Abstract: Abstract 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 proposed approach allows to easily formalize practical mission constraints, such as maximum thrust threshold and eclipses; free and fixed time of flight is manageable as well. The approach is almost completely analytic, which is beneficial as it significantly decreases the computational load. It is well suited for complex mission scenarios and for fast detection near optimal solutions to support the whole mission design. PubDate: 2021-01-11

Abstract: Abstract 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 are induced due to the energy exchange between the pitch and roll–yaw motions, specifically near the 2:1 resonant commensurability regions. A visualization tool is developed to study these pitch oscillations and gain insight into the rigid body motion near internal resonance conditions. A measure of coupling between the pitching and roll–yaw motions is developed to quantify the energy exchange utilizing information from the state transition matrix. PubDate: 2021-01-10

Abstract: Abstract 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 its inclination due to the inclination of the third-body’s orbit. To study this phenomenon, a new method for finding the inclination of the perturbed body is proposed based on a coordinate transformation. This enables the derivation of an analytical solution for the orbital inclination, predicting the long-term evolution thereof when the third body’s orbit is inclined. It is found that the amplitude of the inclination depends on the inclination of the third body, and on the relative angle between the orbital planes of the perturbed orbiter and the perturbing body. Numerical simulations illustrate the accuracy of the proposed methodology for predicting the long-term evolution of the orbital inclination. PubDate: 2021-01-06

Abstract: Abstract 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 \(n=4,5,6\) . PubDate: 2020-12-08

Abstract: Abstract 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 that can be exploited in mission design. Moreover, the dynamics in the vicinity of each resonance can be analytically described by a resonant model that provides the location of the central and hyperbolic invariant manifolds which drive the phase space evolution. The classical tools of the dynamical systems theory can be applied to perform a preliminary mission analysis for practical applications. On this basis, in this work we provide a detailed derivation of the resonant dynamics, also in non-singular variables, and discuss its properties, by studying the main bifurcation phenomena associated with each resonance. Last, the analytical model will provide a simple analytical expression to obtain the area-to-mass ratio required for a satellite to deorbit from a given altitude in a feasible timescale. PubDate: 2020-11-27

Abstract: Abstract 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 show that the system displays three different dynamical scenarios in the neighborhood of the Moon: two mixed ones, with regular and chaotic orbits, and an almost entirely chaotic one in between. We then analyze the transitions between these scenarios using the monodromy matrix theory and determine that they are given by two specific types of bifurcations. After that, we illustrate how the phase space configurations, particularly the shapes of stability regions and stickiness, are intrinsically related to the hyperbolic invariant manifolds of the Lyapunov orbits around \(L_1\) and also to the ones of some particular unstable periodic orbits. Lastly, we define transit time in a manner that is useful to depict dynamical trapping and show that the traced geometrical structures are also connected to the transport properties of the system. PubDate: 2020-11-25

Abstract: Abstract 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 the independence and non-correlation assumption, we assign parametric families of distributions to the physical properties of Apophis that define the Yarkovsky perturbation and in turn the future orbital evolution of the asteroid. We find \({\mathrm{IP}}\le 5\times 10^{-5}\) for the 2036 keyhole and \({\mathrm{IP}}\le 1.6\times 10^{-5}\) for the 2068 keyhole. These upper bounds are largely conservative choices due to the rather wide range of statistical distributions that we explored. PubDate: 2020-11-25

Abstract: Abstract 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 of the five-body problem. Sundman’s inequality is applied to derive boundary surfaces to the allowed real motion of the system. This enables the derivation of a stability criterion valid for all time for the hierarchical stability of the CS5BP. We show that the hierarchical stability depends solely on the Szebehely constant \(C_0\) which is a dimensionless function involving the total energy and angular momentum. We then explore the effect on the stability of the whole system of varying the relative sizes of the masses. The CS5BP is hierarchically stable for \(C_0 > 0.065946\) . This criterion can be applied in the investigation of the stability of quintuple hierarchical stellar systems and symmetrical planetary systems. PubDate: 2020-11-25

Abstract: Abstract 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. For each value of the semi-major axis, we explain the astronomical meaning of the sextuples defining the orbit space \(\mathbb {S}^2\times \mathbb {S}^2\) and its connection with the orbital elements. More precisely, we present alternative sextuple coordinates for the set of bounded Keplerian orbits that ‘separate’ the node of the orbital plane from the argument of perigee giving the Laplace vector in that plane. Still, the reduction of the axial symmetry defined by the third component of the angular momentum is performed. For the thrice reduced space \(\varGamma _{0,L,H}\) we propose the Cushman–Deprit coordinates, a variant to the set given by Cushman. The main feature of these variables is that they are all with the same dimensions, which is convenient for the normalization procedure. As an application of the proposed scheme, we study the spatial lunar problem. PubDate: 2020-11-25

