Noncollinear Equilibrium Points in CRTBP with Yukawa-Like Corrections to Newtonian Potential under an Oblate Primary Model

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Introduction
Te general three-body problem deals with the motion of three arbitrary spherically symmetric bodies considered as a point mass. Te motions of these bodies are related to the Newtonian force of gravity, which are superimposed on each other and have no specifc path. Te closed form of analytical solution to the general three-body problem is yet to be determined.
Te restricted three-body problem (R3BP) is an approximation of the general three-body problem in which one body is treated as having an infnitesimal mass compared to the other two bodies. Te bigger bodies are called primaries which revolve around their common center of mass in circular or elliptical orbits in a rotating coordinate system in which the infnitesimal mass also moves without disturbing the motion of the primaries. Te restricted three-body problem has fve equilibrium points, three collinear, and two noncollinear or triangular. Te collinear equilibria are unstable for all values of mass parameter but triangular equilibria are stable for a critical mass parameter μ 0 = 0.03852 [1].
Te restricted three-body problem has been studied by many researchers in last two decades in diferent aspects. In the classical restricted three-body problem, the primaries are assumed as spherical in shape, but in real situation, several heavenly bodies such as Earth, Saturn, and Jupiter are sufciently oblate. Te oblateness efect in the restricted three-body problem has been investigated by El-Shaboury [2]; Khanna and Bhatnagar [3]; Raheem and Singh [4]; Ammar et al. [5]; Idrisi and Taqvi [6,7], Singh and Umar [8]; Bury and McMahon [9]; Saeed and Zotos [10]; Alrebdi et al. [11] etc. New theories in the contemporary world predict improvements to the theory of gravity. Te Yukawa potential was frst proposed by Yukawa [12] to modify the Newtonian potential. Te strong interactions between particles are well described by the Yukawa potential, a nonrelativistic potential. In a two-body problem, the modifed potential energy may be used to express the gravity efects on the secondary primary m in the presence of the Yukawa correction [13] as where V N (r) is the Newtonian potential between the two bodies m and M, V Y (r) is the Yukawa correction to the Newtonian potential, r is the distance between m and M, G is the Newtonian gravitational constant, α ϵ (− 1, 1) is the coupling constant of the Yukawa force to the Gravitational force, and λ ϵ (0, ∞) is the range of the Yukawa force [14]. Terefore, the corresponding force between m and M can be expressed as As α ⟶ 0, the Newtonian gravitational force can be obtained.
In the restricted three-body problem, Kokubun [14] has included Yukawa-like corrections to Newtonian potential. His fndings difered signifcantly from the purely Newtonian case. Reference [15] provides the minimal values of the Yukawa coupling constant for the artifcial satellites LAGEOS and LAGEOS II. Massa [16] investigated Mach's principle and Yukawa potential within the Sciama linear approach framework. Haranas and Ragos [17] investigated satellite dynamics while taking Yukawa-like corrections into account. Pricopi [18] has investigated the stability of celestial orbits under the efect of the Yukawa potential in the twobody problem. Reference [19] has analyzed the elliptical and circular orbits of the Earth while taking into account the Yukawa potential and Poynting-Robertson efect. Te dynamics and stability of the two-body problem were examined by Cavan et al. [20] while taking the Yukawa corrections to Newtonian potential into account. Idrisi et al. [21] have investigated the triangular equilibria in the framework of Yukawa correction to Newtonian potential in the circular restricted three-body problem.
Te dynamics surrounding noncollinear equilibrium points in a circular restricted three-body problem with a Yukawa-like adjustment to Newtonian potential under an oblate primary model piqued our attention. Te existence and linear stability of noncollinear equilibrium points under an oblate primary model with Yukawa like-corrections to Newtonian potential have been examined in this study.

Yukawa Correction to Newtonian Potential
Te modifed potential between two bodies M and m can be described as follows: where V N (r) � Newtonian potential between the two bodies M and m, V Y (r) � Yukawa correction to the Newtonian potential, r � distance between m and M, G � Newtonian gravitational constant, α ∈ (− 1, 1) is the coupling constant of Yukawa force to the gravitational force, and λ ∈ (0, ∞) is the range of Yukawa force [14]. Terefore, the corresponding force between M and m can be expressed as where F N (r) � Newtonian gravitational force between M and m and F Y (r) � Yukawa correction to Newtonian gravitational force between M and m. From (4), as α ⟶ 0 or λ ⟶ 0, the term F Y (r) vanishes and F(r) � F N (r). If α < 0, F(r) < F N (r) and for α > 0, F(r) > F N (r), Figure 1. Tus, as α increases in the interval (− 1, 1), the force between m and M also increases and vice-versa.
But as λ ⟶ ∞ the force between M and m is given by From (5), it is clear that as α ⟶ − 1, F ∞ (r) ⟶ 0, i.e., the force between m and M reduces as α reduces. For α ⟶ 0, F ∞ (r) ⟶ F N (r) and the Newtonian gravitational force can be obtained. But as α ⟶ 1, F ∞ (r) ⟶ 2F N (r), i.e., the force acting between m and M is twice of the Newtonian gravitational force, as shown in Figure 2.

