# Impacts of Poynting–Robertson Drag and Dynamical Flattening Parameters on Motion around the Triangular Equilibrium Points of the Photogravitational ER3BP

Authors: Aishetu Umar; Aminu Abubakar Hussain
Journal: Advances in Astronomy
 (2021)

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## Abstract

Using an analytical and numerical study, this paper investigates the equilibrium state of the triangular equilibrium pointsL4,5 of the Sun-Earth system in the frame of the elliptic restricted problem of three bodies subject to the radial component of Poynting–Robertson (P–R) drag and radiation pressure factor of the bigger primary as well as dynamical flattening parameters of both primary bodies (i.e., Sun and Earth). The equations of motion are presented in a dimensionless-pulsating coordinate system ξ−η, and the positions of the triangular equilibrium points are found to depend on the mass ratio μ and the perturbing forces involved in the equations of motion. A numerical analysis of the positions and stability of the triangular equilibrium points of the Sun-Earth system shows that the perturbing forces have no significant effect on the positions of the triangular equilibrium points and their stability. Hence, this research work concludes that the motion of an infinitesimal mass near the triangular equilibrium points of the Sun-Earth system remains linearly stable in the presence of the perturbing forces.


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## Body

## 1. Introduction

The study of the equilibrium state of an infinitesimal mass (a test particle) with regard to the dynamical system of the restricted three-body problem (R3BP) remains one of the most important and interesting aspects in the study of celestial mechanics and dynamical astronomy. The elliptic restricted three-body problem (ER3BP) deals with the description-study of the motion of an infinitesimal mass (m3), in the vicinity of the gravitational fields of two dominant bodies m1 and m2 which are called the primaries (bigger primary and smaller primary, respectively), where m1≥m2≫m3. These primary bodies revolve about their common centre of mass in elliptic orbits, under the influence of their mutual gravitational attraction. The ER3BP admits five equilibrium points at which the test particle (infinitesimal mass) would remain fixed if placed there. Three of such points lie on the line joining the two dominant bodies m1 and m2, called the collinear equilibrium points Lii=1,2,3, and are unstable, while the other two points form equilateral triangles with the two dominant bodies, called the triangular equilibrium points L4 and L5, and are stable for 0<μ<μc, where μ is the mass ratio defined by μ=m2/m1+m2 and μc is the critical mass parameter [1].In an attempt to have a much more realistic description of the motion of an infinitesimal mass over the decades, the classical R3BP has been modified in the sense that additional dynamical potentials of the system were considered in different approaches [2–8] and others.Abouelmagd [9] in his study of the R3BP found out that the locations of the triangular points and their linear stability are affected by the oblateness of the more massive primary, up to the second zonal harmonic J4. In his numerical study, he concluded that the existence of J4 sometimes does not affect the stability of the equilibrium points as in the Earth-Moon, Saturn-Phoebe, and Uranus-Caliban systems. Also, Abouelmagd et al. [10] examined the effects of oblateness of the three participating bodies together with small perturbations in Coriolis and centrifugal forces. Afterwards, Abouelmagd et al. [11] studied the effect of the first two even zonal harmonic coefficients of both the primaries as well as the periodic orbits around the equilibrium points. They found out that the triangular equilibrium points are linearly stable for 0<μ<μC and unstable for μC≤μ≤0.5, while the collinear equilibrium points remain linearly unstable.Singh and Tyokyaa [12] examined the stability of the triangular points in the ER3BP with oblateness up to the second zonal harmonic J4 of both primaries. They concluded that the location and linear stability of the triangular points are affected by the oblateness of the primary bodies, eccentricity of the orbits of the primaries, and the semi-major axis of the system and both destabilized the system. In the same year, Singh et al. [13] investigated the influence of the zonal harmonics (J2 and J4) of the primary and the radiation pressure of the secondary on the locations and linear stability of the triangular points. They however claimed that the parameters involved in the system affect the positions of the equilibrium points and destabilize the system as well.The Doppler shift and absorptions and subsequent re-emission of incident radiation, that is, the so-called Poynting–Robertson’s (P–R) drag, are often neglected by many researchers in the estimation of light radiation force. Poynting [14] while studying radiation in the Solar System stated that radiation affects temperature and small bodies. He asserted that particles such as cosmic dust grain or small meteors are affected considerably by gravitational and light radiation force as they approach luminous celestial bodies. Furthermore, infinitesimal bodies in solar orbits suffer a gradual loss of angular momentum and ultimately spiral into the Sun. Later, Robertson [15] in a modified theory of Poynting considered only terms of the first order in the ratio of velocity of the particles to that of light. He investigated the dynamical effects of drag in the Solar System and derived the expression for the times of fall from circular orbits. Thus at a cosmically rapid rate, the P–R effect sweeps small particles of the Solar System into the sun.Researchers like Burns et al. [16], Murray [17], Singh and Simeon [18], Alhussain [19], Chakraborty and Narayan [20], Amuda et al. [21], and others studied the R3BP by taking into account the P–R drag in different views. Mishra et al. [22] examined the stability of triangular points under the assumption that the bigger primary is a source of radiation with the incident P–R drag while the smaller is an oblate spheroid in the frame of the ER3BP. They concluded that the triangular points are unstable. In a recent study, Singh and Amuda [23] investigated the linear stability around L4,5 of a test particle in the field of post-AGB binary system with the effective P–R drag force. They asserted that P–R drag and the mass parameter μ contribute in shifting the locations of the triangular points and the triangular points are unstable in the linear sense due to the presence of complex conjugate roots.The aim of the present paper is to further investigate the effects of radiation pressure and P–R drag of the bigger primary onL4,5 in the ER3BP, taking into account the effects of dynamical flattening parameters of the primary bodies.


