P/2010 TO20 LINEAR-Grauer: A comet in transition from Centaurs to the Jupiter family

The object P/2010 TO20 LINEAR-Grauer, discovered at a heliocentric distance of over 5 AU, and at first classified as a Trojan, is now believed to be a comet. This paper reports special observations of the object that have allowed a significant refinement of its orbit and investigation of its dynamic evolution. It is shown that P/2010 TO20 LINEAR-Grauer is not a Trojan yet demonstrates unusual dynamic features. In particular, the object moves in a temporary satellite orbit relative to Jupiter over the observation interval. The comet has been in the Hill sphere for about two years and has made one revolution around the planet. The jovicentric distance function has two minima, and the smallest distance is 0.075 AU. Our estimates show that, with a probability of 0.76, the comet is likely to move in a Jupiter family orbit with a perihelion distance of less than 2.5 AU. The average time for such a transition is around forty thousand years.     

Short-Period Comets with a High Tisserand Constant: III. Kinematics of Low-Velocity Encounters

Within the framework of a pair two-body problem (Sun–Jupiter, Sun–comet), the kinematics of the encounter of a minor body with a planet is investigated. The notion of points of low-velocity tangency of the orbits of the comet and Jupiter, as well as the point of Jovicentric velocity and the low-velocity tangent section of a cometary orbit, is introduced. The conditions and definitions of low-velocity and high-velocity encounters are proposed. The systems of inequalities relating the aand eparameters, which make it possible to single out those comets that are likely to be objects with low-velocity encounters, are presented. The regions of orbits that have low-velocity tangent sections, i.e., regions of low-velocity tangency of orbits, are singled out on the (a, e) plane. These regions agree well with the corresponding parameters of the orbits of real comets whose evolution contains low-velocity encounters with Jupiter.     

The capture of comets by Jupiter

The capture of comets with parabolic orbits by Jupiter is investigated. The influence of the gravitational force of the Sun on the cometary orbit during the passage of Jupiter's sphere of influence is taken into account. A comparison of the present results with previous calculations demonstrate the importance of the solar perturbations.It is also shown that captures of comets with parabolic orbits and repeated close passages to Jupiter cannot explain all of the observed cometary orbits found in the family of Jupiter.     

From the Oort cloud to observable short-period comets – I. The initial stage of cometary capture

We investigate the first stage of the dynamical evolution of Oort cloud comets entering the planetary region for the first time. To this purpose, we integrate numerically the motions of a large number of fictitious comets pertaining to two samples, both with perihelion distances up to 5.7 au and random inclinations; the first sample is composed of comets whose orbits have at least one node close to 5.2 au, while the second is not subject to this constraint. We examine the orbits when the comets come to aphelion after their first perihelion passage within the planetary region, and find that there is a clear statistical dependence of the energy perturbations on the Tisserand parameter. There appear to be two main processes, of comparable importance, governing the shortening of semimajor axes to values of less than 1000 au, i.e. planetary close encounters, especially with Jupiter, and indirect perturbations due to the shifting of the motion from barycentric to heliocentric and back; the former process mostly affects comets crossing the ecliptic at about 5.2 au, or on low-inclination orbits, while the latter mostly affects comets of small perihelion distance. This last result may help to understand the relative paucity of Halley-type comets with perihelion distances larger than about 1.5 au.     

Searching for the source of short-period comet nuclei: The direction of the spatial migration of comets

An attempt is made to determine the spatial location of the main source of short-period comet nuclei. Numerical calculations for the orbital evolution of Jupiter family comets, medium-period comets, and Centaurs are used to show that the orbits of small solar system bodies tend to evolve in the direction of increasing semimajor axes. This relates to bodies that can experience encounters with planets and whose orbital evolution is shaped by gravitational perturbations. It is concluded that there is good reason to search for the main source of the nuclei of Jupiter family comets at distances of 6 AU or less from the sun.     

