Abstract
The present manuscript reveals the existence and stability of the libration points (LPs) in the axisymmetric restricted five-body problem (AR5BP) in which mass of the fifth particle varies with respect to time as per Jeans’ law. It has been assumed that the four bodies Pi with masses mi,i=1,2,…,4 (when m3=m4=m˜) form an axisymmetric convex configuration. The system of differential equations which govern the motion of the test particle (whose mass is variable) moving under the gravitational influence of the four primaries, have been presented, indeed, these equations are different from those of the test particle with constant mass in the AR5BP. We have determined the in-plane and out-of-plane LPs along with their linear stability. Furthermore, it is observed that the existence of these LPs depends not only on the angle parameters α and β but also on the parameters occur due to variation in mass namely γ(0<γ≤1) and σ(0≤σ≤2.2, a proportionality constant occurs in Jeans’ law). Moreover, we have investigated the evolution of the regions of possible motion as function of variable mass parameters where the fifth body can move freely. The topology of the basins of convergence (BoC) linked with the LPs as function of σ is unveiled by deploying the bivariate version of Newton-Raphson (NR) iterative method, and, in order to measure the uncertainty of the basins, the basin entropy is also evaluated. The co-relations between the domains of basins of convergence (DoBoC) and the required number of iterations to achieve predefined accuracy, and also with the corresponding probability distributions are illustrated.
Highlights
Ij where G is the Gaussian constant of gravitation and ri j = |r j − ri| is the distance between Pi and Pj bodies
The central configuration of four bodies is analyzed by Erdi and Czirjak (2016) where two of the bodies are on the axis of symmetry where the positions of the bodies on the axis of symmetry are shown by the angle co-ordinates with respect to the outer bodies
To determine the linear stability of this nontrivial solution corresponds to the libration points (LPs) depend on the boundedness of the solution of linear, homogeneous system of equations associated to Eq 32
Summary
We can not use Newton’s second law for the equation of motion within a frame of the variable mass system directly as it is true only for the constant mass system (see Plastino and Muzio (1992)). If the loss of mass is taken as non-isotropic, we can apply Newton’s second law in the modified form as used by Abouelmagd et al (2015b) to obtain the equation of motion for the fifth body of the infinitesimal mass m5 in the inertial frame, V. in the case in which the escaping or incoming mass occurs from n points for the body in the form. Eq 4 represents the equation of motion of the fifth body whose mass varies with respect to time t according to Jeans’ law, and θ i = νi − R 5 where θ i and νi are the relative velocity and velocity respectively of the escaping or incoming mass with respect to the body from the points i(i = 1, 2, 3, ..., n), whereas R 5 is the velocity of fifth particle in the inertial frame. Where x, y, and zare components of the velocity , whereas the Jacobian constant is represented by C which is conserved
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