Hill Stability Research Articles

Effects of non-Kozai mutual inclinations on two-planet system stability through all phases of stellar evolution

Previous full-lifetime simulations of single-star multi-planet systems across all phases of stellar evolution have predominately assumed coplanar or nearly-coplanar orbits. Here we assess the consequences of this assumption by removing it and exploring the effect of giant branch mass loss on the stability of two-planet systems with small to moderate non-Kozai (<40 degrees) relative inclinations. We run nearly 10^4 simulations over 14 Gyr for F-star, A-star and B-star planet hosts, incorporating main sequence stellar masses of 1.5, 2.0, 2.5, 3.0 and 5.0 solar masses, and initial planetary semimajor axis ratios that straddle their three-dimensional Hill stability limits. We find that the near-coplanar assumption can approximate well the stability frequencies and critical separations found for higher inclinations, except around strong mean-motion commensurabilities. Late instabilities -- after the star has become a white dwarf -- occur throughout the explored mutual inclination range. Consequently, non-Kozai mutual inclination should not be used as a predictive orbital proxy for determining which white dwarf multi-planet systems discovered by Gaia should represent high-priority follow-up targets for the detection of metal pollution and planetary debris discs.

Hill stability in the AMD framework

In a two-planet system, a topological boundary that is created by Sundman (1912, Acta Math., 36, 105) inequality can forbid close encounters between the two planets for an infinite time. A system is said to be Hill stable if it verifies this topological condition. Hill stability is widely used in the study of extrasolar planet dynamics. However, the coplanar and circular orbit approximation is often used. In this paper, we explain how the Hill stability can be understood in the framework of angular momentum deficit (AMD). In the secular approximation, AMD allows us to discriminate between a priori stable systems and systems for which a more in-depth dynamical analysis is required. We show that the general Hill stability criterion can be expressed as a function of only semimajor axes, masses, and total AMD of the system. The proposed criterion is only expanded in the planets-to-star mass ratio ε and not in the semimajor axis ratio, eccentricities, nor the mutual inclination. Moreover, the expansion in ε remains excellent up to values of about 10−3 even for two planets with very different mass values. We performed numerical simulations in order to highlight the sharp change of behavior between Hill stable and Hill unstable systems. We show that Hill stable systems tend to be very regular, whereas Hill unstable systems often lead to rapid planet collisions. We also note that Hill stability does not provide protection from the ejection of the outer planet.

Hill stability of the satellite in the elliptic restricted four-body problem

We consider an elliptic restricted four-body system including three primaries and a massless particle. The orbits of the primaries are elliptic, and the massless particle moves under the mutual gravitational attraction. From the dynamic equations, a quasi-integral is obtained, which is similar to the Jacobi integral in the circular restricted three-body problem (CRTBP). The energy constant $C$ determines the topology of zero velocity surfaces, which bifurcate at the equilibrium point. We define the concept of Hill stability in this problem, and a criterion for stability is deduced. If the actual energy constant $C_{\mathrm{ac}}\ ( {>} 0 ) $ is bigger than or equal to the critical energy constant $C_{\mathrm{cr}}$ , the particle will be Hill stable. The critical energy constant is determined by the mass and orbits of the primaries. The criterion provides a way to capture an asteroid into the Earth–Moon system.

The Hill stability of triple planets in the Solar system

The Hill stability of the nine known triple asteroid systems in the solar system has been investigated in a framework of the three body system. In this paper, the Sun and triple-asteroid system are treated as a four body system to analyze the influence of the Sun on the Hill stability of the triple subsystem. First, the relationship of the total energy and the angular momentum between the four body system and the triple subsystem is derived. It is found that the total energy of this 1–3 configuration four body system is the sum of the energy of the triple subsystem and the energy of a two-body system composed of the Sun and the mass center of the subsystem; so is the angular momentum. Then, the Hill stability of the triple subsystem is reinvestigated using a previous criterion in the four body problem (Gong and Liu in Mon. Not. R. Astron. Soc. 462:547–553, 2016) and the results are compared to those in the three body problem. Among the nine known triple-asteroid systems, 1995 CC and 1999 TC are Hill stable for both models; the others are stable in the three body model while not stable in the four body model. In addition, the exploration of Pluto by New Horizons has attracted great attention in recent years, the Sun-Pluto-Charon-Hydra four body system is investigated in the paper, and it is found that the system is Hill stable.

