Forced hovering orbit above the primary in the binary asteroid system

Abstract

Different types of orbits in the vicinity of a binary asteroid system have been studied extensively, motivated by the challenging dynamical environment and ongoing space missions. This work addresses a new type of orbit that may have potential application in missions around the BAS—the forced hovering orbit (FHO) above the primary. The dynamics model of the FHO considers two major perturbations, from the secondary’s gravity and from the solar radiation pressure. The primary is modelled as a tri-axial ellipsoid, and its second degree and order-gravity field is adopted. The secondary is taken as a sphere. In the body-fixed frame of the primary, we firstly construct high-order analytical solutions of the FHO around the ‘equilibrium’ point. Then, depending on the relative strength of the SRP and the secondary’s gravity perturbation, the analysis is further carried out in three cases: the SRP-dominating case, the secondary-dominating case, and the competitive case in which the strength of the SRP and the secondary’s gravity perturbation is comparable with each other. Analysing the discovered BAS in the solar system, we find that all three cases exist, with the competitive case the most common. In the secondary-dominating case, a resonance may arise between the secondary’s orbital frequency in the primary’s body-fixed frame and one intrinsic frequency around the ‘equilibrium’ point. In the competitive case, we use the fast Lyapunov indicator (FLI) to characterize the stability of the FHO. Finally, using the analytical solution as an initial seed, we employ the multiple shooting method to design quasi-periodic FHO in (66391) Moshup and Squannit. These orbits may have potential applications in exploiting the BAS.

Appendices

Appendix A Coefficients for Eq. 28

The coefficients of the (1st$\times P_3)$ solution are presented below. The coefficients of the (2nd $\times P_3)$ and (3rd $\times P_3)$ solution are attached in the supplementary material:

$$\begin{aligned} \alpha ^{10}_{\xi _1}{} & {} = -\frac{3}{8} \frac{\mu r_s^{2}\left( \omega ^{2}-6 \omega +{\Omega }^0_{yy}\right) }{r^{\prime 4}\left( \omega ^{4}+\omega ^{2}\Omega ^0_{xx}+\omega ^{2} {\Omega }^0_{yy}-4 \omega ^{2}+{\Omega }^0_{yy}{\Omega }^0_{xx}\right) }r\nonumber \\ r\nonumber \\ \beta ^{20}_{\xi _1}{} & {} = -\frac{3}{2} \frac{\mu r_s\left( 4 \omega ^{2}-4 \omega +{\Omega }^0_{yy}\right) }{r^{\prime 3}\left( 16 \omega ^{4}+4 \omega ^{2} {\Omega }^0_{xx}+4 \omega ^{2} {\Omega }^0_{yy}-16\omega ^{2}+{\Omega }^0_{xx}{\Omega }^0_{yy}\right) }r\nonumber \\ r\nonumber \\ \alpha ^{30}_{\xi _1}{} & {} = \frac{15}{8} \frac{\mu r_s^{2}\left( 9 \omega ^{2}-6 \omega +{\Omega }^0_{yy}\right) }{r^{\prime 4}\left( 81 \omega ^{4}+9 \omega ^{2} {\Omega }^0_{xx}+9 \omega ^{2} {\Omega }^0_{yy}-36 \omega ^{2}+{\Omega }^0_{xx}{\Omega }^0_{yy}\right) }r\nonumber \\ r\nonumber \\ \alpha ^{01}_{\xi _1}{} & {} = \frac{\rho _\odot \cos \theta _\odot (\omega _\odot ^2+{\Omega }^0_{yy}-2\omega _\odot )}{r^2_\odot (\omega _\odot ^4+{\Omega }^0_{xx}\omega ^2_\odot +{\Omega }^0_{yy}\omega ^2_\odot +{\Omega }^0_{yy}{\Omega }^0_{xx}-4\omega ^2_\odot )}r\nonumber \\ r\nonumber \\ \beta ^{10}_{\eta _1}{} & {} = -\frac{3}{8} \frac{\mu r_s^{2}\left( 3 \omega ^{2}-2 \omega +3 {\Omega }^0_{xx}\right) }{r^{\prime 4}\left( \omega ^{4}+\omega ^{2} {\Omega }^0_{xx}+\omega ^{2} {\Omega }^0_{yy}-4 \omega ^{2}+{\Omega }^0_{xx}{\Omega }^0_{yy}\right) } r\nonumber \\ \alpha ^{20}_{\eta _1}{} & {} = \frac{3}{2} \frac{\mu r_s\left( 4 \omega ^{2}-4 \omega +{\Omega }^0_{xx}\right) }{r^{\prime 3}\left( 16 \omega ^{4}+4 \omega ^{2}{\Omega }^0_{xx}+4 \omega ^{2}{\Omega }^0_{yy}-16 \omega ^{2}+{\Omega }^0_{xx}{\Omega }^0_{yy}\right) }r\nonumber \\ r\nonumber \\ \beta ^{30}_{\eta _1}{} & {} = \frac{15}{8} \frac{\mu r_s^{2}\left( 9 \omega ^{2}-6 \omega +{\Omega }^0_{xx}\right) }{r^{\prime 4}\left( 81 \omega ^{4}+9 \omega ^{2} {\Omega }^0_{xx}+9 \omega ^{2} {\Omega }^0_{yy}-36 \omega ^{2}+{\Omega }^0_{xx}{\Omega }^0_{yy}\right) }r\nonumber \\ r\nonumber \\ \beta ^{01}_{\eta _1}{} & {} = \frac{\rho _\odot \cos \theta _\odot (\omega _\odot ^2+{\Omega }^0_{xx}-2\omega _\odot )}{r^2_\odot (\omega _\odot ^4+{\Omega }^0_{xx}\omega ^2_\odot +{\Omega }^0_{yy}\omega ^2_\odot +{\Omega }^0_{yy}{\Omega }^0_{xx}-4\omega ^2_\odot )}r\nonumber \\ r\nonumber \\ C_{\eta _1}{} & {} = -\frac{\mu r_s}{2{\Omega }^0_{yy} r^{\prime 3}}r\nonumber \\ r\nonumber \\ C_{\zeta _1}{} & {} = \frac{\rho _\odot \sin \theta _\odot }{r_\odot ^2{\Omega }^0_{z}}, \end{aligned}$$

