A photometric and dynamic study of comet C/2013 A1 (Siding Spring) from observations at a heliocentric distance of ~4.1 AU

An analysis is presented for the photometric data on comet C/2013 A1 (Siding Spring) from observations at a large heliocentric distance (~4.1 AU). Comet C/2013 A1 (Siding Spring) displays intense activity despite the relatively large heliocentric distance. The morphology of the comet’s coma is analyzed. The following parameters are measured: the color indices V-R, the normalized spectral gradient of the reflectivity of the comet’s dust S', and the dust production rate Afρ. A numerical simulation is performed for the evolution of the comet’s orbit after a close encounter with Mars. The most probable values are obtained for the Keplerian orbital elements of the comet over a hundred-year period. The comet’s orbit remains nearly parabolic after passing the orbits of all the Solar System planets.     

Inner Solar System dynamical analogs of plutinos

By studying orbits of asteroids potentially in 3:2 exterior mean motion resonance with Earth, Venus, and Mars, we have found plutino analogs. We identify at least 27 objects in the inner Solar System dynamically protected from encounter through this resonance. These are four objects associated with Venus, six with Earth, and seventeen with Mars. Bodies in the 3:2 exterior resonance (including those in the plutino resonance associated with Neptune) orbit the Sun twice for every three orbits of the associated planet, in such a way that with sufficiently low libration amplitude close approaches to the planet are impossible. As many as 15% of Kuiper Belt objects share the 3:2 resonance, but are poorly observed. One of several resonance sweeping mechanisms during planetary migration is likely needed to explain the origin and properties of 3:2 resonant Kuiper Belt objects. Such a mechanism likely did not operate in the inner Solar System. We suggest that scattering by the next planet out allows entry to, and exit from, 3:2 resonance for objects associated with Venus or Earth. 3:2 resonators of Mars, on the other hand, do not cross the paths of other planets, and have a long lifetime. There may exist some objects trapped in the 3:2 Mars resonance which are primordial, with our tests on the most promising objects known to date indicating lifetimes of at least tens of millions of years. Identifying 3:2 resonant systems in the inner Solar System permits this resonance to be studied on shorter timescales and with better determined orbits than has been possible to date, and introduces new mechanisms for entry into the resonant configuration.     

Interplanetary dust dynamics: III. Dust released from P/Encke: Distribution with respect to the zodiacal cloud

Numerical simulations of the trajectories of over 200 30-μm-radius dust particles released by Comet P/Encke were designed to study the evolution and redistribution of orbital elements as the dust particles spiral in toward the Sun. The dust assumes Jupiter crossing orbits immediately after release due to radiation pressure, while the comet's orbit remains inside Jupiter's orbital path. By the time the dust particles have spiraled past Jupiter, information on their origin from P/Encke is erased from the distribution in orbital elements. The primary objective of this study is to compare the observed spatial distribution of zodiacal/interplanetary dust with that of the model cloud inside Jupiter's orbit. The observed location of the plane of maximum dust density “symmetry plane” of the zodiacal cloud is compared to a least-square-fit plane of the model cloud. A clear correlation between the two planes is found. The variation of the observed inclination and nodes with heliocentric distance agrees also, at least qualitatively, with that found in the model cloud. The hypothesis that short-period comets may have contributed in a major way to the zodiacal cloud is compatible with these results. The study is directly relevant to, and supports, Whipple's suggestion that Comet P/Encke may have been a major source to the zodiacal cloud.     