Resonance and Capture of Jupiter Comets

A number of Jupiter family comets such as Otermaand Gehrels 3make a rapid transition from heliocentric orbits outside the orbit of Jupiter to heliocentric orbits inside the orbit of Jupiter and vice versa. During this transition, the comet can be captured temporarily by Jupiter for one to several orbits around Jupiter. The interior heliocentric orbit is typically close to the 3:2 resonance while the exterior heliocentric orbit is near the 2:3 resonance. An important feature of the dynamics of these comets is that during the transition, the orbit passes close to the libration points L ₁and L ₂, two of the equilibrium points for the restricted three-body problem for the Sun-Jupiter system. Studying the libration point invariant manifold structures for L ₁and L ₂is a starting point for understanding the capture and resonance transition of these comets. For example, the recently discovered heteroclinic connection between pairs of unstable periodic orbits (one around the L ₁and the other around L ₂) implies a complicated dynamics for comets in a certain energy range. Furthermore, the stable and unstable invariant manifold tubes associated to libration point periodic orbits, of which the heteroclinic connections are a part, are phase space conduits transporting material to and from Jupiter and between the interior and exterior of Jupiter's orbit.     

Long-term evolution of Oort Cloud comets: capture of comets

We test different possibilities for the origin of short-period comets captured from the Oort Cloud. We use an efficient Monte Carlo simulation method that takes into account non-gravitational forces, Galactic perturbations, observational selection effects, physical evolution and tidal splittings of comets. We confirm previous results and conclude that the Jupiter family comets cannot originate in the spherically distributed Oort Cloud, since there is no physically possible model of how these comets can be captured from the Oort Cloud flux and produce the observed inclination and Tisserand constant distributions. The extended model of the Oort Cloud predicted by the planetesimal theory consisting of a non-randomly distributed inner core and a classical Oort Cloud also cannot explain the observed distributions of Jupiter family comets. The number of comets captured from the outer region of the Solar system are too high compared with the observations if the inclination distribution of Jupiter family comets is matched with the observed distribution. It is very likely that the Halley-type comets are captured mainly from the classical Oort Cloud, since the distributions in inclination and Tisserand value can be fitted to the observed distributions with very high confidence. Also the expected number of comets is in agreement with the observations when physical evolution of the comets is included. However, the solution is not unique, and other more complicated models can also explain the observed properties of Halley-type comets. The existence of Jupiter family comets can be explained only if they are captured from the extended disc of comets with semimajor axes of the comets   a <5000 au  . The original flattened distribution of comets is conserved as the cometary orbits evolve from the outer Solar system era to the observed region.     

Orbits of short period comets captured from the Oort cloud

Oort cloud comets occasionally obtain orbits which take them through the planetary region. The perturbations by the planets are likely to change the orbit of the comet. We model this process by using a Monte Carlo method and cross sections for orbital changes, i.e. changes in energy, inclination and perihelion distance, in a single planet-comet encounter. The influence of all major planets is considered. We study the distributions of orbital parameters of observable comets, i.e. those which have perihelion distance smaller than a given value. We find that enough comets are captured from the Oort cloud in order to explain the present populations of short period comets. The median value of cos i for the Jupiter family is 0.985 while it is 0.27 for the Halley types. The results may explain the orbital features of short period comets, assuming that the active lifetime of a comet is not much greater than 400 orbital revolutions.     

Orbits of short period comets captured from the Oort cloud

Oort cloud comets occasionally obtain orbits which take them through the planetary region. The perturbations by the planets are likely to change the orbit of the comet. We model this process by using a Monte Carlo method and cross sections for orbital changes, i.e. changes in energy, inclination and perihelion distance, in a single planet-comet encounter. The influence of all major planets is considered. We study the distributions of orbital parameters of observable comets, i.e. those which have perihelion distance smaller than a given value. We find that enough comets are captured from the Oort cloud in order to explain the present populations of short period comets. The median value of cos i for the Jupiter family is 0.985 while it is 0.27 for the Halley types. The results may explain the orbital features of short period comets, assuming that the active lifetime of a comet is not much greater than 400 orbital revolutions.     

Close encounters of nearly parabolic comets and planets

An overview is given of close encounters of nearly parabolic comets (NPCs; with periods of P > 200 years and perihelion distances of q > 0.1 AU; the number of the comets is N = 1041) with planets. The minimum distances Δ_min between the cometary and planetary orbits are calculated to select comets whose Δ_min are less than the radius of the planet’s sphere of influence. Close encounters of these comets with planets are identified by numerical integration of the comets’ equations of motion over an interval of ±50 years from the time of passing the perihelion. Close encounters of NPCs with Jupiter in 1663–2011 are reported for seven comets. An encounter with Saturn is reported for comet 2004 F2 (in 2001).     

