Equations of motion of the restricted problem of three bodies with variable mass

Das, Rik; Shrivastava, Ajay Kumar; Ishwar, B.

doi:10.1007/bf01232197

Cited by 53 publications

(44 citation statements)

References 3 publications

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“…By using Meshcherskii [10] inverse transformations, and putting (13) As the positions of the primaries are fixed and their distances to the libration points are invariable, the stability of (13) and (5) is consistent with each other. In fact, the original null solution when 0 1  has been disturbed into a non-trivial solution.…”

Section: Stability Of the Liberation Pointsmentioning

confidence: 99%

“…In fact, the original null solution when 0 1  has been disturbed into a non-trivial solution. Thus, the linear stability of this solution depends on the existence of stable region of the libration point, which in turn depends on the boundedness of the solution of linear and homogenous system of equations (13). We have determined the linear stability of the libration points.…”

Section: Stability Of the Liberation Pointsmentioning

confidence: 99%

“…Researchers have also studied about restricted three and four body problem with variable mass ie. Shrivastava [13], Singh [14], [15], [16], [17], Zhang [19], Abouelmagd [1]. After reviewing all the literature, we impressed to do the work on the restricted four body problem with one of the primaries as oblate body and all the three primaries are placed at the vertices of a triangle and also the infinitesimal mass is taken as variable mass.…”

Section: Introductionmentioning

confidence: 99%

“…Here the body P 2 is an oblate body and others are spherical in shape. And let P 1 P 2 = ρ 12 , P 2 P 3 = ρ 23 and P 1 P 3 = ρ 13 . And let the co-ordinates of P i be (X i , Y i ) (i = 1, 2, 3).…”

Section: Introductionmentioning

confidence: 99%

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Stability of the equilibrium points in the circular restricted four body problem with oblate primary and variable mass

Ansari

2016

IJAA

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This paper investigates the liberation points and stability of the restricted four body problem with one of the primaries as oblate body and the infinitesimal body is taken as variable mass. Due to oblateness, the equilateral triangular configuration is no longer exists and becomes an isosceles triangular configuration. Moreover, we have found seven equilibrium points out of which three are asymptotically stable (dark black in the tables) and rest four are unstable.

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Section: Stability Of the Liberation Pointsmentioning

confidence: 99%

Section: Stability Of the Liberation Pointsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stability of the equilibrium points in the circular restricted four body problem with oblate primary and variable mass

Ansari

2016

IJAA

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“…Majorana in [34] examined the linear stability of eight equilibrium points which depends on the values of the mass parameter in the restricted four-body problem. Shrivastava et al in [47] deduced the equations of motion in the restricted three-body problem with decreasing mass by using the Jeans law and Meshcherskii transformation. Shrivastava et al in [46] evaluated the equilibrium points in the Robes restricted problem of three-bodies with effect of perturbations in the coriolis and centrifugal forces.…”

Section: Introductionmentioning

confidence: 99%

The effect of perturbations on the circular restricted four-body problem with variable masses

Ansari¹,

Alhussain²,

Kellil³

2017

J. Math. Computer Sci.

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This paper presents a new investigation of the circular restricted four body problem under the effect of any variation in coriolis and centrifugal forces. Here, masses of all the bodies vary with time. This has been done by considering one of the primaries as oblate body and all the primaries are placed at the vertices of a triangle. Due to the oblateness, the triangular configuration becomes an isosceles triangular configuration which was an equilateral triangle in the classical case. After evaluating the equations of motion, we have determined the equilibrium points, the surfaces of the motion, the time series and the basins of attraction of the infinitesimal body. We note that, when we increase both the coriolis and centrifugal forces, the curves, surfaces of motion, and the basins of attraction are shrinking except when we fix the centrifugal force and increase the value of coriolis force, the curves are expanding and the equilibrium points are away from the origin. The behavior of the surfaces of motion and the basins of attraction in the last case (fixing the centrifugal force and increasing the value of coriolis force) will be studied next. In all the present study, we found that all the equilibrium points are unstable.

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The effect of small perturbations in the Coriolis and centrifugal forces on the existence of libration points in the restricted four‐body problem with variable mass

et al. 2018

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This paper shows the effect of small perturbations in the Coriolis and centrifugal forces in the restricted four‐body problem (R4BP) with variable mass. The existence, location, and stability of the libration points are investigated numerically and graphically under these perturbations. In the present problem, a fourth body with infinitesimal mass is moving under the Newtonian gravitational attraction of three primaries which are moving in a circular orbit around their common center of mass fixed at the origin of the coordinate system. Moreover, according to the solution of Lagrange, the primaries are fixed at the vertices of an equilateral triangle. The fourth body does not affect the motion of three primaries. Furthermore, the fourth body's mass varies according to Jeans' law. The equations of motion of the test particle, i.e., fourth body moving under the gravitational influence of the primaries, are derived. Throughout the paper, we consider the case where the primary body placed along the x‐axis is dominant while the other two small primaries are equal. Further, it is shown that there exist either 8 or 10 libration points out of which 2 or 4 are collinear with the dominating primary and the rest are non‐collinear for fixed values of the parameters. The linear stability of all the libration points under consideration is investigated, and these libration points are found to be unstable. The allowed regions of motion are determined by using the zero‐velocity surface, and the positions of the libration points on the orbital plane are presented. Moreover, by using the Newton–Raphson iterative scheme, we unveiled the effects of the Coriolis and centrifugal forces on the topology of the basins of convergence associated with the libration points.

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Equations of motion of the restricted problem of three bodies with variable mass

Cited by 53 publications

References 3 publications

Stability of the equilibrium points in the circular restricted four body problem with oblate primary and variable mass

Stability of the equilibrium points in the circular restricted four body problem with oblate primary and variable mass

The effect of perturbations on the circular restricted four-body problem with variable masses

The effect of small perturbations in the Coriolis and centrifugal forces on the existence of libration points in the restricted four‐body problem with variable mass

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