensionless controlling equations of the motion of a small, charged body under the gravitational effect of two big bodies with one of them has a significant magnetic field. The equations for the planar problem, z = 0, are

5. Artificial Equilibrium Points

The liberation points are found by letting , thus equations (28) and (29) will give

Equations (30) and (31) represent the controlling equations depending on the position of the spacecraft’s liberation points and the charge (per unit mass) required to generate Lorenz acceleration and the masses of the primaries.

From Equation (31), we haveor,

Equation (32) leads to the collinear solution while Equation (33) gives us the equilateral triangular solution as a function of the charge per unit mass, the magnetic field strength of the smaller primary and the relative masses of the primaries.

5.1. Controlling the Collinear Solution

To control the collinear solution, substituting into Equation (30) to get