Extrasolar Planetary Systems

The discovery of planetary systems around alien stars is an outstanding achievement of recent years. The idea that the Solar System may be representative of planetary systems in the Galaxy in general develops upon the knowledge, current until the last decade of the 20th century, that it is the only object of its kind. Studies of the known planets gave rise to a certain stereotype in theoretical research. Therefore, the discovery of exoplanets, which are so different from objects of the Solar System, alters our basic notions concerning the physics and very criteria of normal planets. A substantial factor in the history of the Solar System was the formation of Jupiter. Two waves of meteorite bombardment played an important role in that history. Ultimately there arose a stable low-entropy state of the Solar System, in which Jupiter and the other giants in stable orbits protect the inner planets from impacts by dangerous celestial objects, reducing this danger by many orders of magnitude. There are even variants of the anthropic principle maintaining that life on Earth owes its genesis and development to Jupiter. Some 20 companions more or less similar to Jupiter in mass and a few infrared dwarfs, have been found among the 500 solar-type stars belonging to the main sequence. Approximately half of the exoplanets discovered are of the hot-Jupiter type. These are giants, sometimes of a mass several times that of Jupiter, in very low orbits and with periods of 3–14 days. All of their parent stars are enriched with heavy elements, [Fe/H] = 0.1–0.2. This may indicate that the process of exoplanet formation depends on the chemical composition of the protoplanetary disk. The very existence of exoplanets of the hot-Jupiter type considered in the context of new theoretical work comes up against the problem of the formation of Jupiter in its real orbit. All the exoplanets in orbits with a semimajor axis of more than 0.15–0.20 astronomical units (AU) have orbital eccentricities of more than 0.1, in most cases of 0.2–0.5. In conjunction with their possible migration into the inner reaches of the Solar System, this poses a threat to the very existence of the inner planets. Recent observations of gas–dust clouds in very young stars show that hydrogen dissipates rapidly, in several million years, and dissipation is completed earlier than, according to the accretion theory, the gas component of such a planet as Jupiter forms. The mass of the remaining hydrogen is usually small, much smaller than Jupiter's mass. However, the giant planets of the Solar System retain a few percent of the amount of hydrogen that should be contained in the early protoplanetary disk, creating difficulties in understanding their formation. A plausible explanation is that gravitational instabilities in the protoplanetary disk could be the mechanism of their rapid formation.     

Evidence for an Extended Scattered Disk

By telescopic tracking, we have established that the transneptunian object (TNO) 2000 CR₁₀₅ has a semimajor axis of 220±1 AU and perihelion distance of 44.14±0.02 AU, beyond the domain which has heretofore been associated with the “scattered disk” of Kuiper Belt objects interacting via gravitational encounters with Neptune. We have also firmly established that the TNO 1995 TL₈ has a high perihelion (of 40.08±0.02 AU). These objects, and two other recent discoveries which appear to have perihelia outside 40 AU, have probably been placed on these orbits by a gravitational interaction which is not strong gravitational scattering off of any of the giant planets on their current orbits. Their existence may thus have profound cosmogonic implications for our understanding of the formation of the outer Solar System. We discuss some viable scenarios which could have produced these objects, including long-term diffusive chaos and scattering off of other massive bodies in the outer Solar System. This discovery implies that there must be a large population of TNOs in an “extended scattered disk” with perihelia above the previously suggested 38 AU boundary. The total population is difficult to estimate due to the ease with which such objects would have been lost. This illustrates the great value of frequent and well time-sampled recovery observations of trans-neptunian objects within their discovery opposition.     

Steady state populations of Apollo-Amor objects

The steady-state distribution of orbits of Apollo-Amor objects is calculated for a variety of possible sources. These include asteroids near the inner edge of the belt, cometary orbits similar to Encke, and hypothetical extinct cometary orbits with perihelia larger than that of Encke. In all but one case, the steady-state distributions are similar for all these sources, and predict Amor/Apollo ratios of 1.5 to 3. These ratios are lower than those predicted by work in which the effects of the ν₆ secular resonance were not considered. These results are in general agreement with observation, although the higher (～3) Amor/Apollo ratios found for many of the sources may turn out to be unacceptably high. The absolute number of Apollo-Amors observed is found to require an injection rate of ～15 objects/(10⁶ years). This rate is easily achieved if the present existence of Encke is assumed to be a reasonably probable event, and if Encke becomes a ～1-km-diameter Apollo object following exhaustion of its volatile material; best estimates of the injection rate from the asteroid belt [～1.5/(10⁶ years)] are too low. Hence a dominant cometary component is suggested. The predicted number of Apollo objects in small (q < 1.0 AU, a < 1.4 AU orbits is in agreement with observation. Predicted lunar and terrestrial cratering rates agree approximately with observation. An unexplained difference between the lunar and terrestrial results is probably caused by uncertainties in the scaling laws or crater counts used. This discrepancy precludes an exact test of these calculations using cratering data.     

