A Mechanism for Producing Short-Period Binaries

When three point masses form a hierarchical triple system, the short-period orbit can be severely modified by the long-period orbit if the two orbits are inclined to each other by more than about 39^deg (). Such an inclination can induce ‘Kozai cycles’ (Kozai, 1962), in which the eccentricity of the inner orbit cycles by a large amount while its period and therefore semimajor axis remains roughly constant. During those periastra when the eccentricity is largest, tidal friction may become important, and this can result in a secular shrinkage of the orbit, until it becomes circularised at a period of a few days.However, apsidal motion due to either GR or to the quadrupolar distortion of the components (if they are no longer treated as point masses) can reduce the range of eccentricity. We explore the limits on outer and inner orbital period that these perturbations imply.If the components are F/G/K/M dwarfs, then rotationally-driven dynamo activity can become important at the short periods that can occur in the right circumstances. It can cause the period to shorten further. The result may be a contact binary, and/or a merger in which the two stars of the inner pair coalesce to form a single rapidly rotating star. We suggest that this may be the origin of AB Dor, a very rapidly rotating K dwarf that is probably about 50 Myr old.     

Öpik-type collision probability for high-inclination orbits

The classical Öpik theory provides an estimate of the collision probability between two bodies on bound, heliocentric or planetocentric orbits under restrictive assumptions of: (i) constant eccentricity and inclination, and (ii) uniform circulation of the longitude of node and argument of pericenter. These assumptions are violated whenever either of the orbits has a large inclination with respect to the local Laplace plane or large eccentricity, and their motion is perturbed by an exterior (tidal) gravitational field of a planet or the Sun. In this situation, known as the Lidov–Kozai regime, the eccentricity and inclination values exhibit large and correlated oscillations. At the same time, the longitude of node and the argument of pericenter may have strongly nonlinear time evolution, with the latter being even bound to a small interval of values. Here we develop a new Öpik-type collision probability theory which is valid even for highly inclined and/or eccentric orbits of the projectile. We assume that the orbit of the target is circular and in the local Laplace plane. Such a generalized setting is necessary, as an example, to correctly estimate the terrestrial impact fluxes of sporadic micrometeoroids on high-inclination orbits (notably those from the toroidal source and the associated helion and anti-helion arcs).     

The origin of the Kuiper Belt high-inclination population

I simulate the orbital evolution of the four major planets and a massive primordial planetesimal disk composed of 10⁴ objects, which perturb the planets but not themselves. As Neptune migrates by energy and angular momentum exchange with the planetesimals, a large number of primordial Neptune-scattered objects are formed. These objects may experience secular, Kozai, and mean motion resonances that induce temporary decrease of their eccentricities. Because planets are migrating, some planetesimals can escape those resonances while in a low-eccentricity incursion, thus avoiding the return path to Neptune close encounter dynamics. In the end, this mechanism produces stable orbits with high inclination and moderate eccentricities. The population so formed together with the objects coming from the classical resonance sweeping process, originates a bimodal distribution for the Kuiper Belt orbits. The inclinations obtained by the simulations can attain values above 30° and their distribution resembles a debiased distribution for the high-inclination population coming from the real classical Kuiper Belt.     

The relativistic factor in the orbital dynamics of point masses

There is a growing population of relativistically relevant minor bodies in the Solar System and a growing population of massive extrasolar planets with orbits very close to the central star where relativistic effects should have some signature. Our purpose is to review how general relativity affects the orbital dynamics of the planetary systems and to define a suitable relativistic correction for Solar System orbital studies when only point masses are considered. Using relativistic formulae for the N body problem suited for a planetary system given in the literature we present a series of numerical orbital integrations designed to test the relevance of the effects due to the general theory of relativity in the case of our Solar System. Comparison between different algorithms for accounting for the relativistic corrections are performed. Relativistic effects generated by the Sun or by the central star are the most relevant ones and produce evident modifications in the secular dynamics of the inner Solar System. The Kozai mechanism, for example, is modified due to the relativistic effects on the argument of the perihelion. Relativistic effects generated by planets instead are of very low relevance but detectable in numerical simulations.     