Capture of comets from the Oort cloud into Halley-type and Jupiter-family orbits

This paper analyzes the capture of comets into Halley-type and Jupiter-family orbits from the nearparabolic flux of the Oort cloud. Two types of capture into Halley-type orbits are found. The first type is the evolution of near-parabolic orbits into short-period orbits (with heliocentric orbital periods P < 200 years) as a result of close encounters with giant planets. This process is followed by a very slow drift of cometary orbits into the inner part of the Solar System. Only those comets may pass from short-period orbits into Halley-type and Jupiter-family orbits, which move in orbits with perihelion distances q < 13 au. In the second type of capture, the perihelion distances of cometary orbits become rather small (< 1.5 au) during the first stage of dynamic evolution under the action of perturbations from the Galaxy, and then their semimajor axes decrease as a result of diffusion. The capture takes place, on average, in 500 revolutions of the comet about the Sun, whereas in the first case, the comet is captured, on average, after 12500 revolutions. The region of initial orbital perihelion distances q > 4 au is found to be at least as important a source of Halley-type comets as the region of perihelion distances q < 4 au. More than half of the Halley-type comets are captured from the nearly parabolic flux with q > 4 au. The analysis of the dynamic evolution of objects moving in short-period orbits shows that the distribution of Centaurs orbits agrees well with the observed distribution corrected for observational selection effects. Hence, the hypothesis associating the origin of Centaurs with the Edgeworth-Kuiper belt and the trans-Neptunian region exclusively should be rejected.     

Tidal gravitational forces: The infall of “new” comets and comet showers

The tidal gravitational field of the Galaxy directed into the galactic plane changes the angular momentum of comets in the Oort cloud. For comet orbits with semimajor axis greater than 2 × 10⁴ AU, the change of angular momentum in one orbit is sufficient to bring comets from the Oort cloud into the visible region, causing the infall of “new” comets. The limiting size orbit is weakly dependent on the angle between the major axis of the comet orbit and the galactic plane. The flux of comets into the inner Solar System caused by the galactic tidal field will be continuous and nearly isotropic. This effect appears to exclude any determination of the trajectories of passing stars by analysis of the angular distribution of new comets. The production of intense comet showers by the tidal field of a solar companion or of an interstellar cloud is considered. We show that the direction of a solar companion cannot be found from the present distribution of observable comets. The frequency of comet showers induced by encounters with interstellar clouds is found to be much lower than that from passing stars, and the tidal fields of interstellar clouds are not strong enough to cause comet showers of sufficient intensity to result in Earth impacts.     

Relationship of comets and planets

We analyze our earlier data on the numerical integration of the equations of motion for 274 short-period comets (with the period P<200 yr) on a time interval of 6000 yr. As many as 54 comets had no close approaches to planets, 13 comets passed through the Saturnian sphere of action, and one comet passed through the Uranian sphere of action. The orbital elements of these 68 comets changed by no more than ±3 percent in a space of 6000 yr. As many as 206 comets passed close to Jupiter. We confirm Everhart’s conclusion that Jupiter can capture long-period comets with q = 4–6 AU and i < 9° into short-period orbits. We show that nearly parabolic comets cross the solar system mainly in the zone of terrestrial planets. No relationship of nearly parabolic comets and terrestrial planets was found for the epoch of the latest apparition of comets. Guliev’s conjecture about two trans-Plutonian planets is based on the illusory excess of cometary nodes at large heliocentric distances. The existence of cometary nodes at the solar system periphery turns out to be a solely geometrical effect.     

A Monte Carlo investigation of Jovian perturbations on short-period comet orbits

We present statistical distributions of Jovian perturbations on short-period comet orbits resulting from accurate numerical integrations. Our sample of 60, 000 cometary orbits with low inclinations and random orientations is characterized by perihelia between 0 and 7 AU and aphelia between 4 and 13 AU. The perturbations considered are those experienced because of Jupiter's gravitation per orbital revolution by the comets. Regularization and accurate step-length control in the numerical integration gives statistical results appreciably different from those computed by Rickman and Vaghi (1978). Their use of a crude method of integration led to erroneous results for close encounters. Strong asymmetries of the $\text{δ(}\text{1}\text{a}\text{)}$ distributions, in particular for the extreme tails, are observed for perihelion- or aphelion-tangent orbits. These orbits are also shown to experience the strongest energy perturbations on the average. Some results concerning the perturbations of Tisserand parameters are indicated. The perturbation distributions for the angular elements are described and discussed. The role of the minimum distance from Jupiter as an indicator of perturbations is investigated.     