On the stability of a planet between Mars and the asteroid belt: Implications for the Planet V hypothesis

The stability of an additional planet between the orbit of Mars and the asteroid belt is examined in the context of the Planet V hypothesis. In this model, the Solar System initially contained a fifth terrestrial planet, “Planet V,” which was removed after ∼700 Myr, a possible trigger for the late heavy bombardment on the inner planets. The model is investigated using 96 N-body integrations of the 8 major planets with an additional body between Mars and the asteroid belt. In more than 1/4 of simulations, Planet V survives for 1000 Myr. In most other cases, Planet V collides with the Sun or hits another planet after several hundred Myr, leaving 4 surviving terrestrial planets. In 24/96 simulations, Planet V is lost by ejection or collision with the Sun while the other four terrestrial planets survive without undergoing a collision. In 18 cases, Planet V is removed at least 200 Myr after the beginning of the simulation. The endstate depends sensitively on the mass of Planet V. Collision with the Sun is likely when Planet V's mass is 0.25 Mars masses or less. When Planet V is more massive than this, collisions involving it and/or other terrestrial planets become commonplace. In unstable systems, the times of first encounter and first collision/ejection depend on the initial aphelion distance of Mars. Reducing Mars's aphelion distance increases these times and also increases the fraction of systems surviving for 1000 Myr. When Mars's current orbit is used, the stability of Planet V increases when these two planets are widely separated initially. Planet V's aphelion distance Q typically begins to cross the asteroid belt within a few tens to a few hundred Myr, and its orbit last leaves the belt several hundred Myr later in most cases. The total time spent with Q>2.1 AU is typically less than 200 Myr.     

Multi-SunSynchronous Orbits in the Solar System

Performances of a planetary observation system are strongly related to the choice of the orbit used. Trajectories with characteristics of periodicity are very useful for the assessment of time-varying phenomena and thus Periodic SunSynchronous and Periodic Multi-SunSynchronous Orbits are particularly suitable to this end. In this paper, the research into these kinds of orbits, previously proposed for the Earth and Mars, has been extended to planets of the Solar System and to their principal moons. In general, these trajectories are typically obtained under the hypothesis that the J₂ harmonic is predominant with respect to the other orbital perturbations, since this allows an analytical solution. However, the hypothesis of J₂ predominant is not always verified in the Solar System and so analytical techniques must be replaced by numerical simulations. Interesting results have been obtained for the planets Mars and Jupiter and for the moons Europa, Callisto and Titan, where periodic trajectories with reduced revisit times and low altitudes have been found. These solutions allow the observation of time-varying phenomena with high spatial and temporal resolution.     

Solar and planetary destabilization of the Earth-Moon triangular Lagrangian points

In the restricted circular three-body problem, two massive bodies travel on circular orbits about their mutual center of mass and gravitationally perturb the motion of a massless particle. The triangular Lagrange points, L₄ and L₅, form equilateral triangles with the two massive bodies and lie in their orbital plane. Provided the primary is at least 27 times as massive as the secondary, orbits near L₄ and L₅ can remain close to these locations indefinitely. More than 2200 cataloged asteroids librate about the L₄ and L₅ points of the Sun-Jupiter system, and five bodies have been discovered around the L₄ point of the Sun-Neptune system. Small satellites have also been found librating about the L₄ and L₅ points of two of Saturn's moons. However, no objects have been discovered around the Earth-Moon L₄ and L₅ points. Using numerical integrations, we show that orbits near the Earth-Moon L₄ and L₅ points can survive for over a billion years even when solar perturbations are included, but the further addition of the far smaller perturbations from other planets destabilize these orbits within several million years. Thus, the lack of observed objects in these regions cannot be used as a constraint on Solar System formation, nor on the tidal evolution of the Moon's orbit.     

