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.     

Quasi-Tangency Points on the Orbits of a Small Body and a Planet at the Low-Velocity Encounter

We propose a method for selecting a low-velocity encounter of a small body with a planet from the evolution of the orbital elements. Polar orbital coordinates of the quasi-tangency point on the orbit of a small body are determined. Rectangular heliocentric coordinates of the quasi-tangency point on the orbit of a planet are determined. An algorithm to search for low-velocity encounters in the evolution of the orbital elements of small bodies is described. The low-velocity encounter of comet 39P/Oterma with Jupiter is considered as an example.     

Dust Trails of 8P/Tuttle and the Unusual Outbursts of the Ursid Shower

We calculate the position of dust trails from comet 8P/Tuttle, in an effort to explain unusual Ursid meteor shower outbursts that were seen when the comet was near aphelion. Comet 8P/Tuttle is a Halley-type comet in a 13.6-year orbit, passing just outside of Earth's orbit. We find that the meteoroids tend to be trapped in the 12:14 mean motion resonance with Jupiter, while the comet librates in a slightly shorter period orbit around the 13:15 resonance. It takes 6 centuries to decrease the perihelion of the meteoroid orbits enough to intersect Earth's orbit, during which time the meteoroids and comet separate in mean anomaly by 6 years, thus explaining the 6-year lag between the comet's return and Ursid outbursts. The resonances also prevent dispersion along the comet orbit and limit viewing to only one year in each return. We identified past dust trail encounters with dust trails from 1392 (Dec. 1945) and 1378 (Dec. 1986) and predicted another outburst on 2000 December 22 at around 7:29 and 8:35 UT, respectively, from dust trails dating to the 1405 and 1392 returns. This event was observed from California using video and photographic techniques. At the same time, five Global-MS-Net stations in Finland, Japan, and Belgium counted meteors using forward meteor scatter. The outburst peaked at 8:06±07 UT, December 22, at zenith hourly rate ∼90 per hour, and the Ursid rates were above half peak intensity during 4.2 h. We find that most Ursid orbits do scatter around the anticipated positions, confirming the link with comet 8P/Tuttle and the epoch of ejection. The 1405 and 1392 dust trails appear to have contributed similar amounts to the activity profile. Some orbits provide a hint of much older debris being present as well. This work is the strongest evidence yet for the relevance of mean motion resonances in Halley-type comet dust trail evolution.     

Relativistic Precession of Elliptical Orbits of a Circumbinary Planet can be Cancelled

In publications presenting analytical results on the non-coplanar motion of a circumbinary planet it was shown that the unperturbed elliptical orbit of the planet undergoes simultaneously two kinds of the precession: the precession of the orbital plane and the precession of the orbit in its own plane. It is also well-known that there is also the relativistic precession of the planetary orbit in its own plane. In the present paper we study a combined effect of the all of the above precessions. For the general case, where the planetary orbit is not coplanar with the stars orbits, we analyzed the dependence of the critical inclination angle i_c, at which the precession of the planetary orbit in its own plane vanishes, on the angular momentum L of the planet. We showed that the larger the angular momentum, the smaller the critical inclination angle becomes. We presented the analytical result for i_c(L) and calculated the value of L, for which the critical inclination value becomes zero. For the particular case, where the planetary orbit is not coplanar with the stars orbits, we demonstrated analytically that at a certain value of the angular momentum of the planet, the elliptical orbit of the planet would become stationary: no precession. In other words, at this value of the angular momentum, the relativistic precession of the planetary orbit and its precession, caused by the fact that the planet revolves around a binary (rather than single) star, cancel each other out. This is a counterintuitive result.     

Precession of the Orbit of a Planet around Stars Revolving along: Low-Eccentricity Orbits in a Binary System: Analytical Solution

In our previous paper (hereafter, paper I) we presented analytical results on the non-planar motion of a planet around a binary star for the cases of the circular orbits of the components of the binary. We found that the orbital plane of the planet (the plane containing the “unperturbed” elliptical orbit of the planet), in addition to precessing about the angular momentum of the binary, undergoes simultaneously the precession within the orbital plane. We demonstrated that the analytically calculated frequency of this additional precession is not the same as the frequency of the precession of the orbital plane about the angular momentum of the binary, though the frequencies of both precessions are of the same order of magnitude. In the present paper we extend the analytical results from paper I by relaxing the assumption that the binary is circular – by allowing for a relatively small eccentricity ε of the stars orbits in the binary. We obtain an additional, ε-dependent term in the effective potential for the motion of the planet. By analytical calculations we demonstrate that in the particular case of the planar geometry (where the planetary orbit is in the plane of the stars orbits), it leads to an additional contribution to the frequency of the precession of the planetary orbit. We show that this additional, ε-dependent contribution to the precession frequency of the planetary orbit can reach the same order of magnitude as the primary, ε-independent contribution to the precession frequency. Besides, we also obtain analytical results for another type of the non-planar configuration corresponding to the linear oscillatory motion of the planet along the axis of the symmetry of the circular orbits of the stars. We show that as the absolute value of the energy increases, the period of the oscillations decreases.     

