[Paper Review] The periodic and chaotic regimes of motion in the exoplanet 2/1 mean-motion resonance

T. A. Michtchenko, S. Ferraz‐Mello+1 more|arXiv (Cornell University)|Dec 6, 2011

Stellar, planetary, and galactic studies3 citations

TL;DR

This paper maps the phase space of the 2/1 mean-motion resonance in exoplanetary systems, identifying two distinct families of periodic orbits—$σ$-family (librating critical angle) and $Δ\varpi$-family (circulating pericentre angle)—and revealing that asymmetric apsidal corotation resonances (ACRs) with two branches can host true secular resonances. The key finding is that branch B of the two-branch asymmetric ACR enables simultaneous libration of both $\sigma_2$ and $\Delta\varpi$, creating a stable, chaotic-free dynamical regime even at high eccentricities.

ABSTRACT

We present the dynamical structure of the phase space of the planar planetary 2/1 mean-motion resonance (MMR). Inside the resonant domain, there exist two families of periodic orbits, one associated to the librational motion of the critical angle ($σ$-family) and the other related to the circulatory motion of the angle between the pericentres ($Δ\varpi$-family). The well-known apsidal corotation resonances (ACR) appear at the intersections of these families. A complex web of secondary resonances exists also for low eccentricities, whose strengths and positions are dependent on the individual masses and spatial scale of the system. Depending on initial conditions, a resonant system is found in one of the two topologically different states, referred to as extit{internal} and extit{external} resonances. The internal resonance is characterized by symmetric ACR and its resonant angle is $2\,λ_2-λ_1-\varpi_1$, where $λ_i$ and $\varpi_i$ stand for the planetary mean longitudes and longitudes of pericentre, respectively. In contrast, the external resonance is characterized by asymmetric ACR and the resonant angle is $2\,λ_2-λ_1-\varpi_2$. We show that systems with more massive outer planets always envolve inside internal resonances. The limit case is the well-known asteroidal resonances with Jupiter. At variance, systems with more massive inner planets may evolve in either internal or external resonances; the internal resonances are typical for low-to-moderate eccentricity configurations, whereas the external ones for high eccentricity configurations of the systems. In the limit case, analogous to Kuiper belt objects in resonances with Neptune, the systems are always in the external resonances characterized by asymmetric equilibria.

Motivation & Objective

• To understand the full dynamical structure of the 2/1 mean-motion resonance (MMR) beyond apsidal corotation resonances (ACRs).
• To identify and characterize the topologically distinct internal and external resonance states in the phase space of 2/1 MMR systems.
• To investigate how planetary mass ratios and eccentricities influence the existence and stability of secondary resonances and periodic orbits.
• To determine under what conditions systems evolve into internal vs. external resonances, particularly regarding mass dominance of inner or outer planets.
• To map the role of separatrix structures in distinguishing libration from circulation regimes near asymmetric ACRs.

Proposed method

• Numerical construction of dynamical maps in the resonant phase space using energy-conserving initial conditions.
• Application of a low-pass filter to smooth the phase space structure and reveal underlying periodic orbit families.
• Identification of periodic orbits via the resonant Hamiltonian, with energy level contours visualizing stable and unstable equilibria.
• Analysis of critical angles $\sigma = 2\lambda_2 - \lambda_1 - \varpi_1$ and $\Delta\varpi = \varpi_1 - \varpi_2$ to classify motion as librating or circulating.
• Use of the averaged Hamiltonian formalism to model long-term dynamics in the 2/1 MMR, valid for arbitrary mass ratios and eccentricities.
• Distinction between internal resonance (symmetric ACR, $2\lambda_2 - \lambda_1 - \varpi_1$) and external resonance (asymmetric ACR, $2\lambda_2 - \lambda_1 - \varpi_2$) based on orbital geometry and stability.

Experimental results

Research questions

• RQ1 What are the topologically distinct dynamical states (internal vs. external resonance) in the 2/1 mean-motion resonance, and how do they differ in their resonant angle definitions?
• RQ2 How do the two-branch asymmetric ACRs influence the stability and nature of orbital motion, particularly in terms of simultaneous libration of $\sigma_2$ and $\Delta\varpi$?
• RQ3 What role do secondary resonances and separatrix structures play in organizing the phase space of low-eccentricity 2/1 MMR systems?
• RQ4 Why do systems with more massive outer planets always evolve into internal resonances, while systems with more massive inner planets may evolve into either internal or external resonances?
• RQ5 How do the dynamical maps and energy level contours reveal the presence of true secular resonance in the vicinity of branch B of the asymmetric ACR?

Key findings

• The 2/1 MMR phase space contains two primary families of periodic orbits: the $\sigma$-family (librating critical angle) and the $\Delta\varpi$-family (circulating pericentre angle), intersecting at apsidal corotation resonances (ACRs).
• Asymmetric ACRs with two branches exist, and only branch B supports true simultaneous libration of both $\sigma_2$ and $\Delta\varpi$, indicating a novel secular resonance regime.
• The separatrix structure in the phase space around branch B ACR separates regions of libration from circulation, with the separatrix itself formed by solutions asymptotic to an unstable ACR.
• For systems with more massive outer planets, internal resonances are the only stable outcome, corresponding to symmetric ACRs and the resonant angle $2\lambda_2 - \lambda_1 - \varpi_1$.
• In contrast, systems with more massive inner planets can evolve into either internal or external resonances, with external resonances dominating at high eccentricities and characterized by asymmetric equilibria and the resonant angle $2\lambda_2 - \lambda_1 - \varpi_2$.
• Horseshoe-like orbits exist outside the cyan curves in the dynamical maps, encompassing multiple ACRs including both stable asymmetric ACRs and unstable symmetric ACRs, indicating a complex, structurally stable region of quasi-periodic motion.