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

Authors: A. Mostafa; M. A. Yousef; M. I. El-Saftawy
Journal: Advances in Astronomy
 (2022)

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## Abstract

The restricted three body problem was outlined. The acceleration due to planetary magnetic field, in terms of space craft’s orbital elements, was analysed. The 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. The 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. The 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. This is equivalent to about hundreds of Kilometers which is considerable.


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## Body

## 1. Introduction

The 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.The 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.The concept of generating AEP in the RTBP was initiated by Dusek [11]. The stability and location of the AEP were investigated since that time, from different aspects, by several authors [12–16]. On the other side, McInnes and others produced an extensive subsequently researches to find the equilibrium positions using solar sail propulsion such as [17–21]. 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. The 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.


## 2. Formulation of the Problem

Consider the problem of three-bodies with the masses of the primariesm0,m1 and the mass of space craft (SC) m in which m is negligibly small compared to mo and m1. Assume that the primaries revolve in circular orbits around their center of mass with mean motion n and constant separation L. Then, n2L3=k2M with k is the Gaussian constant and M=m1+mo.In Figure1, let the XY plane be the plane of motion of mo and m1, with X axis is the line joining mo and m1, 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.(1)X¨−2nY˙=n2X−k2m0X+ar03+m1X−br3,Y¨+2nX˙=n2Y−k2m0Yr03+m1Yr3,Z¨=−k2m0Zr03+m1Zr3.Figure 1
The kinematics of circular planer three body.If we transform the coordinate system to the center of the second primarym1, with the coordinate x, y, and z, the equations of motion will be:(2)x¨−2ny˙=n2x+b−k2m0x+a+br03+m1xr3,y¨+2nx˙=n2y−k2m0yr03+m1yr3,z¨=−k2m0zr03+m1zr3.With,(3)r0=x+L2+y2+z2,(4)r=x2+y2+z2,(5)x=12Lr02−r2−L2,(6)y=±12L4L2r2−r02−r2−L2.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:(7)aL=axLayLazLT,with aL is adopted such as:(8)axL=−n2x+b−k2m0x+a+br03+m1xr3,(9)ayL=−−n2y−k2m0yr03+m1yr3,(10)azL=−−k2m0zr03+m1zr3.


## 3. Lorentz Acceleration

The Lorentz acceleration for charged SC with massm 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]:(11)aL=Fm=qmV−ϑk^×r×B,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^.The general vector model of a dipole magnetic field is [29]:(12)B=B0r33n^⋅r^r^−n^,where n^ a unit is vector along the north magnetic pole and B0 is the strength of the field in Weber-meters. The unit vectors included in Eq. (12) are described in Figure 2.Figure 2
Orbital motion in planet dipole magnetic field.For a tilted magnetic dipole, the angleα 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^. Then, we can write the last equation in the form:(13)B=B0r33Cosbz⌢.rr−k^.Noting that from the spherical triangleZbL, Cosbz⌢ is related to the orbital elements, and equals Sinν+ωSini and we will denote it “Γ”. Substituting Eq. (13) into Eq. (8), (9) and (10) to describe Lorentz acceleration as:(14)aLxaLyaLzT=qmB0r3x˙y˙z˙T−ϑi^j^k^001xyz×3Γr.xyzT−001T.Completing the required vector operations, we get:(15)aLx=Qr33Γry˙z−yz˙−ϑxz+ϑx−y˙,aLy=Qr33Γrxz˙−x˙z−ϑyz+ϑy−x˙,aLz=3ΓQr4x˙y−y˙x+ϑx2+y2,where, Q=qB0/m and Γ=Sinν+ωSini∈.


## 4. 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 thex-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:(16)x¨−2ny˙=n2x+b−k2m0x+Lr03+m1xr3+Qr33Γry˙z−yz˙−ϑxz+ϑx−y˙,(17)y¨+2nx˙=n2y−k2m0yr03+m1yr3+Qr33Γrxz˙−x˙z−ϑyz+ϑy+x˙,(18)z¨=−k2m0zr03+m1zr3+3ΓQr4x˙y−y˙x+ϑx2+y2.Equations (16), (17) and (18) are the governing equations for the circular restricted three body problem with the magnetic field of the primary m1 considered, and the small mass is controlled through a charge q.


