Motion near the triangular points in the elliptic restricted problem of three bodies

In this paper the first variational equations of motion about the triangular points in the elliptic restricted problem are investigated by the perturbation theories of Hori and Deprit, which are based on Lie transforms, and by taking the mean equations used by Grebenikov as our upperturbed Hamiltonian system instead of the first variational equations in the circular restricted problem. We are able to remove the explicit dependence of transformed Hamiltonian on the true anomaly by a canonical transformation. The general solution of the equations of motion which are derived from the transformed Hamiltonian including all the constant terms of any order in eccentricity and up to the periodic terms of second order in eccentricity of the primaries is given.     

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.     

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.     

The dynamics of eccentric accretion discs in superhump systems

We have applied an eccentric accretion disc theory in simplified form to the case of an accretion disc in a binary system, where the disc contains the 3:1 Lindblad resonance. This is relevant to the case of superhumps in SU Ursae Majoris cataclysmic variables and other systems, where it is thought that this resonance leads to growth of eccentricity and a modulation in the light curve due to the interaction of a precessing eccentric disc with tidal stresses. A single differential equation is formulated which describes the propagation, resonant excitation and viscous damping of eccentricity. The theory is first worked out in the simple case of a narrow ring and leads to the conclusion that the eccentricity distribution is locally suppressed by the presence of the resonance, creating a dip in the eccentricity at the resonant radius. Application of this theory to the superhump case confirms this conclusion and produces a more accurate expression for the precession rate of the disc than has been previously accomplished with simple dynamical estimates.     

Literal expressions for the co-ordinates of the moon

This paper describes a method for finding literal expressions for the first order terms in the Moon's co-ordinates. The method is based on the use of rectangular co-ordinates and was originally proposed by Euler. The variation curve and the terms dependent on the first power of the Lunar eccentricity have been obtained. These results are compared with those of Hill and a number of errors in Hill's results have been found.     

An asymptotic solution for the Trojan case of the plane elliptic restricted problem of three bodies

The planar motion of a Trojan asteroid is considered within the framework of the elliptic restricted three-body problem. The solution is derived asymptotically to second order taking the square root of the Jupiter-Sun mass ratio and the orbital eccentricity of Jupiter as first order quantities. The results are given in explicit form for the coordinates as functions of the true anomaly of Jupiter including both short and long periodic terms resulting from the orbital accentricity of Jupiter.     

卫星径向位置摄动计算中的几个问题

为了对卫星轨道径向位置误差进行分析，本文将给出由地球非球形引力位（包括潮汐形变）引起的卫星径向位置摄动表达式，它将同时包含完整的卫星轨道偏心率的0阶和1阶项，并给出径向位置摄动空间分布的一种简单计算方法，它可明显地节省计算机时．     

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.     

Sensitivity study of high eccentricity orbits for Mars gravity recovery

By linear perturbation theory, a sensitivity study is presented to calculate the contribution of the Mars gravity field to the orbital perturbations in velocity for spacecrafts in both low eccentricity Mars orbits and high eccentricity orbits(HEOs). In order to improve the solution of some low degree/order gravity coefficients, a method of choosing an appropriate semimajor axis is often used to calculate an expected orbital resonance, which will significantly amplify the magnitude of the position and velocity perturbations produced by certain gravity coefficients. We can then assess to what degree/order gravity coefficients can be recovered from the tracking data of the spacecraft. However, this existing method can only be applied to a low eccentricity orbit, and is not valid for an HEO. A new approach to choosing an appropriate semimajor axis is proposed here to analyze an orbital resonance. This approach can be applied to both low eccentricity orbits and HEOs. This small adjustment in the semimajor axis can improve the precision of gravity field coefficients and does not affect other scientific objectives.     

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.     

Ptolemaic Transformation in Keplerian Problem

We propose the Ptolemaic transformation: a canonical change of variables reducing the Keplerian motion to the form of a perturbed Hamiltonian problem. As a solution of the unperturbed case, the Ptolemaic variables define an intermediary orbit, accurate up to the first power of eccentricity, like in the kinematic model of Claudius Ptolemy. In order to normalize the perturbed Hamiltonian we modify the recurrent Lie series algorithm of HoriuuMersman. The modified algorithm accounts for the loss of a term's order during the evaluation of a Poisson bracket, and thus can be also applied in resonance problems. The normalized Hamiltonian consists of a single Keplerian term; the mean Ptolemaic variables occur to be trivial, linear functions of the Delaunay actions and angles. The generator of the transformation may serve to expand various functions in Poisson series of eccentricity and mean anomaly.     

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.     

