On the orbits and the early in-situ formation of Pluto's small satellites

Abstract

The formation of Pluto's small satellites --- Styx, Nix, Keberos and Hydra --- remains a mystery. Their orbits are nearly circular, near mean-motion resonances (MMR) and nearly coplanar with Charon's orbit. One scenario suggests that they all formed close to their current locations from a disk of debris, which was ejected from the Charon-forming impact before the tidal evolution of Charon. The validity of this scenario is tested by performing $N$-body simulations with Pluto-Charon evolving tidally from an initial orbit at a few Pluto radii with initial eccentricity $e_{\rm C}$ = 0 or 0.2. After tidal evolution, the free eccentricities $e_{\rm free}$ of the test particles are compared with the current eccentricities of the four small satellites. The only case with $e_{\rm free}$ matching with the eccentricities of the current satellites are those not affected by MMR during the tidal evolution in a model with constant $Q$ and initial $e_{\rm C}$ = 0.2 that is damped down rapidly, where $Q$ is the effective tidal dissipation function. However, the surviving test particles not affected by resonance do not have strong preference to be in or near 4:1, 5:1 and 6:1 MMR with Charon. An alternative scenario may be needed to explain the formation of the four small satellites.

Inspired by the circumbinary nature of the orbits of the small satellites, a better understanding and description of circumbinary orbits is sought. By applying Fast Fourier transformation (FFT) to the cylindrical distance between circumbinary object and the center of mass of the binary as a function of time $R(t)$, the free eccentricity $e_{\rm free}$, the forced eccentricity $e_{\rm forced}$, and the forced oscillation terms of an orbits around a binary can be extracted. An analytic theory for the osculating Keplerian semi-major axis $a_{\rm osc}$ and eccentricity $e_{\rm osc}$ of an orbit around a circular binary is also derived, in terms of $e_{\rm free}$ and the forced oscillation terms. From the analytic solution, the terms responsible for the large variations of $a_{\rm osc}$ and $e_{\rm osc}$ are identified. Due to the large variations in the osculating elements and the fact that the mean of $e_{\rm osc}$ can be significantly different from $e_{\rm free}$, it is more suitable to use $e_{\rm free}$, which can be determined by FFT, to describe a circumbinary orbit.

The Laplace-like three-body resonance between Styx, Nix and Hydra is also studied. By repeating the simulations of \cite{sho15}, but with a smaller time step of 3000 s, none of the cases are found to have the three-body resonant angle $\Phi$ = 3$l_{\rm S}$ $-$ 5$l_{\rm N}$ + 2$l_{\rm H}$ in purely librating state within the measurable range of the masses of the satellites, where $l_{\rm S}$, $l_{\rm N}$ and $l_{\rm H}$ are the true longitudes of Styx, Nix and Hydra, respectively. We also search for (1) the libration of other three-body resonant angles when $\Phi$ is circulating, and (2) the libration of two-body resonant angles that combine to give $\Phi$ when $\Phi$ is librating. However, none of the resonant angles is librating.