## 5. 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.The equations are diveded by (L·n2) 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,r0∗=r0/L, and finally define ω˜=ϑ/n dimensionless angular velocity. We get(19)d2x∗dt∗2−2dy∗dt∗=x∗+bL−k2n2m0x∗+1r03+m1x∗r3+QnL3r∗33Γrdy∗dt∗z∗L−y∗Ldz∗dt∗−ϑnx∗z∗L+ϑnx∗−dy∗dt∗,d2y∗dt∗2+2dx∗dt∗=y∗−k2n2m0y∗r03+m1y∗r3+QnL3r∗33Γrx∗Ldz∗dt∗−dx∗dt∗z∗L−ϑny∗z∗L+ϑny∗+dx∗dt∗,d2z∗dt∗2=−k2n2m0z∗r03+m1z∗r3+3ΓQnL3r∗4dx∗dt∗y∗−dy∗dt∗x∗+ϑnx∗2+y∗2.We note that the quantityQ/nL3 is dimensionless since the units of Q=qB0/m is: CM−1 (MC−1T−1l3) which is after simplifying T−1l3 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/nL3. Next, we substitute k2=n2L3/m0+m1, then introduce the well-known parameter in the three-body problem, μ=m1/m0+m1=a/L. The equations take the simpler dimensionless form,(20)x¨∗−2y˙∗=x∗+1−μ−1−μx∗+1r0∗3+μx∗r∗3+Q∗r∗33Γr∗y˙∗z∗−y∗z˙∗−ω˜x∗z∗+ω˜x∗−y˙∗,y¨∗+2x˙∗=y∗−1−μy∗r0∗3+μy∗r∗3+Q∗r∗33Γr∗x∗z˙∗−x˙∗z∗−ω˜y∗z∗+ω˜y∗+x˙∗,z¨∗=−1−μz∗r0∗3+μz∗r∗3+3ΓQ∗r∗4x˙∗y∗−y˙∗x∗+ω˜x∗2+y∗2.Finally, we omit the stars for simplicity of writing, keeping in mind that all involved quantities are now dimensionless,(21)x¨−2y˙=x+1−μ−1−μx+1r03+μxr3+Qr33Γry˙z−yz˙−ω˜xz+ω˜x−y˙,(22)y¨+2x˙=y−1−μyr03+μyr3+Qr33Γrxz˙−x˙z−ω˜yz+ω˜y+x˙,(23)z¨=−1−μzr03+μzr3+3ΓQr4x˙y−y˙x+ω˜x2+y2.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. The equations for planar problem are:(24)x¨−2y˙=x+1−μ−1−μx+1r03+μxr3+Qr3ω˜x−y˙,(25)y¨+2x˙=y−1−μyr03+μyr3+Qr3ω˜y+x˙.


## 6. Artificial Equilibrium Points

The liberation points are found by lettingx¨=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,(26)1−μ+xr3−1−μ1+xr3r03−μx+Qω˜x=0,(27)yr3−1−μyr3r03−μy+Qω˜y=0.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.From Eq. (27), we get:(28)y=0,Orr3−1−μr3r03−μ+Qω˜=0,y=0, is the solution of collinear liberation points while r3−1−μr3/r03−μ+Qω˜=0. This leads to the equilateral triangle solution.In the equilateral triangle solution, solving forr, we get:(29)r=r0μ−Qω˜r03+μ−13.In other side, solving Eq. (28) for Q we get:(30)Q=1ω˜μ−r3+1−μrr03.We can express another useful relation sinceQ must satisfy both Eqns. (26) and (27), then:(31)1−μ+xr3x−1−μ1+xr3xr03−μ=r3−1−μr3r03−μ⟹r0=1.Simplifying yields,(32)1−μr3x1−1r03=0⟹r0=1.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 L4 while the negative sign is for L5, as shown in Figure 3.Figure 3
The position of artificial triangular points.The charge per unit massq¯, on the SC can be expressed from Eq. (30), as:(33)q¯=nL3B0ω˜μ−r3+1−μrr03.Applying the constraint (32), with r0=1, leads to:(34)q¯=μnL3B0ω˜1−r3,r=1−Qω˜μ3.When the body is not charged (Q = 0), we get the classical case of the Lagrangian triangular points L4 and L5 with r = 1.


## 7. 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), with r0=1, L = 1, a = µ, b = 1−µ, yields:(35)x=−r22=−121−Qω˜μ2/3,(36)y=±124r2−r4=±1241−Qω˜μ2/3−1−Qω˜μ4/3.Equations (35) and (36) give the coordinates of the artificial triangular points in terms of the parameter µ, the magnetic field strength B0 of the primary at m1, the dimensionless angular velocity ω˜ of the primary m1, and the charge per unit mass q¯.