## 2. Equations of Motion

Letm1,m2, and m3 be the three masses, m1 and m2 being the dominant bodies having an elliptic orbit about their common centre of mass, while m3 being the infinitesimal mass which moves in the same plane with the dominant bodies under the influence of their force-field without influencing their motion. Let x1,y1,z1, x2,y2,z2, and x3,y3,z3 denote the coordinates of m1,m2, and m3, respectively, in the sidereal coordinates. Using Newton’s law, the equations of motion of an infinitesimal mass m3 in the sidereal coordinates system can be represented as(1)m3x¨3=∂V∂x3,m3y¨3=∂V∂y3m3z¨3=∂V∂z3,,where V=m3k2m1/R1+m2/R2; R1 and R2 are the distances of an infinitesimal mass from m1 and m2, respectively, and are defined as Ri2=x−xi2+y−yi2+z−zi2; k2 is the Gaussian constant of gravitation and the dot indicates differentiation with respect to time t.Equations (1) can be rewritten as [3](2)x¨3=∂W∂x3,y¨3=∂W∂y3,z¨3=∂W∂z3,where W=V/m3.Now, we choose and rotate the synodic coordinates uniformly with a positive unit rate, which has the same origin at the centre of mass of the two dominant bodies with the sidereal coordinates. The direction of thex−axis is chosen such that the two dominant bodies always lie on it. Therefore, the equations of motion in sidereal coordinates are related to the equations of motion in synodic coordinates with respect to the true anomaly. By transforming the equations of motion from true anomaly to eccentric anomaly in a dimensionless-pulsating (rotating) coordinate system ξ,η,ζ, we have(3)ξ″+eξ′sinEρ−21−e21/2η′ρ=∂Ω∂ξ,η″+eη′sinEρ+21−e21/2ξ′ρ=∂Ω∂η,ζ″+eζ′sinEρ=∂Ω∂ζ,where the prime ′ represents differentiation with respect to eccentric anomaly E and Ωξ,η,ζ=1/n2ρn2/2ξ2+η2−en2/2ρcosE−eζ2+W¯, ρ=1−ecosE, e and n are the eccentricity of the orbits and mean motion of the dominant bodies, while W¯=−V¯/m3,V¯=−k2m3m1/r1+m2/r2, rii=1,2 are the distances of an infinitesimal mass from the bigger primary and smaller primary, respectively, and are defined as ri2=ξ−ξi2+η2+ζ2,i=1,2., ξ1=−m2a/m1+m2, ξ2=m1a/m1+m2, and a is the semi-major axis of m2 around m1. See [3, 24, 25].Now, integrating equations (3) with respect to the eccentric anomaly E and averaging, we get(4)ξ″−2η′=∂Ω¯∂ξ,η″+2ξ′=∂Ω¯∂η,ζ″=∂Ω¯∂ζ,where Ω¯ is the potential-like function defined by Ω¯ξ,η,ζ=1−e2−1/21/2ξ2+η2+W¯/n2.Thus, the defined dynamical system in equations (4) is the required equations of motion for the ER3BP.Letq be the radiation pressure factor of the bigger primary which is given by(5)Fg1−Fp1=Fg11−Fp1Fg1=qFg1.This implies thatq=1−Fp1/Fg1 such that 0<1−q=α≪1 [26], where Fg1 is the gravitational force of m1 and Fp1 is the radiation pressure of m1.Considering the potential theory, the external gravitational potential due to a body that has axial symmetry can be written in terms of Legendre polynomials asV¯0=−k2m0/r01−∑n=2∞JnPncosθR¯0/r0n (see de Pater and Lissauer [27] for more details), where m0 denotes the mass of the body; r0 denotes the radial distance from the centre of the particle to the centre of any other body; θ denotes the angle between the body’s symmetry axis and vector to the particle; R¯0 denotes the mean radius of the body; Jn denotes the dimensionless coefficient that characterizes the degree of nonspherical components of the potential, Jn is zero for odd n, and when n is even, Jn is called a zonal harmonic coefficient; and the term Pncosθ denotes the Legendre polynomials of degree n and is defined by Pnx=1/2nn!dn/dxnx2−1n. We assume that the infinitesimal mass moves in the same plane of motion as the dominant bodies and this plane coincides with the equatorial plane i.e.θ=900. Therefore, the potential energy of the infinitesimal mass under the effects of the dynamical flattening parameters of both dominant bodies can be written as(6)V¯=−k2m3m11r1+A12r13−3A28r15+m21r2+B12r23−3B28r25,where Ai=J2iAR¯12i and Bi=J2iBR¯22i (i=1,2) represent the dynamical flattening parameters of the bigger and smaller primaries, respectively, J2iA and J2iBi=1,2 are the zonal harmonics coefficients; rii=1,2 is defined as in equation (3); and R¯1 and R¯2 are the mean radii of the dominant masses m1 and m2, respectively.The distance between the dominant bodies isr=a1−ecosE in the elliptic orbit, where a,e,andE are semi-major axis between the dominant bodies, common eccentricity of the dominant bodies, and eccentric anomaly, respectively.And the mean distance between them is as follows:(7)12π∫02πrdE=a.Assuming that the dominant bodies are in elliptical motion with constant angular velocityn(mean motion), then the orbits of m1 and m2 with respect to the centre of mass, with semi-major axes, would be a1=m2/m1+m2a and a2=m1/m1+m2a, respectively, having the same eccentricity; thus, the motion of the bigger and smaller primary can be written as [1, 28](8)m1n2a1=k2m1m2r21+3A12r2+3B12r2−15A28r4−15B28r4,m2n2a2=k2m1m2r21+3A12r2+3B12r2−15A28r4−15B28r4,where n, r, and k are the mean motion, distance between the dominant bodies, and Gaussian constant of gravitation, respectively. The distance between the dominant bodies r is defined to be the semi-major axis a of the orbit (i.e., r=a), since the dominant bodies are in elliptic orbits.Hence, adding equations (8) together, we obtain(9)n2a1+a2=k2m1+m2a21+3A12a2+3B12a2−15A28a4−15B28a4.Assume that the sum of the masses of the dominant bodies and the semi-major axis between them are the units of mass and length, respectively; i.e.,m1+m2=1, and a1+a2=a=1. Also, the unit of time is chosen so as to make the Gaussian constant, k2=1. Hence, equation (9) becomes(10)n2=1+3A12+3B12−15A28−15B28.Using equations (4), (5), (6), and (10), the equations of motion of an infinitesimal mass in the frame of the ER3BP can be modified, taking into account the dynamical flattening parameters of both dominant bodies together with the radiation pressure as well as P–R drag due to the bigger primary in a dimensionless-pulsating (rotating) coordinate system ξ,η as(11)ξ″−2η′=Uξ,η″+2ξ′=Uη,where(12)Uξ=∂Ω¯∂ξ−1−e2−1/2W1N1n2r12,Uη=∂Ω¯∂η−1−e2−1/2W1N2n2r12,Ω¯=1−e2−1/2ξ2+η22+1n21−μqr1+1−μA1q2r13−31−μA2q8r15+μr2+μB12r23−3μB28r25,N1=ξ+μξ+μξ′+ηη′r12+ξ′−nη,N2=ηξ+μξ′+ηη′r12+η′+nξ+μ,W1=1−μ1−qcd,and cd is the dimensionless speed of light [18, 22, 29].