Development of a comet as it pursues its orbit

The properties of cometary dust-swarms in almost parabolic long-period orbits are examined. In general their self-gravitation is stronger than the solar disruptive influence for all except the relatively small part of the orbit within planetary distances during which the sun dominates by so great a factor that the individual particles of the swarm pursue independent orbits apart from the possibility of collisions between them. At aphelion the internal relative speeds of particles are only a few centimetres per second, but at and near perihelion they may rise to the order of a kilometre per second. For purely dynamical reasons the extent of the swarm in directions perpendicular to the orbital motion will strongly diminish as perihelion is approached and correspondingly increase thereafter, while the dimension along the orbit will change in direct proportion to the orbital velocity. Every particle must cross through the median orbital plane near perihelion, and collisions between a proportion of the particles will occur at speeds capable of fragmenting them into myriads of smaller dustparticles, also heating them at and near the colliding elements of their surfaces. Increase of reflected sunlight will result and also release of material in gaseous form by solar plus collisional heating. Sufficiently finely divided dust particles will be driven out of the comet by radiation-pressure to form a dust-tail, while suitable gaseous compounds if present will be driven out to give a gas-tail. For Sungrazing comets, complete gasification must occur at and near perihelion, and very considerable extension along the orbit. Such comets would recondense to small solid particles on receding again from the Sun. The effect of passage of the solar system through interstellar gas-clouds is shown to be capable of substantially affecting the angular momentum of a comet about the Sun, thus accounting for the existence of comets with high values of perihelion-distance. This same process would enable cometary particles to adsorb interstellar gases at their surfaces and regenerate their gas-content. The mass-loss by a comet at each return strongly indicates, that comets cannot have originated at the same time as the planets, a result further supported by the rapid expulsion of entire comets through purely dynamical action of the planets. That the quiescent structure of comets consists of a vast widely spaced swarm of minute dust-particles receives circumstantial support from the highly varied and peculiar properties long since recorded for numerous comets. These properties exhibit such erratic diversity as to make clear that only a theory involving considerable range of essential parameters can be capable of accounting for them adequately.     

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.     

The time evolution of periodic comet Russel 3, 1983i

The orbit of the newly-discovered comet P/Russel 3 (1983i) is integrated backward and forward to study its evolution within time. Two close approaches with Jupiter in the limited calculation are found: one on December, 1941 and the other on May, 2024. Since its aphelion lies near the Jupiter orbit the orbit of P/Russel 3 appears to be rather unstable.     

Exploiting Unstable Periodic Orbits of a Chaotic Invariant Set for Spacecraft Control

The purpose of this work is to show that chaos control techniques (OGY, in special) can be used to efficiently keep a spacecraft around another body performing elaborate orbits. We consider a satellite and a spacecraft moving initially in coplanar and circular orbits, with slightly different radii, around a heavy central planet. The spacecraft, which is the inner body, has a slightly larger angular velocity than the satellite so that, after some time, they eventually go to a situation in which the distance between them becomes sufficiently small, so that they start to interact with one another. This situation is called as an encounter. In previous work we have shown that this scenario is a typical situation of a chaotic scattering for some well-defined range of parameters. Considering this scenario, we first show how it is possible to find the unstable periodic orbits that are located in the chaotic invariant set. From the set of unstable periodic orbits, we select the ones that can be combined to provide the desired elaborate orbit. Then, chaos control technique based on the OGY method is used to keep the spacecraft in the desired orbit. Finally, we analyze the results and make considerations regarding a realistic scenario of space exploration.     

The geometry of impacts on a synchronous planetary satellite

We consider a satellite in a circular orbit about a planet that, in turn, is in a circular orbit about the Sun; we further assume that the plane of the planetocentric orbit of the satellite is the same as that of the heliocentric orbit of the planet. The pair planet–satellite is encountered by a population of small bodies on planet-crossing, inclined orbits. With this setup, and using the extension of Öpik’s theory by Valsecchi et al. (Astron Astrophys 408:1179–1196, 2003), we analytically compute the velocity, the elongation from the apex and the impact point coordinates of the bodies impacting the satellite, as simple functions of the heliocentric orbital elements of the impactor and of the longitude of the satellite at impact. The relationships so derived are of interest for satellites in synchronous rotation, since they can shed light on the degree of apex–antapex cratering asymmetry that some of these satellites show. We test these relationships on two different subsets of the known population of Near Earth Asteroids.     