## 8. Stability of Artificial Lorentz Triangular Liberation Points

Recalling the equations of motion in planar case:(37)x¨−2y˙=1−μ+x−1−μx+1r03+μxr3+Qr3ω˜x−y˙,y¨+2x˙=y−1−μyr03+μyr3+Qr3ω˜y+x˙.These equations can be written as(38)x¨−2y˙=∂Ω∂x−Qr3y˙,y¨+2x˙=∂Ω∂y+Qr3x˙,where,(39)Ω=Ω¯−Qω˜r,Ω¯=12x+1−μ2+y2+1−μr0+μr.Then,(40)x¨−2y˙+Qr3y˙=∂Ω¯∂x,y¨+2x˙−Qr3x˙=∂Ω¯∂y.Now, we linearize the equations about the equilibrium points (x0, y0) by letting,(41)x=x0+εξ,y=y0+εη,where. ε≪1.Then, we expand the equations keeping only first order terms. This gives,(42)ξ¨+c−2η˙=a11ξ+a12η,η¨+c−2ξ˙=a12ξ+a22η,where,(43)a11=∂2Ω∂x2|x0,y0,a12=∂2Ω∂x∂y|x0,y0,a22=∂2Ω∂y2|x0,y0,c=Qr3|x0,y0.The characteristic equation for the system (40) is then,(44)λ4+λ24−4c+c2−a11−a22+a11a22−a122=0.This can be written as:(45)λ4+Fλ2+H=0,where,(46)F=4−4c+c2−a11−a22,H=a11a22−a122.The condition forλ to be imaginary is that λ2 is a real negative value. This is guaranteed by the satisfaction of the conditions:(47)F2−4H>0,(48)F>0,(49)H>0.Equations (47), (48) and (49) are the necessary conditions for the artificial liberation points to be stable.


## 9. Numerical Investigation for the Case of the Sun-Jupiter System

The magnetic field of Jupiter has the valuesB0=1.5812×1020 Tesla meter3. The parameter µ = 0.000954, the mean motion n = 0.00006042 hour−1, the sun-Jupiter distance L = 779 × 106 km., and the angular velocity ϑ=0.63301 hour−1 [30]. The dimensionless quantity Q will thus have the value q¯B0/nL3=5.80366×10−9q¯. 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 X4,5 = −0.5 and Y4,5 =  ±3/2. These values will differ by a small amount due to the existence of the Lorentz force. We plot the deviations of r, X4,5, and Y4,5 denoted by δr, δX4 and δY4 respectively against the charge per unit mass q¯ in Figure 4–Figure 6, where we note that X4 = X5 and Y4 = −Y5, so we plot only the graphs for L4. The results show a variation of an order from 10−7 to 10−6 for the dimensionless x, y coordinates and distance r from Jupiter. This is equivalent to about hundreds of kilometers which is considerable. The range of q¯ is from −0.1 to 0.1 C/kg.Figure 4
The variation ofL4 distance with Jupiter from r = 1 against the specific charge.Figure 5
The variation ofL4 y-coordinate y=3/2 against the specific charge.Figure 6
The variation ofL4 x-coordinate x=−1/2 against the specific charge.Concerning the stability, the graphs ofF, H, and F2-4H are plotted in the same range of the specific charge −0.1 to 0.1 to check conditions (47), (48), and (49). The graphs show values greater than zero in the whole range which guarantees the stability of the artificial triangular points as the case of the natural points. The results are given in Figure 7−Figure 9.Figure 7
Condition (47), F2—4H > 0 is satisfied for −0.1≤q¯≤0.1.Figure 8
Condition (48), F > 0 is satisfied for −0.1≤q¯≤0.1 [Note: F=1±O10−8].Figure 9
Condition (49), H > 0 is satisfied for −0.1≤q¯≤0.1.Finally, the Eigen values of equation (44) for different values of q¯ are given in Table 1.Table 1
The Eigen values for equation (44).
kλ (k)−0.1±0.08046382187044038ⅈ±0.9967575415163028ⅈ−0.08±0.08046383300313438ⅈ±0.9967575382901124ⅈ−0.06±0.08046384413583031ⅈ±0.9967575350639152ⅈ−0.04±0.08046385526853787ⅈ±0.9967575318377115ⅈ−0.02±0.08046386640124309ⅈ±0.9967575286115022ⅈ0±0.08046387753395576ⅈ±0.9967575253852867ⅈ0.02±0.0804638886666647ⅈ±0.9967575221590655ⅈ0.04±0.08046389979937688ⅈ±0.9967575189328386ⅈ0.06±0.0804639109321091ⅈ±0.996757515706604ⅈ0.08±0.0804639220648292ⅈ±0.9967575124803647ⅈ0.1±0.0804639331975582ⅈ±0.9967575092541185ⅈ


## 10. Conclusion

The artificial triangular liberation pointsL4,5 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. The 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. The 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.The 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. This is equivalent to about hundreds of kilometers which is considerable. The 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 L4 while the negative sign is for L5. 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.).The results also show the stability of the Lorentz artificial L4,5, 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].



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