Contents lists available at ScienceDirect New Astronomy journal homepage: www.elsevier.com/locate/newast The motion properties of the infinitesimal body in the framework of bicircular Sun perturbed Earth–Moon system Elbaz I. Abouelmagd ⁎ ,a,b , Abdullah A. Ansari c a Celestial Mechanics and Space Dynamics Research Group (CMSDRG), Astronomy Department, National Research Institute of Astronomy and Geophysics (NRIAG), Helwan 11421 - Cairo – Egypt b Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Mathematics Department, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia c International Center for Advanced Interdisciplinary Research (ICAIR), Ratiya Marg, Sangam Vihar, New Delhi, India ARTICLE INFO Keywords: Bicircular model Energy conservation law Stability of equilibrium points Basins attraction of convergence ABSTRACT In this paper, we investigate the cases which admit Jacobian and energy conversation are constants, in the Sun perturbed Earth–Moon system. We prove that the Jacobian integral is a constant in two special cases, which can be used to determine the regions of motion from the zero velocity surfaces. On continuation of our study, we numerically illustrate the equilibrium points and their stability and Poincaré surfaces of section. In addition we reveal the basins of attraction, associated with the points of equilibrium using color–coded diagrams. 1. Introduction In general the models of two, three, four or five bodies can be used to study and analyse the dynamical behavior of solar system objects or other celestial bodies. These models can be used also to study the motion of spacecraft in space missions. But it is well known that each one of these models can provide many complicated and generalized versions. This is due to the existence of many perturbed forces, for example, irregular shape of the most celestial bodies, radiation pressure for radiating bodies (e.g. the sun or the stars), the effect of solar wind of the sun in some cases, atmosphere drag force and variable mass or many other perturbed for ces. The models of restricted three, four or five–bodies with their simple forms (without perturbations) are appropriate and considerable for numerous purposes. Because they provide a good insight on the dyna- mical structures in the field of celestial mechanics or astrodynamics. In addition, they can be used to determine the regions of possible or for- bidden motion. However these models are not effective with the cases which include considerable perturbations. Hence it is necessary to use the generalized versions of these problems, to get a complete and per- fect pictures on motion of dynamics of the systems. The perturbed versions are considered more generalized and com- prehensive models. Which will be used in order to accomplish the studying and investigating of the dynamical structures for the celestial bodies or design the motion of spacecraft with a high accuracy and precision. One of the most important analysis of these models is finding the periodic or numerical solutions. Within the frame of perturbed two–body problem, there are many studies which are introduced by Abouelmagd et al. (2015, 2016a, 2017); Abouelmagd (2018). While within the frame of three bodies, various effective work are studied by Abouelmagd et al. (2016b); Alzahrani et al. (2017); Elshaboury et al. (2016), there are also considerable work in the context of four or five bodies in order to investigate numerically the basins of convergence, associated with the points of equilibrium using color- coded diagrams, see for more details (Ansari, 2016; Suraj et al., 2018; 2019a; 2019b). These studies presented a clear and complete picture on the structures and the features on the existence of equilibria points and their stability as well as the periodic motion around these points or the existence of periodic orbits, Some interesting papers for the readers in this context are also constructed by Abouelmagd (2012, 2013); Pathak et al. (2019); Abouelmagd et al. (2019). One of the most important dynamical systems in both celestial mechanics and astrodynamics is bicircular restricted three–body pro- blem (BCP). This problem (BCP) is considered a four–body problem with a simple form, either a perturbed restricted three–body problem or its extension. It is an interesting problem in the present time that is why it is attracting to the researchers. This model consists of two restricted three–body problems. Firstly, two bodies are moving in circular orbits around their common center of mass (i.e. barycenter) which is taken as origin. Secondly, the center of mass of above stated system and the third body are moving in circular orbits around the barycenter of all system at the same time. In this model the fourth body (the https://doi.org/10.1016/j.newast.2019.101282 Received 6 May 2019; Received in revised form 7 June 2019; Accepted 2 July 2019 ⁎ Corresponding author at: Celestial Mechanics and Space Dynamics Research Group (CMSDRG), Astronomy Department. National Research Institute of Astronomy and Geophysics (NRIAG), Helwan 11421 - Cairo – Egypt. E-mail addresses: elbaz.abouelmagd@nriag.sci.eg, eabouelmagd@gmail.com (E.I. Abouelmagd), icairndin@gmail.com (A.A. Ansari). New Astronomy 73 (2019) 101282 Available online 02 July 2019 1384-1076/ © 2019 Published by Elsevier B.V. T