Confined Chaos and the Chaotic Angular Motion of Atlas, a Saturn’s Inner Satellite

The dynamics of the Solar System exhibit inherent chaos and instability. Mathematical tools, such as the maximum Lyapunov Characteristic Exponent (LCE), and Lyapunov Time (TL), play a crucial role in providing a qualitative understanding of chaos within celestial objects, such as asteroids and moonlets. Celestial bodies with relatively small Lyapunov times have garnered significant research interest due to their stable orbits, a phenomenon referred to as stable or confined chaos. Notable examples include the Saturn’s satellites: Atlas, with a Lyapunov time on the order of 10 years, Prometheus and Pandora. This work aims to study the chaotic behavior of the Atlas satellite and its relatively small. We present three-dimensional model approach designed to isolate the radial contribution from the LCE and assess its influence within the LCE. Our investigation focuses on the Saturn system, comprising Saturn itself, along with its satellites Atlas, Prometheus, Pandora, and Mimas. To estimate the radial contribution of the LCE, we find the projection of the radial vector of a ghost Atlas (a slightly displaced Atlas) onto the Atlas radial vector, which allows us to calculate the difference between the radial vectors. This methodology enables us to estimate the radial contribution of the LCE and calculate the Lyapunov time. Remarkably, our results demonstrate that orbits remain confined even for integration times exceeding. Furthermore, we investigate the temporal behavior of Atlas’s angular position in its orbit, potentially shedding light on chaotic angular dynamics.