# Out of plane equilibrium points locations and the forbidden movement regions in the restricted three-body problem with variable mass

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## Abstract

This work aims to present an analytical study on the dynamics of a third body in the restricted three-body problem. We study this model in the context of the third body having variable-mass changes according to Jeans’ law. The equation of motion is constructed when the variation of the mass is non-isotropic. We find an appropriate approximation for the locations of the out-of-plane equilibrium points in the special case of a non-isotropic variation of the mass. Moreover, some graphical investigations are shown for the effects of the parameters which characterize the variable mass on the locations of the out-of-plane equilibrium points, the regions of possible and forbidden motions of the third body. This model has many applications, especially in the dynamics behavior of small objects such as cosmic dust and grains. It also has interesting applications for artificial satellites, future space colonization or even vehicles and spacecraft parking.

## Keywords

Restricted three-body problem Variable mass Out of plane equilibrium points## Notes

### Acknowledgements

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (857-71-D1435). The authors, therefore, acknowledge with thanks DSR technical and financial support.

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