On linking coefficient of two Keplerian orbits

We define a function of the set of pairs of Keplerian ellipses so that the sign of the function will be a topological invariant of their configuration. The sign is negative if and only if the related ellipses are linked. Two modifications of the coefficient which are more reliable in the case of closed to coplanar orbits are proposed. Explicit formulae representing the linking coefficients as functions of orbital elements are deduced. Extension in the case of unbounded orbits is obtained. We suggest different ways to use these coefficients for determining intersections of pairs of osculating Keplerian orbits. If we study dynamical behaviour of geometric configuration of pairs of Keplerian orbits, we can fix the moments of their intersections. These moments correspond exactly to the vanishing of linking coefficients. This revised version was published online in July 2006 with corrections to the Cover Date.     

On stellar encounters and their effect on cometary orbits in the Oort cloud

We systematically investigate the encounters between the Sun and neighbouring stars and their effects on cometary orbits in the Oort cloud, including the intrinsic one with the star Gl 710 (HIP 89 825), with some implications to stellar and cometary dynamics. Our approach is principally based on the combination of a Keplerian‐rectilinear model of stellar passages and the Hipparcos Catalogue (ESA 1997). Beyond the parameters of encounter, we pay particular attention to the observational errors in parallaxes and stellar velocities, and their propagation in time. Moreover, as a special case of this problem, we consider the collision probability of a star passing very closely to the Sun, taking also into account the mutual gravitational attraction between the stars. In the part dealing with the influence of stellar encounters on the orbital elements of Oort cloud comets, we derive new simple formulae calculating the changes in the cometary orbital elements, expressed as functions of the Jeans impulse formula. These expressions are then applied to calculate numerical values of the element changes caused by close encounters of neighbouring stars with some model comets in the Oort cloud. Moreover, the general condition for an ejection of comets from the cloud effected by a single encounter is derived and discussed.     

Galactic orbits of Hipparcos stars: Classification of stars

The Galactic orbits of 27 440 stars of all classes with accurate coordinates and parallaxes of more than 3 mas from the Hipparcos catalogue, proper motions from the Tycho-2 catalogue, and radial velocities from the Pulkovo Compilation of Radial Velocities (PCRV) are analyzed. The sample obtained is much more representative than the Geneva-Copenhagen survey and other studies of Galactic orbits in the solar neighborhood. An estimation of the influence of systematic errors in the velocities on orbital parameters shows that the errors of the proper motions due to the duplicity of stars are tangible only in the statistics of orbital parameters for very small samples, while the errors of the radial velocities are noticeable in the statistics of orbital parameters for halo stars. Therefore, previous studies of halo orbits may be erroneous. The distribution of stars in selection-free regions of the multidimensional space of orbital parameters, dereddened colors, and absolute magnitudes is considered. Owing to the large number of stars and the high accuracy of PCRV radial velocities, nonuniformities of this distribution (apart from the well-known dynamical streams) have been found. Stars with their peri- and apogalacticons in the disk, perigalacticons in the bulge and apogalacticons in the disk, perigalacticons in the bulge and apogalacticons in the halo, and perigalacticons in the disk and apogalacticons in the halo have been identified. Thus, the bulge and the halo are inhomogeneous structures, each consisting of at least two populations. The radius of the bulge has been determined: 2 kpc.     

Search for kinematic siblings of the sun based on data from the XHIP catalog

From the XHIP catalogue, we have selected 1872 F-G-K stars with relative parallax measurement errors <20% and absolute values of their space velocities relative to the Sun <15 km s^?1. For all these stars, we have constructed their Galactic orbits for 4.5 Gyr into the past using an axisymmetric Galactic potential model with allowance made for the perturbations from the spiral density wave. Parameters of the encounter with the solar orbit have been calculated for each orbit. We have detected three new stars whose Galactic orbits were close to the solar one during a long time interval in the past. These stars are HIP 43852, HIP 104047, and HIP 112158. The spectroscopic binary HIP 112158 is poorly suited for the role of a kinematic sibling of the Sun by its age and spectroscopic characteristics. For the single star HIP 43852 and the multiple system HIP 104047, this role is quite possible. We have also confirmed the status of our previously found candidates for close encounters, HIP 47399 and HIP 87382. The star HIP 87382 with a chemical composition very close to the solar one is currently the most likely candidate, because it persistently shows close encounters with the Sun on time scales of more than 3 Gyr when using various Galactic potential models both without and with allowance made for the influence of the spiral density wave.     

