Search

In the exploration of celestial bodies, including planets, moons, and asteroids, it is useful to identify stable orbits in which the orbital elements remain, on average, constant (referred to as frozen orbits). This paper aims to investigate the feasibility of frozen orbits for a small body, such as a probe, orbiting a main body under the perturbation due to the gravitational attraction of an inner third-body, on a circular (or slightly eccentric) orbit coplanar with the main body equator, by means of an approach based on mean element theory. The disturbing potential has been developed in Legendre polynomials up to order l = 5 (triakontadipole level of approximation). Thus, the double-averaged potential function and the associated constants of motion have been determined up to the same order. Then, by applying the Lagrange Planetary Equations, the secular and long-term orbital element variations have been obtained. Therefore, frozen orbital solutions have been evaluated, and the effects on them of the addition of main body oblateness perturbation have been considered. Furthermore, the procedure has been repeated, and the frozen solutions are re-evaluated in the case of a disturbing body with small eccentricity up to order l = 4 (hexadecapole level of approximation). These solutions offer significant advantages in exploring bodies subject to a notable perturbation from an inner third-body, such as binary asteroid systems. • Frozen orbits for the inner third-body problem. • Inner circular planar third-body: Double-averaged potential up to triakontadipole. • Inner slightly eccentric planar third-body up to hexadecapole. • Orbital solutions for binary asteroid observation. [ABSTRACT FROM AUTHOR]

In the present work, the problem of frozen orbits of a charged satellite around the Earth is treated. The geopotential is considered as oblate body up to J 6 zonal harmonics. The first normalized Hamiltonian is utilized so as to compute families of the long-term frozen orbits for a charged satellite in the concerned model using the Lie transform method. Since the mathematical model of the problem is complicated, the Hamilton canonical equations have been solved numerically. Our numerical investigation of the considered dynamical system reveals no frozen orbits can be obtained for orbits with perigee point lies on the equator g ∈ 0.0 ° , 0.180 ° . But, in the neighbouhood of these orbits, equilibria for a very nearly circular orbits as well as equatorial HEO exist. Our results reflected the remarkable effects of the electromagnetic perturbations on the frozen orbit dynamics, especially for the medium and high Earth orbits. Investigating charged satellites carrying different electronic charges, the phase space (I , e) is noticeably changed near the equilibria g = n (180 °) , n = 0 , 1. It is observed also that a very little change in the amount of charge carried by the satellite, gives rise to a change in the phase space especially for the nearly circular orbits that have inclinations in between the two roots of critical of inclinations. [ABSTRACT FROM AUTHOR]

We propose a methodology to study the bifurcation sequences of frozen orbits when the second-order fundamental model of the satellite problem is augmented with the contribution of octupolar terms and relativistic corrections. The method is based on the analysis of twice-reduced closed normal forms expressed in terms of suitable combinations of the invariants of the Kepler problem, able to provide a clear geometric view of the problem. [ABSTRACT FROM AUTHOR]

Let us consider the restricted three-body problem. Analysis of the orbital motion of a spacecraft around planets or moons is presented taking into account the nonsphericity of the primaries and the perturbations coming from a third body in an elliptical and inclined orbit. In the specific case of a spacecraft designed to explore a planet, moon or asteroid, it is noteworthy the increasing use of the averaging methods. This is a very powerful technique to simulate, very fast, the main effects caused by the disturbers on the dynamics of the spacecraft. In this work, we focus on the averaged methods applied in different conditions. Some comparisons are presented between the single-averaged, double-averaged models and the complete model, that is, the unaveraged model based on direct integration of the Cartesian (x, y, z) coordinates. This unaveraged model is quite necessary as it provides all the requirements to validate the performance and evaluate the usefulness of the averaged models for each specific problem. In the first part of this paper, we describe briefly some well-known techniques to obtain the averaged model considering the nonsphericity of the primary as well as the perturbation due to the third body. On the other hand, this is a opportunity to mention some misprints and typos problems, in the literature related to this subject. We compared the performance of single- and double-averaging models, keeping the x–y–z unaveraged model as the baseline of reference. The case of a high lunar orbit (Nie et al. in Celest Mech Dyn Astron 131(29):1–31, 2019) considering the perturbation of the Earth seems to be instructive. Single-average model is more accurate than the double-average model in the analysis of the eccentricity evolution, but in some cases of the inclination evolution, the three models agree and the average models are both very accurate. When comparing the results, eventual typos were detected in some works related to the literature of this subject. In the second part of this paper, we detached some aspects of the dynamics of a probe around Mercury (Sect. 5) involved in frozen orbit (FO) and in "quasi-frozen orbit", (quasi-FO). Due to the interesting gravitational field of the planet and its proximity to the Sun, this is an important problem. Recently, many papers, not only on pure dynamics but on gravitational field of Mercury, have been published, according to references listed in this work. An exhaustive investigation on FO using double-averaging model was reported in Tresaco et al. (Celest Mech Dyn Astron 130(9):1–26, 2018). In this paper we revisit this problem, using x–y–z-model as a primary source of results. After a number of experiments, it was possible to use confidently the single averaging in many cases, for instance, in searching "quasi-FO" for Mercury planet. Although we do not include the effect of the radiation pressure, a number of our simulations can be compared with those given in Tresaco et al. (2018). [ABSTRACT FROM AUTHOR]

