Artificial equilibrium points in binary asteroid systems with continuous low-thrust

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Abstract

The positions and dynamical characteristics of artificial equilibrium points (AEPs) in the vicinity of a binary asteroid with continuous low-thrust are studied. The restricted ellipsoid–ellipsoid model of binary system is employed for the binary asteroid system. The positions of AEPs are obtained by this model. It is found that the set of the point \(L_{1}\) or \(L_{2}\) forms a shape of an ellipsoid while the set of the point \(L_{3}\) forms a shape like a “banana”. The effect of the continuous low-thrust on the feasible region of motion is analyzed by zero velocity curves. Because of using the low-thrust, the unreachable region can become reachable. The linearized equations of motion are derived for stability’s analysis. Based on the characteristic equation of the linearized equations, the stability conditions are derived. The stable regions of AEPs are investigated by a parametric analysis. The effect of the mass ratio and ellipsoid parameters on stable region is also discussed. The results show that the influence of the mass ratio on the stable regions is more significant than the parameters of ellipsoid.

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Acknowledgements

The work described in this paper was supported by the National Natural Science Foundation of China (Grant No. 11672126), Innovation Funded Project of Shanghai Aerospace Science and Technology (Grant No. SAST2015036), the Opening Grant from the Key Laboratory of Space Utilization, Chinese Academy of Sciences (LSU-2016-07-01). The authors fully appreciate their financial supports.

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Correspondence to Shuang Li.