Hill stability of the coplanar four-body problem with a binary subsystem

Hill stability of the coplanar four-body problem with a binary subsystem

On the Existence of Regular and Irregular Outer Moons Orbiting the Pluto–Charon System

Abstract The dwarf planet Pluto is known to host an extended system of five co-planar satellites. Previous studies have explored the formation and evolution of the system in isolation, neglecting perturbative effects by the Sun. Here we show that secular evolution due to the Sun can strongly affect the evolution of outer satellites and rings in the system, if such exist. Although precession due to extended gravitational potential from the inner Pluto–Charon binary quench such secular evolution up to a crit ∼ 0.0035 au (∼0.09 R Hill the Hill radius; including all of the currently known satellites), outer orbits can be significantly altered. In particular, we find that co-planar rings and satellites should not exist beyond a crit; rather, satellites and dust particles in these regions secularly evolve on timescales ranging between 104 and 106 years, and quasi-periodically change their inclinations and eccentricities through secular evolution (Lidov–Kozai oscillations). Such oscillations can lead to high inclinations and eccentricities, constraining the range where such satellites (and dust particles) can exist without crossing the orbits of the inner satellites or crossing the outer Hill stability range. Outer satellites, if such exist are therefore likely to be irregular satellites, with orbits limited to be non-circular and/or highly inclined. Current observations, including the recent data from the New-Horizons mission explored only inner regions (&lt;0.0012 au) and excluded the existence of additional satellites; however, the irregular satellites discussed here should reside farther, in the yet uncharted regions around Pluto.

Stability of multiplanet systems in binaries

When exploring the stability of multiplanet systems in binaries, two parameters are normally exploited: the critical semimajor axis ac computed by Holman and Wiegert (1999) within which planets are stable against the binary perturbations, and the Hill stability limit Delta determining the minimum separation beyond which two planets will avoid mutual close encounters. Our aim is to test whether these two parameters can be safely applied in multiplanet systems in binaries or if their predictions fail for particular binary orbital configurations. We have used the frequency map analysis (FMA) to measure the diffusion of orbits in the phase space as an indicator of chaotic behaviour. First we revisited the reliability of the empirical formula computing ac in the case of single planets in binaries and we find that, in some cases, it underestimates by 10-20% the real outer limit of stability. For two planet systems, the value of Delta is close to that computed for planets around single stars, but the level of chaoticity close to it substantially increases for smaller semimajor axes and higher eccentricities of the binary orbit. In these configurations ac also begins to be unreliable and non linear secular resonances with the stellar companion lead to chaotic behaviour well within ac, even for single planet systems. For two planet systems, the superposition of mean motion resonances, either mutual or with the binary companion, and non linear secular resonances may lead to chaotic behaviour in all cases. We have developed a parametric semiempirical formula determining the minimum value of the binary semimajor axis, for a given eccentricity of the binary orbit, below which stable two planet systems cannot exist.

Hill stability of the satellites in coplanar four-body problem

Hill stability of the satellites in coplanar four-body problem

Linear stability of Hill’s vortex to axisymmetric perturbations

We consider the linear stability of Hill’s vortex with respect to axisymmetric perturbations. Given that Hill’s vortex is a solution of a free-boundary problem, this stability analysis is performed by applying methods of shape differentiation to the contour dynamics formulation of the problem in a three-dimensional axisymmetric geometry. This approach allows us to systematically account for the effect of boundary deformations on the linearized evolution of the vortex under the constraint of constant circulation. The resulting singular integro-differential operator defined on the vortex boundary is discretized with a highly accurate spectral approach. This operator has two unstable and two stable eigenvalues complemented by a continuous spectrum of neutrally stable eigenvalues. By considering a family of suitably regularized (smoothed) eigenvalue problems solved with a range of numerical resolutions, we demonstrate that the corresponding eigenfunctions are in fact singular objects in the form of infinitely sharp peaks localized at the front and rear stagnation points. These findings thus refine the results of the classical analysis by Moffatt &amp; Moore (J. Fluid Mech., vol. 87, 1978, pp. 749–760).

Parametric resonance of submerged floating pipelines with bi-frequency parametric and vortex-induced oscillations excitations

ABSTRACTThis study investigates the parametric resonance of submerged floating pipelines during vortex-induced excitations. The submerged floating pipeline is considered as a potential solution for deep-water fluid transportation. However, the parametric resonance due to vortex-induced oscillations must be a major concern for its design, because it can lead to the huge motion and fatigue damage of the pipeline. Thus it is essential to predict the parametric resonance of the submerged floating pipeline system. Bi-frequency Hill's equations of the dynamic responses of the pipeline are derived, and vortex-induced forces are main causes of parametric excitations. Hill stability diagrams are used to analyse stability of solutions of the equations. The differences between the parametric resonance and vortex-induced resonance are also compared. The results show that the parametric resonance may take place in rigid body motion and transverse vibration of the submerged floating pipeline. The pipeline will have large vibration in both cross-line and in-line directions at the same time when the parametric resonance occurs, which, however, cannot happen in just the vortex-induced resonance. Some effective measures are given to designers to avoid the parametric resonance for the submerged floating pipeline design.

