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

Aishetu Umar,Aminu Abubakar Hussain,Elbaz Abouelmagd

doi:10.1155/2021/6657500

Aishetu Umar, Aminu Abubakar Hussain + Show 1 more

Open Access

https://doi.org/10.1155/2021/6657500

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Abstract

Using an analytical and numerical study, this paper investigates the equilibrium state of the triangular equilibrium points L 4 , 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.

Highlights

Using an analytical and numerical study, this paper investigates the equilibrium state of the triangular equilibrium points L4, 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). e 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
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
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 [2,3,4,5,6,7,8] and others

Summary

Equations of Motion

Let m1, 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. Erefore, the potential energy of the infinitesimal mass under the effects of the dynamical flattening parameters of both dominant bodies can be written as. E distance between the dominant bodies is r a(1 − e cos E) in the elliptic orbit, where a, e , and E are semi-major axis between the dominant bodies, common eccentricity of the dominant bodies, and eccentric anomaly, respectively. 185rB42􏼣, where n, r, and k are the mean motion, distance between the dominant bodies, and Gaussian constant of gravitation, respectively. 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 ξ′′ − 2η′ Uξ,. Equations (24) are the required locations of the triangular equilibrium points L4, 5 denoted by (ξ, ± η)

Stability of the Triangular Equilibrium Points

Numerical Application