# Effect of the Planetesimal Belt on the Dynamics of the Restricted Problem of 2 + 2 Bodies
## 5 Conclusions
In the framework of the perturbed restricted problem of 2 + 2 bodies by a planetesimal belt, our obtained results can be outlined as:  
* The existence of equilibrium points was examined for both collinear and noncollinear points;
* The stability of motion around these points was studied;
* We compared the corresponding results of the perturbed and unperturbed models;
* The impact of a planetesimal belt was observed on collinear and noncollinear equilibrium points;
* All equilibrium points were found to be unstable, whereas the noncollinear equilibrium points were investigated numerically. Furthermore, the restricted 2 + 2 bodies problem can be used as a credible model to describe the capture of small bodies by a planet. If the small bodies are temporarily captured in the Hill sphere of a smaller primary, they may become near enough to each other to exchange energy so that one of them becomes regularly and permanently captured. **Author Contributions:** Formal analysis, G.M., A.K.P., S.A., E.I.A. and B.S.K. ; Investigation, G.M., A.K.P. and B.S.K. ; Methodology, G.M., A.K.P. and E.I.A. ; Project administration, S.A. and E.I.A. ; Software, G.M., A.K.P. and E.I.A. ; Validation, G.M., A.K.P., S.A. and E.I.A. ; Visualization, E.I.A. ; Writing--original draft, G.M., A.K.P. and E.I.A. ; Writing--review & editing, G.M., A.K.P., E.I.A. and S.A. All authors have read and agreed to the published version of the manuscript. **Funding:** This research received no external funding  
**Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** The data analyzed in this study were generated by MATLAB R2018b software. The mass parameter \(\mu\) of the different planetary system, the mass of the planetesimal belt \(M_{b}\), and the parameter \(T\) were taken from various sources [16; 23; 24] and are available at the following: URL: [[https://nssdc.gsfc.nasa.gov/planetary/factsheet/index.html](https://nssdc.gsfc.nasa.gov/planetary/factsheet/index.html)]([https://nssdc.gsfc.nasa.gov/planetary/factsheet/index.html](https://nssdc.gsfc.nasa.gov/planetary/factsheet/index.html)), [[https://ssd.jpl.nasa.gov/?phys_data](https://ssd.jpl.nasa.gov/?phys_data)]([https://ssd.jpl.nasa.gov/?phys_data](https://ssd.jpl.nasa.gov/?phys_data)) (accessed on 26 December 2021). **Acknowledgments:** We thank the DST(SERB) Government of India (project no. -DST(SERB)/(163)/ 2016-2017/506/AM) for the use of the computation facility. The first author thanks the Council of Scientific and Industrial Research (CSIR), Government of India (file no. 09/085(0126)/2019-EMR-1) for financial support. The second author thanks Manipal University Jaipur for the financial supported from the Enhanced Seed Grant through the Endowment Fund reference no. EF/2021-22/QE04-07. This work was funded by the National Research Institute of Astronomy and Geophysics (NRIAG), Helwan, Cairo, Egypt. The fourth author, therefore, thanks NRIAG for technical and financial support. **Conflicts of Interest:** The authors declare that there are no conflict of interest regarding the publication of this article. **Ethics Statement:** The authors state that this research complies with all ethical standards.