The Architectural Design Rules of Solar Systems Based on the New Perspective

In this paper I present a new perspective of the birth and evolution of Planetary Systems. This new perspective presents an all encompassing and self consistent Paradigm of the birth and evolution of the solar systems. In doing so it redefines astronomy and rewrites astronomical principles. Kepler and Newton defined a stable and non-evolving elliptical orbits. While this perspective defines a collapsing or expanding spiral orbit of planets except for Brown Dwarfs. Brown Dwarfs are significant fraction of the central star. Hence they rapidly evolve from non-Keplerian state to the end point which is a Keplerian state where it is in stable elliptical orbits. On the basis of the Lunar Laser Ranging Data released by NASA on the Silver Jubilee Celebration of Man’s Landing on Moon on 21st July 1969–1994, theoretical formulation of Earth-Moon tidal interaction was carried out and Planetary Satellite Dynamics was established. It was found that this mathematical analysis could as well be applied to Star and Planets system and since every star could potentially contain an extra-solar system, hence we have a large ensemble of exo-planets to test our new perspective on the birth and evolution of solar systems. Till date 403 exo-planets have been discovered in 390 extra-solar systems by radial velocity method, by transiting planet method, by gravitational lensing method, by direct imaging method and by timing method. I have taken 12 single planet systems, four Brown Dwarf—Star systems and two Brown Dwarf pairs. Following architectural design rules are corroborated through this study of exo-planets. All planets are born at inner Clarke’s Orbit what we refer to as inner geo-synchronous orbit in case of Earth-Moon System. The inner Clarke’s Orbit is an orbit of unstable equilibrium. By any perturbative force such as cosmic particles or radiation pressure, the planet gets tipped long of a_G1 or short of a_G1. Here a_G1 is inner Clarke’s Orbit. If planet is long of a_G1 then it is said to be in extra-synchronous orbit. Here Gravitational Sling Shot effect is in play. In gravity assist planet fly-by maneuver in space flights, gravitational sling shot is routinely used to boost the space craft to its destination. The exo-planet can either be launched on death spiral as CLOSE HOT JUPITERS or can be launched on an expanding spiral path as the planets in our Solar System are. In death spiral, exo-planet less than 5 m_J will get pulverized and vaporized in close proximity to the host star. If the mass is between 5 and 7.5 m_J then it will be partially vaporized and partially engulfed by the host star and if it is greater than 7.5 m_J, then it will be completely ingested by the host star. In the process the planet will deposit all its material and angular momentum in the Host Star. This will leave tell-tale imprints of ingestion: in such cases host Star will have higher ⁷Li, host star will become a rapidly rotating progenitor and the host star will have excess IR. All these have been confirmed by observations of Transiting Planets. It was also found that if the exo-planet are significant fraction of the host star then those exo-planets rapidly migrate from a_G1 to a_G2 and have very short Time Constant of Evolution as Brown Dwarfs have. But if exo-planets are insignificant fraction of the host star as our terrestrial planets are then they are stay put in their original orbit of birth. By corollary this implies that Giant exo-planets reach nearly Unity Evolution Factor in a fraction of the life span of a solar system. This is particularly true for brown dwarfs orbiting main sequence stars. In this study four star systems hosting Brown Dwarfs, two Brown Dwarf pairs and 12 extrasolar systems hosting Jupiter sized planets are selected. In Brown Dwarfs evolution factor is invariably UNITY or near UNITY irrespective of their respective age and Time Constant of Evolution is very short of the order of year or tens of years. In case of 12 exo-planets system with increasing mass ratio evolution factor increases and time constant of evolution shortens from Gy to My though there are two exceptions. TW Hydrae is a special case. This Solar System is newly born system which is only 9 million years old. Hence its exo-planet has just been born and it is very near its birth place just as predicted by my hypothesis. In fact it is only slightly greater than a_G1. This vindicates our basic premise that planets are always born at inner Clarke’s Orbit. This study vindicates the design rules which had been postulated at 35th COSPAR Scientific Assembly in 2004 at Paris, France, under the title “New Perspective on the Birth & Evolution of Solar Systems”.     

Secular perihelion advances of the inner planets and asteroid Icarus

A small effect expected from a recently proposed gravitational impact model (Wilhelm et al., 2013) is used to explain the remaining secular perihelion advance rates of the planets Mercury, Venus, Earth, Mars, and the asteroid (1566) Icarus—after taking into account the disturbances related to Newton’s Theory of Gravity. Such a rate was discovered by Le Verrier (1859) for Mercury and calculated by Einstein (1915, 1916) in the framework of his General Theory of Relativity (GTR). Accurate observations are now available for the inner Solar System objects with different orbital parameters. This is important, because it allowed us to demonstrate that the quantitative amount of the deviation from an 1/r potential is—under certain conditions—only dependent on the specific mass distribution of the Sun and not on the characteristics of the orbiting objects and their orbits. A displacement of the effective gravitational from the geometric centre of the Sun by about 4400 m towards each object is consistent with the observations and explains the secular perihelion advance rates.     