Model Description and Equations of Motion
Let us consider two primaries P 1 and P 2 having masses m 1 and m 2 (m 1 > m 2 ) moving around their common center of mass in circular orbits. Te more massive primary m 1 is considered to be an oblate body while less massive primary m 2 is spherical in shape. Te equations of motion of the infnitesimal mass in a barycentric synodic co-ordinate system (x, y) and dimensionless variables are and the potential function U can be expressed as σ � (r 2 e − r 2 p )/5r 2 is the oblateness factor due to bigger primary m 1 , r e and r p are the equatorial and polar radii respectively of m 1 , r is the distance between m 1 and m 2 considered as unity, n is the mean-motion of the primaries, and defned as |α| < 1 is the coupling constant of Yukawa force to gravitational force, λ ∈ (0, ∞) is the range of Yukawa force.
We can defne a mass parameter β > 0 as Terefore, the distances of infnitesimal mass from the primaries P 1 and P 2 , are given by Te Jacobi integral associated to the problem is given by where v is the velocity of infnitesimal mass and C is Jacobi constant.

Noncollinear Equilibrium Points
Te noncollinear equilibrium points are the solution of the equations U x � 0 and U y � 0, y ≠ 0, i.e.,

Advances in Astronomy
On eliminating r 1 and r 2 from the (12) and (13), respectively, we have Te solution of (15) is r 1 � 1. To solve (14), we assume that r 2 � 1 + δ, δ << 1. On substituting r 2 � 1 + δ in (14) and considering only linear terms in α and δ and then solving it for δ, we obtain Thus Now, solving r 1 � 1 and (17), we have the coordinates of noncollinear equilibrium points E 4, 5 (x 0 , y 0 ), i.e., For a nonoblate case, i.e., σ � 0 we obtain r i � 1 which is the classic case of restricted three-body problem [1], and hence in the nonoblate case, the noncollinear equilibrium points are not afected by the Yukawa force [21]. For α � 0, the results are agreed with [22].
For α ∈ α + , as λ increases in the interval 0 < λ < λ 0 , the abscissa x 0 of E 4 moves toward the center of mass of the system and the ordinate y 0 moves vertically upward and vice-versa. In the interval λ 0 < λ < ∞, x 0 and y 0 decrease and hence approach to x * and y * , respectively, as λ increases and vice versa. For α ∈ α − , as λ increases in the interval 0 < λ < λ 0 , the abscissa x 0 moves away from x * and y 0 moves vertically downward and vice-versa. In the interval λ 0 < λ < ∞, x 0 and y 0 increase and hence approach to x * and y * , respectively, as λ increases and vice-versa (Figures 4 and 5).
Te noncollinear equilibrium points E 4, 5 at the critical point λ � 1/2 have maximum or minimum values according to α ∈ α + or α ∈ α − , respectively, are given as Advances in Astronomy
Te shaded region in Figure 8 corresponds to stable region for the noncollinear equilibrium points, and it is seen that as alpha increases the stability region also increases and vice-versa.

Real Application to the Earth-Moon System
From astrophysical data [23], mass of Earth � 5.972 × 10 24 kg, mass of moon � 7.348 × 10 22 kg, axes of the Earth: r e � 6378.140 km, r p � 6356.755 km, and average distance between Earth and moon � 382500 km.
(28) Table 1 lists the numerical locations of noncollinear equilibrium points E 4, 5 (x 0 , y 0 ) for the aforementioned values of β, σ and for |α| < 1. For all possible values of α, it has been found that the numerical values of x 0 and y 0 are identical up to six decimal places.

. Conclusion
We studied the dynamics around noncollinear equilibrium points in the circular restricted three-body problem under the considerations of oblateness of more massive primary and Yukawa-like corrections to Newtonian potential. Te modifed gravitational force between the two masses M and m, therefore, can be written as F(r) � F N (r) + F Y (r), where F N (r) is Newtonian gravitational force between M and m, and F Y (r) is Yukawa correction to Newtonian gravitational force between M and m. It is found that as α ⟶ 0 or λ ⟶ 0, the term F Y (r) vanishes and F(r) � F N (r), where α ∈ (− 1, 1) is the coupling constant of Yukawa force to gravitational force and λ ∈ (0, ∞) is the range of Yukawa force. If α < 0, F(r) < F N (r) and for α > 0, F(r) > F N (r), Figure 1. Tus, as α increases in the interval (− 1, 1), the force between m and M also increases and vice-versa. But as λ ⟶ ∞, the force between M and m is given by F ∞ (r) and F ∞ (r) ⟶ 0 as α ⟶ − 1, i.e., the force between m and M reduces as α reduces. For α ⟶ 0, F ∞ (r) ⟶ F N (r) and the Newtonian gravitational force can be obtained. But as α ⟶ 1, F ∞ (r) ⟶ 2F N (r), i.e., the force acting between m and M is twice of the Newtonian gravitational force, as shown in Figure 2.

Data Availability
Te data used to support the fndings of this study are included within this research article. For simulation, we have used data from other research papers which are properly cited.

Conflicts of Interest
Te authors declare that they have no conficts of interest.