(A1)

Appendix B The matrix of $G_{3\times 3}$

Elements of matrix $G_{3\times 3}$ are presented below:

$$\begin{aligned} \Omega ^0_{xx}= & {} \frac{(2 x^2-y^2-z^2)}{r^5} - \frac{3}{2}\frac{C_{20}(4 x^4+3 x^2 y^2-27 x^2 z^2-y^4+3 y^2 z^2+4 z^4)}{r^9} \nonumber \\{} & {} \qquad + \frac{3 C_{22}(12 x^4-51 x^2 y^2-21 x^2 z^2+7 y^4+9 y^2 z^2+2 z^4)}{r^9}\nonumber \\ \Omega ^0_{xy}= & {} \frac{3 xy}{r^5} - \frac{15}{2}\frac{C_{20}xy(x^2+y^2-6D0 z^2)}{r^9} + \frac{105 C_{22}(x^2-y^2)xy}{r^9} \nonumber \\ \Omega ^0_{xz}= & {} \frac{3 xz}{r^5} - \frac{15}{2}\frac{C_{20}xz(3 x^2+3 y^2-4 z^2)}{r^9} + \frac{15 C_{22}xz(5D0 x^2-9 y^2-2 z^2)}{r^9}\nonumber \\ \Omega ^0_{yy}= & {} -\frac{(x^2-2 y^2+z^2)}{r^5} + \frac{3}{2}\frac{C_{20}(x^4-3 x^2 y^2-3 x^2 z^2-4 y^4+27 y^2 z^2-4 z^4)}{r^9} \\{} & {} \qquad - \frac{3 C_{22}(7 x^4-51 x^2 y^2+9 x^2 z^2+12 y^4-21 y^2 z^2+2 z^4)}{r^9}\nonumber \\ \Omega ^0_{yz}= & {} \frac{3 yz}{r^5} - \frac{15}{2}\frac{C_{20}yz(3 x^2+3 y^2-4 z^2)}{r^9}+\frac{15 C_{22}yz(9 x^2-5D0 y^2+2 z^2)}{r^9}\nonumber \\ \Omega ^0_{zz}= & {} - \frac{(x^2+y^2-2 z^2)}{r^5} + \frac{3}{2}\frac{C_{20}(3 x^4+6D0 x^2 y^2-24D0 x^2 z^2+3 y^4-24D0 y^2 z^2+8D0 z^4)}{r^9}\nonumber \\{} & {} \qquad -\frac{15 C_{22}(x^2-y^2)(x^2+y^2-6D0 z^2)}{r^9}\nonumber \Omega ^0_{yx} = \Omega ^0_{xy}, \Omega ^0_{zx} = \Omega ^0_{xz}, \Omega ^0_{zy} = \Omega ^0_{yz} , \end{aligned}$$

(B2)

Appendix C The matrix of $K_{3\times 3}$

Elements of matrix $K_{3\times 3}$ are presented below:

$$\begin{aligned} \begin{array}{l} \Omega ^\epsilon _{xx} = -\frac{(r^{\prime 2}-2 x^2+4 x x^\prime -3 x^{\prime 2}+y^2-2 y y^{\prime 2}+z^2)}{\mid \Delta \mid ^5}\\ \\ \Omega ^\epsilon _{xy} = \frac{3(-y^{\prime }+y)(x-x^\prime )}{\mid \Delta \mid ^5}\\ \\ \Omega ^\epsilon _{xz} = \frac{3 z(x-x^\prime )}{\mid \Delta \mid ^5}\\ \\ \Omega ^\epsilon _{yy} = -\frac{(r^{\prime 2}+x^2-2 x x^\prime -2 y^2+4 y y^{\prime }-3 y^{\prime 2}+z^2)}{\mid \Delta \mid ^5}\\ \\ \Omega ^\epsilon _{yz} = \frac{3 z (y-y^{\prime })}{\mid \Delta \mid ^5}\\ \\ \Omega ^\epsilon _{zz} = -\frac{(r^{\prime 2}+x^2-2 x x^\prime +y^2-2 y y^{\prime }-2 z^2)}{\mid \Delta \mid ^5}\\ \\ \Omega ^\epsilon _{yx} = \Omega ^\epsilon _{xy}, \Omega ^\epsilon _{zx} = \Omega ^\epsilon _{xz}, \Omega ^\epsilon _{zy} = \Omega ^\epsilon _{yz} \end{array}, \end{aligned}$$

(C3)