Orbital Stability of Planet-hosting Triple-star Systems according to Hill: Applications to Alpha Centauri and 16 Cygni

In this study we investigate aspects of orbital stability for the Alpha Centauri and 16 Cygni systems. They are planet-hosting triple star systems of highly hierarchic nature. For each system, orbital stability of the outlying stellar component and the observed exoplanet(s) are explored through assessing Hill stability. Orbital stability is identified for all components, including the observed system planets.

Orbital stability of S-type circumbinary planets

Nowadays, more than 4,000 exoplanets have been discovered, including a hundred of circumbinary planets. In the following work, the orbital variations of 67 S-type circumbinary planets have been studied. Their orbital evolutions for a thousand years are simulated using the REBOUND package. The published physical and orbital parameters of the systems are used to computed the systems’ orbital instability limits: Roche limit and Hill’s sphere. From 67 systems, there are two unstable circumbinary systems: Kepler-420 and GJ 86. Kepler-420 Ab orbit passes into the system’s Roche limit due to its high orbital eccentricity. For GJ 86 Ab, the planet orbits outside its Hill’s sphere. The instability of GJ 86 Ab might be caused by an inaccurate measurement of GJ 86 A physical parameters. Using the GJ 86 A mass obtained from Farihi et al., the planet orbits in the stable orbit zone.

Orbital stability analysis of trajectories of highly nonlinear dynamic systems with feedback coupling

Orbital stability and qualitative study of the oscillations of a highly nonlinear dynamic system with feedback coupling are considered. For a highly nonlinear dynamic system with feedback coupling that satisfies Liénard’s theorem (on the existence and uniqueness of a periodic solution), a complete study of the phase pattern of the system is conducted. Applying the Poincaré criterion, the conditions for the existence of limit cycles and their Lyapunov stability are determined. The diagrams of phase trajectories are constructed numerically using the Mathcad 15 software package. Limit cycles are established, which are consistent with the limit cycles obtained by the Poincaré method. The behavior of trajectories outside the limit cycles is investigated. Recurrent homogeneous Pfaff equations are obtained, which determine the behavior of the systems “at infinity”. It was determined that the infinitely distant point of the horizontal axis is the only singular point for these equations. Linear approximations of recurrent homogeneous equations are obtained, which make it possible to determine the nature of the singular points. It was found that the trajectories then wind like a spiral on the limit cycles. Images of trajectories on the phase plane outside the limit cycles for the cases of degrees of nonlinearity under consideration are constructed.

Deep Exploration of the Planets HR 8799 b, c, and d with Moderate-resolution Spectroscopy

The four directly imaged planets orbiting the star HR 8799 are an ideal laboratory to probe atmospheric physics and formation models. We present more than a decade’s worth of Keck/OSIRIS observations of these planets, which represent the most detailed look at their atmospheres to date by its resolution and signal-to-noise ratio. We present the first direct detection of HR 8799 d, the second-closest known planet to the star, at moderate spectral resolution with Keck/OSIRIS (K band; R ≈ 4000). Additionally, we uniformly analyze new and archival OSIRIS data (H and K band) of HR 8799 b, c, and d. First, we show detections of water (H2O) and carbon monoxide (CO) in the three planets and discuss the ambiguous case of methane (CH4) in the atmosphere of HR 8799 b. Then, we report radial-velocity (RV) measurements for each of the three planets. The RV measurement of HR 8799 d is consistent with predictions made assuming coplanarity and orbital stability of the HR 8799 planetary system. Finally, we perform a uniform atmospheric analysis on the OSIRIS data, published photometric points, and low-resolution spectra. We do not infer any significant deviation from the stellar value of the carbon-to-oxygen ratio (C/O) of the three planets, which therefore does not yet yield definitive information about the location or method of formation. However, constraining the C/O for all the HR 8799 planets is a milestone for any multiplanet system, and particularly important for large, widely separated gas giants with uncertain formation processes.

Semi-Analytical Estimates for the Orbital Stability of Earth’s Satellites

Normal form stability estimates are a basic tool of Celestial Mechanics for characterizing the long-term stability of the orbits of natural and artificial bodies. Using high-order normal form constructions, we provide three different estimates for the orbital stability of point-mass satellites orbiting around the Earth. (i) We demonstrate the long-term stability of the semimajor axis within the framework of the $$J_2$$ J 2 problem, by a normal form construction eliminating the fast angle in the corresponding Hamiltonian and obtaining $${\mathcal {H}}_{J_2}$$ H J 2 . (ii) We demonstrate the stability of the eccentricity and inclination in a secular Hamiltonian model including lunisolar perturbations (the ‘geolunisolar’ Hamiltonian $${\mathcal {H}}_\mathrm{gls}$$ H gls ), after a suitable reduction of the Hamiltonian to the Laplace plane. (iii) We numerically examine the convexity and steepness properties of the integrable part of the secular Hamiltonian in both the $${\mathcal {H}}_{J_2}$$ H J 2 and $${\mathcal {H}}_\mathrm{gls}$$ H gls models, which reflect necessary conditions for the holding of Nekhoroshev’s theorem on the exponential stability of the orbits. We find that the $${\mathcal {H}}_{J_2}$$ H J 2 model is non-convex, but satisfies a ‘three-jet’ condition, while the $${\mathcal {H}}_\mathrm{gls}$$ H gls model restores quasi-convexity by adding lunisolar terms in the Hamiltonian’s integrable part.