The Dynamics of Objects in the Inner Edgeworth–Kuiper Belt

Objects in 3:2 mean motion resonance with Neptune are protected from close encounters with Neptune by the resonance. Bodies in orbits with semi-major axis between 39.5 and about 42 AU are not protected by the resonance; indeed due to overlapping secular resonances, the eccentricities of orbits in this region are driven up so that a close encounter with Neptune becomes inevitable. It is thus expected that such orbits are unstable. The list of known Trans-Neptunian objects shows a deficiency in the number of objects in this gap compared to the 43–50 AU region, but the gap is not empty. We numerically integrate models for the initial population in the gap, and also all known objects over the age of the Solar System to determine what fraction can survive. We find that this fraction is significantly less than the ratio of the population in the gap to that in the main belt, suggesting that some mechanism must exist to introduce new members into the gap. By looking at the evolution of the test body orbits, we also determine the manner in which they are lost. Though all have close encounters with Neptune, in most cases this does not lead to ejection from the Solar System, but rather to a reduced perihelion distance causing close encounters with some or all of the other giant planets before being eventually lost from the system, with Saturn appearing to be the cause of the ejection of most of the objects.     

Building the terrestrial planets: Constrained accretion in the inner Solar System

To date, no accretion model has succeeded in reproducing all observed constraints in the inner Solar System. These constraints include: (1) the orbits, in particular the small eccentricities, and (2) the masses of the terrestrial planets - Mars’ relatively small mass in particular has not been adequately reproduced in previous simulations; (3) the formation timescales of Earth and Mars, as interpreted from Hf/W isotopes; (4) the bulk structure of the asteroid belt, in particular the lack of an imprint of planetary embryo-sized objects; and (5) Earth’s relatively large water content, assuming that it was delivered in the form of water-rich primitive asteroidal material. Here we present results of 40 high-resolution (N = 1000-2000) dynamical simulations of late-stage planetary accretion with the goal of reproducing these constraints, although neglecting the planet Mercury. We assume that Jupiter and Saturn are fully-formed at the start of each simulation, and test orbital configurations that are both consistent with and contrary to the “Nice model”. We find that a configuration with Jupiter and Saturn on circular orbits forms low-eccentricity terrestrial planets and a water-rich Earth on the correct timescale, but Mars’ mass is too large by a factor of 5-10 and embryos are often stranded in the asteroid belt. A configuration with Jupiter and Saturn in their current locations but with slightly higher initial eccentricities (e = 0.07-0.1) produces a small Mars, an embryo-free asteroid belt, and a reasonable Earth analog but rarely allows water delivery to Earth. None of the configurations we tested reproduced all the observed constraints. Our simulations leave us with a problem: we can reasonably satisfy the observed constraints (except for Earth’s water) with a configuration of Jupiter and Saturn that is at best marginally consistent with models of the outer Solar System, as it does not allow for any outer planet migration after a few Myr. Alternately, giant planet configurations which are consistent with the Nice model fail to reproduce Mars’ small size.     

The relationship of active comets, “extinct” comets,and dark asteroids

Spectrophotometric data on groups of asteroids in different types of orbits reveal different distributions of spectral properties, depending on whether the orbits are cometary or noncometary. In a list of 10 asteroids frequently suggested on purely dynamical grounds to be extinct or dormant comets, all have properties suggestive of spectral classes D, P, or C. Preliminary IRAS albedo results support this. Objects in these classes are very dark, reddish-black to neutral-black, and prevalent among the Trojans and outer belt. Two comets observed at low activity (visible nuclei) also have properties more consistent with D asteroids than any other class (very low reported geometric albedos of 0.02 and red colors). Consistent with these results are very low albedos reported for materials in more than a dozen comets; they average 0.05. Also, sampled cometary dust particles appear to consist of dark carbonaceous materials. Dramatically different are a control group of 13 Aten/Apollo/Amor objects selected from noncometary orbits. Most are in moderate-albedo classes: 8 or 9 appear to be of class S, and only 1 is in a low-albedo class (C). These are probably mostly objects perturbed out of the inner asteroid belt. The preponderence of S's in the noncometary group, together with the preponderence of ordinary chondrites among meteorites, may be evidence that such meteorites came from S asteroids. The data indicate that extinct, dormant, inactive, and minimally active comet nuclei have low albedos (p_v=a few percent) and very red to moderately red colors. As a group, their spectra are more similar to those of outer Solar System asteroids of classes D, P, and C, than to those of inner belt classes, though the observations are frequently not yet complete enough to assign definitively a spectral class. The results, taken together, support the view that dynamically identified “extinct comet candidates” are indeed outer Solar System objects probably of cometary origin. The results also support a scenario of Solar System formation in which dark carbonaceous dust dominated the spectrophotometric properties of planetesimals formed from about 2.7 AU out to at least the Trojan region at 5.2 AU. From 2.7 to at least 5.2 AU, and from class C to class D, the color of this dust reddens, apparently due to increasing amounts of red organic condensates. Comets are probably also colored to different degrees, by dust of this type, and may in some cases be even redder than D asteroids.     