The shape and dynamics of a heliotropic dusty ringlet in the Cassini Division

The so-called “Charming Ringlet” (R/2006 S3) is a low-optical-depth, dusty ringlet located in the Laplace gap in the Cassini Division, roughly 119,940 km from Saturn center. This ringlet is particularly interesting because its radial position varies systematically with longitude relative to the Sun in such a way that the ringlet’s geometric center appears to be displaced away from Saturn’s center in a direction roughly toward the Sun. In other words, the ringlet is always found at greater distances from the planet’s center at longitudes near the sub-solar longitude than it is at longitudes near Saturn’s shadow. This “heliotropic” behavior indicates that the dynamics of the particles in this ring are being influenced by solar radiation pressure. In order to investigate this phenomenon, which has been predicted theoretically but not observed this clearly, we analyze multiple image sequences of this ringlet obtained by the Cassini spacecraft in order to constrain its shape and orientation. These data can be fit reasonably well with a model in which both the eccentricity and the inclination of the ringlet have “forced” components (that maintain a fixed orientation relative to the Sun) as well as “free” components (that drift around the planet at steady rates determined by Saturn’s oblateness). The best-fit value for the eccentricity forced by the Sun is 0.000142 ± 0.000004, assuming this component of the eccentricity has its pericenter perfectly anti-aligned with the Sun. These data also place an upper limit on a forced inclination of 0.0007°. Assuming the forced inclination is zero and the forced eccentricity vector is aligned with the anti-solar direction, the best-fit values for the free components of the eccentricity and inclination are 0.000066 ± 0.000003 and 0.0014 ± 0.0001°, respectively. While the magnitude of the forced eccentricity is roughly consistent with theoretical expectations for radiation pressure acting on 10-to-100-μm-wide icy grains, the existence of significant free eccentricities and inclinations poses a significant challenge for models of low-optical-depth dusty rings.     

Improved equations for eccentricity generation in hierarchical triple systems

In a series of papers, we developed a technique for estimating the inner eccentricity in hierarchical triple systems, with the inner orbit being initially circular. However, for certain combinations of the masses and the orbital elements, the secular part of the solution failed. In this paper, we derive a new solution for the secular part of the inner eccentricity, which corrects the previous weakness. The derivation applies to hierarchical triple systems with coplanar and initially circular orbits. The new formula is tested numerically by integrating the full equations of motion for systems with mass ratios from 10⁻³ to 10³. We also present more numerical results for short-term eccentricity evolution, in order to get a better picture of the behaviour of the inner eccentricity.     

Analysis of the orbit of cosmos 387 (1970-111A) near 15th-order resonance

Cosmos 387 (1970-111A) was launched on 16 December 1970 into a near-circular orbit with an average height of 540 km and an inclination of 74.0°. On 5 November 1971 the orbit, in its slow contraction under the influence of air drag, passed through 15th-order resonance, when the ground track repeats after 15 revolutions. The orbit has been determined with the aid of the RAE orbit refinement program PROP at 19 epochs between May 1971 and June 1972, using 1500 optical and radar observations. The average accuracy is about 70 m in perigee height and 0.001° in inclination.The variation of orbital inclination while the satellite was experiencing 15th-order resonance, as given by these 19 orbits and 55 U.S. Navy orbits, has been analysed to obtain equations accurate to 4 per cent for the geopotential coefficients of order 15 and odd degree (15, 17, 19 …). These equations have subsequently been used (with others) in determining individual coefficients of order 15 and odd degree.The variation of eccentricity with argument of perigee showed unexpected complexity, including a tight loop near resonance (Fig. 4). Analysis of the variation in eccentricity has yielded, for the first time, accurate equations for the geopotential coefficients of order 15 and even degree (16, 18 …), thus opening the way to the evaluation of individual coefficients of this type. The variations in the argument of perigee and right ascension of the node have also been analysed.     

Trajectories of satellites under the combined influences of earth oblateness and air drag

Trajectories of satellites under the influences of earth oblateness and air drag are derived by the asymptotic method in nonlinear mechanics. Based on the assumptions: (1) the dominant oblateness factor of the earth is the second harmonic (J ₂), (2) a non-rotating, spherically symmetric atmosphere and an exponential distribution of atmospheric density, (3) original elliptical orbits being of small eccentricity, closed-form solutions for the improved first order approximation are obtained. After finding the osculating orbital elements of the resulting trajectories, we expose the behavior of osculating orbits at various inclinations.     

The elimination of the 1:2 critical terms of a first order theory of Uranus perturbed by Neptune through Von Zeipel method

In this paper we eliminate in a first order U-N theory the 1 : 2 critical terms up to the third degree with respect to eccentricity — inclination in both parts, main and indirect of the U-N planetary Hamiltonian. We operate the Von Zeipel technique. We adopt, in this theory, the Jacobi-Radau coordinates, and the Poincaré canonical variables. We neglect powers higher than the third in the eccentricity-inclination. This paper is related to the two previous articles (Kamel, 1982; 1983).     

The galactic distribution of globular clusters

The spatial distribution of galactic globular clusters has been studied by means of Walsh-Hadamard transforms. Consistent indications have been collected suggesting the existence, at least in a first approximation, of two disting populations of clusters — the metal poor clusters showing a spherical distribution whereas metal rich clusters are flattened along the disk.Simulated cluster-halos have been computed in order to study the dependence of observed radial velocities on the eccentricity of cluster orbits, showing that only a general high value of eccentricity seems able to reproduce the observational features.