Secular resonances between bodies on close orbits: a case study of the Himalia prograde group of jovian irregular satellites

Abstract

The gravitational interaction between two objects on similar orbits can effect noticeable changes in the orbital evolution even if the ratio of their masses to that of the central body is vanishingly small. Christou (Icarus 174:215–229, 2005) observed an occasional resonant lock in the differential node $\varDelta \varOmega$ between two members in the Himalia irregular satellite group of Jupiter in the N-body simulations (corresponding mass ratio $\sim 10^{-9}$). Using a semianalytical approach, we have reproduced this phenomenon. We also demonstrate the existence of two additional types of resonance, involving angle differences $\varDelta \omega$ and $\varDelta (\varOmega +\varpi )$ between two group members. These resonances cause secular oscillations in eccentricity and/or inclination on timescales $\sim$1 Myr. We locate these resonances in (a, e, i) space and analyse their topological structure. In subsequent N-body simulations, we confirm these three resonances and find a fourth one involving $\varDelta \varpi$. In addition, we study the occurrence rates and the stability of the four resonances from a statistical perspective by integrating 1000 test particles for 100 Myr. We find $\sim$10 to 30 librators for each of the resonances. Particularly, the nodal resonance found by Christou is the most stable: 2 particles are observed to stay in libration for the entire integration.

Appendices

Appendix 1: Derivation of the “averaged” Kozai Hamiltonian

We derive the Kozai Hamiltonian (11) from its original form (4) here. First, we substitute e and i in Eq. (4) in terms of the angular momenta G and H. Then we solve for G so $G=G(F,H,\omega ,const.)$, where const. refers to combinations of k, $m_\odot$, a and $a_\odot$. Note that in this expression, F, H and const. are all constant and only $\omega$ and G are variable. Furthermore, it can be shown with this equation that the variation in eccentricity during a Kozai cycle, $\varDelta e^2$ is a small quantity (e.g., for Himalia’s nominal orbit, $\varDelta e^2\lesssim 0.03$). Therefore, we assume that the angular momentum $G\propto \sqrt{1-e^2}$ can be expressed as

$$\begin{aligned} G^2=b_0+b_1 \cos 2 \omega + \cdots , \end{aligned}$$

(29)

where the coefficients $b_1\ll b_0$. We can then express $b_0$ and $b_1$ as functions of F, H and const. by comparing the above equation with $G=G(F,H,\omega ,const.)$. Now, if we further omit the term $b_1 \cos 2 \omega$ we have

$$\begin{aligned} G^2=b_0=b_0(F,H,const.)\propto {1-\langle e\rangle ^2}. \end{aligned}$$

(30)

Here, $\langle e\rangle$ is not the e in the original Hamiltonian (4) but the averaged e over a Kozai cycle. Meanwhile, we also have $H^2\propto (1-\langle e\rangle ^2)\cos ^2\langle i\rangle$ and $\langle i\rangle$ is the averaged i. Then we solve Eq. (30) for F. So now F is a function of H, $b_0=G^2$ and const.. After substituting $\langle e\rangle$ and $\langle i\rangle$ into it, we arrive at the “averaged” Kozai Hamiltonian (11). The new eccentricity and inclination are constant under the solar forcing; for simplicity, we still use the old notations e and i.

Appendix 2: The non-canonical property of coorbital potential

As stated in Sect. 3.3, the coorbital potential (6) (and hence (17)) is not canonical in the variables used in Eq. (16). To prove this, we presume it is canonical and introduce the following canonical transformations. For the first transformation, we name the variables on the left side as “absolute Cartesian” variables.

$$\begin{aligned} {\bar{G}}_{\mathrm {y},\mathrm {H}}= & {} \sqrt{2 {{\bar{G}}}_{\mathrm {H}}} \sin {\bar{g}_{\mathrm {H}}} ,\quad {\bar{G}}_{\mathrm {x},\mathrm {H}}= \sqrt{2 {{\bar{G}}}_{\mathrm {H}}} \cos {\bar{g}_{\mathrm {H}}} ;\nonumber \\ \bar{H}_{\mathrm {y},\mathrm {H}}= & {} \sqrt{2 {\bar{H}}_{\mathrm {H}}} \sin {\bar{h}_{\mathrm {H}}} ,\quad \bar{H}_{\mathrm {x},\mathrm {H}}= \sqrt{2 {\bar{H}}_{\mathrm {H}}} \cos {\bar{h}_{\mathrm {H}}} ;\nonumber \\ {\bar{G}}_\mathrm {y,P}= & {} \sqrt{2 {{\bar{G}}}_{\mathrm {P}}} \sin {\bar{g}_{\mathrm {P}}} ,\quad {\bar{G}}_\mathrm {x,P}= \sqrt{2 {{\bar{G}}}_{\mathrm {P}}} \cos {\bar{g}_{\mathrm {P}}} ;\nonumber \\ \bar{H}_\mathrm {y,P}= & {} \sqrt{2 {\bar{H}}_{\mathrm {P}}} \sin {\bar{h}_{\mathrm {P}}} ,\quad \bar{H}_\mathrm {x,P}= \sqrt{2 {\bar{H}}_{\mathrm {P}}} \cos {\bar{h}_{\mathrm {P}}} . \end{aligned}$$

