We describe the families of periodic orbits in the 2-dimensional 1/2 retrograde resonance at mass ratio \(10^{-3}\), analyzing their stability and bifurcations into 3-dimensional periodic orbits. We explain the role played by periodic orbits in adiabatic resonance capture, in particular how the proximity between a stable family and an unstable family with a nearly critical segment, associated with

Quasi-satellite orbits (QSOs) are considered by JAXA’s MMX mission, in which CNES is involved, for the scientific observation of the Martian moon Phobos prior to landing and sample return operations. These periodic orbits, originally defined in the Mars-Phobos circular restricted three-body problem, generically lose periodicity once the eccentricity of Phobos’ orbit is taken into account. In this case

Constellation design theory has been studied extensively. However, analysis of longitude-dependent perturbation effects on inter-satellite distance drift has not received much attention. In addition to oblateness-related perturbations, sectoral and tesseral perturbations are non-negligible for low Earth orbits, due to their effect on inter-satellite distance evolution. This paper introduces the idea

Loosely captured orbits with circulating and pulsating eccentricity vectors have a variety of attractive mission design properties, including low insertion costs, near-circular and highly eccentric phases, mid- to high inclinations, long-term stability, and spatially distributed close approaches. Such orbits are the known result of averaging third-body perturbations over the spacecraft and system motions

Since Poincaré, the three-body problem is known to be chaotic and is believed to lack a general deterministic solution. Instead, decades ago a statistical solution was marked as a goal. Yet, despite considerable progress, all extant approaches display two flaws. First, probability was equated with phase space volume, thereby ignoring the fact that significant regions of phase space describe regular

The Bicircular Problem (BCP) is a periodic time dependent perturbation of the Earth-Moon Restricted Three-Body Problem that includes the direct gravitational effect of the Sun. In this paper we use the BCP to study the existence of Halo-like orbits around \(L_2\) in the Earth-Moon system taking into account the perturbation of the Sun. By means of computing families of 2D invariant tori, we show that

In this paper we look forward to obtain a global picture of simple transfer opportunities from Lunar to Sun-Earth libration orbits, useful for a preliminary design of these mission scenarios. Considering the trajectory of the Chinese Change’2 spacecraft as a reference case, the main transfer families are characterized and classified. In a second step, the results are analyzed and extended to departures

The first pair of satellites belonging to the European Global Navigation Satellite System (GNSS)—Galileo—has been accidentally launched into highly eccentric, instead of circular, orbits. The final height of these two satellites varies between 17,180 and 26,020 km, making these satellites very suitable for the verification of the effects emerging from general relativity. We employ the post-Newtonian

This paper presents a collection of analytical formulae that can be used in the long-term propagation of the motion of a spacecraft subject to low-thrust acceleration and orbital perturbations. The paper considers accelerations due to: a low-thrust profile following an inverse square law, gravity perturbations due to the central body gravity field and the third-body gravitational perturbation. The

We describe a comprehensive model for systems locked in the Laplace resonance. The framework is based on the simplest possible dynamical structure provided by the Keplerian problem perturbed by the resonant coupling truncated at second order in the eccentricities. The reduced Hamiltonian, constructed by a transformation to resonant coordinates, is then submitted to a suitable ordering of the terms

The present work studies the continuation class of the regular n-gon solution of the n-body problem. For odd numbers of bodies between \(n = 3\) and \(n = 15\), we apply one parameter numerical continuation algorithms to the energy/frequency variable and find that the figure eight choreography can be reached starting from the regular n-gon. The continuation leaves the plane of the n-gon and passes

This paper defines a set of six non-singular orbital elements designed specifically for the characterization of uncertainty in the state of a resident space object in circular or elliptic orbit and demonstrates their use for uncertainty propagation in the context of the perturbed two-body problem of orbital mechanics. As evidenced by the time evolution of the Cramér–von Mises test statistic, representation

The second of Eq shows that the coefficient of fifth-order term for the general precession in longitude.

In the present effort, we revisit the Shannon entropy approach for the study of both the extent and the rate of diffusion in a given dynamical system. In particular, we provide a theoretical and numerical study of the dependence of the formulation on the parameters of the method. We succeed in deriving not only a diffusion coefficient, \(D_{S}\), but also an estimate of the macroscopical instability

The present work studies the robustness of certain basic homoclinic motions in an equilateral restricted four-body problem. The problem can be viewed as a two-parameter family of conservative autonomous vector fields. The main tools are numerical continuation techniques for homoclinic and periodic orbits, as well as formal series methods for computing normal forms and center stable/unstable manifold

In this paper, we study the problem of determining whether a global family of even periodic solutions of a generalized Sitnikov problem, which emerges from a circular generalized Sitnikov problem, continues for all values of eccentricity in [0, 1) or ends in the equilibrium.

