Study of a Particular Model of Encounter Between a Comet and a Planet

The objective of this paper is to develop a simple model of an encounter between a comet and a planet, with a subsequent capture or an escape, and to study the potential consequences. The hypothetical scenario is as follows: a comet with a conic orbit meets close to one of its vertices (located near the ecliptic plane), a jovian planet, and transforms its orbit. There are two hypotheses which are made for the shock: this vertex becomes one of the final vertices and the orbital plane of the comet is unchanged during the encounter as it was the case for Brooks 2 in 1886. In this model, it was able to find an equation which was then used to obtain the pre‐ and post‐encounter orbits elements and the kind of orbit (ellipse, hyperbola, parabola) with respect to the initial inclination. The numerical experiments with the observed comets often provide pre‐encounter orbits with an aphelion point located near another jovian planet farther from the Sun, and so on with sometimes several planets, or with an aphelion point located beyond the Pluto orbit. This revised version was published online in July 2006 with corrections to the Cover Date.     

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 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.     

Qualitative features of the evolution of some polar satellite orbits

Two special cases of the problem of the secular perturbations in the orbital elements of a satellite with a negligible mass produced by the joint influence of the oblateness of the central planet and the attraction by its most massive (or main) satellites and the Sun are considered. These cases are among the integrable ones in the general nonintegrable evolution problem. The first case is realized when the plane of the satellite orbit and the rotation axis of the planet lie in its orbital plane. The second case is realized when the plane of the satellite orbit is orthogonal to the line of intersection between the equatorial and orbital planes of the planet. The corresponding particular solutions correspond to those polar satellite orbits for which the main qualitative features of the evolution of the eccentricity and pericenter argument are described here. Families of integral curves have been constructed in the phase plane of these elements for the satellite systems of Jupiter, Saturn, and Uranus.     

Tidal evolution of Dysnomia, satellite of the dwarf planet Eris

The past tidal evolution of the satellite Dysnomia of the dwarf planet Eris can be inferred from the current physical and orbital properties of the system. Preliminary considerations, which assumed a circular orbit for the satellite, suggested that the satellite formed close to the planet, perhaps as a result of a giant impact, and that it is thus unlikely that smaller satellites lie further out. However, if the satellite's orbit is eccentric, even if the eccentricity is very small, a qualitatively different past tidal evolution may be indicated. Early in the Solar System's history, the satellite may have been on a highly eccentric orbit much farther from the planet than it is now, suggestive of a capture origin. Additional satellites farther out cannot be ruled out.     

Generation of a secular system in the analytical theory for the motion of the moon

In order to generate an analytical theory of the motion of the Moon by considering planetary perturbations, a procedure of general planetary theory (GPT) is used. In this case, the Moon is considered as an addition planet to the eight principal planets. Therefore, according to the GPT procedure, the theory of the Moon’s orbital motion can be presented in the form of series with respect to the evolution of eccentric and oblique variables with quasi-periodic coefficients, which are the functions of mean longitudes for principal planets and the Moon. The relationship between evolution variables and the time is determined by a trigonometric solution for the independent secular system that describes the secular motion of a perigee and the Moon node by considering secular planetary inequalities. Principal planetary coordinates required for generating the theory of the motion of the Moon includes only Keplerian terms, the intermediate orbit, and the linear theory with respect to eccentricities and inclinations in the first order relative to the masses. All analytical calculations are performed by means of the specialized echeloned Poisson Series Processor EPSP.     

A numerical-analytical method for studying the orbital evolution of distant planetary satellites

We describe an approximate numerical-analytical method for calculating the perturbations of the elements of distant satellite orbits. The model for the motion of a distant satellite includes the solar attraction and the eccentricity and ecliptic inclination of the orbit of the central planet. In addition, we take into account the variations in planetary orbital elements with time due to secular perturbations. Our work is based on Zeipel’s method for constructing the canonical transformations that relate osculating satellite orbital elements to the mean ones. The corresponding transformation of the Hamiltonian is used to construct an evolution system of equations for mean elements. The numerical solution of this system free from rapidly oscillating functions and the inverse transformation from the mean to osculating elements allows the evolution of distant satellite orbits to be studied on long time scales on the order of several hundred or thousand satellite orbital periods.     

Eccentricity evolution of giant planet orbits due to circumstellar disk torques

The extrasolar planets discovered to date possess unexpected orbital elements. Most orbit their host stars with larger eccentricities and smaller semi-major axes than similarly sized planets in our own Solar System do. It is generally agreed that the interaction between giant planets and circumstellar disks (Type II migration) drives these planets inward to small radii, but the effect of these same disks on orbital eccentricity, ?, is controversial. Several recent analytic calculations suggest that disk-planet interactions can excite eccentricity, while numerical studies generally produce eccentricity damping. This paper addresses this controversy using a quasi-analytic approach, drawing on several preceding analytic studies. This work refines the current treatment of eccentricity evolution by removing several approximations from the calculation of disk torques. We encounter neither uniform damping nor uniform excitation of orbital eccentricity, but rather a function d?/dt that varies in both sign and magnitude depending on eccentricity and other Solar System properties. Most significantly, we find that for every combination of disk and planet properties investigated herein, corotation torques produce negative values of d?/dt for some range in ? within the interval [0.1, 0.5]. If corotation torques are saturated, this region of eccentricity damping disappears, and excitation occurs on a short timescale of less than 0.08 Myr. Thus, our study does not produce eccentricity excitation on a timescale of a few Myr—we obtain either eccentricity excitation on a short time scale, or eccentricity damping on a longer time scale. Finally, we discuss the implications of this result for producing the observed range in extrasolar planet eccentricity.     

