Artificial Triangular Points by Lorentz Force in the Restricted Three-Body Problem

'e restricted three body problem was outlined. 'e acceleration due to planetary magnetic field, in terms of space craft’s orbital elements, was analysed. 'e conditions for calculating the liberation points including the mutual gravitational attraction and the effect of Lorentz acceleration were derived for the case of circular planer restricted three bodies.'e stability of the solution for the artificial Lorentz triangular liberation points was studied. Finally, numerical investigation for the case of Sun-Jupiter system was calculated as case study. 'e results show the ability of changing the position of the triangular liberation points by an order from 10 7 to 10 6 for the dimensionless x, y coordinates and distance r from Jupiter. 'is is equivalent to about hundreds of Kilometers which is considerable.


Introduction
e history of the RTBP begins in 1772 by Euler and Lagrange, continues in 1836 by Hill [1], and is followed by Poincare [2], Levi-Civita [3], Birkhoff [4], and then by Szebhley [5]. Till now there are great names and important contributions to the RTBP.
Most perturbation techniques are applied to solve the equations of motion of the RTBP. Delva [6], used a Lie series to determine the orbits of massless bodies in the elliptic RTBP. Also Sandor et al. [7], applied the method of the short time Lyapunov indicators to the RTBP in order to study the structure of the phase space in some selected regions. e idea of the effect of geomagnetic field on the rotation and motion of Earth's artificial satellite was initiated by Fain and Greer [8]. Westerman, [9], used their idea to study the perturbation on circular orbit in the plane of the geomagnetic equator. Sehnal [10], concluded, at that time, no precise information available to evaluation of the effects of the Earth's magnetic field on the orbit of a charged satellite. e concept of generating AEP in the RTBP was initiated by Dusek [11]. e stability and location of the AEP were investigated since that time, from different aspects, by several authors [12][13][14][15][16]. On the other side, McInnes and others produced an extensive subsequently researches to find the equilibrium positions using solar sail propulsion such as [17][18][19][20][21]. Such work lead to find infinite equilibrium surfaces depending on the magnitude of the propulsive acceleration.
In the direction to finding the AEP using low-thrust systems Morimoto, et al. [22], proved the existence of infinite equilibrium surfaces depending on the magnitude of the propulsive acceleration.
However, only a subset of the potentially achievable AEPs turns out to be stable and, as thus, could not be exploited by a spacecraft without the use of a suitable control system. e topology of such subset of stable AEPs is strictly dependent on the propulsion system type employed by the spacecraft. In fact, as was recently pointed out by Bombardelli and Peláez [23], if the available propulsive acceleration is low, the stable AEPs are confined to a very restricted region around the classical Lagrange points.
In the last decade, several important articles worked on the effects of Lorentz force to control both orbit motion and attitude of a spacecraft whether alone or combined with other forces, for example, in 2011 Pollock et al. [24] proposed analytical solutions for the relative motion of a Lorentz spacecraft, while in the year, 2020, Mostafa et al. [25], studied the use of Lorentz force to control the perturbations due to solar radiation pressure, and Sun et al. [26], studied the coupled effect of Lorentz force with aerodynamics on controlling a spacecraft. Huang et al. [27], introduced a hybrid control system that contains both the Lorentz force and torque to control the orbit and attitude of a satellite.
In this article we study the effect of Lorentz force on the position and stability of the triangular liberation points in the restricted three body problem with an application given to the Sun-Jupiter system.

Formulation of the Problem
Consider the problem of three-bodies with the masses of the primaries m 0 , m 1 and the mass of space craft (SC) m in which m is negligibly small compared to m o and m 1 . Assume that the primaries revolve in circular orbits around their center of mass with mean motion n and constant separation L. en, In Figure 1, let the XY plane be the plane of motion of m o and m 1 , with X axis is the line joining m o and m 1 , Y axis is normal to X in their orbital plane while Z is perpendicular to the XY plane. Let (X, Y, Z) be the coordinates of the SC in this coordinate system (the synodic coordinate system), then the equations of motion of SC in the framework of the circular restricted three body problem (CRTBP), using dimensionless units, become.
If we transform the coordinate system to the center of the second primary m 1 , with the coordinate x, y, and z, the equations of motion will be: With, x � 1 2L Equations (2) and (3) give the base to calculate the liberation points (Lagrange points). To change the position of such points, additional acceleration, artificially, must be added to the equations. Let such acceleration be denoted by: a L � a xL a yL a zL T , with a L is adopted such as:

Lorentz Acceleration
e Lorentz acceleration for charged SC with mass m experienced by a charge q (Coulombs) moving through a planet magnetic field B, in plant center coordinate system i, j and k, is given by Brett Jordan Streetman [28]: where i and j are unit vectors in the equatorial plane of the planet while k in the direction of rotational axis. With q/m is the charge-to-mass ratio of the SC. in Coulombs per kilogram (C/kg). V is the inertial spacecraft velocity while ϑ × r is the velocity of the magnetic field which is rotate with the spin speed ϑ, of the planet and can written as ϑ � ϑk. e general vector model of a dipole magnetic field is [29]: where n a unit is vector along the north magnetic pole and B 0 is the strength of the field in Weber-meters. e unit vectors included in Eq. (12) are described in Figure 2. For a tilted magnetic dipole, the angle α between n and k, in the Earth case is 11.7°and is independent of time.
If we neglect the tilt angle of the magnetic field (i.e. α � 0), the unit vector n will be coincidence with the unit vector k. en, we can write the last equation in the form: Noting that from the spherical triangle ZbL, Cosbz ⌢ is related to the orbital elements, and equals Sin(] + ω)Sin i and we will denote it "Γ". Substituting Eq. (13) into Eq. (8), (9) and (10) to describe Lorentz acceleration as: Completing the required vector operations, we get: where, Q � qB 0 /m and Γ � Sin(] + ω)Sin i ∈.

Controlling Equations
To control the position of liberation points using Lorentz force, we must add the Lorentz acceleration to the equations of motion and applying the stationary conditions. If we neglect the inclination of the equatorial plane of the planet on its orbit, and choose the x-axis to be the line joining the Sun with planet, y-axis normal to it in the planet's orbital plane and z-axis normal to the orbital plane, then Eqns. (3), (4) and (15) will be described in the same coordinate system.
Substituting Eqns. (15) into Eqns. (3), (4) the equations of motion are derived in the form: Equations (16), (17) and (18) are the governing equations for the circular restricted three body problem with the magnetic field of the primary m 1 considered, and the small mass is controlled through a charge q.

Dimensionless Form of Controlling Equations
In what follows, we put equations (16), (17) and (18) in the usual dimensionless form of the restricted three body problem so that it is easy to compare the case of artificial points by the addition of the Lorentz force with the natural points when q is set equal zero. e equations are diveded by (L·n 2 ) which is clearly the dimensions of acceleration to have it in dimensionless form, we further define the quantities: t * � n·t dimensionless time, and x * � x/L, y * y/L, z * � z/L dimensionless distances. Also let r * � r/L, r * 0 � r 0 /L, and finally define ω � ϑ/n dimensionless angular velocity. We get Advances in Astronomy We note that the quantity Q/nL 3 is dimensionless since the units of Q � qB 0 /m is: CM − 1 (MC − 1 T − 1 l 3 ) which is after simplifying T − 1 l 3 where we use C for unit of electric charge, M for unit of mass, T for time and l for length. So, we also define the dimensionless quantity Q * � Q/nL 3 . Next, we substitute k 2 � n 2 L 3 /(m 0 + m 1 ), then introduce the wellknown parameter in the three-body problem, μ � m 1 /(m 0 + m 1 ) � a/L. e equations take the simpler dimensionless form, Finally, we omit the stars for simplicity of writing, keeping in mind that all involved quantities are now dimensionless, Equations (21), (22) and (23) are the dimensionless controlling equations of the motion of a small body with an electric charge, under the effect of two big bodies one of them has a significant magnetic field. e equations for planar problem are:

Artificial Equilibrium Points
e liberation points are found by letting € x � _ x � € y � _ y � 0. We also take L as the unit of distance � 1, and 1/n the unit of time, hence n � 1, thus Eqns (24) and (25) will give,

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Equations (26) and (27) represent the controlling equations depending on the position of the space craft's liberation pointes and the charge (per unit mass) required to generate Lorenz acceleration.
In the equilateral triangle solution, solving for r, we get: In other side, solving Eq. (28) for Q we get: We can express another useful relation since Q must satisfy both Eqns. (26) and (27), then: Simplifying yields, Which leads to conclude that the artificial liberation points due to Lorentz force generated from the small primary exist on a circular arc with center at the big primary with radius the unit distance, with angles ±(π/3) ± ε where ε is the small deviation from the original point of equilateral triangular case, and the positive sign is for L 4 while the negative sign is for L 5 , as shown in Figure 3. e charge per unit mass q, on the SC can be expressed from Eq. (30), as: Applying the constraint (32), with r 0 � 1, leads to: When the body is not charged (Q � 0), we get the classical case of the Lagrangian triangular points L 4 and L 5 with r � 1.

