Spin-orbit coupling dynamics in a planar synchronous binary asteroid

Abstract

Purpose: The 1:1 spin-orbit resonance phenomenon is widely observed in binary asteroid systems. We aim to investigate the intrinsic dynamic mechanism behind the phenomenon under the coupled influence of the secondary’s rotation and orbital motion. Methods: The planar sphere–ellipsoid model is used to approximate the synchronous binary asteroid. The Lindstedt–Poincaré method is applied on the spin-orbit problem to find its explicit quasi-periodic solution. Results: Numerical simulations demonstrate that analytical solutions truncated at high orders are accurate enough to describe the orbital and rotational motions of the synchronous binary asteroid. With the help of the solution, we are able to identify in a more accurate way the stable region for the synchronous state by using the Lyapunov characteristic exponent. Moreover, the resonances that determine the boundary of the stability region are identified. Conclusion: The stable synchronous state requires a small eccentricity $e$ of the mutual orbit but permits a large libration angle $\theta$ of the secondary. The anti-correlation of $\theta$ and $e$ is confirmed. The stable region for a very elongated secondary is small, which helps explain the lack of such secondaries in observations (see Table 1 in Pravec et al. in Icarus 267:267–295, 2016). Findings of this study provide insights into the inherent dynamics that determine the rotational states of a synchronous binary asteroid.

Appendix:  The coefficients in Equation (10)

$$\begin{aligned} & {c_{1}} = - \frac{{3\left ( {{A_{1}} + {A_{2}}} \right )}}{{2r_{0}^{4}}} - \frac{1}{{r_{0}^{2}}} + \frac{{{K^{2}}{r_{0}}}}{{{\Upsilon ^{2}}}}, \\ &{c_{2}} = \frac{{6\left ( {{A_{1}} + {A_{2}}} \right )}}{{r_{0}^{5}}} + \frac{2}{{r_{0}^{3}}} + \frac{{{K^{2}}\left ( {{I_{\mathrm{{S}}}} - 3\mu r_{0}^{2}} \right )}}{{{\Upsilon ^{3}}}}, \\ & {c_{3}} = \frac{{2{r_{0}}{I_{\mathrm{{S}}}}K}}{{{\Upsilon ^{2}}}},\quad {c_{4}} = \frac{{{r_{0}}I_{\mathrm{{S}}}^{2}}}{{{\Upsilon ^{2}}}},\quad {c_{7}} = - \frac{{2{A_{2}}\Upsilon }}{{{I_{\mathrm{{S}}}}r_{0}^{5}}}, \\ &{c_{8}} = - \frac{{2K}}{{{r_{0}}\Upsilon }},\quad {c_{9}} = - \frac{{2{I_{\mathrm{{S}}}}}}{{{r_{0}}\Upsilon }}, \\ & {c_{5}} = 2\frac{{{I_{\mathrm{{S}}}}K}}{{{\mu ^{2}}}}\left [ {{r_{0}}F \left ( i \right ) + F\left ( {i - 1} \right )} \right ], \\ & {c_{10}} = \frac{{2K}}{{{I_{\mathrm{{S}}}}}}\left [ {{{\left ( { - {r_{0}}} \right )}^{ - i - 1}} + {r_{0}}f\left ( i \right ) + f\left ( {i - 1} \right )} \right ], \\ & {c_{6}} = \frac{{I_{\mathrm{{S}}}^{2}}}{{{\mu ^{2}}}}\left [ {{r_{0}}F \left ( i \right ) + F\left ( {i - 1} \right )} \right ], \\ & {c_{11}} = 2\left [ {{{\left ( { - {r_{0}}} \right )}^{ - i - 1}} + {r_{0}}f \left ( i \right ) + f\left ( {i - 1} \right )} \right ] \end{aligned}$$