A new method for the determination of an intermediate orbit using three positions of a small body on the celestial sphere

We propose a new method for the determination of the preliminary orbit of a small celestial body using three pairs of its angular coordinates in three moments of time. The method is based on the use of the intermediate orbit we constructed earlier using three position vectors and the corresponding time moments. This intermediate orbit accounts for the main part of the perturbations of the motion of the body under study. We compare the results obtained by the classical Lagrange-Gauss method, Herrick-Gibbs method, generalized Herrick-Gibbs method, and the new method by the examples of the determination of the orbit of the small planet 1566 Icarus. The comparison showed that the new method is a highly efficient tool for the study of perturbed motion. It is especially efficient when applied to high-precision observational data covering short arcs of the orbit.     

Cartography of the <Emphasis Type="Italic">b</Emphasis>-plane of a close encounter I: semimajor axes of post-encounter orbits

Close planetary encounters play an important role in the evolution of the orbits of small Solar system bodies and are usually studied with the help of numerical integrations. Here we study close encounters in the framework of an analytic theory, focusing on the so-called b-plane, which is the plane centred on the planet and perpendicular to the planetocentric velocity at infinity of the small body. As shown in previous papers, it is possible to identify the initial conditions on the b-plane that lead to post-encounter orbits of given semimajor axis. In this paper we exploit analytical relationships between b-plane coordinates and pre-encounter orbital elements and compute the probability of transition to these post-encounter states, and numerically check the validity of the analytic approach.     

The effect on the Edgeworth–Kuiper Belt of a large distant tenth planet

We investigate the orbital evolution of both real and hypothetical Edgeworth–Kuiper Objects in order to determine whether any conclusions can be drawn regarding the existence, or otherwise, of the tenth planet postulated by Murray. We find no qualitative difference in the orbital evolution, and so conclude that the hypothetical planet has been placed on an orbit at such a large heliocentric distance that no evidence for the existence, or non-existence, can be found from a study of the known Edgeworth–Kuiper Objects.     

Eccentric Extrasolar Planets: The Jumping Jupiter Model

Most extrasolar planets discovered to date are more massive than Jupiter, in surprisingly small orbits (semimajor axes less than 3 AU). Many of these have significant orbital eccentricities. Such orbits may be the product of dynamical interactions in multiplanet systems. We examine outcomes of such evolution in systems of three Jupiter-mass planets around a solar-mass star by integration of their orbits in three dimensions. Such systems are unstable for a broad range of initial conditions, with mutual perturbations leading to crossing orbits and close encounters. The time scale for instability to develop depends on the initial orbital spacing; some configurations become chaotic after delays exceeding 10⁸ y. The most common outcome of gravitational scattering by close encounters is hyperbolic ejection of one planet. Of the two survivors, one is moved closer to the star and the other is left in a distant orbit; for systems with equal-mass planets, there is no correlation between initial and final orbital positions. Both survivors may have significant eccentricities, and the mutual inclination of their orbits can be large. The inner survivor's semimajor axis is usually about half that of the innermost starting orbit. Gravitational scattering alone cannot produce the observed excess of “hot Jupiters” in close circular orbits. However, those scattered planets with large eccentricities and small periastron distances may become circularized if tidal dissipation is effective. Most stars with a massive planet in an eccentric orbit should have at least one additional planet of comparable mass in a more distant orbit.     

A method of intermediate orbit determination based on four positions of a minor body on the celestial sphere

A new method of computing the preliminary orbit of a celestial body based on four pairs of angle measurements has been suggested. The method makes use of preliminary orbit previously constructed by the author based on two position vectors and a corresponding time interval, taking into account the main part of the perturbations in the motion of the body under study. Using the example of constructing the orbit of the minor planet 1383 Limburgia, the results obtained using a four-position procedure of the Gaussian type based on the solution of a two-body problem have been compared with those of the new method. The comparison showed the new method to be highly efficient for perturbed motion studies. It is especially advantageous in the case of high-accuracy observation data on small orbital arcs.     

Perturbation of a satellite orbiter by the oblateness of the primary planet

The perturbation of an orbiter around a large satellite of a giant planet (Jupiter, Saturn, Uranus or Neptune) produced by the oblateness of the planet is investigated. The perturbing force of theJ ₂-term (general case) and theJ ₄-term (special case of small eccentricity and inclination) is expanded in an appropriate form and the main term and the parallactic term are given explicitly. The variations of the orbital elements are derived using the stroboscopic method. An example shows that the perturbation of the orbit cannot be neglected.     

