Öpik-type collision probability for high-inclination orbits

The classical Öpik theory provides an estimate of the collision probability between two bodies on bound, heliocentric or planetocentric orbits under restrictive assumptions of: (i) constant eccentricity and inclination, and (ii) uniform circulation of the longitude of node and argument of pericenter. These assumptions are violated whenever either of the orbits has a large inclination with respect to the local Laplace plane or large eccentricity, and their motion is perturbed by an exterior (tidal) gravitational field of a planet or the Sun. In this situation, known as the Lidov–Kozai regime, the eccentricity and inclination values exhibit large and correlated oscillations. At the same time, the longitude of node and the argument of pericenter may have strongly nonlinear time evolution, with the latter being even bound to a small interval of values. Here we develop a new Öpik-type collision probability theory which is valid even for highly inclined and/or eccentric orbits of the projectile. We assume that the orbit of the target is circular and in the local Laplace plane. Such a generalized setting is necessary, as an example, to correctly estimate the terrestrial impact fluxes of sporadic micrometeoroids on high-inclination orbits (notably those from the toroidal source and the associated helion and anti-helion arcs).     

Long-Period Variations of the Eccentricity Vector Valid also for Near Circular Orbits around a Non-Spherical Body

This paper studies the long period variations of the eccentricity vector of the orbit of an artificial satellite, under the influence of the gravity field of a central body. We use modified orbital elements which are non-singular at zero eccentricity. We expand the long periodic part of the corresponding Lagrange equations as power series of the eccentricity. The coefficients characterizing the differential system depend on the zonal coefficients of the geopotential, and on initial semi-major axis, inclination, and eccentricity. The differential equations for the components of the eccentricity vector are then integrated analytically, with a definition of the period of the perigee based on the notion of “free eccentricity”, and which is also valid for circular orbits. The analytical solution is compared to a numerical integration. This study is a generalization of (Cook, Planet. Space Sci., 14, 1966): first, the coefficients involved in the differential equations depend on all zonal coefficients (and not only on the very first ones); second, our method applies to nearly circular orbits as well as to not too eccentric orbits. Except for the critical inclination, our solution is valid for all kinds of long period motions of the perigee, i.e., circulations or librations around an equilibrium point.     

Balanced earth satellite orbits

An analytic model for third-body perturbations and for the second zonal harmonic of the central body's gravitational field is presented. A simplified version of this model applied to the Earth-Moon-Sun system indicates the existence of high-altitude and highly-inclined orbits with their apsides in the equator plane, for which the apsidal as well as the nodal motion ceases. For special positions of the node, secular changes of eccentricity and inclination disappear too (balanced orbits). For an ascending node at vernal equinox, the inclination of balanced orbits is 94.56°, for a node at autumnal equinox 85.44°, independent of the eccentricity of the orbit. For a node perpendicular to the equinox, there exist circular balanced orbits at 90° inclination. By slightly adjusting the initial inclination as suggested by the simplified model, orbits can be found — calculated by the full model or by different methods — that show only minor variations in eccentricity, inclination, argument of perigee, and longitude of the ascending node for 10⁵ revolutions and more. Orbits near the unstable equilibria at 94.56° and 85.44° inclination show very long periodic librations and oscillations between retrogade and prograde motion.Retired from IBM Vienna Software Development Laboratory.     

Oblateness effect of Saturn on periodic orbits in the Saturn-Titan restricted three-body problem

We explore the effect of oblateness of Saturn (more massive primary) on the periodic orbits and the regions of quasi-periodic motion around both the primaries in the Saturn-Titan system in the framework of planar circular restricted three-body problem. First order interior and exterior mean motion resonances are located. The effect of oblateness is studied on the location, nature and size of periodic and quasi-periodic orbits, using the numerical technique of Poincare surface of sections. Some of the periodic orbits change to quasi-periodic orbits due to the effect of oblateness and vice-versa. The stability of the orbits around Saturn, Titan and both varies with the inclusion of oblateness. The centers of the periodic orbits around Titan move towards Saturn, whereas those around Saturn move towards Titan. For the orbit around Titan at C=2.9992, x=0.959494, the apocenter becomes pericenter. By incorporating oblateness effect, the orbit around Titan at C=2.99345, x=0.924938 is captured by Saturn, remains in various trajectories around Saturn, and as time progresses it spirals away around both the primaries.     