Vaporization of comet nuclei: Light curves and life times

We have examined the effects of vaporization from the nucleus of a comet and show that a latitude dependence of vaporization can, in some cases, explain asymmetries in cometary light curves. We also find that a non-uniform distribution of solar radiation over a comet can considerably shorten the vaporization lifetime compared to the results normally obtained by assuming that the nuclear surface is isothermal.Independent of any latitude effects, comets with CO₂-dominated nuclei and with perihelion distances less than 0.5 AU have vaporization lifetimes less than or comparable to their dynamical ejection times. This may explain the observed deficit of comets with small perihelion distances. Similarly comets with CO₂-dominated nuclei and perihelia near Jupiter's orbit have vaporization lifetimes that are shorter than the time for capture into short-period orbits. We suggest, therefore, that at least some new comets are composed in large part of CO₂, while only H₂O-dominated comets, with lower vaporization rates, can survive to be captured into short-period orbits.     

Chaotic dynamics of planet‐encountering bodies

The dynamics of two families of minor inner solar system bodies that suffer frequent close encounters with the planets is analyzed. These families are: Jupiter family comets (JF comets) and Near Earth Asteroids (NEAs). The motion of these objects has been considered to be chaotic in a short time scale,and the close encounters are supposed to be the cause of the fast chaos. For a better understanding of the chaotic behavior we have computed Lyapunov Characteristic Exponents (LCEs) for all the observed members of both populations. LCEs are a quantitative measure of the exponential divergence of initially close orbits. We have observed that most members of the two families show a concentration of Lyapunov times (inverse of LCE) around 50–100yr. The concentration is more pronounced for JF comets than for NEAs, among which a lesser spread is observed for those that actually cross the Earth's orbit (mean perihelion distance q < 1.05 AU). It is also observed that a general correspondence exists between Lyapunov times and the time between consecutive encounters. A simple model is introduced to describe the basic characteristics of the dynamical evolution. This model considers an impulsive approach, where the particles evolve unperturbedly between encounters and suffer ‘kicks’ in semimajor axis at the encounters. It also reproduces successfully the short Lyapunov times observed in the numerical integrations and is able to estimate the dynamical lifetimes of comets during a stay in the Jupiter family in correspondence with previous estimates. It has been demonstrated with the model that the encounters with the largest effect on the exponential growth of the distance between initially nearby orbits are neither the infrequent deep encounters, nor the frequent and far ones; instead, the intermediate approaches have the most relevant contribution to the error growth. Such encounters are at a distance a few times the radius of the Hill's sphere of the planet (e.g. 3). An even simpler model allows us to get analytical estimates of the Lyapunov times in good agreement with the values coming from the model above and the numerical integrations. The predictability of the medium‐term evolution and the hazard posed to the Earth by those objects are analysed in the Discussion section. This revised version was published online in July 2006 with corrections to the Cover Date.     

Orbital evolution of long-period comets I. Trivariate Monte Carlo simulation

The mean orbital evolution of long-period comets for 16 representative initial orbits to short-period comets is calculated by a Monte Carlo method. First, trivariate perturbation distributions of barycentric Kepler energy, total angular momentum, and its z component in single encounters of comets with Jupiter are obtained numerically. Their characteristics are examined in detail and the distributions are found to be simple, symmetric, and easy to handle. Second, utilizing these distributions, we have done trivariate Monte Carlo simulations of the orbital evolution of long-period comets, with special emphasis on high-inclination orbits. About half of the 16 initial orbits are traced up to 5000 returns. For each of these orbits, the mean values of semimajor axis, perihelion distance, and inclination; their standard deviations, survival, and capture rates; as well as time scales of orbital evolution are calculated as functions of return number. Survival rates of the initial orbits with high inclination (～90°) and small perihelion distance (～1–2 AU) have been found to be only two or three times smaller than those of the main-source orbits of short-period comets established quantitatively by Everhart. The time scales of orbitsl evolution of the former, however, are nearly 10 times longer than the latter. There is a general trend that, for smaller perihelion distance, the survival efficiency becomes higher. The results of this paper should be considered a basis for a succeeding paper (Paper II) in which the physical lifetime of comets will be determined, and a comparison with the orbital data will be done.     

