Expansion of the disturbing function for high eccentricity and large amplitude of libration

Following some ideas, developed by Woltjer (1928), Message (1989), Yokoyama (1988, 1989) and Duriez (1990) an expansion of the disturbing function is given for high values of the eccentricity and large amplitude of libration. The classical expansion can be obtained as a particular case of the present model. Several asteroids with high eccentricity and large amplitude of libration are tested and the results are much better than those obtained from the classical theory.     

The geopotential in nonsingular orbital elements

The geopotential expansion is givenentirely in terms of nonsingular orbital elements. The expansion and its derivatives are valid for zero eccentricity and inclination. The development begins with the geopotential expansion in singular, classical elements as given by Izsak (1964), Allan (1965) and Kaula (1966). The singular geopotential is then transformed into a nonsingular set of elements     

Analytical direct planetary perturbations on the orbital motion of satellites

An analytical expansion of the disturbing function arising from direct planetary perturbations on the motion of satellites is derived. As a Fourier series, it allows the investigation of the secular effects of these direct perturbations, as well as of every argument present in the perturbation. In particular, we construct an analytical model describing the evection resonance between the longitude of pericenter of the satellite orbit and the longitude of a planet, and study briefly its dynamic. The expansion developed in this paper is valid in the case of planar and circular planetary orbits, but not limited in eccentricity or inclination of the satellite orbit.     

A numerical investigation of secular terms of the planetary disturbing function

The secular terms of the planetary disturbing function are given, after elimination of short period terms by von Zeipel's transformation. The adequacy of this expansion up to terms of eighth order in the inclination and eccentricity is investigated by numerical processes, as a function of the Keplerian elementsa, e andi. The eccentricityé of the outer planet, is taken equal to zero. It is concluded that for values ofi which are not small the inclusion of additional terms in the expression for the disturbing function, results to drastic changes in its values, while larger values ofe do not have an equaly large effect on the disturbing function.     

Variations of perturbations in perigee height with eccentricity for artificial Earth's satellites due to air drag

The variations of perturbations in perigee distance for different values of the orbital eccentricity for artificial Earth's satellites due to air drag have been studied. The analytical solution of deriving these perturbations, using the TD model (Total Density) have been applied, Helali (1987). The Theory is valid for altitudes ranging from 200 to 500 km above the Earth's surface and for solar 10.7 cm flux. Numerical examples are given to illustrate the variations of the perturbations in perigee distance with changing eccentricity (e < 0.2). A stronge perturbations in the perigee distance have been shown when the eccentricity in the range 0.001 <e < 0.05, especially for perigee distance 200 km.     

Linear stability of the triangular equilibrium points in the photogravitational elliptic restricted problem,II

The linear stability of the triangular equilibrium points in the photogravitational elliptic restricted three-body problem is examined and the stability regions are determined in the space of the parameters of mass, eccentricity, and radiation pressure, in the case of equal radiation factors of the two primaries. The full range of values of the common radiation factor is explored, from the gravitational caseq ₁ =q ₂ =q = 1 down to the critical value ofq = 1/8 at which the triangular equilibria disappear by coalescing on the rotating axis of the primaries. It is found that radiation pressure exerts a significant influence on the stability regions. For certain intervals of radiation values these regions become qualitatively different from the gravitational case as well as the solar system case considered in Paper I. There exist values of the common radiation factor, in the range considered, for which the triangular equilibrium points are stable for the entire range of mass distribution among the primaries and for large eccentricities of their orbits.     

An approximate integral and solution for eccentric planetary type orbits in the restricted problem of three bodies

The circular restricted problem of three bodies is investigated analytically with respect to the problem of deriving a second integral of motion besides the well known Jacobian Integral. The second integral is searched for as a correction the angular momentum integral valid in the two body case. A partial differential equation equivalent to the problem is derived and solved approximately by an asymptotic Fourier method assuming either sufficiently small values for the dimensionless mass parameter or sufficiently large distances from the barycentre. The solution of the partial equation then leads to a function of the coordinates, velocities and time being nearly constant, which means that its variation with time is about 40–300 times less than that of the pure angular momentum. By averaging over the remaining fluctuating part of the quasi-integral we are able to integrate the first order equations using a renormalization transformation. This leads to an explicit expression for the approximate solution of the circular problem which describes the motion of the third body orbiting both primaries with nonvanishing initial eccentricity (eccentric planetary type orbits). One of the main results is an explicit formula for the frequency of the perihelion motion of the third body which depends on the mass parameter, the initial distance of the third body from the barycentre and the initial eccentricity. Finally we study orbits of the P-Type, being defined as solutions of the restricted problem with circular initial conditions (vanishing initial eccentricity).     