On the stability of Earth-like planets in multi-planet systems

We present a continuation of our numerical study on planetary systems with similar characteristics to the Solar System. This time we examine the influence of three giant planets on the motion of terrestrial-like planets in the habitable zone (HZ). Using the Jupiter–Saturn–Uranus configuration we create similar fictitious systems by varying Saturn’s semi-major axis from 8 to 11 AU and increasing its mass by factors of 2–30. The analysis of the different systems shows the following interesting results: (i) Using the masses of the Solar System for the three giant planets, our study indicates a maximum eccentricity (max-e) of nearly 0.3 for a test-planet placed at the position of Venus. Such a high eccentricity was already found in our previous study of Jupiter–Saturn systems. Perturbations associated with the secular frequency g ₅ are again responsible for this high eccentricity. (ii) An increase of the Saturn-mass causes stronger perturbations around the position of the Earth and in the outer HZ. The latter is certainly due to gravitational interaction between Saturn and Uranus. (iii) The Saturn-mass increased by a factor 5 or higher indicates high eccentricities for a test-planet placed at the position of Mars. So that a crossing of the Earth’ orbit might occur in some cases. Furthermore, we present the maximum eccentricity of a test-planet placed in the Earth’ orbit for all positions (from 8 to 11 AU) and masses (increased up to a factor of 30) of Saturn. It can be seen that already a double-mass Saturn moving in its actual orbit causes an increase of the eccentricity up to 0.2 of a test-planet placed at Earth’s position. A more massive Saturn orbiting the Sun outside the 5:2 mean motion resonance (a _S  ≥9.7 AU) increases the eccentricity of a test-planet up to 0.4.     

Orbital stability of systems of closely-spaced planets,II: configurations with coorbital planets

We numerically investigate the stability of systems of 1 \({{\rm M}_{\oplus}}\) planets orbiting a solar-mass star. The systems studied have either 2 or 42 planets per occupied semimajor axis, for a total of 6, 10, 126, or 210 planets, and the planets were started on coplanar, circular orbits with the semimajor axes of the innermost planets at 1 AU. For systems with two planets per occupied orbit, the longitudinal initial locations of planets on a given orbit were separated by either 60° (Trojan planets) or 180°. With 42 planets per semimajor axis, initial longitudes were uniformly spaced. The ratio of the semimajor axes of consecutive coorbital groups in each system was approximately uniform. The instability time for a system was taken to be the first time at which the orbits of two planets with different initial orbital distances crossed. Simulations spanned virtual times of up to 1 × 10⁸, 5 × 10⁵, and 2 × 10⁵ years for the 6- and 10-planet, 126-planet, and 210-planet systems, respectively. Our results show that, for a given class of system (e.g., five pairs of Trojan planets orbiting in the same direction), the relationship between orbit crossing times and planetary spacing is well fit by the functional form log(t _c /t ₀) = b β + c, where t _c is the crossing time, t ₀ = 1 year, β is the separation in initial orbital semimajor axis (in terms of the mutual Hill radii of the planets), and b and c are fitting constants. The same functional form was observed in the previous studies of single planets on nested orbits (Smith and Lissauer 2009). Pairs of Trojan planets are more stable than pairs initially separated by 180°. Systems with retrograde planets (i.e., some planets orbiting in the opposite sense from others) can be packed substantially more closely than can systems with all planets orbiting in the same sense. To have the same characteristic lifetime, systems with 2 or 42 planets per orbit typically need to have about 1.5 or 2 times the orbital separation as orbits occupied by single planets, respectively.     