Satellite relative motion around the Earth has been thoroughly studied during the last two decades. However, considerably less attention has been given to the study of satellite relative motion around Mars. As the cost of space technologies decreases and more space missions are within reach, formation flying missions around Mars have the potential to benefit future exploration missions launched to the Red Planet. A key parameter in such missions will be the frequency at which the spacecraft need to perform formation-keeping maneuvers to compensate for unwanted drifts due to differential perturbations. The Martian J 3 and J 4 gravitational harmonics are significant enough to warrant a dedicated investigation of bounded satellite relative motion configurations. In this study, we derive conditions for bounded satellite relative motion in non-critical inclinations around Mars, while considering its gravitational harmonics up to J 4 . We first introduce a family of stable frozen orbits facilitating the implementation of formation flying and then apply differential nodal precession negation and differential periapsis rotation negation methods while considering the gravitational harmonics up to J 4 . Using this procedure, we demonstrate how the secular growth of the relative distance can be arrested during long time intervals. [ABSTRACT FROM AUTHOR]

Lunar frozen orbits, characterized by fixed eccentricity and argument of perigee on average, have been previously studied using different dynamical models. In this work, frozen orbits about the Moon are investigated on the basis of an averaged Hamiltonian. The gravitational field of the Moon is considered up to the seven zonal harmonic plus the third body perturbation (Earth). The third body is assumed to move in an elliptic inclined orbit. The averaging procedure is performed through Lie transformation method. We used the eccentricity-inclination diagrams to obtain a deep insight about the evolution of frozen orbits. For the reduced system, we found two frozen solutions which correspond to argument of perigee ω=π2,3π2. Moreover, we studied the evolution of the eccentricity and inclination as a function of time. The results showed that, for moderate altitude orbits, the eccentricity oscillates with small amplitudes around its initial value, while the inclination almost remains constant. In higher altitudes case, we observed that variations in eccentricity and inclination are larger than the moderate ones. The present dynamical model gives acceptable results for low initial eccentricity and inclination.

Nearly-circular Frozen Orbits (FOs) around axisymmetric bodies – or, quasi-circular Periodic Orbits (POs) around non-axisymmetric bodies – are of primary concern in the design of low-altitude survey missions. Here, we study very low-altitude orbits (down to 50 km) in a high-degree and order model of the Martian gravity field. We apply Prony's Frequency Analysis (FA) to characterize the time variation of their orbital elements by computing accurate quasi-periodic decompositions of the eccentricity and inclination vectors. An efficient, iterative filtering algorithm, previously applied to lunar orbiters, complements the method and is used to accurately compute the locations of POs/FOs, for a wide range of initial conditions. By defining the 'distance' of any orbit from the family of POs and using the relative amplitudes of the different components of the motion, we can build 'dynamical fate maps' that graphically depict the survivability of low-eccentricity, low-altitude orbits at every inclination, and can be used for efficient mission planning. While lowering the altitude generally enhances the effect of tesseral and sectorial gravity harmonics, we find this to have less consequence for low altitude Martian satellites, in contrast with the Lunar case. Hence, a high-degree (≃ 20 th) axisymmetric model is adequate for preliminary mission design at moderate altitudes, but should be complemented at low altitudes by the methods described here. All families of POs and their spectral decompositions can be accurately and effectively computed by continuation in arbitrarily complex Martian gravity models, as our filtering algorithm requires only short integration arcs. [ABSTRACT FROM AUTHOR]