Stability of a planet in the HD 41004 binary system

AbstractThe 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. (© 2016 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

The hill stability of planetary motion under stellar approaches

The hill stability of planetary motion under stellar approaches

The resonance overlap and Hill stability criteria revisited

We review the orbital stability of the planar circular restricted three-body problem in the case of massless particles initially located between both massive bodies. We present new estimates of the resonance overlap criterion and the Hill stability limit and compare their predictions with detailed dynamical maps constructed with N-body simulations. We show that the boundary between (Hill) stable and unstable orbits is not smooth but characterized by a rich structure generated by the superposition of different mean-motion resonances, which does not allow for a simple global expression for stability. We propose that, for a given perturbing mass $$m_1$$ and initial eccentricity e, there are actually two critical values of the semimajor axis. All values $$a < a_\mathrm{Hill}$$ are Hill-stable, while all values $$a > a_\mathrm{unstable}$$ are unstable in the Hill sense. The first limit is given by the Hill-stability criterion and is a function of the eccentricity. The second limit is virtually insensitive to the initial eccentricity and closely resembles a new resonance overlap condition (for circular orbits) developed in terms of the intersection between first- and second-order mean-motion resonances.

Hill Stability of Configurations in the Full N-Body Problem

AbstractRigorous results on Hill Stability for the classical N-body problem are in general unknown for N ≥ 3, due to the complex interactions that may occur between bodies and the many different outcomes which may occur. However, the addition of finite density for the bodies along with a rigidity assumption on their mass distribution allows for Hill stability to be easily established. In this note we generalize results on Hill stability developed for the Full 3-body problem and show that it can be applied to the Full N-body problem. Further, we find that Hill Stability concepts can be applied to identify types of configurations which can escape and types which cannot as a function of the system energy.

Long-term orbital stability of exosolar planetary systems with highly eccentric orbits

AbstractNowadays, many extrasolar planetary systems possessing at least one planet on a highly eccentric orbit have been discovered. In this work, we study the possible long-term stability of such systems. We consider the general three body problem as our model. Highly eccentric orbits are out of the Hill stability regions. However, mean motion resonances can provide phase protection and orbits with long-term stability exist. We construct maps of dynamical stability based on the computation of chaotic indicators and we figure out regions in phase space, where the long-term stability is guaranteed. We focus on regions where at least one planet is highly eccentric and attempt to associate them with the existence of stable periodic orbits. The values of the orbital elements, which are derived from observational data, are often given with very large deviations. Generally, phase space regions of high eccentricities are narrow and thus, our dynamical analysis may restrict considerably the valid domain of the system's location.

Analytical criteria of Hill stability in the elliptic restricted three body problem

Due to the time-dependent Jacobi integral in the elliptic restricted three body problem, it is difficult to develop analytical criteria of Hill stability. In this paper, the Hill stability of the orbit around the one primary is concerned. Several analytical criteria are established based on the bifurcation of the extremum of the Jacobi integral. One criterion is used to judge the Hill stability of the orbit with orbit size known. One criterion is used to judge the Hill stability of the orbit with orbital element completely known. These criteria are applied to the judge the Hill stability of Jupiter’s moons and planets in three stellar binaries.

Hill stability of natural planet satellites in the restricted elliptic three-body problem

The existence of the Jacobi integral in a restricted circular problem of three bodies allows the construction of Hill surfaces of zero velocity by means of which the Hill stability of the motion is established. In a restricted noncircular three-body problem, the creation of zero-velocity surfaces is impossible as the Jacobi integral does not exist. However, by means of an invariant integral relation, a so-called Jacobi quasi-integral, containing one unknown function, Lukyanov (2005) constructed the regions of possible motion changing over time and surfaces of the minimum energy limiting them. At zero value of eccentricity these surfaces will be transformed to zero-velocity surfaces for a restricted circular three-body problem. In the framework of a restricted elliptic three-body problem, the Hill stability of the motion of all planetary satellites is investigated. The surfaces of minimum energy limiting regions of possible motion and defining the Hill stability of the motion in an elliptic restricted three-body problem are constructed.

On the Hill stable motions in the three-body problem

In the three-body problem, we consider the Hill stability of motions. We prove a theorem on Hill stability, which is an essential strengthening of Theorem 2 from Sosnitskii (2014a). Our theorem can be applied in the case where the model of restricted three-body problem is usually considered. To analyze the stability, we use only the energy integral.

The equilibrium of systems with dry friction

The properties of the equilibrium positions of mechanical systems with Coulomb friction are discussed. A comparative analysis of various definitions of the concept of equilibrium is given. It is shown that the principles of virtual displacements and least constraint can be extended to the problem of statics with friction. The definitions of Lyapunov stability and Hill stability are examined; the latter approach has definite advantages in certain problems. Several mechanical examples are considered to illustrate the results obtained.

Connection between Hill stability and weak stability in the elliptic restricted three-body problem

This paper provides a study on the connection between Hill stability and weak stability in the framework of the spatial elliptic restricted three-body problem. We determine a necessary condition for weak stability by giving an upper and a lower bound of qualitative measure of the Hill stability. The sufficient condition for weak stability and the symmetry of weak stable regions around the planets of the Solar System is also investigated.