Stability of the coplanar planetary four-body system

We consider the coplanar planetary four-body problem, where three planets orbit a large star without the cross of their orbits. The system is stable if there is no exchange or cross of orbits. Starting from the Sundman inequality, the equation of the kinematical boundaries is derived. We discuss a reasonable situation, where two planets with known orbits are more massive than the third one. The boundaries of possible motions are controlled by the parameter c~2E. If the actual value of c~2E is less than or equal to a critical value(c~2 E)cr, then the regions of possible motions are bounded and therefore the system is stable.The criteria obtained in special cases are applied to the Solar System and the currently known extrasolar planetary systems. Our results are checked using N-body integrator.     

Migration of Trans-Neptunian Objects to the Terrestrial Planets

The orbital evolution of more than 22000 Jupiter-crossing objects under thegravitational influence of planets was investigated. We found that the meancollision probabilities of Jupiter-crossing objects (from initial orbits close tothe orbit of a comet) with the terrestrial planets can differ by more than twoorders of magnitude for different comets. For initial orbital elements close tothose of some comets (e.g., 2P and 10P), about 0.1% of objects got Earth-crossingorbits with semi-major axes a < 2 AU and moved in such orbits for more than a Myr (up to tens or even hundreds of Myrs).Results of our runs testify in favor of at least one of these conclusions: (1) the portionof 1-km former trans-Neptunian objects (TNOs) among near-Earth objects (NEOs)can exceed several tens of percent, (2) the number of TNOs migrating inside the solarsystem could be smaller by a factor of several than it was earlier considered, (3) mostof 1-km former TNOs that had got NEO orbits disintegrated into mini-comets and dustduring a smaller part of their dynamical lifetimes if these lifetimes are not small.     

The 1/1 resonance in extrasolar planetary systems

We study orbits of planetary systems with two planets, for planar motion, at the 1/1 resonance. This means that the semimajor axes of the two planets are almost equal, but the eccentricities and the position of each planet on its orbit, at a certain epoch, take different values. We consider the general case of different planetary masses and, as a special case, we consider equal planetary masses. We start with the exact resonance, which we define as the 1/1 resonant periodic motion, in a rotating frame, and study the topology of the phase space and the long term evolution of the system in the vicinity of the exact resonance, by rotating the orbit of the outer planet, which implies that the resonance and the eccentricities are not affected, but the symmetry is destroyed. There exist, for each mass ratio of the planets, two families of symmetric periodic orbits, which differ in phase only. One is stable and the other is unstable. In the stable family the planetary orbits are in antialignment and in the unstable family the planetary orbits are in alignment. Along the stable resonant family there is a smooth transition from planetary orbits of the two planets, revolving around the Sun in eccentric orbits, to a close binary of the two planets, whose center of mass revolves around the Sun. Along the unstable family we start with a collinear Euler–Moulton central configuration solution and end to a planetary system where one planet has a circular orbit and the other a Keplerian rectilinear orbit, with unit eccentricity. It is conjectured that due to a migration process it could be possible to start with a 1/1 resonant periodic orbit of the planetary type and end up to a satellite-type orbit, or vice versa, moving along the stable family of periodic orbits.     

The Common Origin of the High Inclination TNO's

Numerical integrations of the four major planets orbits inside a primordialplanetesimals disk show that a fraction of Neptune primordial scatteredobjects are deposited into the classical Kuiper Belt at Solar System age. Theseobjects exhibit inclinations as high as 40° and can account forpresent high inclinations population in the classical Kuiper Belt. The samemechanism can also originate high perihelion scattered objects like 2000 CR₁₀₅. The process that in the end produced such objects can be divided into two phases, a migration phase where nonconservative dynamics acted to producesome stable objects already at 10⁸ years and a nonmigrating phase that helped to establish some other objects as stable TNO's. Low inclination CKBO's have inprinciple an origin through the resonance sweeping process, although someresults from numerical integrations at least suggest a possible origin also fromthe primordial Neptune scattered population.