A new analytic approach to the Sitnikov problem

A new analytic approach to the solution of the Sitnikov Problem is introduced. It is valid for bounded small amplitude solutions (z _max = 0.20) (in dimensionless variables) and eccentricities of the primary bodies in the interval (–0.4 < e < 0.4). First solutions are searched for the limiting case of very small amplitudes for which it is possible to linearize the problem. The solution for this linear equation with a time dependent periodic coefficient is written up to the third order in the primaries eccentricity. After that the lowest order nonlinear amplitude contribution (being of order z ³) is dealt with as perturbation to the linear solution. We first introduce a transformation which reduces the linear part to a harmonic oscillator type equation. Then two near integrals for the nonlinear problem are derived in action angle notation and an analytic expression for the solution z(t) is derived from them. The so found analytic solution is compared to results obtained from numeric integration of the exact equation of motion and is found to be in very good agreement. CERN SL/AP     

The Parabolic Three-Body Problem

In this communication we present an analytical model for the restricted three-body problem, in the case where the perturber is in a parabolic orbit with respect to the central mass. The equations of motion are derived explicitly using the so-called Global Expansion of the disturbing function, and are valid for any eccentricity of the massless body, as well as in the case where both secondary masses have crossing orbits. Integrating the equations of motion over the complete passage of the perturber through the system, we are then able to construct a first-order algebraic mapping for the change in semimajor axis, eccentricity and inclination of the perturbed body.Comparisons with numerical solutions of the exact equations show that the map yields precise results, as long as the minimum distance between both bodies is not too small. Finally, we discuss several possible applications of this model, including the evolution of asteroidal satellites due to background bodies, and simulations of passing stars on extra-solar planets.     

Numerical results to the Sitnikov-problem

We present numerical results of the so-called Sitnikov-problem, a special case of the three-dimensional elliptic restricted three-body problem. Here the two primaries have equal masses and the third body moves perpendicular to the plane of the primaries' orbit through their barycenter. The circular problem is integrable through elliptic integrals; the elliptic case offers a surprisingly great variety of motions which are until now not very well known. Very interesting work was done by J. Moser in connection with the original Sitnikov-paper itself, but the results are only valid for special types of orbits. As the perturbation approach needs to have small parameters in the system we took in our experiments as initial conditions for the work moderate eccentricities for the primaries' orbit (0.33e _primaries 0.66) and also a range of initial conditions for the distance of the 3 ^rd body (= the planet) from very close to the primaries orbital plane of motion up to distance 2 times the semi-major axes of their orbit. To visualize the complexity of motions we present some special orbits and show also the development of Poincaré surfaces of section with the eccentricity as a parameter. Finally a table shows the structure of phase space for these moderately chosen eccentricities.     

Extension of the solution of Kepler's equation to high eccentricities

The classic Lagrange's expansion of the solutionE(e, M) of Kepler's equation in powers of eccentricity is extended to highly eccentric orbits, 0.6627 ... <e<1. The solutionE(e, M) is developed in powers of (e–e*), wheree* is a fixed value of the eccentricity. The coefficients of the expansion are given in terms of the derivatives of the Bessel functionsJ _n (ne). The expansion is convergent for values of the eccentricity such that |e–e*|<(e*), where the radius of convergence (e*) is a positive real number, which is calculated numerically.     

Effect of Planetary Eccentricity on Ballistic Capture in the Solar System

Ballistic capture of spacecraft and celestial bodies by planets of the solar system is studied considering the elliptic restricted three body model. A preferential region, due to the eccentricity of the planet and the Sun-gravity-gradient effect is found for the capture phenomenon. An analytical formula is derived which determines the limiting value of the satellite capture eccentricity e_c as a function of the pericenter distance x_p and planet’s true anomaly. The analytic values e_c are tested by a numerical propagator, which makes use of planetary ephemeris, and only a small difference with respect to numerical integration is found. It turns out that lower values of e_c occur when the planet anomaly is close to zero; that is, capture is easier when the planet is at its perihelion. This fact is confirmed by the capture of celestial bodies. It is shown that Jupiter comets are generally captured when Jupiter is in its perihelion region. Ballistic capture is also important in interplanetary missions. The propellant saved using the minimum ballistic capture eccentricity is evaluated for different missions and compared with respect to the case in which the insertion orbit is a parabola: a significant saving can be accomplished.     

Numerical investigation of the planar restricted three-body problem

The critical orbits, corresponding to bifurcations of the generating family and its branches, are considered more closely and the part off is investigated that has branches of very high order only. Three families of periodic solutions of the elliptic problem are also determined in an effort to follow the evolution of the stability region aroundf when the eccentricity of the primaries is increased to non-zero values.     

A Perturbative Treatment of The Co-Orbital Motion

We develop a formalism of the non-singular evaluation of the disturbing function and its derivatives with respect to the canonical variables. We apply this formalism to the case of the perturbed motion of a massless body orbiting the central body (Sun) with a period equal to that of the perturbing (planetary) body. This situation is known as the co-orbital motion, or equivalently, as the 1/1 mean motion commensurability. Jupiter's Trojan asteroids, Earth's co-orbital asteroids (e.g., (3753) Cruithne, (3362) Khufu), Mars' co-orbital asteroids (e.g., (5261) Eureka), and some Jupiter-family comets are examples of the co-orbital bodies in our solar system. Other examples are known in the satellite systems of the giant planets. Unlike the classical expansions of the disturbing function, our formalism is valid for any values of eccentricities and inclinations of the perturbed and perturbing body. The perturbation theory is used to compute the main features of the co-orbital dynamics in three approximations of the general three-body model: the planar-circular, planar-elliptic, and spatial-circular models. We develop a new perturbation scheme, which allows us to treat cases where the classical perturbation treatment fails. We show how the families of the tadpole, horseshoe, retrograde satellite and compound orbits vary with the eccentricity and inclination of the small body, and compute them also for the eccentricity of the perturbing body corresponding to a largely eccentric exoplanet's orbit.This revised version was published online in October 2005 with corrections to the Cover Date.     