(A.1)

$$ \begin{aligned} {\varphi _{ij}} & = {\delta _{j,0}} \left\{ \frac{{{K^{2}}}}{{{\mu ^{2}}}}\left [ {{r_{0}}F\left ( i \right ) + F \left ( {i - 1} \right )} \right ] \right.\\ &\left.\quad {}+ {{\left ( { - 1} \right )}^{i + 1}} \frac{{\left ( {i + 1} \right )}}{{r_{0}^{i + 2}}}\left [ {1 + \frac{{\left ( {i + 2} \right )\left ( {i + 3} \right )\left ( {{A_{1}} + {A_{2}}} \right )}}{{4r_{0}^{2}}}} \right ] \right\} \\ &\quad {} + {\Delta _{i,1}}{\Delta _{i,2}}{\Delta _{j,0}}{\Delta _{j,\lfloor \left ( {i + 1} \right )/2 \rfloor}}{\left ( { - 1} \right )^{i - j + 1}} \\ &\quad {}\times\frac{{{2^{2j - 2}}\left ( {i - 2j + 1} \right )\left ( {i - 2j + 2} \right )\left ( {i - 2j + 3} \right ){A_{2}}}}{{\left ( {2j} \right )!r_{0}^{i - 2j + 4}}} \\ &\quad {} + {\delta _{i \bmod 2,0}}{\delta _{j,\lfloor \left ( {i + 1} \right )/2 \rfloor}}{\left ( { - 1} \right )^{j + 1}} \frac{{3{A_{2}}}}{{2r_{0}^{4}}} \frac{{{2^{2j}}}}{{\left ( {2j} \right )!}} \end{aligned} $$

(A.2)

$$ \begin{aligned} {\psi _{ij}} &= {\left ( { - 1} \right )^{i - j + 1}} \frac{{{2^{2j - 2}}\left ( {i - 2j + 2} \right )\left ( {i - 2j + 3} \right )}}{{\left ( {2j - 1} \right )!r_{0}^{i - 2j + 4}}} \\ &\quad {}\times\left [ { \frac{{\left ( {i - 2j + 4} \right )\left ( {i - 2j + 5} \right )}}{{12r_{0}^{2}}} + \frac{\mu }{{{I_{\mathrm{{S}}}}}}} \right ]{A_{2}} \end{aligned} $$

(A.3)

Note that we introduce the following functions in the above expressions,

$$ \begin{aligned} f\left ( i \right ) &= \frac{1}{{2\sqrt { - {I_{\mathrm{{S}}}}/\mu } }} \left [ {{\left ( { \frac{{\sqrt { - {I_{\mathrm{{S}}}}/\mu } - {r_{0}}}}{{{I_{\mathrm{{S}}}}/\mu + r_{0}^{2}}}} \right )}^{i + 1}}\right.\\ &\quad {}\left. + {{\left ( { - 1} \right )}^{i}}{{\left ( { \frac{{\sqrt { - {I_{\mathrm{{S}}}}/\mu } + {r_{0}}}}{{{I_{\mathrm{{S}}}}/\mu + r_{0}^{2}}}} \right )}^{i + 1}} \right ], \\ F\left ( i \right ) &=\! \left \{ \textstyle\begin{array}{l} \left [ {{r_{0}}\sqrt { - {I_{\mathrm{{S}}}}/\mu } - {I_{\mathrm{{S}}}}/\mu \left ( {2 + i} \right )} \right ]{\left ( {\sqrt { - {I_{\mathrm{{S}}}}/ \mu } - {r_{0}}} \right )^{i + 2}} \\ - {\left ( { - 1} \right )^{i}}\left [ {r{{\sqrt { - {I_{\mathrm{{S}}}}/ \mu } }_{0}} + {I_{\mathrm{{S}}}}/\mu \left ( {2 + i} \right )} \right ] \\\quad {}\times{\left ( {\sqrt { - {I_{\mathrm{{S}}}}/\mu } + {r_{0}}} \right )^{i + 2}} \end{array}\displaystyle \hspace{-3pt} \right \} \\ &\quad {\left [ {4{{\left ( {{I_{\mathrm{{S}}}}/\mu } \right )}^{2}}{{\left ( {{I_{ \mathrm{{S}}}}/\mu + r_{0}^{2}} \right )}^{i + 2}}} \right ]^{ - 1}}, \\ {\delta _{i,j}}& = \left \{ \textstyle\begin{array}{cc} {1,} & {i = j} \\ {0,} & {i \ne j} \end{array}\displaystyle \right ., \quad {\Delta _{i,j}} = \left \{ \textstyle\begin{array}{cc} {0,} & {i = j} \\ {1,} & {i \ne j} \end{array}\displaystyle \right ., \quad \Upsilon = \mu r_{0}^{2} + {I_{\mathrm{{S}}}}. \end{aligned} $$

(A.4)