Solar sails are a type of propulsion that uses solar radiation pressure to generate acceleration. The fundamental goal for any solar sail design is to provide a large and flat reflective film which requires a minimum of structural support mass. This research takes into account the non-sphericity of the central body, the perturbation of the third body and the solar radiation pressure to analyze the behavior of the orbit of a spacecraft when it has a solar sail around Mercury. We present an approach where we plot maps to analyze frozen orbits with longer lifetimes around Mercury. A set of initial conditions, which may contribute with the scientific missions planned to visit the planet Mercury in the next few years, are presented. Frozen orbits were found, i.e., orbits with smaller variation of the orbital elements. An approach is also presented to analyze the effect of the non-sphericity of Mercury on the motion of the spacecraft. In addition, the J2 and J3 zonal terms are also considered, as well as the C22 sectorial term. [ABSTRACT FROM AUTHOR]

Abstract In this research paper, new families of frozen orbits of a satellite orbiting the oblate as well as triaxial Moon are investigated. The Hamiltonian of the problem is constructed including the zonal harmonic coefficients of the Moon's gravity field up to J 4 and its triaxiality term J 2 , 2 . Using two successive canonical Lie transforms, the short and long periodic terms are eliminated from the Hamiltonian. The secular terms are retained up to first order plus the coupling quadrupole nonlinearity. New families of the critical roots of inclination are revealed, one of them is very close the polar orbits and the other is near to the usual critical inclination. The variations in the critical inclination due to the change in the eccentricity, in the semi-major axis and in the argument of periapsis are studied. A family of frozen apsidal line orbits is obtained and then is represented graphically. To guarantee these orbits, the solution for the periapsis argument is solved. This actually set out some restrictions on choosing the inclination satisfying the frozen argument of periapsis orbits. The perturbations in the critical inclination become significant for the high lunar orbits. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Taking into consideration a probe moving in an elliptical orbit around a celestial body, the possibility of determining conditions which lead to constant values on average of all the orbit elements has been investigated here, considering the influence of the planetary oblateness and the long-term effects deriving from the attraction of several perturbing bodies. To this end, three equations describing the variation of orbit eccentricity, apsidal line and angular momentum unit vector have been first retrieved, starting from a vectorial expression of the Lagrange planetary equations and considering for the third-body perturbation the gravity-gradient approximation, and then exploited to demonstrate the feasibility of achieving the above-mentioned goal. The study has led to the determination of two families of solutions at constant mean orbit elements, both characterised by a co-planarity condition between the eccentricity vector, the angular momentum and a vector resulting from the combination of the orbital poles of the perturbing bodies. As a practical case, the problem of a probe orbiting the Moon has been faced, taking into account the temporal evolution of the perturbing poles of the Sun and Earth, and frozen solutions at argument of pericentre 0 $$^{\circ }$$ or 180 $$^{\circ }$$ have been found. [ABSTRACT FROM AUTHOR]

An analysis to determine solutions with constant orbit elements has been carried out through a vectorial formulation of the perturbation equations, under the long-term influence due to the attraction of a disturbing body moving over an inclined elliptical orbit. After having gained a frozen orbital plane by assuming an orbital pole parallel or perpendicular to the perturbing body pole, the feasibility to get a frozen condition also on eccentricity or argument of pericentre has been demonstrated and several solutions have been proposed. Moreover, when the orbital pole is perpendicular to the perturbing body pole, a prime integral of motion, linking orbit eccentricity and argument of pericentre, has been retrieved. This prime integral has permitted the identification of solutions characterised by slow variations of eccentricity. A study to obtain orbits at constant eccentricity or argument of pericentre has also been carried out, regardless of the orbital plane evolution. This has highlighted how, while the solutions with a frozen apsidal line have to be determined by means of numerical methods, not pursued in this paper, the ones characterised by a null variation of eccentricity can be retrieved analytically. Examples, for a probe orbiting Mercury, have also been presented. [ABSTRACT FROM AUTHOR]