## 3. Locations of the Triangular Equilibrium PointsL4,5

To obtain the equilibrium positions of an infinitesimal mass, the equationsξ′=η′=ξ″=η″=0 must be satisfied in the equations of motion (11); i.e., they are the solutions of the equations Uξ=Uη=0, and thus resulting in(13)ξn2−1−μξ+μqr13−31−μξ+μqA12r15+151−μξ+μqA28r17−μξ+μ−1r23−3μξ+μ−1B12r25+15μξ+μ−1B28r27+Wnηr12=0,(14)n2η−1−μqηr13−31−μqηA12r15+151−μqηA28r17−μηr23−3μηB12r25+15μηB28r27−Wnξ+μr12=0,which can be rewritten as(15)n2−1−μqr13−31−μqA12r15+151−μqA28r17−μr23−3μB12r25+15μB28r27η=Wnξ+μr12.Multiplying equations (13) and (14) by η and ξ+μ, respectively, we obtain(16)ξηn2−1−μξ+μηqr13−31−μξ+μηqA12r15+151−μξ+μηqA28r17−μξ+μ−1ηr23−3μξ+μ−1ηB12r25+15μξ+μ−1ηB28r27+Wnη2r12=0,(17)n2ξ+μη−1−μξ+μqηr13−31−μξ+μqηA12r15+151−μξ+μqηA28r17−μξ+μηr23−3μξ+μηB12r25+15μξ+μηB28r27−Wnξ+μ2r12=0.Subtracting equation (17) from equation (16), we obtain(18)n2=Wnμη+1r23+3B12r25−15B28r27=0.Using equation (18) in equation (15), we have(19)n2=qr13+3qA12r15−15qA28r17+Wnξ+μr12η1−μ−Wnη1−μ.In the absence of the dynamical flattening parametersAi=Bi=0,i=1,2 and P–R drag (i.e., W=0), the solutions of equations (18) and (19) are r1=r2=1. Then, considering the above parameters, the solutions of equations (18) and (19) would change slightly by(20)r1=1+ε1,r2=1+ε2,where εii=1,2≪1.Substituting equations (20) in equations (18) and (19) together with help of equation (10), we obtain the series equations in terms of εii=1,2≪1. Solving these equations by holding the expressions which contain Ai,Bi,A12,B12, A1B1i=1,2 (since Ai=J2iAR¯12i and Bi=J2iBR¯22ii=1,2, then A12,B12,A1B1,A2, and B2 have the same powers of mean radii of the dominant bodies), α (where α=1−q) and W also by restricting ourselves only to the quadratic terms in εii=1,2≪1, we have(21)ε1=−α3−B12+5B28+5A1B14+B122−W331−μ,ε2=−A12+5A28+5A1B14+A122+2W33μ.Substituting equations (21) in equations (20), we have(22)r1=1−α3−B12+5B28+5A1B14+B122−W331−μ,r2=1−A12+5A28+5A1B14+A122+2W33μ.Usingr12=ξ+μ2+η2&r22=ξ+μ−12+η2 defined in equations (3), then the exact solutions of the triangular points L4,5 are(23)ξ=12−μ+12r12−r22,η=±r12+r222−r22−r1222−14.Substituting equation (22) in (23), we obtain(24)ξ=12−μ−α3+12A1−B1−58A2−B2−58A12−B12−W2−μ33μ1−μ,η=±321−2α9−13A1+B1+512A2+B2+736A12+B12+17A1B19+2W2−3μ93μ1−μ.Hence, equations (24) are the required locations of the triangular equilibrium points L4,5 denoted by ξ,±η.