The young long-period comet family of Saturn

The distributions of long-period comets with respect to the minimum distance Δ between their orbits and the orbit of Saturn or Jupiter, constructed by Konopleva using data up to 1972, exhibit a sharp peak at Δ<0.5 au for the Saturnian family, while being fairly monotonic for Jupiter. Hence, in view of the appreciable eccentricity of Saturn's orbit and the rotation of its perihelion longitude with a period of 47 kyr, the conclusion was drawn by Drobyshevski that the objects belonging to this peak are young (10 kyr).
Similar distributions constructed using more recent data show less pronounced differences between one another. Analysis of the distributions for various epochs shows that the initially noted difference is due to observational selection, being inherent to brighter comets. Since on average the cometary activity fades with age, the conclusion that the Saturnian family comets, forming the peak at Δ<0.5 au, are young is all the more substantiated. The question concerning the origin of these comets, which in all likelihood were ejected over a period of a decade from deep inside the Saturnian sphere of influence , is still open. The only self-consistent hypothesis that we see now is that of their appearance as a result of an explosion of the electrolysed ice envelope of Titan. We encourage the development of other explanations.     

A former asteroidal planet as the origin of comets

The idea of a missing planet between Mars and Jupiter has been with us since the formulation of the Titius-Bode law. The discovery of the asteroid belt in that location led to speculation about a planetary breakup event. Both ideas remained conjectures until Ovenden's finding in 1972, from which it could be derived that the mass of the missing planet was about 90 Earth masses and that its breakup was astronomically recent. Apparently much of that mass was blown out of the solar system during the disruption of the planet. Because of the action of planetary perturbations, only two types of orbits of surviving fragments could remain at present-asteroid orbits and once-around very-long-period elliptical orbits. Objects in the latter type of orbit are known to exist-the very-long-period comets. A large number of these are on elliptical trajectories with periods of revolution of 5 million years; yet they are known to have made no more than one revolution in an orbit passing close to the Sun. By direct calculation it is possible to predict the distribution of the orbital elements of objects moving on long-period ellipses which might have originated in a breakup event in the asteroid belt 5 million years ago. The comet orbits have the predicted distribution in every case where a measure is possible. Some of the distribution anomalies, such as a bias in the directions of perihelion passage, are statistically strong and would be difficult to explain in any other uncontrived way. In addition, a relative deficiency of orbits with perihelia less than 1 AU indicates that the comets must have had small perihelion distances since their origin, rather than that they have been perturbed into small perihelion orbits from a distant “cloud” of comets by means of stellar encounters. The comet orbital data lead to the conclusion that all comets originated in a breakup event in the asteroid belt (5.5±0.6) × 10⁶ years ago. Asteroid and meteoritic evidence can now be interpreted in a way which not only is supportive but also provides fresh insights into understanding their physical, chemical, and dynamical properties. Particularily noteworthy are the young cosmic-ray exposure ages of meteorites, evidence of a previous high-temperature/pressure environment and of chemical differentiation of the parent body, and compositional similarities among comets, asteroids, and meteorites. Certain “explosion signatures” in asteroid orbital element distributions are likewise indicative. Tektites may also have originated in the same event; but if so, there are important implications regarding the absolute accuracy of certain geological dating methods. Little is known about possible planetary breakup mechanisms of the requisite type, though some speculations are offered. In any case, the asteroid belt is an existing fact; and the arguments presented here that a large planet did disintegrate 5 million years ago must be judged on their merits, even in the absence of a suitable theory of planetary explosions.     

Dynamical evolution of NEAs: Close encounters, secular perturbations and resonances

We discuss the main mechanisms affecting the dynamical evolution of Near-Earth Asteroids (NEAs) by analyzing the results of three numerical integrations over 1 Myr of the NEA (4179) Toutatis. In the first integration the only perturbing planet is the Earth. So the evolution is dominated by close encounters and looks like a random walk in semimajor axis and a correlated random walk in eccentricity, keeping almost constant the perihelion distance and the Tisserand invariant. In the second integration Jupiter and Saturn are present instead of the Earth, and the 3/1 (mean motion) and v ₆ (secular) resonances substantially change the eccentricity but not the semimajor axis. The third, most realistic, integration including all the three planets together shows a complex interplay of effects, with close encounters switching the orbit between different resonant states and no approximate conservation of the Tisserand invariant. This shows that simplified 3-body or 4-body models cannot be used to predict the typical evolution patterns and time scales of NEAs, and in particular that resonances provide some “fast-track” dynamical routes from low-eccentricity to very eccentric, planet-crossing orbits.