We offer a comprehensive description for the dynamics of a spacecraft affected by solar radiation pressure (SRP) orbiting a small body. Constrains are given for regions, in which stable motion is possible. For short and long time scales two different analytical frameworks are summarized and applied. (1) For time scales well below one heliocentric revolution we examine the "static" case, involving SRP fixed in both magnitude and direction. We demonstrate a closed-form solution for quasi terminator orbits using parabolic coordinates. (2) Next, we study the "dynamic" case where the asteroid is in an eccentric orbit about the Sun, involving changing solar aspect angle and distance. To solve this Augmented Hill Three-Body Problem (AH3BP), SRP effects are averaged over the anomaly of the orbiter. From this approximation we derive constrains for Sun-synchronous orbits in size and eccentricity. The findings of the analysis (1) and (2) are then applied to small- and medium-sized spacecraft orbiting specific asteroids, comets, dwarf planets and (for comparison) planets. We consider ranges of orbiter mass and surface area exposed to the Sun, as well as small body parameters, including mass and orbit. We show the resulting constrains on orbit size as well as parameters of Sun-synchronous orbits and frozen orbits in tables. While terminator orbits may only vary in size, quasi terminator orbits can cover wide regions best described in the parabolic coordinates of case (1). This region has four parameters for our orbit options. As alternative application for orbit stability we calculate constraints on orbit and particle sizes for dust particles. Numerical integration is used to validate the resilience of these solutions to further perturbation by third bodies or the small body's non-spherical shape. • We estimated orbital distances for spacecrafts in Sun-synchronous orbits around 9 small bodies. • Orbital shapes deviate greatly from Keplerian conic sections due to solar radiation pressure. • Quasi-terminator orbit dynamics are confined and described by up to four parabolas. • For orbiting dust a frozen state is assumed, limits to orbital sizes are given. • Numeric simulations add perturbing effects and validate predictions about orbit stability. [ABSTRACT FROM AUTHOR]

Generally, any initially-close satellites-chief and deputy-moving on orbits with slightly different orbital elements, will depart each other on locally unbounded relative trajectories. Thus, constraints on the initial conditions must be imposed to mitigate the chief-deputy mutual departure. In this paper, it is analytically proven that choosing the chief's orbit to be a frozen orbit can mitigate the natural relative drift of the satellites. Using mean orbital element variations, it is proven that if the chief's orbit is frozen, then the mean differential eccentricity is periodic, leading to a periodic variation of the differential mean argument of latitude. On the other hand, if the chief's orbit is non-frozen, a secular growth in the differential mean argument of latitude leads to a concomitant along-track separation of the deputy from the chief, thereby considerably increasing the relative distance evolution over time. Long-term orbital simulation results indicate that the effect of choosing a frozen orbit vis-à-vis a non-frozen orbit can reduce the relative distance drift by hundreds of meters per day. [ABSTRACT FROM AUTHOR]

Abstract: This paper investigates the effect of planetary oblateness and solar radiation pressure on the orbits of high area-to-mass spacecraft. A planar Hamiltonian model shows the existence of equilibrium orbits with the orbit apogee pointing towards or away from the Sun. These solutions are numerically continued to non-zero inclinations and considering the obliquity of the ecliptic plane relative to the equator. Quasi-frozen orbits are identified in eccentricity, inclination and the angle between the Sun-line and the orbit perigee. The long-term evolution of these orbits is then verified through numerical integration. A set of ‘heliotropic’ orbits with apogee pointing in the direction of the Sun is proposed for enhancing imaging and telecommunication on the day side of the Earth. The effects of J 2 and solar radiation pressure are exploited to obtain a passive rotation of the apsides line following the Sun; moreover the effect of solar radiation pressure enables such orbits at higher eccentricities with respect to the J 2 only case. [Copyright &y& Elsevier]

Low-altitude, near-polar orbits are very desirable for scientific missions to study the natural satellites of the planets of the Solar System, such as Europa, that is one of the natural satellites of Jupiter. The problem is analyzed considering that an artificial satellite is orbiting Europa and that this spacecraft is perturbed by the nonuniform distribution of mass of the planetary satellite (J2, J3, C22) and by the gravitational attraction of the third-body. We present an analytical theory using the averaged model and applications were done by performing numerical integrations of the analytical equations developed. Using the averaged method, several frozen orbits were obtained. Some of them has low inclination, low altitude and long lifetime. Numerical simulations are performed using the software Mercury, to compare the results obtained using the analytical theory. [ABSTRACT FROM AUTHOR]