## 4. Stability of the Triangular Equilibrium Points

To examine the stability of the triangular equilibrium points, we place the infinitesimal mass at one of the equilibrium points and give it a small velocity. The point is stable for the oscillatory solutions with small amplitude and unstable for exponentially diverging solutions.Assume thatξ0,η0 are the coordinates of the equilibrium points under consideration and let x,y be the small displacements from these coordinates of the equilibrium points.These can be written as(25)ξ=ξ0+x,η=η0+y.Now, using equations (25) and holding only the linear terms of Taylor’s theorem as the second and higher power of xandy being very small, equation (11) become(26)x″−2y′=Uξ0+xUξξ0+yUξη0+x′Uξξ′0+y′Uξη′0,y″+2x′=Uη0+xUηξ0+yUηη0+x′Uηξ′0+y′Uηη′0,where the superscript 0 of equations (26) indicates that the partial derivatives are evaluated at the equilibrium points ξ0,η0. At equilibrium points, Uξ0=Uη0=0. Hence, equations (26) become(27)x″−2y′=xUξξ0+yUξη0+x′Uξξ′0+y′Uξη′0,y″+2x′=xUηξ0+yUηη0+x′Uηξ′0+y′Uηη′0.Supposex=Aeλt and y=Beλt are the trial solution of equations (27) (variational equations). Then, by using these values of the trial solutions in equations (27), we get(28)λ2−Uξξ0−λUξξ′0A+−2λ−Uξη0−λUξη′0B=0,λ2−Uηη0−λUηη′0B+2λ−Uηξ0−λUηξ′0A=0.Equations (28) have a nontrivial solution if(29)λ2−Uξξ0−λUξξ′0−2λ−Uξη0−λUξη′02λ−Uηξ0−λUηξ′0λ2−Uηη0−λUηη′0=0.Hence, the required characteristic equation of this dynamical system is given by [3, 22](30)λ4+a0λ3+a1λ2+a2λ+a3=0,where(31)a0=−Uξξ′0+Uηη′0,a1=4−Uξξ0+Uηη0+2Uξη′0+Uηξ′0+Uξξ′0Uηη′0−Uξη′0Uηξ′0,a2=Uξξ0Uηη′0+Uξξ′0Uηη0+2Uξη0−Uηξ0−Uξη′0Uηξ0−Uξη0Uηξ′0,a3=Uξξ0Uηη0−Uξη0Uηξ0.Now, the second partial derivatives of the modified potential-like functionU at triangular equilibrium points are(32)Uξξ0=34+3e28−12−3μ2α+94−3μA1−154−75μ16A2−34−3μB1+1516−75μ16B2−34−27μ16A12+4516−27μ16B12−15A1B18−Wμ2−13μ+843μ1−μ,Uηη0=94+9e28+12−3μ2α+3A14−154−45μ16A2+3B14−1516+45μ16B2−154−45μ16A12−1516+45μ16B12+39A1B18+W5μ2−17μ+83μ1−μ,Uξη0=33212−μ+12−μe22−19+μ9α+51−μ6+μ6A1−51−μ3+5μ24A2−1−μ6+5μ6B1+51−μ24+5μ3B2−109−73μ72A12+772+73μ72B12+131−μ36−13μ36A1B1−W27μ2−31μ+8183μ1−μ,Ωηξ0=Ωξη0,Uξξ′0=−5W4,Uηη′0=−7W4Uξη′0=Uηξ′0=−3W4,Uξη′02=0.Hence, the general expressions for the roots of the characteristic equation (30) are(33)λ1⟶−12a022−4a13−M4R−P−Q−a04−R2,λ2⟶12a022−4a13−M4R−P−Q−a04−R2,λ3⟶−12a022−4a13+M4R−P−Q−a04+R2,λ4⟶12a022−4a13+M4R−P−Q−a04+R2,where(34)K=a12−3a0a2+12a3,L=2a13−9a0a1a2+27a22+27a02a3−72a1a3,M=−a03+4a0a1−8a2,p=L+L2−4K31/33×21/3,Q=21/3K3L+L2−4K31/3,R=P+a024−2a13_Q.