Phased formations of satellites provide an important means of keeping multiple satellites in close proximity without becoming dangerously close to one another. In order to minimize the amount of propellant necessary to keep a set of satellites in a phased formation, this paper presents a new condition for in-plane linearized secular \(J_2\)-invariance in highly eccentric orbits. A maintenance strategy

The paper presents a novel 3-dimensional shape-based algorithm which extends the domain of analytical solutions to planeto-centric mission scenarios, which classically entail even thousands revolutions to transfer to the final orbit. Thanks to the strong physical meaning the proposed method keeps while shaping the trajectory, the method succeeds in outputting a solution close to the real optimum. The

This paper studies the attitude dynamics of a rigid body in a Keplerian orbit. We show that the use of Classical Rodrigues Parameters for the attitude motion of the rigid body subject to gravity-gradient torques enables us to characterize the equilibria associated with the rotational motion about its mass center. A parametric study of the stability of equilibria is conducted to show that large oscillations

The effects of third-body perturbations on natural and artificial satellite orbits have been studied extensively. However, much less attention has been given to the case considering the orbital inclination of the third body. In this paper, it is shown that a perturbed orbit, even with a small initial inclination—much less than the critical Kozai inclination of 39.23 degrees—may significantly increase

We present a computer assisted proof of the full listing of central configurations for spatial n-body problem for \(n=5\) and 6, with equal masses. For each central configuration, we give a full list of its Euclidean symmetries. For all masses sufficiently close to the equal masses case, we give an exact count of configurations in the planar case for \(n=4,5,6,7\) and in the spatial case for \(n=4

Recent works demonstrated that the dynamics caused by the planetary oblateness coupled with the solar radiation pressure can be described through a model based on singly averaged equations of motion. The coupled perturbations affect the evolution of the eccentricity, inclination and orientation of the orbit with respect to the Sun–Earth line. Resonant interactions lead to non-trivial orbital evolution

In this work, we investigate the Earth–Moon system, as modeled by the planar circular restricted three-body problem, and relate its dynamical properties to the underlying structure associated with specific invariant manifolds. We consider a range of Jacobi constant values for which the neck around the Lagrangian point \(L_1\) is always open, but the orbits are bounded due to Hill stability. First,

We present an approach to estimate an upper bound for the impact probability of a potentially hazardous asteroid when part of the force model depends on unknown parameters whose statistical distribution needs to be assumed. As case study, we consider Apophis’ risk assessment for the 2036 and 2068 keyholes based on information available as of 2013. Within the framework of epistemic uncertainties, under

In this paper, we develop an analytical stability criterion for a five-body symmetrical system, called the Caledonian Symmetric Five-Body Problem (CS5BP), which has two pairs of equal masses and a fifth mass located at the centre of mass. The CS5BP is a planar problem that is configured to utilise past–future symmetry and dynamical symmetry. The introduction of symmetries greatly reduces the dimensions

We study the reduction and regularization processes of perturbed Keplerian systems from an astronomical point of view. Our approach connects axially symmetric perturbed 4-DOF oscillators with Keplerian systems, including the case of rectilinear solutions. This is done through a preliminary reduction recently studied by the authors. Then, the reduction program continues by removing the Keplerian energy

This paper presents a derivative-free method for computing approximate solutions to the uncertain Lambert problem (ULP) and the reachability set problem (RSP) while utilizing higher-order sensitivity matrices. These sensitivities are analogous to the coefficients of a Taylor series expansion of the deterministic solution to the ULP and RSP, and are computed in a derivative-free and computationally

When an asteroid has a few observations over a short time span the information contained in the observational arc could be so little that a full orbit determination may be not possible. One of the methods developed in recent years to overcome this problem is based on the systematic ranging and combined with the Admissible Region theory to constrain the poorly-determined topocentric range and range-rate

The beta angle, an angle formed by the sunlight and a spacecraft orbital plane, is an important parameter for science orbit design of orbiter missions. This angle defines lighting conditions and eclipse occurrences, and is used for science observation planning. Not only is this parameter perturbed by the irregular gravity field of the primary body, it varies with the body’s motion around the Sun. Investigating

The nutations of Mars are about to be estimated to a few milliarcseconds accuracy with the radioscience experiments onboard InSight and ExoMars 2022. The contribution to the nutations due to the liquid core and tidal deformations will be detected, allowing to constrain the interior of Mars. To avoid introducing systematic errors in the determination of the core properties, an accurate precession and