Secular and tidal evolution of circumbinary systems

We investigate the secular dynamics of three-body circumbinary systems under the effect of tides. We use the octupolar non-restricted approximation for the orbital interactions, general relativity corrections, the quadrupolar approximation for the spins, and the viscous linear model for tides. We derive the averaged equations of motion in a simplified vectorial formalism, which is suitable to model the long-term evolution of a wide variety of circumbinary systems in very eccentric and inclined orbits. In particular, this vectorial approach can be used to derive constraints for tidal migration, capture in Cassini states, and stellar spin–orbit misalignment. We show that circumbinary planets with initial arbitrary orbital inclination can become coplanar through a secular resonance between the precession of the orbit and the precession of the spin of one of the stars. We also show that circumbinary systems for which the pericenter of the inner orbit is initially in libration present chaotic motion for the spins and for the eccentricity of the outer orbit. Because our model is valid for the non-restricted problem, it can also be applied to any three-body hierarchical system such as star–planet–satellite systems and triple stellar systems.     

The role of cluster evolution in disrupting planetary systems and discs: the Kozai mechanism

We examine the effects of dynamical evolution in clusters on planetary systems or protoplanetary discs orbiting the components of binary stars. In particular, we look for evidence that the companions of host stars of planetary systems or discs could have their inclination angles raised from zero to between the threshold angles (39.23° and 140.77°) that can induce the Kozai mechanism. We find that up to 20 per cent of binary systems have their inclination angles increased to within the threshold range. Given that half of all extrasolar planets could be in binary systems, we suggest that up to 10 per cent of extrasolar planets could be affected by this mechanism.     

The Hill stability of inclined small mass binary systems in three-body systems with special application to triple star systems, extrasolar planetary systems and Binary Kuiper Belt systems

The dynamical stability of a bound triple system composed of a small binary or minor planetary system moving on a orbit inclined to a central third body is discussed in terms of Hill stability for the full three-body problem. The situation arises in the determination of stability of triple star systems against disruption and component exchange and the determination of stability of extrasolar planetary systems and minor planetary systems against disruption, component exchange or capture. The Hill stability criterion is applied to triple star systems and extrasolar planetary systems, the Sun-Earth-Moon system and Kuiper Belt binary systems to determine the critical distances for stable orbits. It is found that increasing the inclination of the third body decreases the Hill regions of stability. Increasing the eccentricity of the binary also produces similar effects.These type of changes make exchange or disruption of the component masses more likely. Increasing the eccentricity of the binary orbit relative to the third body substantially decreases stability regions as the eccentricity reaches higher values. The Kuiper Belt binaries were found to be stable if they move on circular orbits. Taking into account the eccentricity, it is less clear that all the systems are stable.     

Eccentricities of Planets in Binary Systems

The most puzzling property of the extrasolar planets discovered by recent radial velocity surveys is their high orbital eccentricities, which are very difficult to explain within our current theoretical paradigm for planet formation. Current data reveal that at least 25% of these planets, including some with particularly high eccentricities, are orbiting a component of a binary star system. The presence of a distant companion can cause significant secular perturbations in the orbit of a planet. At high relative inclinations, large-amplitude, periodic eccentricity perturbations can occur. These are known as “Kozai cycles” and their amplitude is purely dependent on the relative orbital inclination. Assuming that every planet host star also has a (possibly unseen, e.g., substellar) distant companion, with reasonable distributions of orbital parameters and masses, we determine the resulting eccentricity distribution of planets and compare it to observations? We find that perturbations from a binary companion always appear to produce an excess of planets with both very high (?0.6) and very low (e ? 0.1) eccentricities. The paucity of near-circular orbits in the observed sample implies that at least one additional mechanism must be increasing eccentricities. On the other hand, the overproduction of very high eccentricities observed in our models could be combined with plausible circularization mechanisms (e.g., friction from residual gas) to create more planets with intermediate eccentricities (e? 0.1–0.6).     