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.     

任意倾角的共轨运动研究

共轨运动天体与摄动天体的半长径相同,处于1:1平运动共振中.太阳系内多个行星的特洛伊天体即为处于蝌蚪形轨道的共轨运动天体,其中一些高轨道倾角特洛伊天体的轨道运动与来源仍未被完全理解.利用一个新发展的适用于处理1:1平运动共振的摄动函数展开方式,对三维空间中的共轨运动进行考察,计算不同初始轨道根数情况下共轨轨道的共振中心、共振宽度,分析轨道类型与初始轨道根数的关系.并将分析方法所得结果与数值方法的结果相互比较验证,得到了广阔初始轨道根数空间内共轨运动的全局图景.     

New orbits of wide visual double stars

Based on photographic and CCD observations with the Pulkovo 26-inch refractor, radial velocity measurements with the 1.5-m RTT-150 telescope (TUBITAK National Observatory, Turkey), and highly accurate observations published in the WDS catalog, we have obtained the orbits of ten wide visual double stars by the apparent motion parameter method. The orientation of the orbits in the Galactic coordinate system has been determined. For the outer pair of the multiple star HIP 12780 we have calculated a family of orbits with a minimum period P = 4634 yr. Two equivalent solutions with the same period have been obtained for the stars HIP 50 (P = 949 yr) and HIP 66195 (P = 3237 yr). We have unambiguously determined the orbits of six stars: HIP 12777 (P = 3327 yr), HIP 15058 (P = 420 yr), HIP 33287 (P = 1090 yr), HIP 48429 (P = 1066 yr), HIP 69751 (P = 957 yr), and HIP 73846 (P = 1348 yr). The orbit of HIP 55068 is orientated perpendicularly to the plane of the sky, P >1000 yr. The star HIP 48429 is suspected to have an invisible companion.     

Orbital evolution of Saturn’s new outer satellites and their classification

Based on data for twelve recently discovered outer satellites of Saturn, we investigate their orbital evolution on long time scales. For our analysis, we use the previously obtained general solution of Hill’s double-averaged problem, which was refined for libration orbits, and numerical integration of the averaged system of equations in elements with allowance for Saturn’s orbital evolution. The following basic quantitative parameters of evolving orbits are determined: extreme eccentricities and inclinations, as well as circulation periods of the pericenter arguments and of the longitudes of the ascending nodes. For four new satellite orbits, we have revealed the libration pattern of variations in pericenter arguments and determined the ranges and periods of their variations. Based on characteristic features of the orbits of Saturn’s new satellites, we propose their natural classification.     

Mean elements of GEOS 1 and GEOS 2

A combined analytical-numerical procedure for determining mean orbital elements is presented and applied to the orbits of GEOS 1 and GEOS 2. The precision of the mean semi-major axes of these orbits is a few tens of centimeters when optical flash data are used to determine 2 day orbital arcs. Four day Minitrack orbits give mean semi-major axes of a few meters precision. The mean orientation parameters (i, ) determined from the optical data are obtained to a precision of about 0.1.     

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.     

A method for calculating the orbits of interferometer/spectroscopic binaries from combined data

An extension of Docobo's analytical method for calculating the orbits of visual double stars allows all orbital elements to be obtained from three interferometer observations (θ,ρt) and three spectroscopic observations (v;t).     