Ringlet–satellite interactions

We study the interaction of a satellite and a nearby ringlet on eccentric and inclined orbits. Secular torques originate from mean motion resonances and the secular interaction potential which represents the m  = 1 global modes of the ring. The torques act on the relative eccentricity and inclination. The resonances damp the relative eccentricity. The inclination instability owing to the resonances is turned off by a finite differential eccentricity of the order of 0.27 for nearly coplanar systems. The secular potential torque damps the eccentricity and inclination and does not affect the relative semi-major axis; also, it suppresses the inclination instability that persists at small differential eccentricities. The damping of the relative eccentricity and inclination forces an initially circular and planar small mass ringlet to reach the eccentricity and inclination of the satellite. When the planet is oblate, the interaction of the satellite damps the proper precession of a small mass ringlet so that it precesses at the satellite's rate independently of their relative distance. The oblateness of the primary modifies the long-term eccentricity and inclination magnitudes and introduces a constant shift in the apsidal and nodal lines of the ringlet with respect to those of the satellite. These results are applied to Saturn's F-ring, which orbits between the moons Prometheus and Pandora.     

The dynamics of Neptune Trojan – I. The inclined orbits

The stability of Trojan type orbits around Neptune is studied. As the first part of our investigation, we present in this paper a global view of the stability of Trojans on inclined orbits. Using the frequency analysis method based on the fast Fourier transform technique, we construct high-resolution dynamical maps on the plane of initial semimajor axis a ₀ versus inclination i ₀. These maps show three most stable regions, with i ₀ in the range of  (0°, 12°), (22°, 36°)  and  (51°, 59°),  respectively, where the Trojans are most probably expected to be found. The similarity between the maps for the leading and trailing triangular Lagrange points L ₄ and L ₅ confirms the dynamical symmetry between these two points. By computing the power spectrum and the proper frequencies of the Trojan motion, we figure out the mechanisms that trigger chaos in the motion. The Kozai resonance found at high inclination varies the eccentricity and inclination of orbits, while the  ν₈  secular resonance around   i ₀∼ 44°  pumps up the eccentricity. Both mechanisms lead to eccentric orbits and encounters with Uranus that introduce strong perturbation and drive the objects away from the Trojan like orbits. This explains the clearance of Trojan at high inclination  (>60°)  and an unstable gap around  44°  on the dynamical map. An empirical theory is derived from the numerical results, with which the main secular resonances are located on the initial plane of  ( a ₀, i ₀)  . The fine structures in the dynamical maps can be explained by these secular resonances.     

月球重力场对绕月低轨卫星的长期影响

从解析形式出发,利用月球重力场模型JGL165P1,分析了月球重力场(带谐项)对绕月低轨卫星的长期影响。为了减少计算误差,保证计算精度,在分析解中使用循环公式来计算倾角函数。结果指出对于一个高度为100km的极月轨道卫星,冻结轨道存在的可能性不大,但是当轨道倾角在i=90°附近或者高度再高一些,则有可能存在冻结轨道;对于100km高的初始圆轨道,卫星在无控的情况下半年内将会坠落到月球表面,如果高度增加到200km,则不进行轨道控制也不会坠落到月面上。利用仿真软件GEODYN解算出来的结果证实了上述结论。     

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.     

On the Stability Regions of the Trojan Asteroids

The orbits of fictitious bodies around Jupiter’s stable equilibrium points L ₄ and L ₅ were integrated for a fine grid of initial conditions up to 100 million years. We checked the validity of three different dynamical models, namely the spatial, restricted three body problem, a model with Sun, Jupiter and Saturn and also the dynamical model with the Outer Solar System (Jupiter to Neptune). We determined the chaoticity of an orbit with the aid of the Lyapunov Characteristic Exponents (=LCE) and used also a method where the maximum eccentricity of an orbit achieved during the dynamical evolution was examined. The goal of this investigation was to determine the size of the regions of motion around the equilibrium points of Jupiter and to find out the dependance on the inclination of the Trojan’s orbit. Whereas for small inclinations (up to i=20°) the stable regions are almost equally large, for moderate inclinations the size shrinks quite rapidly and disappears completely for i>60^°. Additionally, we found a difference in the dynamics of orbits around L ₄ which – according to the LCE – seem to be more stable than the ones around L ₅.     