(31)

On the right-hand sides of the above expressions, $({\bar{g}},\,{{\bar{G}}})$ and $({\bar{h}},\,{\bar{H}})$ are conjugate Poincaré variables with subscripts “H” and “P” referring to Himalia and a particle, respectively. Then another transformation is introduced. The variables on the left are referred to as “relative Cartesian” variables.

$$\begin{aligned} {\bar{G}}_\mathrm {y,t}= & {} {\bar{G}}_\mathrm {y,P}/\sqrt{2}+ {\bar{G}}_{\mathrm {y},\mathrm {H}}/\sqrt{2} ,\quad {\bar{G}}_\mathrm {x,t}= {\bar{G}}_\mathrm {x,P}/\sqrt{2}+{\bar{G}}_{\mathrm {x},\mathrm {H}}/\sqrt{2} ;\nonumber \\ \bar{H}_\mathrm {y,t}= & {} \bar{H}_\mathrm {y,P}/\sqrt{2}+\bar{H}_{\mathrm {y},\mathrm {H}}/\sqrt{2} ,\quad \bar{H}_\mathrm {x,t}=\bar{H}_\mathrm {x,P}/\sqrt{2}+\bar{H}_{\mathrm {x},\mathrm {H}}/\sqrt{2} ;\nonumber \\ {\bar{G}}_\mathrm {y,r}= & {} {\bar{G}}_\mathrm {y,P}/\sqrt{2}-{\bar{G}}_{\mathrm {y}, \mathrm {H}}/\sqrt{2} ,\quad {\bar{G}}_\mathrm {x,r} {\bar{G}}_\mathrm {x,P}/\sqrt{2}-{\bar{G}}_{\mathrm {x}, \mathrm {H}}/\sqrt{2} ;\nonumber \\ \bar{H}_\mathrm {y,r}= & {} \bar{H}_\mathrm {y,P}/\sqrt{2}-\bar{H}_{\mathrm {y}, \mathrm {H}}/\sqrt{2} ,\quad \bar{H}_\mathrm {x,r}= \bar{H}_\mathrm {x,P}/\sqrt{2}-\bar{H}_{\mathrm {x},\mathrm {H}}/\sqrt{2} . \end{aligned}$$

(32)

This relates the variables with subscript “r” to the variables used in the coorbital theory as given by Namouni (1999); see the paragraph containing Eqs. (6)–(9).

Now note that, since we assume the coorbital potential is canonical in the original Poincaré variables, it should preserve the Hamiltonian property in the variables above. However, since the coorbital potential contains only the relative elements, i.e., in the relative Cartesian variables, it only depends on the variables with subscript “r”. Hence, if we consider the Jupiter–Himalia-particle system. The quantities with subscript “t” will be conserved. However, this is not possible. Taking ${\bar{G}}_\mathrm {y,t}={\bar{G}}_\mathrm {y,P}/\sqrt{2}+{\bar{G}}_{\mathrm {y},\mathrm {H}}/\sqrt{2}$ for instance. Since the test particle is massless, it should have no influence on Himalia and thus the second term on the right hand side ${\bar{G}}_{\mathrm {y},\mathrm {H}}/\sqrt{2}$ is constant. From the above reasoning, we know that, ${\bar{G}}_\mathrm {y,t}$ should be conserved. This means that, the first term on the right hand side, ${\bar{G}}_\mathrm {y,P}/\sqrt{2}=\sqrt{{{\bar{G}}}_{\mathrm {P}}} \sin {\bar{g}_{\mathrm {P}}}$ will no change. Similarly, $\sqrt{{{\bar{G}}}_{\mathrm {P}}} \cos {\bar{g}_{\mathrm {P}}}$, $\sqrt{{\bar{H}}_{\mathrm {P}}} \sin {\bar{h}_{\mathrm {P}}}$ and $\sqrt{{\bar{H}}_{\mathrm {P}}} \cos {\bar{h}_{\mathrm {P}}}$ should all be constant. If so, the orbital elements of the particle are not evolving under the perturbation of Himalia. Of course this is not true—the coorbital potential is not a Hamiltonian in the variables used here.

However, the level-curve like structure in the phase diagrams Figs. 8, 10 and 11 of the three types of resonance suggests the existence of a conserved quantity (a Hamiltonian) for these two-dimensional systems. We have tried to construct one of the form of $F_\mathrm {T}=F_{\mathrm {S}}+A \times R_{\mathrm {S}}$ (see Eqs. (16) and (17)), where A is a constant depending on some non-changing parameters. For instance, for the case of nodal resonance, A could be a function of the masses, the semimajor axes and the eccentricities (but not a function of inclinations). However, the solution for A seems to be nonexistent (at least for the nodal resonance). Thus we argue that, the conserved quantity of the two-dimensional systems might be of more complicated forms.