Abstract: For long enough tethers, the coupling of the attitude and orbital dynamics may show non-negligible effects in the orbital motion of a tethered satellite about a central body. In the case of fast rotating tethers the attitude remains constant, on average, up to second order effects. Besides, for a tether rotating in a plane parallel to the equatorial plane of the central body, the attitude–orbit coupling effect is formally equal to the perturbation of the Keplerian motion produced by the oblateness of the central body and, therefore, may have a stabilizing effect in the orbital dynamics. In the case of a tethered satellite in a low lunar orbit, it is demonstrated that feasible tether lengths can help in modifying the actual map of lunar frozen orbits. [Copyright &y& Elsevier]

Abstract: The dynamics of orbits around planetary satellites, taking into account the gravitational attraction of a third-body and the non-uniform distribution of mass of the planetary satellite, is studied. The Hamiltonian considered is explicitly time-dependent. Conditions for frozen orbits are presented. Low-altitude, near-polar orbits, very desirable for scientific missions to study planetary satellites such as the Jupiter’s moon Europa, are analyzed. Lifetimes for these orbits are computed through the single and double averaged method. Comparison between the results obtained by the single and double averaged method is presented. The single-averaged model is more realistic, since it does not eliminate the term due to the equatorial ellipticity of the planetary satellite as done by the double-averaged problem. Considering the single-averaged method, we found unstable frozen orbits where the satellite does not impact with the surface of Europa for at least 200days. We present an approach using the unaveraged disturbing potential to analyze the effects of these terms in the amplitude of the eccentricity. [Copyright &y& Elsevier]

Abstract: The behavior of low altitude near-circular lunar orbits is a key design issue for some missions related to the physical exploration of the Moon. Because of its masconian character, the gravity field of the Moon requires higher order truncations to give a realistic description of the long-term behavior of low-lunar orbits. We show that the required understanding of the dynamical behavior in the vicinity of the Moon can be reached through the combination of analytical techniques and periodic orbits computation. A model that consists of a high degree, zonal truncation of the Selenopotential superimposed to the Earth mass-point attraction is used to explore the existence and orbital characteristics of long-lifetime orbits close to the Moon at any inclination. The averaging provides a global view on the frozen orbit''s geometry and local descriptions of the averaged flow. But it also makes available the short-period terms of the transformation from mean to osculating elements. A refinement of the osculating elements by means of differential corrections allows to compute lunar repeat ground-trace orbits in high fidelity potentials without restricting to zonal models. [Copyright &y& Elsevier]

Frozen orbits are always important foci of orbit design because of their valuable characteristics that their eccentricity and argument of pericentre remain constant on average. This study investigates quasi-circular frozen orbits and examines their basic nature analytically using two different methods. First, an analytical method based on Lagrangian formulations is applied to obtain constraint conditions for Martian frozen orbits. Second, Lie transforms are employed to locate these orbits accurately, and draw the contours of the Hamiltonian to show evolutions of the equilibria. Both methods are verified by numerical integrations in an 80 × 80 Mars gravity field. The simulations demonstrate that these two analytical methods can provide accurate enough results. By comparison, the two methods are found well consistent with each other, and both discover four families of Martian frozen orbits: three families with small eccentricities and one family near the critical inclination. The results also show some valuable conclusions: for the majority of Martian frozen orbits, argument of pericentre is kept at 270° because J has the same sign as J; while for a minority of ones with low altitude and low inclination, argument of pericentre can be kept at 90° because of the effect of the higher degree odd zonals; for the critical inclination cases, argument of pericentre can also be kept at 90°. It is worthwhile to note that there exist some special frozen orbits with extremely small eccentricity, which could provide much convenience for reconnaissance. Finally, the stability of Martian frozen orbits is estimated based on the trace of the monodromy matrix. The analytical investigations can provide good initial conditions for numerical correction methods in the more complex models. [ABSTRACT FROM AUTHOR]

In this paper, Mathematica module frozen orbits characteristics is developed and illustrated numerically. [ABSTRACT FROM AUTHOR]

In this paper we present an analytical theory with numerical simulations to study the orbital motion of lunar artificial satellites. We consider the problem of an artificial satellite perturbed by the non-uniform distribution of mass of the Moon and by a third-body in elliptical orbit (Earth is considered). Legendre polynomials are expanded in powers of the eccentricity up to the degree four and are used for the disturbing potential due to the third-body. We show a new approximated equation to compute the critical semi-major axis for the orbit of the satellite. Lie-Hori perturbation method up to the second-order is applied to eliminate the terms of short-period of the disturbing potential. Coupling terms are analyzed. Emphasis is given to the case of frozen orbits and critical inclination. Numerical simulations for hypothetical lunar artificial satellites are performed, considering that the perturbations are acting together or one at a time. [ABSTRACT FROM AUTHOR]