Nonlinear dynamical analysis for displaced orbits above a planet

Nonlinear dynamical analysis and the control problem for a displaced orbit above a planet are discussed. It is indicated that there are two equilibria for the system, one hyperbolic (saddle) and one elliptic (center), except for the degenerate h _z ^max, a saddle-node bifurcation point. Motions near the equilibria for the nonresonance case are investigated by means of the Birkhoff normal form and dynamical system techniques. The Kolmogorov–Arnold–Moser (KAM) torus filled with quasiperiodic trajectories is measured in the τ ₁ and τ ₂ directions, and a rough algorithm for calculating τ ₁ and τ ₂ is proposed. A general iterative algorithm to generate periodic Lyapunov orbits is also presented. Transitions in the neck region are demonstrated, respectively, in the nonresonance, resonance, and degradation cases. One of the important contributions of the paper is to derive necessary and sufficiency conditions for stability of the motion near the equilibria. Another contribution is to demonstrate numerically that the critical KAM torus of nontransition is filled with the (1,1)-homoclinic orbits of the Lyapunov orbit.     

Frequency of hyperbolic and interstellar meteoroids

Hyperbolic meteor orbits from the catalog of 64,650 meteors observed by the multistation video meteor network located in Japan (SonotaCo 2009) have been investigated with the aim of determining the relation between the frequency of hyperbolic and interstellar meteors. The proportion of hyperbolic meteors in the data decreased significantly (from 11.58% to 3.28%) after a selection of quality orbits, which shows its dependence on the quality of observations. Initially, the hyperbolic orbits were searched for meteors unbound due to planetary perturbation. It was determined that 22 meteors from the 7489 hyperbolic orbits in the catalog (and 2 from the selection of the orbits with the highest quality) had had a close encounter with a planet, none of which, however, produced essential changes in their orbits. Similarly, the fraction of hyperbolic orbits in the data, which could be hyperbolic by reason of a meteor's interstellar origin, was determined to be at most 3.9 × 10^?2. From the statistical point of view, the vast majority of hyperbolic meteors in the database have definitely been caused by inaccuracy in the velocity determination. This fact does not necessarily assume great measurement errors, since, especially near the parabolic limit, a small error in the value of the heliocentric velocity of a meteor can create an artificial hyperbolic orbit that does not really exist. The results show that the remaining 96% of meteoroids with apparent hyperbolic orbits belong to the solar system meteoroid population. This is also supported by their high abundance (about 50%) among the meteor showers.     

Families of periodic orbits in the general three-body problem for the Sun-Jupiter-Saturn mass-ratio and their stability

Several families of planar planetary-type periodic orbits in the general three-body problem, in a rotating frame of reference, for the Sun-Jupiter-Saturn mass-ratio are found and their stability is studied. It is found that the configuration in which the orbit of the smaller planet is inside the orbit of the larger planet is, in general, more stable.We also develop a method to study the stability of a planar periodic motion with respect to vertical perturbations. Planetary periodic orbits with the orbits of the two planets not close to each other are found to be vertically stable. There are several periodic orbits that are stable in the plane but vertically unstable and vice versa. It is also shown that a vertical critical orbit in the plane can generate a monoparametric family of three-dimensional periodic orbits.     

The recent capture of the periodic comet P/Scotti (2000 Y3)

Numerical integrations of 99 orbits centered on that of comet P/Scotti (P/2000 Y3), and of the nominal orbit, were made 4000 days backwards in time, and 73000 days into the future. The integrations show that this comet has been transferred into its present orbit as recently as 1998. The future orbital evolution indicates a stable period for almost 150 years, when another close encounter with Jupiter may lead to further drastic changes of the present orbit.     

Chaotic zone boundary for low free eccentricity particles near an eccentric planet

We consider particles with low free or proper eccentricity that are orbiting near planets on eccentric orbits. Through collisionless particle integration, we numerically find the location of the boundary of the chaotic zone in the planet's corotation region. We find that the distance in semimajor axis between the planet and boundary depends on the planet mass to the 2/7 power and is independent of the planet eccentricity, at least for planet eccentricities below 0.3. Our integrations reveal a similarity between the dynamics of particles at zero eccentricity near a planet in a circular orbit and with zero free eccentricity particles near an eccentric planet. The 2/7th law has been previously explained by estimating the semimajor at which the first-order mean motion resonances are large enough to overlap. Orbital dynamics near an eccentric planet could differ due to first-order corotation resonances that have strength proportional to the planet's eccentricity. However, we find that the corotation resonance width at low free eccentricity is small; also the first-order resonance width at zero free eccentricity is the same as that for a zero-eccentricity particle near a planet in a circular orbit. This accounts for insensitivity of the chaotic zone width to planet eccentricity. Particles at zero free eccentricity near an eccentric planet have similar dynamics to those at zero eccentricity near a planet in a circular orbit.     