Calculation of the intermediate perturbed orbit from three or more positions of a small body on the celestial sphere

Two new methods are described for finding the orbit of a small celestial body from three or more pairs of angular measurements and the corresponding time points. The methods are based on, first, the approach that has been developed previously by the author to the determination, from a minimum number of observations, of intermediate orbit considering most of the perturbations in the bodies’ motion and, second, Herget’s algorithmic procedure enabling the introduction of additional observations. The errors of orbital parameters calculated by the proposed methods are two orders of magnitude smaller than the corresponding errors of the traditional approach based on the construction of an unperturbed Keplerian orbit. The thus-calculated orbits of the minor planets 1566 Icarus, 2002 EC1, and 2010 TO48 are used to compare the results of Herget’s multiposition procedure and the new methods. The comparison shows that the new methods are highly effective in the study of perturbed motion. They are particularly beneficial if high-precision observational data covering short orbital arcs are available.     

Averaged model to study long-term dynamics of a probe about Mercury

This paper provides a method for finding initial conditions of frozen orbits for a probe around Mercury. Frozen orbits are those whose orbital elements remain constant on average. Thus, at the same point in each orbit, the satellite always passes at the same altitude. This is very interesting for scientific missions that require close inspection of any celestial body. The orbital dynamics of an artificial satellite about Mercury is governed by the potential attraction of the main body. Besides the Keplerian attraction, we consider the inhomogeneities of the potential of the central body. We include secondary terms of Mercury gravity field from \(J_2\) up to \(J_6\), and the tesseral harmonics \(\overline{C}_{22}\) that is of the same magnitude than zonal \(J_2\). In the case of science missions about Mercury, it is also important to consider third-body perturbation (Sun). Circular restricted three body problem can not be applied to Mercury–Sun system due to its non-negligible orbital eccentricity. Besides the harmonics coefficients of Mercury’s gravitational potential, and the Sun gravitational perturbation, our average model also includes Solar acceleration pressure. This simplified model captures the majority of the dynamics of low and high orbits about Mercury. In order to capture the dominant characteristics of the dynamics, short-period terms of the system are removed applying a double-averaging technique. This algorithm is a two-fold process which firstly averages over the period of the satellite, and secondly averages with respect to the period of the third body. This simplified Hamiltonian model is introduced in the Lagrange Planetary equations. Thus, frozen orbits are characterized by a surface depending on three variables: the orbital semimajor axis, eccentricity and inclination. We find frozen orbits for an average altitude of 400 and 1000 km, which are the predicted values for the BepiColombo mission. Finally, the paper delves into the orbital stability of frozen orbits and the temporal evolution of the eccentricity of these orbits.     

A Specific Nongravitational Effect in the Asteroid Belt

The evolution of orbits of asteroids found in the IRAS and WISE albedo databases was calculated numerically from 2005 to 2016. It follows from the analysis of the obtained results that a certain nongravitational effect (NGE) currently affects the motion of a considerable fraction of main-belt asteroids with diameters up to 40 km. This conclusion agrees with the available data regarding the axial rotation of asteroids. The NGE manifests itself in an increase in the semimajor axes of orbits of low-albedo asteroids relative to the semimajor axes of orbits of high-albedo bodies. The NGE-induced rate of elongation of semimajor axes of asteroids with albedos р_v < 0.1 may be as high as (2–8) × 10^–8 AU/year. Errors in orbital elements of asteroids (unrelated to the accuracy of observational data used to determine these orbital elements) were found in one of the MPC catalogues for 2003 in the process of estimation of the accuracy of calculations.     

Eight new and three recalculated orbits for binaries

In this paper new orbital elements are given for eleven binaries. For eight of them, WDS 00003–4417 = I 1477, WDS 00106–7313 = I 43 AB, WDS 00366+5609 = A 914, WDS 00519–4343 = I 47, WDS 01315+1521 = BU 506, WDS 01577+4434 = A 1526, WDS 08144–4550 = FIN 113 AB and WDS 08291–4756 = FIN 315 Aa‐Ab, the orbital elements are calculated for the first time. For three of them, WDS 04422+2257 = MCA 16 Aa‐Ab, WDS 08275–5501 = FIN 116 and WDS 14567–6247 = FIN 372, the orbital elements are recalculated. One of the eleven binaries, MCA 16 Aa‐Ab, was discovered by McAlister in 1980 by speckle interferometry and four pairs were discovered by Finsen between 1929 and 1960. For these five pairs, all measured separations are less than 0″.4 and most of the observations were done by using the interferometric techniques. The orbital periods calculated here are between 39 and 270 years. The remaining six pairs were discovered between 1878 and 1926 and most of the observations are visual. They have longer orbital periods, between 384 and 1637 years. In addition to the orbital elements the masses, dynamical parallaxes, absolute magnitudes and ephemerides for the next five years are also given in this paper (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The reality of comet groups and pairs

