scholarly journals On Robe's Circular Restricted Problem of Three Variable Mass Bodies

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Jagadish Singh ◽  
Oni Leke
Keyword(s):  
Equilibrium Points ◽  
Mass Parameter ◽  
Variable Mass ◽  
Density Ratio ◽  
Density Parameter ◽  
Out Of Plane ◽  
Nonautonomous Equations ◽  
The Masses ◽  
Three Body ◽  
Plane Equilibrium

This paper investigates the motion of a test particle around the equilibrium points under the setup of the Robe’s circular restricted three-body problem in which the masses of the three bodies vary arbitrarily with time at the same rate. The first primary is assumed to be a fluid in the shape of a sphere whose density also varies with time. The nonautonomous equations are derived and transformed to the autonomized form. Two collinear equilibrium points exist, with one positioned at the center of the fluid while the other exists for the mass ratio and density parameter provided the density parameter assumes value greater than one. Further, circular equilibrium points exist and pairs of out-of-plane equilibrium points forming triangles with the centers of the primaries are found. The out-of-plane points depend on the arbitrary constant , of the motion of the primaries, density ratio, and mass parameter. The linear stability of the equilibrium points is studied and it is seen that the circular and out-of-plane equilibrium points are unstable while the collinear equilibrium points are stable under some conditions. A numerical example regarding out-of-plane points is given in the case of the Earth, Moon, and submarine system. This study may be useful in the investigations of dynamic problem of the “ocean planets” Kepler-62e and Kepler-62f orbiting the star Kepler-62.

scholarly journals Generalized Out-of-Plane Equilibrium Points in the Frame of Elliptic Restricted Three-Body Problem: Impact of Oblate Primary and Luminous-Triaxial Secondary

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Aminu Abubakar Hussain ◽  
Aishetu Umar
Keyword(s):  
Binary Systems ◽  
Equilibrium Points ◽  
Semimajor Axis ◽  
Three Body Problem ◽  
Out Of Plane ◽  
Elliptic Restricted Problem ◽  
The Stability ◽  
Zero Velocity Curves ◽  
Three Body ◽  
Plane Equilibrium

This paper studies the motion of a third body near the 1st family of the out-of-plane equilibrium points, L6,7, in the elliptic restricted problem of three bodies under an oblate primary and a radiating-triaxial secondary. It is seen that the pair of points (ξ0,0,±ζ0) which correspond to the positions of the 1st family of the out-of-plane equilibrium points, L6,7, are affected by the oblateness of the primary, radiation pressure and triaxiality of the secondary, semimajor axis, and eccentricity of the orbits of the principal bodies. But the point ±ζ0 is unaffected by the semimajor axis and eccentricity of the orbits of the principal bodies. The effects of the parameters involved in this problem are shown on the topologies of the zero-velocity curves for the binary systems PSR 1903+0327 and DP-Leonis. An investigation of the stability of the out-of-plane equilibrium points, L6,7 numerically, shows that they can be stable for 0.32≤μ≤0.5 and for very low eccentricity. L6,7 of PSR 1903+0327 and DP-Leonis are however linearly unstable.

scholarly journals The Motion of Out-of-Plane Equilibrium Points in the Elliptic Restricted Three-Body Problem at J4

2021 ◽  
Vol 17 ◽  
pp. 1-11
Author(s):  
Jagadish Singh ◽  
Tyokyaa K. Richard
Keyword(s):  
Binary Systems ◽  
Body Problem ◽  
Equilibrium Points ◽  
Major Axis ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Semi Major Axis ◽  
Out Of Plane ◽  
Three Body ◽  
Plane Equilibrium

We have investigated the motion of the out-of-plane equilibrium points within the framework of the Elliptic Restricted Three-Body Problem (ER3BP) at J4 of the smaller primary in the field of stellar binary systems: Xi- Bootis and Sirius around their common center of mass in elliptic orbits. The positions and stability of the out-of-plane equilibrium points are greatly affected on the premise of the oblateness at J4 of the smaller primary, semi-major axis and the eccentricity of their orbits. The positions L6, 7 of the infinitesimal body lie in the xz-plane almost directly above and below the center of each oblate primary. Numerically, we have computed the positions and stability of L6, 7 for the aforementioned binary systems and found that their positions are affected by the oblateness of the primaries, the semi-major axis and eccentricity of their orbits. It is observed that, for each set of values, there exist at least one complex root with positive real part and hence in Lyapunov sense, the stability of the out-of-plane equilibrium points are unstable.

scholarly journals Comment on “Out-of-plane equilibrium points in the restricted three-body problem with oblateness (Research Note)”

2018 ◽  
Vol 614 ◽  
pp. A67 ◽  
Author(s):  
Nan Wu ◽  
Xuefeng Wang ◽  
Li-Yong Zhou
Keyword(s):  
Potential Function ◽  
Body Problem ◽  
Equilibrium Points ◽  
Point Of View ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Physical Point ◽  
Out Of Plane ◽  
Three Body ◽  
Plane Equilibrium

Douskos & Markellos (2006, A&A, 446, 357) first reported the existence of the out-of-plane equilibrium points in restricted three-body problem with oblateness. This result deviates significantly from the intuitive physical point of view that there is no other force that can balance the combined gravitation in Z direction. In fact, the out-of-plane equilibrium in that model is illusory and we prove here that such equilibrium points arise from the improper application of the potential function.

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