## 5. Numerical Application

In this section, we study numerically the locations and stability of the triangular equilibrium pointsL4,5 of the Sun-Earth system, by taking into account the dynamical flattening parameters of both the Sun (m1) and Earth (m2), the radiation pressure factor, and P–R drag of the Sun.For the purpose of computation in this paper, the astrophysical data of the Sun-Earth system are borrowed from NASA ADS, Ragos et al. [29], Mecheri et al. [30], and Singh and Umar [31].The first two even zonal harmonics of the Sun areJ2A∼2.2×10−7&J4A∼−4.5×10−9 and those of the Earth are J2B∼1.0×10−3&J4B∼−1.6×10−6. Also, the orbital eccentricity of the Earth is e∼0.0167. Now, the dynamical flattening parameters of the primary bodies are given by A1∼5.0×10−12, A2∼−2.0×10−18,B1∼2.0×10−12,andB2∼−5.2×10−24.The radiation pressure factor of the Sunq is defined as q=1−α such that α can be expressed as α=L⊙/2πGm1cκ¯ [28], where L⊙ is the luminosity of the Sun, G is the gravitational constant, c is the speed of light, and κ¯ is the mass per unit area. By using Stefan–Boltzmann’s law, the luminosity of the primary can be expressed as L⊙=4πR¯⊙2σT⊙4, where σ is the Stefan–Boltzmann’s constant (see [28]). Also, the dimensionless velocity of light and the mass ratio of the Sun-Earth system are given by cd∼10064.84 and μ∼3.0035×10−6, respectively [29].