Preliminary mission design for planetary satellite orbiters requires a deep knowledge of the long term dynamics that is typically obtained through averaging techniques. The problem is usually formulated in the Hamiltonian setting as a sum of the principal part, which is given through the Kepler problem, plus a small perturbation that depends on the specific features of the mission. It is usually derived from a scaling procedure of the restricted three body problem, since the two main bodies are the Sun and the planet whereas the satellite is considered as a massless particle. Sometimes, instead of the restricted three body problem, the spatial Hill problem is used. In some cases the validity of the averaging is limited to prohibitively small regions, thus, depriving the analysis of significance. We find this paradigm at Enceladus, where the validity of a first order averaging based on the Hill problem lies inside the body. However, this fact does not invalidate the technique as perturbation methods are used to reach higher orders in the averaging process. Proceeding this way, we average the Hill problem up to the sixth order obtaining valuable information on the dynamics close to Enceladus. The averaging is performed through Lie transformations and two different transformations are applied. Firstly, the mean motion is normalized whereas the goal of the second transformation is to remove the appearance of the argument of the node. The resulting Hamiltonian defines a system of one degree of freedom whose dynamics is analyzed. [ABSTRACT FROM AUTHOR]

Abstract: In the framework of the elliptic restricted three-body problem we develop an analytical theory for spacecraft motion close to Mercury. Besides the perturbations due to the gravity of the Sun and Mercury and the eccentricity of Mercury''s orbit around the Sun, i.e., the elliptic restricted three-body problem, the theory includes the effects of the oblateness and the possible latitudinal asymmetry of Mercury, and is valid for any eccentricity of the spacecraft''s orbit. The initial Hamiltonian defines a non-autonomous but periodic dynamical system of two degrees of freedom. The mean motion of the spacecraft and the time are averaged using two successive Lie–Deprit transformations. The resulting Hamiltonian defines a one degree of freedom system and depends upon three essential parameters. When the latitudinal asymmetry coefficient vanishes the flow of this system is entirely analyzed through the discussion of the occurrence of its (relative) equilibria and bifurcations in accordance with the parameters the problem depends upon. Frozen orbits of the initial system together with their stability are obtained related to the relative equilibria. If the latitudinal asymmetry of Mercury is taken into account, the equatorial symmetry of the problem is broken and introduces important changes in the dynamics. A variety of tests show a very good agreement between averaged and non-averaged models, and the reliability of the theory is further checked by performing long-term integrations in ephemeris. [Copyright &y& Elsevier]

• In this work we investigate moderate altitude frozen orbits about the Moon. • The perturbations considered are due to lunar gravity field and a third body. • The Lie method is utilized to carry out the required transformations. • We found two frozen solutions for the reduced system. Lunar frozen orbits, characterized by fixed eccentricity and argument of perigee on average, have been previously studied using different dynamical models. In this work, frozen orbits about the Moon are investigated on the basis of an averaged Hamiltonian. The gravitational field of the Moon is considered up to the seven zonal harmonic plus the third body perturbation (Earth). The third body is assumed to move in an elliptic inclined orbit. The averaging procedure is performed through Lie transformation method. We used the eccentricity-inclination diagrams to obtain a deep insight about the evolution of frozen orbits. For the reduced system, we found two frozen solutions which correspond to argument of perigee ω = π 2 , 3 π 2 . Moreover, we studied the evolution of the eccentricity and inclination as a function of time. The results showed that, for moderate altitude orbits, the eccentricity oscillates with small amplitudes around its initial value, while the inclination almost remains constant. In higher altitudes case, we observed that variations in eccentricity and inclination are larger than the moderate ones. The present dynamical model gives acceptable results for low initial eccentricity and inclination. [ABSTRACT FROM AUTHOR]

In this work, the periodic orbits' phase portrait of the zonal J 2 + J 3 problem is studied. In particular, we center our attention on the periodic orbits of the third kind in the Poincaré sense using the averaging theory of dynamical systems. We find three families of polar periodic orbits and four families of inclined periodic orbits for which we are able to state their explicit expressions. [ABSTRACT FROM AUTHOR]