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.     

The nature of the orbits of charged dust injected into the Jovian magnetosphere during the tidal break-up of comet Shoemaker-Levy 9

Assuming that the spin and magnetic axis of Jupiter are strictly parallel and that the grain charge remains constant we have derived two integrals of the 3D equations of motion of charged dust grains moving within the co-rotating regions of the Jovian magnetosphere taking into account both planetary gravitation and magnetospheric rotation. We then apply this model to study the fate of fine dust injected into the Jovian magnetosphere as a result of the tidal disruption of comet Shoemaker-Levy 9 during its first encounter with Jupiter in July 1992. This analysis, which uses the integrals of the equation of motion rather than the equation of motion itself as was done by Horanyi (1994), does not allow us to calculate the orbits or the orbital evolution of the grains. But it does allow us to construct the spatial regions to which the grains are confined, at least initially before evolutionary effects take over. We have chosen three points along the path of the disintegrating comet for the injection of dust and used two values for the uncertain floating potential of the dust in the inner Jovian magnetosphere. Grains can have three different fates, depending on their size, their acquired potential and their point of injection. While the smallest grains are quickly lost by collision with the planet at high latitudes independent of the sign of their charge, those in an intermediate but narrow size range, injected near the equatorial plane can be trapped in a region close to it, this being true for both positive and negative grains. While somewhat larger positive grains may be initially ejected outward by the co-rotational electric force, similar negative grains, pulled inward by this force collide with the planet at low latitudes. In all cases the largest grains, which are dominated by planetary gravity, initially escape from the inner magnetosphere by following in the path of the comet.Using a detailed time dependent numerical calculation of the jovicentric orbits of the charged dust debris of the disintegrating comet, that allows for variation in the grain potential, while also allowing for perturbations of the grain orbits due to solar radiation pressure and solar gravity Horanyi (1994) found that grains in the size range (1.5m <a < 2.5m) which initially make large excursions from the planet, will eventually form a ring in the radial range 4.5R _J <r < 6R _J . Our present analytical calculation cannot make such a prediction about the evolutionary fate of the dust debris. It can, however, estimate the size of the grains that are initially confined to regions near the points of injection, before evolutionary effects become important.     

Interplanetary dust particles near Jupiter

We calculate the expected counting rate of a flat micrometeoroid detector of finite sensitivity passing in hyperbolic orbit near a planet. We assume that the distribution of particle sizes, s, can be expressed as a power law spectrum of index p, i.e. dN(s) = Cs^?pds, and also that the particles encounter the sphere of influence of the planet with a certain speed v_∞. The results of the calculations are then compared with the results returned by Pioneer 10 in its flyby of Jupiter. The observed increase in impact rate near Jupiter can be completely explained in terms of gravitational “focusing” of particles which are in heliocentric orbits; i.e., they are not in orbit about Jupiter. The absolute concentration of particles near the orbit of Jupiter is of the same order as at 1 AU: the exact ratio being a function of particle speed and spectral index. Data from one flyby are insufficient to determine a unique value for both the spectral index, p, and the particle velocity, v_∞, but limits can be set. For reasonable encounter speeds (corresponding to eccentricities and inclinations of dust particles experienced near the Earth), the particles near Jupiter are characterized by a spectrum of index p ～ 3. The spectral index which best fits the data increases with increasing encounter speeds.     

Orbital Distribution Arbitrarily Close to the Homothetic Equilateral Triple Collision in the Free‐Fall Three‐Body Problem with Equal Masses

The existence of escape and nonescape orbits arbitrarily close to the homothetic equilateral triplecollision orbit is considered analytically in the threebody problem with zero initial velocities and equal masses. It is proved that escape orbits in the initial condition space are distributed around three kinds of isosceles orbits. It is also proved that nonescape orbits are distributed in between the escape orbits where different particles escape. In order to show this, it is proved that the homotheticequilateral orbit is isolated from other triplecollision orbits as far as the collision at the first triple encounter is concerned. Moreover, the escape criterion is formulated in the planarisosceles problem and translated into the words of regularizing variables. The result obtained by us explains the orbital structure numerically.     

Application of the restricted hyperbolic three-body problem to a star-sun-comet system

The equations of motion for a third body of small mass are developed in the problem where the two primary bodies are in hyperbolic orbits about each other. The equations are applied to a hypothetical star-sun-comet system to determine the effect of the stellar encounter on the orbit of the comet.This paper is part of a doctoral thesis completed at the University of Illinois at Urbana-Champaign.