Eccentricity Generation in Hierarchical Triple Systems with Non-coplanar and Initially Circular Orbits

In a previous paper, we developed a technique for estimating the inner eccentricity in coplanar hierarchical triple systems on initially circular orbits, with comparable masses and with well-separated components, based on an expansion of the rate of change of the Runge-Lenz vector. Now, the same technique is extended to non-coplanar orbits. However, it can only be applied to systems with I ₀ < 39.23° or I ₀ > 140.77°, where I is the inclination of the two orbits, because of complications arising from the so-called ‘Kozai effect’. The theoretical model is tested against results from numerical integrations of the full equations of motion. This revised version was published online in July 2006 with corrections to the Cover Date.     

Stability of inclined orbits of terrestrial planets in habitable zones

In long-term stability studies of terrestrial planets moving in the habitable zone (HZ) of a sun-like star, we distinguish four different configurations: (i) planets moving in binary star systems, (ii) the inner type (where the gas giant moves outside the HZ), (iii) the outer type (where the gas giant is closer to the star, than the HZ) and (iv) the Trojan type (where the gas giant moves in the HZ). Since earlier calculations indicated, that the stability of the motion in the HZ also depends on the inclination of the terrestrial planet orbits, we present a detailed numerical investigation to show correlations between the eccentricity, the mass and the distance of the giant planet for various inclinations of the terrestrial planets. The orbital stability of the HZ was examined for all four configurations stated above. While we could find hardly any stable orbits for the first three types for inclinations higher than 40^°, the Trojan planets can be stable up to an inclination of 60^°. Additionally, we could also find some stabilizing effects of the inclination for the first three types. As dynamical model we used the elliptic restricted three-body problem, which consists of two massive and one mass-less body. This allows an application to all detected and future extrasolar single planet systems.     

Stability of a planet in the HD 41004 binary system

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 M_J) 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 M_J 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 & Co. KGaA, Weinheim)     

Third Body Perturbations of Double Stars

We report on the different results concerning the stability of the hierarchical triple systems where a close binary is accompanied by a third star. There are different possible approaches to answer the question of the stability limits for such triple stars: the most direct investigations can be undertaken in integrating numerically the respective equations of motion for many different initial conditions. It is then difficult to take into account all the important parameters like eccentricities, inclination, phases and masses. Analytical approaches and qualitative methods are more approriate to deal with this problem; the respective results confirm the numerically found results that: 1. for prograde orbits the ratio semimajor axis of the inner orbits to the periastron position of the outer orbit is approximately 3.2 2. for retrograde orbits this ratio is just some 10 percents smaller 3. the results are not sensitive in what concerns the masses involved 4. There is a tendency that the inclinations and eccentricities change slightly the stability limits mentioned above. This revised version was published online in August 2006 with corrections to the Cover Date.     

The occurrence of high-order resonances and Kozai mechanism in the scattered disk

By means of numerical methods we explore the relevance of the high-order exterior mean motion resonances (MMR) with Neptune that a scattered disk object (SDO) can experience in its diffusion to the Oort cloud. Using a numerical method for estimate the strength of these resonances we show that high-eccentricity or high-inclination resonant orbits should have evident dynamical effects. We investigate the properties of the Kozai mechanism (KM) for non-resonant SDO's and the conditions that generate the KM inside a MMR associated with substantial changes in eccentricity and inclination. We found that the KM inside a MMR is typical for SDO's with Pluto-like or greater inclinations and is generated by the oscillation of ω inside the mixed (e,i) resonant terms of the disturbing function. A SDO diffusing to the Oort cloud should experience temporary captures in MMR, preferably of the type 1:N, and when evolving inside a MMR and experiencing the KM it can reach regions where the strength of the resonance drops and consequently there is a possibility of being decoupled from the resonance generating by this way a long-lived high-perihelion scattered disk object (HPSDO).     