To describe the rotation of a “rigid mantle + liquid core” system, the Poincaré–Hough–Zhukovsky equations are used. An analysis is made of the previously obtained (Ol’shanskii in Celest Mech Dyn Astron 131(12):Article number:57, 2019) conditions for regular precession of a system that does not have an axial symmetry. Upon receipt of the conditions, it is considered that the external torque can be neglected

The circular restricted three-body model is widely used for astrodynamical studies in systems where two major bodies are present. However, this model relies on many simplifications, such as point-mass gravity and planar, circular orbits of the bodies, and limiting its accuracy. In an effort to achieve higher-fidelity results while maintaining the autonomous simplicity of the classic model, we employ

Over the course of the last decade, observations of highly inclined (orbital inclination i > 60\(^\circ \)) trans-Neptunian objects (TNOs) have posed an important challenge to current models of solar system formation (Levison et al. in Icarus 196(1):258–273, 2008; Nesvorný in Astron J 150:73, 2015). These remarkable minor planets necessitate the presence of a distant reservoir of strongly out-of-plane

A previous study showed that a fingerprint of the initial shape of synthetic Oort clouds was detectable in the flux of “new” long-period comets. The present study aims to explain in detail how such a fingerprint is propagated by different classes of observable comets to improve the detection of fingerprints. It appears that three main long-term behaviors of observable comets are involved in this propagation:

This work introduces two Monte Carlo (MC)-based sampling methods, known as line sampling and subset simulation, to improve the performance of standard MC analyses in the context of asteroid impact risk assessment. Both techniques sample the initial uncertainty region in different ways, with the result of either providing a more accurate estimate of the impact probability or reducing the number of required

The stable distant retrograde orbits (DROs) around the Moon are considered as potential parking orbits for cislunar stations that are important facilities in cislunar space. Transfer orbits from DROs to lunar orbits will be fundamental and routine for operations of the cislunar stations. This paper studies transfer orbits from DROs to low lunar orbits with inclinations between 0° and 90°. Ten DROs

We deal with the orbit determination problem for a class of maps of the cylinder generalizing the Chirikov standard map. The problem consists of determining the initial conditions and other parameters of an orbit from some observations. A solution to this problem goes back to Gauss and leads to the least squares method. Since the observations admit errors, the solution comes with a confidence region

We prove the existence of periodic orbits of the two fixed centers problem bifurcating from the Kepler problem. We provide the analytical expressions of these periodic orbits when the mass parameter of the system is sufficiently small.

The set of transverse homoclinic intersections for a saddle-focus equilibrium in the planar equilateral restricted four-body problem admits certain simple homoclinic orbits which form the skeleton of the complete homoclinic intersection—or homoclinic web. In the present work, the planar restricted four-body problem is viewed as an invariant subsystem of the spatial problem, and the influence of this

In the first paragraph of the Introduction, the last sentence should read as “ Vinti.

Spacecraft and natural objects orbiting an active comet are perturbed by gas drag from the coma. These gases expand radially at about 0.5 km/s, much faster than orbital velocities that are on the order of meters per second. The coma has complex gas distributions and is difficult to model. Accelerations from gas drag can be on the same order of gravity and are currently poorly understood. Semi-analytical

Luna 3 (or Lunik 3 in Russian sources) was the first spacecraft to perform a flyby of the Moon. Launched in October 1959 on a translunar trajectory with large semimajor axis and eccentricity, it collided with the Earth in late March 1960. The short, 6-month dynamical lifetime has often been explained through an increase in eccentricity due to the Lidov–Kozai effect. However, the classical Lidov–Kozai

We study planar and three-dimensional retrograde periodic orbits, using the model of the restricted three-body problem (RTBP) with the Sun and Neptune as primaries and focusing on the dynamics of resonant trans-Neptunian objects (TNOs). The position and the stability character of the periodic orbits can provide important piece of information on the stability and long-term evolution of small TNOs in

We perform statistical wavelet analysis of the Main Belt asteroids, seeking statistically significant asteroid families. The goal is to test the new wavelet analysis algorithm and to compare its results with more traditional methods like the hierarchic clustering. We first consider several 1D distributions for various physical and orbital parameters of asteroids. Then, we consider three bivariate distributions

The current giant planet region is a transitional zone where transneptunian objects (TNOs) cross in their way to becoming Jupiter Family Comets. Their dynamical behavior is conditioned by the intrinsic dynamical features of TNOs and also by the encounters with the giant planets. We address the Giant Planet Crossing (GPC) population (those objects with 5.2 au \(< q < 30\,\hbox {au}\)) studying their

We consider, after Albouy–Moeckel, the inverse problem for collinear central configurations: Given a collinear configuration of n bodies, find positive masses which make it central. We give some new estimates concerning the positivity of Albouy–Moeckel Pfaffians: We show that for any homogeneity \(\alpha \) and \(n\le 6\) or \(n\le 10\) and \(\alpha =1\) (computer assisted) the Pfaffians are positive

