Dynamics of two planets in co-orbital motion (1004.0726v1)

Abstract: We study the stability regions and families of periodic orbits of two planets locked in a co-orbital configuration. We consider different ratios of planetary masses and orbital eccentricities, also we assume that both planets share the same orbital plane. Initially we perform numerical simulations over a grid of osculating initial conditions to map the regions of stable/chaotic motion and identify equilibrium solutions. These results are later analyzed in more detail using a semi-analytical model. Apart from the well known quasi-satellite (QS) orbits and the classical equilibrium Lagrangian points L4 and L5, we also find a new regime of asymmetric periodic solutions. For low eccentricities these are located at $(\sigma,\Delta\omega) = (\pm 60\deg, \mp 120\deg)$, where \sigma is the difference in mean longitudes and \Delta\omega is the difference in longitudes of pericenter. The position of these Anti-Lagrangian solutions changes with the mass ratio and the orbital eccentricities, and are found for eccentricities as high as ~ 0.7. Finally, we also applied a slow mass variation to one of the planets, and analyzed its effect on an initially asymmetric periodic orbit. We found that the resonant solution is preserved as long as the mass variation is adiabatic, with practically no change in the equilibrium values of the angles.

• The paper demonstrates that co-orbital planets can exhibit stable periodic orbits, including new Anti-Lagrangian solutions with eccentricities up to 0.7.
• Using numerical simulations complemented by a semi-analytical Hamiltonian model, the study maps stable and chaotic zones across various initial conditions.
• The findings offer a new theoretical framework for understanding resonance interactions and guide the search for undiscovered co-orbital exoplanets.

Dynamics of Two Planets in Co-Orbital Motion

The paper presents a comprehensive paper on the dynamics and stability of two planets in co-orbital configuration, specifically examining periodic orbits and stability regions within a 1/1 mean-motion resonance (MMR). Utilizing both numerical simulations and a semi-analytical model, the research explores various configurations, including quasi-satellite (QS) orbits and traditional Lagrangian points ($L_4$ and $L_5$), while identifying a novel regime termed Anti-Lagrangian solutions.

The paper first approaches the problem through numerical simulations, mapping stable and chaotic zones over a range of initial conditions. Notably, it identifies not only the known QS orbits and Lagrangian points but also introduces new asymmetric periodic solutions located around $$(\Delta \lambda,\Delta \varpi) = (\pm 60^\circ,\mp 120^\circ)$$. These Anti-Lagrangian solutions extend the traditional understanding of co-orbital dynamics, showing stability for eccentricities up to approximately $0.7$.

Furthermore, the simulations are supported by a semi-analytical model, adopting a Hamiltonian approach to understand the underlying dynamics. This model verifies the periodic nature of the Anti-Lagrangian solutions, demonstrating their existence as stable configurations across a broad parameter space in terms of planetary masses and eccentricities. Especially intriguing is the observed linear relationship in the eccentricities for particular mass ratios, supporting a hypothesis that could inform future studies on co-orbital groups.

In terms of practical implications, this work enhances the understanding of possible co-orbital configurations in multiplanet systems, including both established systems and potentially undiscovered exoplanetary bodies exhibiting similar dynamics. From a theoretical perspective, the paper provides a new framework for examining resonance interactions, bridging classical celestial mechanics with modern computational techniques.

Looking forward, the research sets a foundation for exploring how such configurations evolve over time and under varying conditions, such as adiabatic mass changes. The adaptability of the Anti-Lagrangian solutions to such changes suggests robustness, warranting further inquiry into their potential observability in real planetary systems. This could be particularly relevant as observational techniques continue to improve, potentially unveiling new co-orbital exoplanetary systems.