On the stability of circumbinary planetary systems

The dynamics of circumbinary planetary systems (the systems in which the planets orbit a central binary) with a small binary mass ratio discovered to date is considered. The domains of chaotic motion have been revealed in the “pericentric distance–eccentricity” plane of initial conditions for the planetary orbits through numerical experiments. Based on an analytical criterion for the chaoticity of planetary orbits in binary star systems, we have constructed theoretical curves that describe the global boundary of the chaotic zone around the central binary for each of the systems. In addition, based on Mardling’s theory describing the separate resonance “teeth” (corresponding to integer resonances between the orbital periods of a planet and the binary), we have constructed the local boundaries of chaos. Both theoretical models are shown to describe adequately the boundaries of chaos on the numerically constructed stability diagrams, suggesting that these theories are efficient in providing analytical criteria for the chaoticity of planetary orbits.     

Dynamics of the Jupiter Trojans with Saturn’s perturbation in the present configuration of the two planets

The dynamics of the two Jupiter triangular libration points perturbed by Saturn is studied in this paper. Unlike some previous works that studied the same problem via the pure numerical approach, this study is done in a semianalytic way. Using a literal solution, we are able to explain the asymmetry of two orbits around the two libration points with symmetric initial conditions. The literal solution consists of many frequencies. The amplitudes of each frequency are the same for both libration points, but the initial phase angles are different. This difference causes a temporary spatial asymmetry in the motions around the two points, but this asymmetry gradually disappears when the time goes to infinity. The results show that the two Jupiter triangular libration points should have symmetric spatial stable regions in the present status of Jupiter and Saturn. As a test of the literal solution, we study the resonances that have been extensively studied in Robutel and Gabern (Mon Not R Astron Soc 372:1463–1482, 2006). The resonance structures predicted by our analytic theory agree well with those found in Robutel and Gabern (Mon Not R Astron Soc 372:1463–1482, 2006) via a numerical approach. Two kinds of chaotic orbits are discussed. They have different behaviors in the frequency map. The first kind of chaotic orbits (inner chaotic orbits) is of small to moderate amplitudes, while the second kind of chaotic orbits (outer chaotic orbits) is of relatively larger amplitudes. Using analytical theory, we qualitatively explain the transition process from the inner chaotic orbits to the outer chaotic orbits with increasing amplitudes. A critical value of the diffusion rate is given to separate them in the frequency map. In a forthcoming paper, we will study the same problem but keep the planets in migration. The time asymmetry, which is unimportant in this paper, may cause an observable difference in the two Jupiter Trojan groups during a very fast planet migration process.     

Five new and three improved mutual orbits of transneptunian binaries

We present three improved and five new mutual orbits of transneptunian binary systems (58534) Logos-Zoe, (66652) Borasisi-Pabu, (88611) Teharonhiawako-Sawiskera, (123509) 2000 WK₁₈₃, (149780) Altjira, 2001 QY₂₉₇, 2003 QW₁₁₁, and 2003 QY₉₀ based on Hubble Space Telescope and Keck II laser guide star adaptive optics observations. Combining the five new orbit solutions with 17 previously known orbits yields a sample of 22 mutual orbits for which the period P, semimajor axis a, and eccentricity e have been determined. These orbits have mutual periods ranging from 5 to over 800 days, semimajor axes ranging from 1600 to 37,000 km, eccentricities ranging from 0 to 0.8, and system masses ranging from 2 × 10¹⁷ to 2 × 10²² kg. Based on the relative brightnesses of primaries and secondaries, most of these systems consist of near equal-sized pairs, although a few of the most massive systems are more lopsided. The observed distribution of orbital properties suggests that the most loosely-bound transneptunian binary systems are only found on dynamically cold heliocentric orbits. Of the 22 known binary mutual orbits, orientation ambiguities are now resolved for 9, of which 7 are prograde and 2 are retrograde, consistent with a random distribution of orbital orientations, but not with models predicting a strong preference for retrograde orbits. To the extent that other perturbations are not dominant, the binary systems undergo Kozai oscillations of their eccentricities and inclinations with periods of the order of tens of thousands to millions of years, some with strikingly high amplitudes.     