Evolution of the obliquities of the giant planets in encounters during migration

Tsiganis et al. [Tsiganis, K., Gomes, R., Morbidelli, A., Levison, H.F., 2005. Nature 435, 459-461] have proposed that the current orbital architecture of the outer Solar System could have been established if it was initially compact and Jupiter and Saturn crossed the 2:1 orbital resonance by divergent migration. The crossing led to close encounters among the giant planets, but the orbital eccentricities and inclinations were damped to their current values by interactions with planetesimals. Brunini [Brunini, A., 2006. Nature 440, 1163-1165] has presented widely publicized numerical results showing that the close encounters led to the current obliquities of the giant planets. We present a simple analytic argument which shows that the change in the spin direction of a planet relative to an inertial frame during an encounter between the planets is very small and that the change in the obliquity (which is measured from the orbit normal) is due to the change in the orbital inclination. Since the inclinations are damped by planetesimal interactions on timescales much shorter than the timescales on which the spins precess due to the torques from the Sun, especially for Uranus and Neptune, the obliquities should return to small values if they are small before the encounters. We have performed simulations using the symplectic integrator SyMBA, modified to include spin evolution due to the torques from the Sun and mutual planetary interactions. Our numerical results are consistent with the analytic argument for no significant remnant obliquities.     

Influence of fast interstellar gas flow on the dynamics of dust grains

The orbital evolution of a dust particle under the action of a fast interstellar gas flow is investigated. The secular time derivatives of Keplerian orbital elements and the radial, transversal, and normal components of the gas flow velocity vector at the pericentre of the particle’s orbit are derived. The secular time derivatives of the semi-major axis, eccentricity, and of the radial, transversal, and normal components of the gas flow velocity vector at the pericentre of the particle’s orbit constitute a system of equations that determines the evolution of the particle’s orbit in space with respect to the gas flow velocity vector. This system of differential equations can be easily solved analytically. From the solution of the system we found the evolution of the Keplerian orbital elements in the special case when the orbital elements are determined with respect to a plane perpendicular to the gas flow velocity vector. Transformation of the Keplerian orbital elements determined for this special case into orbital elements determined with respect to an arbitrary oriented plane is presented. The orbital elements of the dust particle change periodically with a constant oscillation period or remain constant. Planar, perpendicular and stationary solutions are discussed. The applicability of this solution in the Solar System is also investigated. We consider icy particles with radii from 1 to 10 μm. The presented solution is valid for these particles in orbits with semi-major axes from 200 to 3000 AU and eccentricities smaller than 0.8, approximately. The oscillation periods for these orbits range from 10⁵ to 2 × 10⁶ years, approximately.     

An Asteroid in a Earth-like Orbit

The dynamical evolution of an asteroid with orbital elements strikingly similar to the Earth is analysed. The object, 1991 VG, was discovered by the Spacewatch telescope during a particular encounter with the planet. 1991 VG experienced a temporary satellite capture by the Earth, a phenomena that is recurrent in its dynamical history. The possible origin of this puzzling object is discussed, including the suggestion that 1991 VG could be a piece of lunar ejecta after a great impact. This revised version was published online in July 2006 with corrections to the Cover Date.     

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)     

A new analysis of the GJ581 extrasolar planetary system

We have done a new analysis of the available observations of the GJ581 exoplanetary system. Today this system is controversial due to choices that can be done in the orbital determination. The main ones are the occurrence of aliases and the additional bodies??the planets f and g??announced in Vogt et?al. (Astrophys J 723:954?C965, 2010). Any dynamical study of exoplanets requires the good knowledge of the orbital elements and the investigations involving the planet g are particularly interesting, since this body would lie in the habitable zone (HZ) of the star GJ581. This region, for this system, is very attractive of the dynamical point of view due to several resonances of two and three bodies present there. In this work, we investigate the conditions under which the planet g may exist. We stress the fact that the planet g is intimately related with the orbital elements of the planet d; more precisely, we conclude that it is not possible to disconnect its existence from the determination of the eccentricity of the planet d. Concerning the planet f, we have found one solution with period ??450?days, but we are judicious about any affirmation concerning this body because its signal is in the threshold of detection and the high period is in a spectral region where the occurrence of aliases is very common. Besides, we outline some dynamical features of the HZ with the dynamical map and point out the role played by some resonances laying there.     

On the Coplanar Integrable Case of the Twice-Averaged Hill Problem with Central Body Oblateness

The twice-averaged Hill problem with the oblateness of the central planet is considered in the case where its equatorial plane coincides with the plane of its orbital motion relative to the perturbing body. A qualitative study of this so-called coplanar integrable case was begun by Y. Kozai in 1963 and continued by M.L. Lidov and M.V. Yarskaya in 1974. However, no rigorous analytical solution of the problem can be obtained due to the complexity of the integrals. In this paper we obtain some quantitative evolution characteristics and propose an approximate constructive-analytical solution of the evolution system in the form of explicit time dependences of satellite orbit elements. The methodical accuracy has been estimated for several orbits of artificial lunar satellites by comparison with the numerical solution of the evolution system.