Astronomy and Astrophysics
Vol.06 No.01(2018), Article ID:23371,10 pages
10.12677/AAS.2018.61001

Numerical Study on the Species of Transfer Orbit in the Circular Restricted Three-Body Problem

Yu Huang, Fan Lin, Yiqing Shen, Ruifang Wang, Fabao Gao*

College of Mathematical Science, Yangzhou University, Yangzhou Jiangsu

Received: Dec. 19th, 2017; accepted: Jan. 3rd, 2018; published: Jan. 10th, 2018

ABSTRACT

Based on the dynamical equations of spatial circular restricted three-body problem, the bifurcation diagram of the system with the mass ratio as the bifurcation parameter is demonstrated, and it is found that when the masses of the two main bodies are considerable equivalent (i.e., the mass ratio is around 0.5), the third body’s dynamic behavior is more complicated than at other mass ratios. When the mass ratio is in the interval (0.4, 0.6), we further simulate 32,000 transfer orbits and make a brief classification of the found orbits.

Keywords:CR3BP, Bifurcation Diagram, Transfer Orbit, Interplanetary Interchange Station

1. 引言

2. 限制性三体系统建模及分析

Figure 1. Coordinate system

$\begin{array}{l}{\stackrel{˙}{x}}_{1}={x}_{4},\\ {\stackrel{˙}{x}}_{2}={x}_{5},\\ {\stackrel{˙}{x}}_{3}={x}_{6},\\ {\stackrel{˙}{x}}_{4}={x}_{1}+2{x}_{5}-\frac{\left(1-\mu \right)\left({x}_{1}+\mu \right)}{{\left[{\left({x}_{1}+\mu \right)}^{2}+{x}_{2}^{2}+{x}_{3}^{2}\right]}^{\frac{3}{2}}}-\frac{\mu \left({x}_{1}-1+\mu \right)}{{\left[{\left({x}_{1}-1+\mu \right)}^{2}+{x}_{2}^{2}+{x}_{3}^{2}\right]}^{\frac{3}{2}}},\\ {\stackrel{˙}{x}}_{5}={x}_{2}-2{x}_{4}-\frac{\left(1-\mu \right){x}_{2}}{{\left[{\left({x}_{1}+\mu \right)}^{2}+{x}_{2}^{2}+{x}_{3}^{2}\right]}^{\frac{3}{2}}}-\frac{\mu {x}_{2}}{{\left[{\left({x}_{1}-1+\mu \right)}^{2}+{x}_{2}^{2}+{x}_{3}^{2}\right]}^{\frac{3}{2}}},\\ {\stackrel{˙}{x}}_{6}=-\frac{\left(1-\mu \right){x}_{3}}{{\left[{\left({x}_{1}+\mu \right)}^{2}+{x}_{2}^{2}+{x}_{3}^{2}\right]}^{\frac{3}{2}}}+\frac{\mu {x}_{3}}{{\left[{\left({x}_{1}-1+\mu \right)}^{2}+{x}_{2}^{2}+{x}_{3}^{2}\right]}^{\frac{3}{2}}}.\end{array}$

3. 系统的分岔图及转移轨道的数值研究

(a)(b)

Figure 2. (a) Bifurcation diagram for $\mu \in \left(0,1\right)$ ; (b) Bifurcation diagram for $\mu \in \left(0.4,0.6\right)$

4. 结论

Figure 3. Mass ratio of $\mu =0.43559$

Figure 4. Mass ratio of $\mu =0.45859$

Figure 5. Mass ratio of $\mu =0.48505$

Figure 6. Mass ratio of $\mu =0.49999$

Figure 7. The initial value in the X direction is −0.986

Figure 8. The initial value in the X direction is −0.193

Figure 9. The initial value in the X direction is −0.004

Figure 10. The initial value in the X direction is 0.051

Figure 11. The initial value in the Y direction is 0.515

Figure 12. The initial value in the Z direction is −0.107

Figure 13. The initial velocity in the X direction is −0.48

Figure 14. The initial velocity in the Y direction is −0.3

Figure 15. The initial velocity in the Z direction is −0.44

Figure 16. The initial velocity in the Z direction is −0.5

Numerical Study on the Species of Transfer Orbit in the Circular Restricted Three-Body Problem[J]. 天文与天体物理, 2018, 06(01): 1-10. http://dx.doi.org/10.12677/AAS.2018.61001

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11. NOTES



*通讯作者。