The current orbit of Methone (S/2004 S 1)

Abstract

The Cassini spacecraft discovered many close-in small satellites in Saturnian system, and several of them exhibit exotic orbital states due to interactions with Mimas and the oblateness of the planet. This work is devoted to Methone, which is currently involved in a 15:14 mean-motion resonance with Mimas. We give an in-deep study on the current orbit of Methone by analyzing and identifying the short, resonant and long-term gravitational perturbations on its orbit. In addition, we perform numerical integrations of full equations of motion of ensembles of close-in small bodies orbiting the non-central field of Saturn. Spectral analyses of the orbits and interpretation of them in dynamical maps allow us to describe the orbit and the dynamics of Methone in view of resonant and long-term dynamics. We show that the current geometric orbit of Methone is aligned with Mimas’ due to a forced resonant component in eccentricity, leading to simultaneous oscillations of several critical angles of the expanded disturbing function. Thus, we explain the simultaneous oscillations of four critical arguments associated to the resonance. The mapping of the Mimas–Methone resonance shows that the domains of the 15:14 Mimas–Methone resonance are dominated by regular motions associated to the Corotation resonance located at eccentricities lower than $\sim \, 0.015$ and osculating semi-major axis in the interval 194,660–194,730 km. Methone is currently located deeply within this site.

Appendices

Appendix 1: Initial conditions and parameters

Table 3 gives the initial osculating elements and masses of the mid-sized satellites of Saturn and Methone provided by Horizons system of ephemerides at date January 01, 2016.

The physical data for Saturn are $M_S=5.6834\times 10^{26}$ (kg) (mass), $R_S=60,268\pm 4$ km (equatorial radius), $J_2=0.01629071$ and $J_4=-0.0009358$ (Jacobson et al. 2006b).

Table 3 Orbital elements and masses of Saturnian satellites taken from the Horizons system referred to date January 01, 2016, 00:00

Appendix 2: Critical arguments $\psi _k$, $5\le k\le 14$

In this Appendix, we extend the interpretation of the critical arguments of the disturbing given in Sect. 3.1 in the case $\psi _k$, $k\ge 5$ (Fig. 10).

The plots of the critical arguments of the disturbing $\psi _k$, $k\ge 5$ are given in Fig. 10. All of these angles but $\psi _7$, $\psi _8$, $\psi _{10}$, $\psi _{11}$ do not have any physical relevance since their time variations are dominated by short-term perturbations in both osculating and geometric variables.

From Table 2, we have:

$$\begin{aligned} \psi _7= & {} \alpha -\varpi _{\mathrm{M}}-\varOmega _{\mathrm{Me}}+\varOmega _{\mathrm{M}}=\psi _1-\Delta \varOmega _{\mathrm{Me}-\mathrm{M}}, \end{aligned}$$

(11)

$$\begin{aligned} \psi _8= & {} \alpha -\varpi _{\mathrm{M}}+\varOmega _{\mathrm{Me}}-\varOmega _{\mathrm{M}}=\psi _1+\Delta \varOmega _{\mathrm{Me}-\mathrm{M}}, \end{aligned}$$

(12)

where $\alpha =15\lambda _{\mathrm{Me}}-14\lambda _{\mathrm{M}}$. $\psi _7$, $\psi _8$ are driven by the coupling of the oscillation of the corotation angle $\psi _1=\alpha -\varpi _{\mathrm{M}}$ with a slower long-term circulation of $\Delta \varOmega _{\mathrm{Me}-\mathrm{M}}$. Both osculating and geometric $\Delta \varOmega _{\mathrm{Me}-\mathrm{M}}$ circulate in prograde direction with a period of $P_{\Delta \varOmega }\sim \, 2443.28$ days (recall from Sect. 3.2 that $\Delta \varOmega _{\mathrm{Me}-\mathrm{M}}$ is not significantly affected by the induced short-term circulations due to $J_2$). The different signs of $\Delta \varOmega _{\mathrm{Me}-\mathrm{M}}$ in (11), (12) cause the long-term circulation of $\psi _7$, $\psi _8$ to occur in the opposite directions.

From Table 2, we have:

$$\begin{aligned} \psi _{10}= & {} \psi _1-\Delta \varpi _{\mathrm{Me}-\mathrm{M}}-\Delta \varOmega _{\mathrm{Me}-\mathrm{M}}, \end{aligned}$$

(13)

$$\begin{aligned} \psi _{11}= & {} \psi _1-\Delta \varpi _{\mathrm{Me}-\mathrm{M}}+\Delta \varOmega _{\mathrm{Me}-\mathrm{M}}. \end{aligned}$$

(14)

Since the geometric $\Delta \varpi _{\mathrm{Me}-\mathrm{M}}$ oscillates around zero, the evolutions of the geometric $\psi _{10}$ and $\psi _{7}$, and $\psi _{11}$ and $\psi _{8}$, are, respectively, very similar.

Fig. 10

Figure 3, continued. $\psi _k$, $5\le k\le 14$ are defined in Table 2