Analytical orbit predictions with air drag using KS uniformly regular canonical elements

A new nonsingular analytical theory for the motion of near Earth satellite orbits with the air drag effect is developed for long term motion in terms of the KS uniformly regular canonical elements by a series expansion method, by assuming the atmosphere to be symmetrically spherical with constant density scale height. The series expansions include up to third order terms in eccentricity. Only two of the nine equations are solved analytically to compute the state vector and change in energy at the end of each revolution, due to symmetry in the equations of motion. Numerical comparisons of the important orbital parameters semi major axis and eccentricity up to 1000 revolutions, obtained with the present solution, with KS elements analytical solution and Cook, King-Hele and Walker's theory with respect to the numerically integrated values, show the superiority of the present solution over the other two theories over a wide range of eccentricity, perigee height and inclination.     

On the Stability of High Inclined L4 and L5Trojans

The orbits of real asteroids around the Lagrangian points L₄ and L ₅of Jupiter with large inclinations (i > 20°) were integrated for 50 Myrs. We investigated the stability with the aid of the Lyapunov characteristic exponents (LCE) but tested also two other methods: on one hand we integrated four neighbouring orbits for each asteroid and computed the maximum distance in every group, on the other hand we checked the variation of the Delaunay element H of the asteroid. In a second simulation – for a grid of initial eccentricity versus initial inclination – we examined the stability of the orbits around both Lagrangian points for 20° < i < 55° and 0.0 < e < 0.20. For the initial semimajor axes we have chosen the one ofJupiter(a = 5.202 AU). We determined the stability with the aid of the LCEs and also the maximum eccentricity of the orbits during the whole integration time. The region around L₄ turned out to be unstable for large inclinations and eccentricities (i > 55° and e > 0.12). The stable region shrinks for orbits around L₅: we found that they become unstable already for i > 45° and e > 0.10. We interpret it as a first hint why we observe more Trojans around the leading Lagrangian point. The results confirm the stability behaviour of the real Trojans which we computed in the first part of the paper.     

用切比雪夫曲线拟合导航卫星广播星历

提出用一种基于切比雪夫曲线多项式拟合的方法,来拟合导航卫星广播星历。该方法对解决 GEO导航卫星的小倾角、小偏心率问题特别有效,并具有精度高、使用方便等特点。     

On the convergence of elliptic motion

The convergence of Lagrange series is studied on a part of the elliptical orbit for values of eccentricity exceeding the Laplace limit. The regions in the vicinity of the two apses of the orbit are identified in which the Lagrange series converge absolutely and uniformly for the values of the eccentricity greater than the Laplace limit. The obtained results are of practical interest for astronomy when studying motions of stellar bodies in orbits with high eccentricity. In particular, these series may be used to calculate the orbits of comets or asteroids with high eccentricity as they pass through the neighborhood of perihelion or to calculate the orbits of artificial satellites with high eccentricity “hanging” in the vicinity of apogee. In stellar dynamics, these series may be used in cases of close binary stars, many of which move in orbits with an eccentricity greater than the Laplace limit.     

Origin of the ten degree Solar System dust bands

The Solar System dust bands discovered by IRAS are toroidal distributions of dust particles with common proper inclinations. It is impossible for particles with high eccentricity (approximately 0.2 or greater) to maintain a near constant proper inclination as they precess, and therefore the dust bands must be composed of material having a low eccentricity, pointing to an asteroidal origin. The mechanism of dust band production could involve either a continual comminution of material associated with the major Hirayama asteroid families, the equilibrium model (Dermott et al. (1984) Nature 312, 505–509) or random disruptions in the asteroid belt of small, single asteroids (Sykes and Greenberg (1986) Icarus 65, 51–69). The IRAS observations of the zodiacal cloud from which the dust band profiles are isolated have excellent resolution, and the manner in which these profiles change around the sky should allow the origin of the bands, their radial extent, the size-frequency distribution of the material and the optical properties of the dust itself to be determined. The equilibrium model of the dust bands suggests Eos as the parent of the 10° band pair. Results from detailed numerical modeling of the 10° band pair are presented. It is demonstrated that a model composed of dust particles having mean semimajor axis, proper eccentricity and proper inclination equal to those of the Eos family member asteroids, but with a dispersion in proper inclination of 2.5°, produces a convincing match with observations. Indeed, it is impossible to reproduce the observed profiles of the 10° band pair without imposing such a dispersion on the dust band material. Since the dust band profiles are matched very well with Eos, Themis and Koronis type material alone, the result is taken as strong evidence in favor of the equilibrium model. The effects of planetary perturbations are included by imposing the appropriate forced elements on the dust particle orbits (these forced elements vary with heliocentric distance). A subsequent model in which material is allowed to populate the inner solar system by a Poynting-Robertson drag distribution is also constructed. A dispersion in proper inclination of 3.5° provides the best match with observations, but close examination of the model profiles reveals that they are slightly broader than the observed profiles. If the variation of the number density of asteroidal material with heliocentric distance r is given by an expression of the form 1/r^τ then these results indicate that γ < 1 compared with γ = 1 expected for a simple Poynting-Robertson drag distribution. This implies that asteroidal material is lost from the system as it spirals in towards the Sun, owing to interparticle collisions.     

