Unclear on my part of this solution really worked. Here is info from the paper. Reference paper, Pluto near the edge of chaos,
https://www.pnas.org/doi/abs/10.1073/pnas.2118692119, 31-March-2022. "Significance The dwarf planet, Pluto, has stirred the imagination of the public and of planetary scientists due to its many unusual properties. Among these properties is its Neptune-crossing orbit whose stability is owed to an orbital resonance with Neptune. Less well understood is the role of the other planets. We demonstrate that the orbital architecture of the giant planets lies within a narrow niche in which Pluto-like orbits are practically stable on gigayear timescales, whereas nearby are strongly chaotic orbits. Pluto is witness to the dynamical history of the solar system; quantifying its proximity to strong chaos and dynamical instability can enable quantitative constraints on its own dynamical history as well as that of the solar system." "Abstract Many of the unusual properties of Pluto’s orbit are widely accepted as evidence for the orbital migration of the giant planets in early solar system history. However, some properties remain an enigma. Pluto’s long-term orbital stability is supported by two special properties of its orbit that limit the location of its perihelion in azimuth and in latitude. We revisit Pluto’s orbital dynamics with a view to elucidating the individual and collective gravitational effects of the giant planets on constraining its perihelion location. While the resonant perturbations from Neptune account for the azimuthal constraint on Pluto’s perihelion location, we demonstrate that the long-term and steady persistence of the latitudinal constraint is possible only in a narrow range of additional secular forcing which arises fortuitously from the particular orbital architecture of the other giant planets. Our investigations also find that Jupiter has a largely stabilizing influence whereas Uranus has a largely destabilizing influence on Pluto’s orbit. Overall, Pluto’s orbit is rather surprisingly close to a zone of strong chaos."
My observation. Numerical simulations of Pluto's orbit indicate it is on the edge of chaos and the goal is to show Pluto can remain stable for at least 4.5 Gyr matching with the age of the solar system using meteorite dates. Showing Pluto is stable is difficult as well as showing it is completely stable over giga year periods. Other reports indicate long age problems with Pluto and its moons remaining stable.
A Pluto--Charon Sonata IV. Improved Constraints on the Dynamical Behavior and Masses of the Small Satellites
ref paper,
https://arxiv.org/pdf/2204.04226.pdf, 12-April-2022, 24-pages. "1. INTRODUCTION...Within the set of completed simulations, they show that a ‘heavy’ satellite system – where the mass of Kerberos is roughly one third the mass of Hydra and the total system mass is MSNKH ∼ 1.15 × 10^20 g (Brozovi´c et al. 2015) – is unstable on time scales <~ 1 Gyr…"
My observation. Ongoing efforts to show dynamical stability for Pluto and its satellites is a work in progress. Various simulation efforts continue to eject Kerberos and Styx on timescales 2 Gyr or less. In 4.5 Gyr, Pluto and satellites could complete 1.8144E+07 revolutions around the Sun. MS BING reports, "Kerberos orbits between Nix and Hydra and makes a complete orbit around Pluto roughly every 32.167 days. Its orbital period is close to a 1:5 orbital resonance with Charon, with the timing discrepancy being about 0.7%."
Using 32.167 day period and applying to Pluto over 4.5 Gyr period, Kerberos could orbit Pluto 5.1136E+10 revolutions or more than 51 billion revolutions. Demonstrating the long term stability of Pluto and its satellites over 4.5 Gyr, looks difficult to show.
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