Although the common genetic origin of the Kreutz family of Sun-grazing comets has generally been accepted, there remains uncertainty with regard to genetic identity among other groups of comets whose orbital elements are nearly alike. Porter has listed a number of such grouds and Öpik has made a statistical study of the orbits of 472 comets with aphelion distance beyond Saturn. He lists 97 groups that show similarities among their three angular elements. He calculates an overall probability of some 10^?39 that these similarities could have occurred by chance, and thus concludes that 60% or more of such comets fall into genetic groups containing from two to seven members. This paper explores the statistical reality of Öpik's groups utilizing the Monte Carlo method of statistics as well as ordinary probability theory. The conclusion is reached that except for a few pairs the similarity among orbital elements within the groups is no greater than random expectation.     

Method of determining the orbits of the small bodies in the solar system based on an exhaustive search of orbital planes

A universal method of determining the orbits of newly discovered small bodies in the Solar System using their positional observations has been developed. The proposed method suggests determining geocentric distances of a small body by means of an exhaustive search for heliocentric orbital planes and subsequent determination of the distance between the observer and the points at which the chosen plane intersects with the vectors pointing to the object. Further, the remaining orbital elements are determined using the classical Gauss method after eliminating those heliocentric distances that have a fortiori low probabilities. The obtained sets of elements are used to determine the rms between the observed and calculated positions. The sets of elements with the least rms are considered to be most probable for newly discovered small bodies. Afterwards, these elements are improved using the differential method.     

TV Observation of the Daytime Meteor Shower; The Arietids

We have carried out multi-station TV observations since 1994 in order to determine the orbit of the Arietid daytime meteor stream. In 1999, one possible Arietid meteor was recorded by our simultaneous observations and its orbit was determined. In 2003, two Arietid meteors were observed from two stations of our observing site, those orbits were determined precisely, the orbital elements were in good agreement with each other. This is the first time that determination of the precise orbit of the Arietids has been made from optical observations. The orbit of these Arietid meteors, and comparison with the orbit obtained from radar observations are discussed.     

Orbital evolution of new distant Neptunian satellites and ω-librators in the satellite systems of Saturn and Jupiter

We used data on the recently discovered three outer Neptunian satellites to analyze the long-period evolution of their orbits. We estimated the ranges of eccentricities and inclinations as well as approximate circulation periods of the pericenter arguments and the longitudes of the ascending nodes. The results were mainly obtained by using two different versions of the averaged Hill problem. Plane sections of the phase space of satellite orbital elements are given. We discuss the peculiarity of the evolution of several satellite orbits related to the librational variation of the pericenter argument ω. The ω-librators of Saturn’s system were found to qualitatively differ from the libration orbit in the system of Jupiter.     

Analytical method for the effects of the asteroid belt on planetary orbits

Analytic expressions are derived for the perturbation of planetary orbits due to a thick constant density asteroid belt. The derivations include extensions and adaptations of Plakhov's analytic expressions for the perturbations in five of the orbital elements for closed orbits around Saturn's rings. The equations of Plakhov are modified to include the effect of ring thickness and additional equations are derived for the perturbations in the sixth orbital element, the mean anomaly. The gravitational potential and orbital perturbations are derived for the asteroid belt with and without thickness, and for a hoop approximation to the belt. The procedures are also applicable to Saturn's rings and the newly discovered rings of Uranus.The effects of the asteroid belt thickness on the gravitational potential coefficients and the orbital motions are demonstrated. Comparisons between the Mars orbital perturbations obtained using the analytic expressions and those obtained using numerical integration are discussed. The effects of the asteroid belt on the Earth based ranging to Mars are also demonstrated.