Contraction of satellite orbits using KS uniformly regular canonical elements in an oblate diurnally varying atmosphere

A new non-singular analytical theory for the motion of near Earth satellite orbits with the air drag effect is developed in terms of the Kustaanheimo and Stiefel (KS) uniformly regular canonical elements, by assuming the atmosphere to be oblate diurnally varying with constant density scale height. The series expansions include up to third-order terms in eccentricity and c (a small parameter dependent on the flattening of the atmosphere). Only two of the nine equations are solved analytically to compute the state vector and change in energy at the end of each revolution, due to symmetry in the equations of motion. Numerical comparisons of the important orbital parameters semimajor axis and eccentricity up to 1000 revolutions, obtained with the present solution, with the third-order analytical theories of Swinerd and Boulton and in terms of the KS elements, with respect to the numerically integrated values, show the superiority of the present solution over the other two theories over a wide range of eccentricity, perigee height and inclination.     

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.     

Proper elements for highly inclined asteroidal orbits

Based on Williams' work and rewritten in action angle variables, a method for the calculation of proper elements is here presented. The averaging over the long periodic terms is performed by the semi numerical method developed by Henrard (1990); no series expansion in eccentricity or inclination of the asteroid is used which allows calculating proper elements for highly inclined orbits. Conversely, the theory is truncated at the first degree in the eccentricity and the inclination of the perturbing planets. A few tests about accuracy and consistency are presented.     

Generalized Averaging Principle and the Secular Evolution of Planet Crossing Orbits

Planet crossing orbits give rise to mathematical singularities that make it not possible to apply the classical averaging principle to study the qualitative evolution of Near Earth Asteroids (NEAs). Recently this principle has been generalized to deal with crossings in a mathematical model with the planets on circular coplanar orbits. More accuracy is needed to compute the averaged evolution of planet crossing orbits for different purposes: computing reliable crossing times for the averaged motion, writing more precise proper elements and frequencies for NEAs, etc. In this paper we present the generalization of the averaging principle using a model where the eccentricity and the inclination of the planets are taken into account.     

Aegaeon (Saturn LIII), a G-ring object

Aegaeon (Saturn LIII, S/2008 S1) is a small satellite of Saturn that orbits within a bright arc of material near the inner edge of Saturn’s G-ring. This object was observed in 21 images with Cassini’s Narrow-Angle Camera between June 15 (DOY 166), 2007 and February 20 (DOY 051), 2009. If Aegaeon has similar surface scattering properties as other nearby small saturnian satellites (Pallene, Methone and Anthe), then its diameter is approximately 500 m. Orbit models based on numerical integrations of the full equations of motion show that Aegaeon’s orbital motion is strongly influenced by multiple resonances with Mimas. In particular, like the G-ring arc it inhabits, Aegaeon is trapped in the 7:6 corotation eccentricity resonance with Mimas. Aegaeon, Anthe and Methone therefore form a distinctive class of objects in the Saturn system: small moons in corotation eccentricity resonances with Mimas associated with arcs of debris. Comparisons among these different ring-arc systems reveal that Aegaeon’s orbit is closer to the exact resonance than Anthe’s and Methone’s orbits are. This could indicate that Aegaeon has undergone significant orbital evolution via its interactions with the other objects in its arc, which would be consistent with the evidence that Aegaeon’s mass is much smaller relative to the total mass in its arc than Anthe’s and Methone’s masses are.