## 6. Discussion

The modified equations of motion of an infinitesimal mass in the framework of the elliptic restricted three-body problem under the effects of dynamical flattening parameters of both primaries, radiation pressure factor, and P–R drag of the bigger primary (i.e., the Sun) are given in equations (11), while the locations and characteristic equation of the triangular points L4,5 are given in equations (24) and (34), respectively.Figure1 shows the effects of the perturbing forces involved in the problem under consideration for the three different cases and classical case as well on the locations of triangular equilibrium points L4,5. Graph (a) is the classical case, graph (b) shows the effects of radiation pressure factor together with P–R drag, and graph (c) shows the effects of dynamical flattening parameters while graph (d) shows the combine effects of radiation pressure factor, P–R drag, and dynamical flattening parameters. This clearly shows that the perturbing forces under consideration have no significant effect on the locations of the triangular equilibrium points L4,5 in the vicinity of the Sun-Earth system. These effects can only be seen in the table (see Table 1).Figure 1
Effects of radiation pressure factor, P–R drag, and dynamical flattening parameters on the locations of triangular equilibrium pointsL4,5 of the Sun-Earth system. (a) Blue triangle is for the classical case. (b) Purple triangle is for Ai=Bi=0,W≠0,α≠0 case. (c) Brown triangle is for Ai≠0,Bi≠0,W=α=0 case. (d) Turquoise triangle is for Ai≠0,Bi≠0,W≠0,α≠0 case.
(a)(b)(c)(d)Table 1
Effects of radiation pressure factor, P–R drag, and dynamical flattening parameters on the locations of triangular equilibrium pointsL4,5 of the Sun-Earth system.
Casei=1,2ξ±η1Classical0.49999699650000000.8660254037844392Ai=Bi=0,W≠0,α≠00.49999699649993470.8660254037844743Ai≠0,Bi≠0,W=α=00.49999699650150000.8660254037824184Ai≠0,Bi≠0,W≠0,α≠00.49999699650143480.866025403782454Table2 shows numerical roots of the characteristic equation (30) for the classical case, as well as three other cases. In all cases, the characteristic roots reveal that all the roots are purely imaginary. This shows that the perturbing forces under consideration have no significant effect on the stability of the triangular equilibrium points L4,5. Hence, the motion of an infinitesimal mass near the triangular equilibrium points L4,5 of the Sun-Earth system is stable in the linear sense under the influence of these perturbations.Table 2
Effects of radiation pressure factor, P–R drag, and dynamical flattening parameters on the stability of triangular equilibrium pointsL4,5 of the Sun-Earth system.
Casei=1,2λ1,2λ3,4Remark1Classical±0.004504233279349719i±0.999780664387228iStable2Ai=Bi=0,W≠0,α≠0±0.004504233327067630i±0.999780664386725iStable3Ai≠0,Bi≠0,W=α=0±0.004504233279348517i±0.999780664379725iStable4Ai≠0,Bi≠0,W≠0,α≠0±0.004504233327101427i±0.999780664379223iStableForA12=B12=A1B1=0 and W=0, the present results of the triangular points in the circular case are in conformity with Singh and Taura [5] for p2=Mb=0 in their results and for α=W=0, which also agrees with those