The Stability of Tidal Equilibrium for Hierarchical Star-Planet-Moon Systems

The Stability of Tidal Equilibrium for Hierarchical Star-Planet-Moon Systems

Authors:

Adams et al

Abstract:

Motivated by the current search for exomoons, this paper considers the stability of tidal equilibrium for hierarchical three-body systems containing a star, a planet, and a moon. In this treatment, the energy and angular momentum budgets include contributions from the planetary orbit, lunar orbit, stellar spin, planetary spin, and lunar spin. The goal is to determine the optimized energy state of the system subject to the constraint of constant angular momentum. Due to the lack of a closed form solution for the full three-body problem, however, we must use use an approximate description of the orbits. We first consider the Keplerian limit and find that the critical energy states are saddle points, rather than minima, so that these hierarchical systems have no stable tidal equilibrium states. We then generalize the calculation so that the lunar orbit is described by a time-averaged version of the circular restricted three-body problem. In this latter case, the critical energy state is a shallow minimum, so that a tidal equilibrium state exists. In both cases, however, the lunar orbit for the critical point lies outside the boundary (roughly half the Hill radius) where (previous) numerical simulations indicate dynamical instability. These results suggest that star-planet-moon systems have no viable long-term stable states analogous to those found for two-body systems.

Stability of a planet in the HD 41004 binary system

Stability of a planet in the HD 41004 binary system

Authors:

Satyal et al

Abstract:

The Hill stability criterion is applied to analyse the stability of a planet in the binary star system of HD 41004 AB, with the primary and secondary separated by 22 AU, and masses of 0.7 M⊙ and 0.4 M⊙, respectively. The primary hosts one planet in an S-type orbit, and the secondary hosts a brown dwarf (18.64 MJ) on a relatively close orbit, 0.0177 AU, thereby forming another binary pair within this binary system. This star-brown dwarf pair (HD 41004 B+Bb) is considered a single body during our numerical calculations, while the dynamics of the planet around the primary, HD 41004 Ab, is studied in different phase-spaces. HD 41004 Ab is a 2.6 MJ planet orbiting at the distance of 1.7 AU with orbital eccentricity 0.39. For the purpose of this study, the system is reduced to a three-body problem and is solved numerically as the elliptic restricted three-body problem (ERTBP). The Hill stability function is used as a chaos indicator to configure and analyse the orbital stability of the planet, HD 41004 Ab. The indicator has been effective in measuring the planet's orbital perturbation due to the secondary star during its periastron passage. The calculated Hill stability time series of the planet for the coplanar case shows the stable and quasi-periodic orbits for at least ten million years. For the reduced ERTBP the stability of the system is also studied for different values of planet's orbital inclination with the binary plane. Also, by recording the planet's ejection time from the system or collision time with a star during the integration period, stability of the system is analysed in a bigger phase-space of the planet's orbital inclination, ≤ 90°, and its semimajor axis, 1.65–1.75 AU. Based on our analysis it is found that the system can maintain a stable configuration for the planet's orbital inclination as high as 65° relative to the binary plane. The results from the Hill stability criterion and the planet's dynamical lifetime map are found to be consistent with each other.

Behaviour of Multi Exoplanetary Systems Near Their Stability Limit

Dynamical behaviour of multiplanet systems close to their stability limit

Authors:

Marzari et al

Abstract:

The dynamics of systems of two and three planets, initially placed on circular and nearly coplanar orbits, is explored in the proximity of their stability limit. The evolution of a large number of systems is numerically computed and their dynamical behaviour is investigated with the frequency map analysis as chaos indicator. Following the guidance of this analysis, it is found that for two--planet systems the dependence of the Hill limit on the planet mass, usually made explicit through the Hill's radius parametrization, does not appear to be fully adequate. In addition, frequent cases of stable chaos are found in the proximity of the Hill limit. For three--planet systems, the usual approach adopted in numerical explorations of their stability, where the planets are initially separated by multiples of the mutual Hill radius, appears too reducing. A detailed sampling of the parameter space reveals that systems with more packed inner planets are stable well within previous estimates of the stability limit. This suggests that a two--dimensional approach is needed to outline when three--planet systems are dynamically stable.