[Paper Review] The exosystems about HD169830 and HD12661: are they dynamical twins?

Krzysztof Goździewski, M. Konacki|arXiv (Cornell University)|Jul 29, 2003

Stellar, planetary, and galactic studiesPhysics and Astronomy17 citations

TL;DR

This paper investigates the dynamical stability and similarities between the exoplanetary systems HD 169830 and HD 12661 using N-body simulations and fast dynamical indicators. Despite differing orbital solutions, both systems exhibit long-term stability with large eccentricity variations, and the HD 169830 system with the updated orbital solution lies between the 9:1 and 10:1 mean motion resonances, suggesting potential dynamical twins with HD 12661 due to similar mass ratios and hierarchical configurations.

ABSTRACT

The new 2-planetary system around HD169830 has been announced during the XIX-th IAP Colloquium "Extrasolar Planets: Today & Tomorrow" (Paris, June 30 - July 4, 2003) by the Geneva Extrasolar Planet Search team. We study the orbital dynamics of this system in the framework of the $N$-body problem. The analysis of its orbital stability is performed using the long-term integrations and the fast indicators, the Mean Exponential Growth factor of Nearby Orbits and the Frequency Map Analysis. The HD169830 appears to be located in a wide stable region of the phase space. The ratio of the mean motions of the planets HD169830b and c is between low-order mean motion resonances, 9:1 and 10:1. The long-term integration of the coplanar configurations, conducted over 1Gyr, reveals that the eccentricities of the companions vary with a large amplitude about 0.4-0.5 but there is no sign of instability. The orbital parameters of the planets resemble those of another 2-planetary system, around HD12661. Both of them can be classified as hierarchical planetary systems. We investigate whether these two exosystems are dynamically similar. Such similarities may be important for finding out if the formation and subsequent orbital evolution of exoplanetary systems obey common rules.

Motivation & Objective

• To assess whether the exoplanetary systems HD 169830 and HD 12661 are dynamically similar despite differing orbital solutions.
• To evaluate the long-term orbital stability of the HD 169830 system using N-body integrations and fast dynamical indicators.
• To determine if the systems' similarities in mass ratios and orbital architecture suggest a common formation or evolutionary mechanism.
• To investigate the role of mean motion resonances and secular apsidal resonances in maintaining stability in hierarchical planetary systems.
• To compare numerical results with predictions from the secular octupole-level theory of Lee & Peale (2003).

Proposed method

• Conducting long-term N-body integrations over 1 Gyr for coplanar configurations of HD 169830 using two distinct orbital solutions (ICI and ICII).
• Applying fast dynamical indicators, including the Mean Exponential Growth factor of Nearby Orbits (MEGAK) and Frequency Map Analysis, to assess orbital stability.
• Using Jacobi coordinates for orbital elements to avoid spurious effects from astrocentric representations in radial velocity fits.
• Performing genetic algorithm optimization to locate the global minimum of the reduced chi-squared statistic for the 2-Keplerian fit to radial velocity data.
• Varying the inclination of the outer planet to 89.9° to enable three-dimensional numerical integration.
• Comparing results from 2-Keplerian and full N-body models to validate the stability and consistency of the orbital solutions.

Experimental results

Research questions

• RQ1 Are the orbital configurations of HD 169830 and HD 12661 dynamically similar, suggesting a shared formation or evolutionary pathway?
• RQ2 Does the updated orbital solution for HD 169830 place it in a stable region of phase space, particularly near the 9:1 to 10:1 mean motion resonance?
• RQ3 To what extent do secular apsidal resonances with antialigned apsides stabilize the HD 169830 system, as seen in the ICII solution?
• RQ4 How do the results from N-body simulations compare with predictions from the secular octupole-level theory of Lee & Peale (2003)?
• RQ5 Could the initial orbital solution for HD 169830 be a local minimum in the chi-squared space, misleading the dynamical interpretation?

Key findings

• The updated orbital solution (ICII) for HD 169830 places the system between the 9:1 and 10:1 mean motion resonances, with a stable configuration over 1 Gyr.
• Long-term N-body integrations show that eccentricities of both planets in HD 169830 vary with amplitudes up to 0.5, yet no signs of instability are observed.
• The ICII solution does not support a secular apsidal resonance with antialigned apsides, unlike the earlier ICII solution, which exhibited such a resonance.
• The dynamical features of HD 169830 with the ICII solution are more similar to systems like HD 38529, HD 74156, and HD 168443, which have large separations and massive outer companions.
• The initial orbital solution (ICI) was likely a local minimum in the chi-squared space, as the genetic algorithm found a better-fitting solution with similar inner planet parameters but significantly altered outer planet parameters.
• The secular octupole-level theory agrees with N-body results for low eccentricities, but discrepancies emerge in high-eccentricity, resonance-adjacent regions, indicating the need for direct integrations in such cases.