Triangular Liberation Points Coordinates
We proved in the previous section that the liberation points exist. In this section we will calculate their position as a function of the charge, of the SC, per unit mass.
Using Eqns. (5) and (6), Equations (35) and (36) give the coordinates of the artificial triangular points in terms of the parameter µ, the magnetic field strength B 0 of the primary at m 1 , the dimensionless angular velocity ω of the primary m 1 , and the charge per unit mass q.

Stability of Artificial Lorentz Triangular Liberation Points
Recalling the equations of motion in planar case: ese equations can be written as

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where, Now, we linearize the equations about the equilibrium points (x 0 , y 0 ) by letting, where. ε ≪ 1. en, we expand the equations keeping only first order terms. is gives, where, e characteristic equation for the system (40) is then, is can be written as: where, H � a 11 a 22 − a 2 12 .
(46) e condition for λ to be imaginary is that λ 2 is a real negative value. is is guaranteed by the satisfaction of the conditions: Equations (47), (48) and (49) are the necessary conditions for the artificial liberation points to be stable.   e parameter µ � 0.000954, the mean motion n � 0.00006042 hour − 1 , the sun-Jupiter distance L � 779 × 10 6 km., and the angular velocity ϑ � 0.63301 hour − 1 [30]. e dimensionless quantity Q will thus have the value qB 0 /nL 3 � 5.80366 × 10 − 9 q. Also, in this case ω � ϑ/n � 10476.829. If the distance from the original triangular liberation points to the small primary (Jupiter in this case) be r � 1, the x and y coordinates X 4,5 � − 0.5 and Y 4,5 � ±( � 3 √ /2). ese values will differ by a small amount due to the existence of the Lorentz force. We plot the deviations of r, X 4,5, and Y 4,5 denoted by δr, δX 4 and δY 4 respectively against the charge per unit mass q in Figure 4- Figure 6, where we note that X 4 � X 5 and Y 4 � − Y 5, so we plot only the graphs for L 4. e results show a variation of an order from 10 − 7 to 10 − 6 for the dimensionless x, y coordinates and distance r from Jupiter. is is equivalent to about hundreds of kilometers which is considerable. e range of q is from − 0.1 to 0.1 C/kg.
Concerning the stability, the graphs of F, H, and Finally, the Eigen values of equation (44) for different values of q are given in Table 1.

Conclusion
e artificial triangular liberation points L 4,5 in the circular restricted three body problem in the case when the small primary has a significant magnetic field are studied. In this work the control for the motion is taken the charge per unit mass of the charged space craft under study. e origin of the coordinates was taken at the small primary which has the center of the magnetic field. A numerical application is taken for the case of Sun-Jupiter system. e choice of this system is because of the small eccentricity of Jupiter's orbit around the sun and the weak effects of other perturbing bodies on the motion, and most important for this study is because of the strong magnetic field of Jupiter.
e results show the ability of variating the triangular liberation points by an order from 10 − 7 to 10 − 6 for the dimensionless x, y coordinates and distance r from Jupiter.
is is equivalent to about hundreds of kilometers which is considerable. e artificial points lie on a circular arc with center at the big primary with radius the unit distance, with angles ±(π/3 ± ε) where ε is the small deviation from the   original point of equilateral triangular case, and the positive sign is for L 4 while the negative sign is for L 5 . It should be mentioned that the results are made for a range of charge per unit mass up to order 0.1 C/kg., where the existing values till now for the specific charge are 0.001-0.01 C/kg. However, it is predicted to overcome the technical problems of this issue to reach values greater than 0.1 C/kg in the future [31] (Peck et al.). e results also show the stability of the Lorentz artificial L 4,5, in the range chosen for the specific charge, as the original points for Sun-Jupiter system.
With the improvement of the studies with artificial liberation points, future studies may generalize the work either to stronger kinds of control or to work with the 5 body problem or even the n-body problem [32,33].

SC:
Space craft TBP: ree body problem RTBP: Restricted three body problem CRTBP: Circular restricted three body problem AEP: Artificial equilibrium points LA: Lorentz acceleration LF: Lorentz force (X, Y, Z): e center of mass coordinates system (synodic coordinate system) (x, y, z): e planet centered coordinates system x * , y * , z * and r * : Dimensionless distances n: Unit vector along the north magnetic pole of the planet k: Unit in the direction of rotational axis of the planet a L : Lorentz acceleration vector q: Charge-to-mass ratio of the SC. in coulombs per kilogram (C/kg) B: Planet magnetic field vector B o : e strength of the field in weber-meters m 0 , m 1 : e masses of the primaries m: Mass of space craft (SC) V: e inertial spacecraft velocity vector ϑ: e rotational velocity vector of the planet ε: e small deviation from the original point N: Mean motion of the primaries around their center of mass ω: Dimensionless angular velocity of the smaller primary.

Data Availability
Data available on request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.