Appendix

Appendix

$$\begin{aligned} &{\frac{\partial V}{\partial x} = \frac{\partial U_{s}}{\partial x} + \frac{\partial U_{e}}{\partial x} + \omega^{2}x + u_{x}} \\ &{\phantom{\frac{\partial V}{\partial x}}=\frac{5C_{20s} \{ y^{2} - 2z^{2} + [ x - r ( 1 - \mu ) ]^{2} \} [ x - r ( 1 - \mu ) ]\mu}{ 2 \{ y^{2} + z^{2} + [ x - r ( 1 - \mu ) ]^{2} \}^{\frac{7}{2}}} }\\ &{\phantom{\frac{\partial V}{\partial x}=} - \frac{C_{20s} [ x - r ( 1 - \mu ) ]\mu}{ \{ y^{2} + z^{2} + [ x - r ( 1 - \mu ) ]^{2} \}^{\frac{5}{2}}} }\\ &{\phantom{\frac{\partial V}{\partial x}=} - \frac{ [ x - r ( 1 - \mu ) ]\mu}{ \{ y^{2} + z^{2} + [ x - r ( 1 - \mu ) ]^{2} \}^{\frac{3}{2}}} }\\ &{\phantom{\frac{\partial V}{\partial x}=} - \frac{15C_{22e} ( 1 - \mu ) ( x + r\mu ) [ - y^{2} + ( x + r\mu )^{2} ]}{ [ y^{2} + z^{2} + ( x + r\mu )^{2} ]^{\frac{7}{2}}} }\\ &{ \phantom{\frac{\partial V}{\partial x}=}+ \frac{5C_{20e} ( 1 - \mu ) ( x + r\mu ) [ y^{2} - 2z^{2} + ( x + r\mu )^{2} ]}{2 [ y^{2} + z^{2} + ( x + r\mu )^{2} ]^{\frac{7}{2}}} }\\ &{\phantom{\frac{\partial V}{\partial x}=} - \frac{C_{20e} ( 1 - \mu ) ( x + r\mu )}{ [ y^{2} + z^{2} + ( x + r\mu )^{2} ]^{\frac{5}{2}}} }\\ &{\phantom{\frac{\partial V}{\partial x}=} + \frac{6C_{22e} ( 1 - \mu ) ( x + r\mu )}{ [ y^{2} + z^{2} + ( x + r\mu )^{2} ]^{\frac{5}{2}}} }\\ &{\phantom{\frac{\partial V}{\partial x}=} - \frac{ ( 1 - \mu ) ( x + r\mu )}{ [ y^{2} + z^{2} + ( x + r\mu )^{2} ]^{\frac{3}{2}}} }\\ &{\phantom{\frac{\partial V}{\partial x}=}+ \omega^{2}x + u\cos \theta \cos \varphi }\\ &{\phantom{\frac{\partial V}{\partial x}}= 0, }\\ &{\frac{\partial V}{\partial y} = \frac{\partial U_{s}}{\partial y} + \frac{\partial U_{e}}{\partial y} + \omega^{2}y + u_{y} }\\ &{\phantom{\frac{\partial V}{\partial y}}= \frac{5C_{20s}y \{ y^{2} - 2z^{2} + [ x - r ( 1 - \mu ) ]^{2} \}\mu}{2 \{ y^{2} + z^{2} + [ x - r ( 1 - \mu ) ]^{2} \}^{\frac{7}{2}}} }\\ &{\phantom{\frac{\partial V}{\partial y}=} - \frac{C_{20s}y\mu}{ \{ y^{2} + z^{2} + [ x - r ( 1 - \mu ) ]^{2} \}^{\frac{5}{2}}} }\\ &{\phantom{\frac{\partial V}{\partial y}=} - \frac{y\mu}{ \{ y^{2} + z^{2} + [ x - r ( 1 - \mu ) ]^{2} \}^{\frac{3}{2}}} }\\ &{\phantom{\frac{\partial V}{\partial y}=} - \frac{15C_{22e}y ( 1 - \mu ) [ - y^{2} + ( x + r\mu )^{2} ]}{ [ y^{2} + z^{2} + ( x + r\mu )^{2} ]^{\frac{7}{2}}} }\\ &{\phantom{\frac{\partial V}{\partial y}=} + \frac{5C_{20e}y ( 1 - \mu ) [ y^{2} - 2z^{2} + ( x + r\mu )^{2} ]}{2 [ y^{2} + z^{2} + ( x + r\mu )^{2} ]^{\frac{7}{2}}} }\\ &{\phantom{\frac{\partial V}{\partial y}=} - \frac{C_{20e}y ( 1 - \mu )}{ [ y^{2} + z^{2} + ( x + r\mu )^{2} ]^{\frac{5}{2}}} }\\ &{\phantom{\frac{\partial V}{\partial y}=} - \frac{6C_{22e}y ( 1 - \mu )}{ [ y^{2} + z^{2} + ( x + r\mu )^{2} ]^{\frac{5}{2}}} }\\ &{\phantom{\frac{\partial V}{\partial y}=} - \frac{y ( 1 - \mu )}{ [ y^{2} + z^{2} + ( x + r\mu )^{2} ]^{\frac{3}{2}}} }\\ &{\phantom{\frac{\partial V}{\partial y}=} + \omega^{2}y + u\cos \theta \sin \varphi }\\ &{\phantom{\frac{\partial V}{\partial y}}= 0, }\\ &{\frac{\partial V}{\partial z} = \frac{\partial U_{s}}{\partial z} + \frac{\partial U_{e}}{\partial z} + u_{z} }\\ &{\phantom{\frac{\partial V}{\partial z}}= \frac{5C_{20s}z \{ y^{2} - 2z^{2} + [ x - r ( 1 - \mu ) ]^{2} \}\mu}{2 \{ y^{2} + z^{2} + [ x - r ( 1 - \mu ) ]^{2} \}^{\frac{7}{2}}} }\\ &{\phantom{\frac{\partial V}{\partial z}=} + \frac{2C_{20s}z\mu}{ \{ y^{2} + z^{2} + [ x - r ( 1 - \mu ) ]^{2} \}^{\frac{5}{2}}} }\\ &{\phantom{\frac{\partial V}{\partial z}=} - \frac{z\mu}{ \{ y^{2} + z^{2} + [ x - r ( 1 - \mu ) ]^{2} \}^{\frac{3}{2}}} }\\ &{\phantom{\frac{\partial V}{\partial z}=} - \frac{15C_{22e}z ( 1 - \mu ) [ - y^{2} + ( x + r\mu )^{2} ]}{ [ y^{2} + z^{2} + ( x + r\mu )^{2} ]^{\frac{7}{2}}} }\\ &{\phantom{\frac{\partial V}{\partial z}=} + \frac{5C_{20e}z ( 1 - \mu ) [ y^{2} - 2z^{2} + ( x + r\mu )^{2} ]}{2 [ y^{2} + z^{2} + ( x + r\mu )^{2} ]^{\frac{7}{2}}} }\\ &{\phantom{\frac{\partial V}{\partial z}=}+ \frac{2C_{20e}z ( 1 - \mu )}{ [ y^{2} + z^{2} + ( x + r\mu )^{2} ]^{\frac{5}{2}}} }\\ &{\phantom{\frac{\partial V}{\partial z}=} - \frac{z ( 1 - \mu )}{ [ y^{2} + z^{2} + ( x + r\mu )^{2} ]^{\frac{3}{2}}} }\\ &{\phantom{\frac{\partial V}{\partial z}=} + u\sin \theta }\\ &{\phantom{\frac{\partial V}{\partial z}}= 0.} \end{aligned}$$

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Bu, S., Li, S. & Yang, H. Artificial equilibrium points in binary asteroid systems with continuous low-thrust. Astrophys Space Sci 362, 137 (2017). https://doi.org/10.1007/s10509-017-3119-7

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Keywords

  • Artificial equilibrium points
  • Continuous low-thrust
  • Binary asteroid system
  • Stability and stable region