of Abouelmagd et al. [11]; the difference in configuration of the primary bodies is responsible for the difference in sign.Their results are [5, 11](35)r12=x+μ2+y2,r22=x+μ−12+y2,ξ=121−2μ−2p13+2p23+A1−54A2−B1+54B2,η=±321−2p1+p29−A13+5A212−B13+5B212−4Mb2rC−19rC2+T22/3,(36)r12=x−μ2+y2,r22=x−μ+12+y2,x4,5=μ−12−12A1−B1+58A2−B2+58A12−B12,y4,5=±321−13A1−B1+512A2+B2+1367A12+B12+68A1B1.In the caseAi=Bi=0i=1,2 in the present work, the obtained results of the triangular points in the circular case are in agreement with those of Singh and Simeon [18] by taking σi=σi′=δ2=W2=0i=1,2,andW1⟶W in their results.Their results are(37)x=12−μ−δ13+δ23−W12−μ3μ1−μ3−W21+μ3μ1−μ3−4+μσ18μ+4+3μσ1′81−μ+−7+3μσ2′81−μy=±321−2δ19−2δ29−13234−11−μσ1′+13194−11−μσ2′+13−234+1μσ1+13194−1μσ2+W12−3μ9μ1−μ+W21−3μ9μ1−μ.Our results for the second partial derivatives and the characteristic equation differ from those of Singh and Taura [5], Abouelmagd et al. [11], and Singh and Simeon [18] due to the elliptic nature of our potential-like function. However, the P–R drag parts of the partial derivatives coincide with those of Singh and Simeon [18], that is, for δ2=W2=0,W⟶W1,andα⟶δ1. Their results for the second partial derivatives are(38)Ωxx0=34+δ13μ2−12+δ212−3μ2+5716+45μ16−32μσ1+316−93μ16+32μσ2+398−69μ16−3μ221−μσ1′+−92+117μ16+3μ221−μσ2′−W1μ2−13μ+84μ1−μ3+W2μ2+11μ−44μ1−μ3,Ωyy0=94+δ112−3μ2+δ2−1+3μ2+8716−45μ16+32μσ1+−216+45μ16−32μσ2+338+135μ16−33μ81−μ+45μ281−μσ1′+−135μ16+33μ81−μ−45μ281−μσ2′+W15μ2−17μ+84μ1−μ3−W25μ2+7μ−44μ1−μ3,Ωxy=33212−μ−δ11+μ9+δ22−μ9+4724−89μ24−13μσ1+924−37μ24+13μσ2+2512−85μ24+μ61−μ+μ261−μσ1′+−32+11μ8−μ61−μ−μ261−μσ2′−W127μ2−31μ+818μ1−μ3−W227μ2−23μ+418μ1−μ3,Ωxy0=Ωyx0,Ωxx˙0=−5W1−5W24,Ωyy˙0=−7W1−7W24,Ωxy˙0=−3W14+3W24,Ωxy˙02=0.Thus, forA2=B2=W=0 in the elliptic case of the triangular points, the obtained results coincide with those of Narayan and Shrivastava [6]; that is, for α⟶∈1,A1⟶A1,B1⟶A2,A1∈1=0&∈2=0. Their results are (39)x0=12−μ+A12−A22+∈23−∈13−A1∈12+A2∈22y0=±321−A13−A23−2∈19−2∈29−A1∈13−A2∈23.In the elliptic case, the obtained results do not agree with those of Singh and Umar [4] and Singh and Tyokyaa [12]. This is because we have used the modified mean motion n(equation (10)) which does not contain the eccentricity e and semi-major axis a. However, by substituting e=0 and a=1, their results in the triangular case coincide fully with ours upon relaxing some parameters in our problem. The same applies to the second partial derivatives.Their results are [12, 14](40)n2=1a1+3A12+3B12−15A28−15B28+3e22,ξ=12−μ+12A1−B1−54A2a−2/3+54B2a−2/3,η2=a2/3−14−a2/3e2+A1+B1−54A2−54B2+A12+B12−58A2a−2/3+58B2a−2/3,(41)n2=1a1+3A12+3A22+3e22,ξ=12−μ+12aq12/31−e2−A1−A2+A1aq12/3−aq22/31−e2−A1−A2+A2aq22/3,η=±aq12/31−e2−A1−A2+A1aq12/3−141+2aq12/31−e2−A1−A2+A1aq12/3−2aq22/31−e2−A1−A2+A2aq22/31/2.The P–R drag finds importance in the investigation of the stability of zodiacal cloud, orbital evolution of cometary meteor steams, asteroidal particles, and dust rings around planets. This model is applicable not only to the Sun-Earth system but also to other systems in both the solar and stellar systems as well.



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