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.     

Asteroid motion near the 3 : 1 resonance

A mapping model is constructed to describe asteroid motion near the 3 : 1 mean motion resonance with Jupiter, in the plane. The topology of the phase space of this mapping coincides with that of the real system, which is considered to be the elliptic restricted three body problem with the Sun and Jupiter as primaries. This model is valid for all values of the eccentricity. This is achieved by the introduction of a correcting term to the averaged Hamiltonian which is valid for small values of the ecentricity.We start with a two dimensional mapping which represents the circular restricted three body problem. This provides the basic framework for the complete model, but cannot explain the generation of a gap in the distribution of the asteroids at this resonance. The next approximation is a four dimensional mapping, corresponding to the elliptic restricted problem. It is found that chaotic regions exist near the 3 : 1 resonance, due to the interaction between the two degrees of freedom, for initial conditions close to a critical curve of the circular model. As a consequence of the chaotic motion, the eccentricity of the asteroid jumps to high values and close encounters with Mars and even Earth may occur, thus generating a gap. It is found that the generation of chaos depends also on the phase (i.e. the angles andv) and as a consequence, there exist islands of ordered motion inside the sea of chaotic motion near the 3 : 1 resonance. Thus, the model of the elliptic restricted three body problem cannot explain completely the generation of a gap, although the density in the distribution of the asteroids will be much less than far from the resonance. Finally, we take into account the effect of the gravitational attraction of Saturn on Jupiter's orbit, and in particular the variation of the eccentricity and the argument of perihelion. This generates a mixing of the phases and as a consequence the whole phase space near the 3 : 1 resonance becomes chaotic. This chaotic zone is in good agreement with the observations.     

Secular resonances in the asteroid belt: Theoretical perturbation approach and the problem of their location

In this paper a theoretical perturbation approach to the problem of the dynamics in secular resonance is exposed. This approach avoids any expansion of the main term of the Hamiltonian (linear term in the masses) with respect to the eccentricity or the inclination of the asteroid, in order to achieve results valid for any value of these variables. Moreover suitable action-angle variables are introduced to take properly into account the dynamics related to the motion of the argument of perihelion of the asteroid, which is relevant at high inclination. A class of secular resonances wider than that usually considered is found. An explicit computation of the location of the main secular resonances, estimating also the contribution of the quadratic term in the masses by means of classical series expansion, is reported in the last sections. The accuracy of computations obtained by series expansion is discussed in the paper.     

Extension of the critical inclination

The critical inclination is of special interest in artificial satellite theory. The critical inclination can maintain minimal deviations of eccentricity and argument of pericentre from the initial values, and orbits at this inclination have been applied to some space missions. Most previous researches about the critical inclination were made under the assumption that the oblateness term J ₂ is dominant among the harmonic coefficients. This paper investigates the extension of the critical inclination where the concept of the critical inclination is different from that of the traditional sense. First, the study takes the case of Venus for instance, and provides some preliminary results. Then for general cases, given the values of argument of pericentre and eccentricity, the relationship between the multiplicity of the solutions for the critical inclination and the values of J ₂ and J ₄ is analyzed. Besides, when given certain values of J ₂ and J ₄, the relationship between the multiplicity of the solutions for the critical inclination and the values of semimajor axis and eccentricity is studied. The results show that for some cases, the value of the critical inclination is far away from that of the traditional sense or even has multiple solutions. The analysis in this paper could be used as starters of correction methods in the full gravity field of celestial bodies.     

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).     

Grain abundance in the primordial atmosphere of the Earth

For models of planetary accumulation in the presence of solar nebular gas, the initial surface temperature of the Earth is controlled by the grain opacity of the atmosphere. The surface temperature in turn controls the quantity of neon dissolved and trapped within the interior of the Earth. To compare accumulation theory with observation calculations have been made of the grain opacity expected to be associated with accumulation in a gaseous nebula. There are two parameters that are in principle determined by the theory, but actually are at present uncertain: the mean eccentricity e of the planetesimal swarm, and the fraction ξ of the accretional energy that is expended in the release of grains into the atmosphere by ablation of the incoming planetesimal. It is found that if the eccentricity of swarm is low (10^?3), rather low values of ξ(10^?5) are required to match the observed neon data. In contrast higher values of ξ are required (10^?1) for the most probable case, intermediate eccentricity (10^?2). For the high eccentricity case (e ～ 0.1) ξ must be >10^?2. The results show that avoidance of excess trapped neon of solar composition places restrictive, but not necessarily impossible, conditions on the parameters of the accumulation theory.