Third-body perturbations have been extensively studied in recent years. Almost all previous works, however, assumed that the perturbations are caused by a third body that orbits the primary on a radius larger than the semimajor axis of the perturbed object. This assumption is justified as long as the primary is not accompanied by a third body in close orbit. In this work, we present an analytic model

The exploration of small bodies in the Solar system and the ability to perform remote or in situ science is tied to the understanding of the dynamical environment of such objects. As such, the evaluation of the gravity field arising from small bodies is key to this understanding. However, remote observations can only produce shape estimates, from which only uncertain gravity fields can be computed

Near rectilinear halo orbits (NRHOs), a subset of the \(L_1\) and \(L_2\) halo orbit families, are strong candidates for a future inhabited facility in line with the goals for NASA’s long-term cislunar transportation network. Higher-period dynamical structures that bifurcate from the NRHO region of the \(L_2\) halo family offer additional insight into motion departing from or arriving into the vicinity

We examine a 2-DOF Hamiltonian system modeling some previously unexplored dynamical effects in first-order mean motion resonances in the spatial circular restricted three-body problem “star-planet-asteroid.” In distinction from the well-known integrable model of Sessin and Ferraz-Mello (Celest Mech Dyn Astron 32:307–332, 1984) and Wisdom (Celest Mech Dyn Astron 38:175–180, 1986), in our analysis, we

The existence of Lagrange coefficients under two-body dynamics is well known, and the concept forms the basis of robust algorithms for solving a variety of fundamental astrodynamics problems. The Lagrange coefficients are generalized to Vinti theory in the present work, where the Vinti potential describes the dynamics of small objects like spacecraft orbiting an oblate body. Exact expressions for the

Having 50 years of unique observations, lunar laser ranging (LLR) is used to test central elements of Einstein’s theory of relativity, like a possible temporal variation of the gravitational constant or metric parameters. Here, we focused on a possible violation of the equivalence principle (EP) due to assumed dark matter in the galactic center. According to the EP, Earth and Moon experience the same

The Colombo top is a basic model in the rotation dynamics of a celestial body moving on a precessing orbit and perturbed by a gravitational torque. The paper presents a detailed study of analytical solution to this problem. By solving algebraic equations of degree 4, we provide the expressions for the extreme points of trajectories as functions of their energy. The location of stationary points (known

The increasing interest towards exploring small bodies in the Solar system has paved the way for the investigation of different gravity field models allowing robust orbit design and dynamical environment characterization. Among these, the spherical harmonics representation and the polyhedral gravity model are the most employed and studied—the former for historical reasons and the latter due to the

The Kepler potential \(\propto {-1/r}\) and the harmonic potential \(\propto r^2\) share the following remarkable property: In either of these potentials, a bound test particle orbits with a radial period that is independent of its angular momentum. For this reason, the Kepler and harmonic potentials are called isochrone. In this paper, we solve the following general problem: Are there any other isochrone

The purpose of this work is to determine the location and stability of the Cassini states of a celestial body with an inviscid fluid core surrounded by a perfectly rigid mantle. Both situations where the rotation speed is either non-resonant or trapped in a \(p\!:\!1\) spin–orbit resonance where p is a half integer are addressed. The rotation dynamics is described by the Poincaré–Hough model which

We propose an adaptation of the semilinear algorithm for the prediction of the impact corridor on ground of an Earth-impacting asteroid. The proposed algorithm provides an efficient tool, able to reliably predict the impact regions at fixed altitudes above ground with 5 orders of magnitudes less computations than Monte Carlo approaches. Efficiency is crucial when dealing with imminent impactors, which

This work focuses on the definition of satellite constellations whose secular relative distributions are invariant under the perturbation produced by the Earth gravitational potential. This is done by defining the satellite distribution directly in the Earth-Centered–Earth-Fixed frame of reference and using the along-track time distances between satellites to define the satellite constellation configuration

In this work, we consider the Kepler problem with a family of singular dissipations of the form \(-\frac{k}{|x|^\beta }\dot{x},\quad k>0, \beta >0.\) We present some results about the qualitative dynamics as \(\beta \) increases from zero (linear drag) to infinity. In particular, we detect some threshold values of \(\beta \), for which qualitative changes in the global dynamics occur. In the case \(\beta

Designing optimal spacecraft trajectories is a critical task for any mission design. In particular, mission designers seek to exploit from the combined effects of planetary gravity-assist maneuvers and electric propulsions systems to reduce both the flight time and propellant consumption. In order to obtain more realistic results, disturbances such as (1) gravitational force of secondary bodies, (2)