The cross-correlation between galaxies and groups: probing the galaxy distribution in and around dark matter haloes

Some massive binaries should contain energetic pulsars which inject relativistic leptons from their inner magnetospheres and/or pulsar wind regions. If the binary system is compact enough, then these leptons can initiate inverse Compton (IC) e^± pair cascades in the anisotropic radiation field of a massive star. γ-rays can be produced in the IC cascade during its development in a pulsar wind region and above a shock in a massive star wind region where the propagation of leptons is determined by the structure of a magnetic field around the massive star. For a binary system with specific parameters, we calculate phase-dependent spectra and fluxes of γ-rays escaping as a function of the inclination angle of the system and for different assumptions on injection conditions of the primary leptons (their initial spectra and location of the shock inside the binary). We conclude that the features of γ-ray emission from such massive binaries containing energetic pulsars should allow us to obtain important information on the acceleration of particles by the pulsars, and on interactions of a compact object with the massive star wind. Predicted γ-ray light curves and spectra in the GeV and TeV energy ranges from such binary systems within our Galaxy and Magellanic Clouds should be observed by future AGILE and GLAST satellites and low-threshold Cherenkov telescopes, such as MAGIC, HESS, VERITAS or CANGAROO III.     

Limits on the Orbits of Possible Eccentric and Inclined Moons of Extrasolar Planets Orbiting Single Stars

Limits are placed on the range of orbits and masses of possible moons orbiting extrasolar planets which orbit single central stars. The Roche limiting radius determines how close the moon can approach the planet before tidal disruption occurs; while the Hill stability of the star–planet–moon system determines stable orbits of the moon around the planet. Here the full three-body Hill stability is derived for a system with the binary composed of the planet and moon moving on an inclined, elliptical orbit relative the central star. The approximation derived here in Eq. (17) assumes the binary mass is very small compared with the mass of the star and has not previously been applied to this problem and gives the criterion against disruption and component exchange in a closed form. This criterion was applied to transiting extrasolar planetary systems discovered since the last estimation of the critical separations (Donnison in Mon Not R Astron Soc 406:1918, 2010a) for a variety of planet/moon ratios including binary planets, with the moon moving on a circular orbit. The effects of eccentricity and inclination of the binary on the stability of the orbit of a moon is discussed and applied to the transiting extrasolar planets, assuming the same planet/moon ratios but with the moon moving with a variety of eccentricities and inclinations. For the non-zero values of the eccentricity of the moon, the critical separation distance decreased as the eccentricity increased in value. Similarly the critical separation decreased as the inclination increased. In both cases the changes though very small were significant.     

On the chaotic orbital dynamics of the planet in the system 16 Cyg

The chaotic orbital dynamics of the planet in the wide visual binary star system 16 Cyg is considered. The only planet in this system has a significant orbital eccentricity, e = 0.69. Previously, Holman et al. suggested the possibility of chaos in the orbital dynamics of the planet due to the proximity of 16 Cyg to the separatrix of the Lidov–Kozai resonance. We have calculated the Lyapunov characteristic exponents on the set of possible orbital parameters for the planet. In all cases, the dynamics of 16 Cyg is regular with a Lyapunov time of more than 30 000 yr. The dynamics is considered in detail for several possible models of the planetary orbit; the dependences of Lyapunov exponents on the time of their calculation and the time dependences of osculating orbital elements have been constructed. Phase space sections for the system dynamics near the Lidov–Kozai resonance have been constructed for all models. A chaotic behavior in the orbital motion of the planet in 16 Cyg is shown to be unlikely, because 16 Cyg in phase space is far from the separatrix of the Lidov–Kozai resonance at admissible orbital parameters, with the chaotic layer near the separatrix being very narrow.