Restricted Problem of Three Bodies With Newtonian + Yukawa Potential

Journal of Applied Mathematics

Eshetie

Tilahun

et al. 2022

We study the triangular equilibrium points in the framework of Yukawa correction to Newtonian potential in the circular restricted three-body problem. The effects of α and λ on the mean-motion of the primaries and on the existence and stability of triangular equilibrium points are analyzed, where α ∈ − 1 , 1 is the coupling constant of Yukawa force to gravitational force, and λ ∈ 0 , ∞ is the range of Yukawa force. It is observed that as λ ⟶ ∞ , the mean-motion of the primaries n ⟶ 1 + α 1 / 2 and as λ ⟶ 0 , n ⟶ 1 . Further, it is observed that the mean-motion is unity, i.e., n = 1 for α = 0 , n > 1 if α > 0 and n < 1 when α < 0 . The triangular equilibria are not affected by α and λ and remain the same as in the classical case of restricted three-body problem. But, α and λ affect the stability of these triangular equilibria in linear sense. It is found that the triangular equilibria are stable for a critical mass parameter μ c = μ 0 + f α , λ , where μ 0 = 0.0385209 ⋯ is the value of critical mass parameter in the classical case of restricted three-body problem. It is also observed that μ c = μ 0 either for α = 0 or λ = 0.618034 , and the critical mass parameter μ c possesses maximum ( μ c max ) and minimum ( μ c min ) values in the intervals − 1 < α < 0 and 0 < α < 1 , respectively, for λ = 1 / 3 .

“…The equations of motion of the infinitesimal mass in a barycentric synodic coordinate system ðx, yÞ and dimensionless variables are [18]:…”

Section: Equations Of Motionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Triangular Equilibria in R3BP under the Consideration of Yukawa Correction to Newtonian Potential

Journal of Applied Mathematics

Eshetie

Tilahun

et al. 2022

“…Te restricted three-body problem under the consideration of Yukawa correction is studied by Kokubun [3]. Te main aim of this research was to study the efect of modifed potential on the important aspects of restricted three-body problem.…”

Section: Introductionmentioning

confidence: 99%

The Predictions of Noncollinear Equilibria Positions in ER3BP with Yukawa-Like Corrections

Anitha¹,

Tilahun

2023

Advances in Astronomy

The existence and stability of noncollinear equilibrium points in the elliptic restricted three-body problem under the consideration of Yukawa correction to Newtonian potential are studied in this paper. The effects of various parameters (μ, ê, α, and λ) on the noncollinear equilibrium points are discussed briefly, and it is found that only ordinate of noncollinear equilibria E4,5 is affected by Yukawa correction while abscissa is affected by only mass parameter μ. The noncollinear equilibria was found linearly stable for a critical mass parameter μc. A critical point λ = ½ is also obtained for the critical mass parameter μc, and at this point, the critical mass parameter μc has maximum or minimum values according to α < 0 or α > 0, respectively.

“…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…”

Section: Introductionmentioning

confidence: 99%

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

Haruna²,

Eshetie

2023

Advances in Astronomy

This study is about the effects of Yukawa-like corrections to Newtonian potential on the existence and stability of noncollinear equilibrium points in a circular restricted three-body problem when bigger primary is an oblate spheroid. It is observed that ∂x0/∂λ = 0 = ∂y0/∂λ at λ0 = 1/2, so we have a critical point λ0 = 1/2 at which the maximum and minimum values of x0 and y0 can be obtained, where λ ∈ (0, ∞) is the range of Yukawa force and (x0, y0) are the coordinates of noncollinear equilibrium points. It is found that x0 and y0 are increasing functions in λ in the interval 0 < λ < λ0 and decreasing functions in λ in the interval λ0 < λ < ∞ for all α ∈ α+. On the other hand, x0 and y0 are decreasing functions in λ in the interval 0 < λ < λ0 and increasing functions in λ in the interval λ0 < λ < ∞ for all α ∈ α−, where α ∈ (−1, 1) is the coupling constant of Yukawa force to gravitational force. The noncollinear equilibrium points are found linearly stable for the critical mass parameter β0, and it is noticed that ∂β0/∂λ = 0 at λ ∗ = 1/3; thus, we got another critical point which gives the maximum and minimum values of β0. Also, ∂β0/∂λ > 0 if 0 < λ < λ ∗ and ∂β0/∂λ < 0 if λ ∗ < λ < ∞ for all α ∈ α−, and ∂β0/∂λ < 0 if 0 < λ < λ ∗ and ∂β0/∂λ > 0 if λ ∗ < λ < ∞ for all α ∈ α+. Thus, the local minima for β0 in the interval 0 < λ < λ ∗ can also be obtained.