Abstract: Abstract 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 tractable manner. The coefficients are computed by minimizing least squared error over the domain of the input probability density function (PDF), and represent the nonlinear mapping of the input PDF to the output PDF. A non-product quadrature method known as the conjugate unscented transform is used to compute the multidimensional expectation values necessary to determine these coefficients with the minimal number of full model propagations. Numerical simulations for both the ULP and the RSP are provided to validate the developed methodology and illustrate potential applications. The benefits and limitations of the presented method are discussed. PubDate: 2020-11-01

Abstract: Abstract 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 result is a set of orbits compatible with the observations, the Manifold Of Variations, a two-dimensional compact manifold parametrised over the Admissible Region. Such a set of orbits represents the asteroid confidence region and is used for short-term hazard predictions. In this paper we present the Manifold Of Variations method and make a detailed analysis of the related probabilistic formalism. PubDate: 2020-10-30

Abstract: Abstract 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 evolution of the beta angle is therefore critical for science orbit design. This paper analyzes the J2-perturbed beta angle evolution via orbit averaging and year averaging, and derives conditions to constrain the short- and long-term evolutions of the beta angle. The orbit averaging analysis is further extended to provide a relaxed orbit condition that allows flexible science orbit design while naturally maintaining the beta angle evolution within science requirements. These analyses are carried out for an arbitrary rotation pole direction as it defines the orientation of the irregular gravity field seen in an inertial frame. The analytical work is numerically demonstrated with science orbit design for the Psyche mission, a recently selected mission of the NASA’s Discovery Program. PubDate: 2020-10-11

Abstract: Abstract 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 nutation model for a rigidly behaving Mars is needed. Here, we develop such a model with adequate accuracy based on the Torque approach and compare it to previous models. We include in the model the forcings by the Sun, Phobos, Deimos, and the other planets of the solar system. We also include the geodetic precession and nutations. We use semi-analytical developments for the solar and planetary torques, and analytical solutions for the effect of Phobos and Deimos and for the geodetic precession and nutations. With a truncation criterion of 0.025 milliarcseconds in prograde and/or retrograde amplitude, we identify 43 nutation terms. The uncertainty on our solution mainly derives from the observational uncertainty on the current determination of the precession rate of Mars. Uncertainties related to our modeling choices are negligible in comparison. Given the current determination of the precession rate ( \(7608.3\pm 2.1\) mas/yr, Konopliv et al. in Icarus 274:253–260, 2016. https://doi.org/10.1016/j.icarus.2016.02.052), our model predicts a dynamical flattening \(H_{{D}}=0.00538017\pm 0.00000148\) and a normalized polar moment of inertia \(C/\mathrm{MR}^2=0.36367\pm 0.00010\) for Mars. PubDate: 2020-09-29

Abstract: Abstract 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 as, for example, for free-floating planetary bodies. In the case when the axis of proper rotation is one of the principal axes of inertia, the formulas for the rates of precession and proper rotation have been simplified. For a particular case, when the shape of the core differs little from spherical, it is shown that the precession and proper rotation rates differ from the rates of empty axisymmetric rigid mantle by values of the first order of smallness. The ratio of these rates differs from the ratio for the rigid mantle by the second-order value. For a system with the axis of proper rotation deviated from a principal axis of inertia, the expression of the principal moments of inertia of the mantle through the moments of inertia of the core and one arbitrary parameter is found. The formulas for finding the angular velocities and for determining the position of the axis of proper rotation relative to the mantle are written in simple parametric form. The possibility of regular precession with the axis of proper rotation, which does not coincide with any principal axes, is studied in the case when the shape of the core differs little from the spherical one. Examples of elongated non-axisymmetric systems that allow precession with the axis of proper rotation deviated from principal axis of inertia are given. PubDate: 2020-09-28

Abstract: Abstract 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 zonal harmonic perturbations since they are symmetric about the z-axis, thus bearing no time-dependent terms. In this study, we focus on how these perturbations affect the dynamic environment near the secondary body in real systems. Concise, easily implementable equations for gravitational potential, particle motion, and modified Jacobi constant in the perturbed model are presented. These perturbations cause a change in the normalized mean motion, and two different formulations are addressed for assigning this new value. The shifting of collinear equilibrium points in many real systems due to \(J_2\) of each body is reported, and we study how families of common periodic orbits—Lyapunov, vertical, and southern halo—shift and distort when \(J_2\) , \(J_4\) , and \(J_6\) of the primary and \(J_2\) of the secondary body are accounted for in the Jupiter–Europa and Saturn–Enceladus systems. It is found that these families of periodic orbits change shape, position, and energy, which can lead to dramatically different dynamical behavior in some cases. The primary focus is on moons of the outer planets, many of which have very small odd zonal harmonic terms, or no measured value at all, so while the developed equations are meant for any and all zonal harmonic terms, only even terms are considered in the simulations. Early utilization of this refined CR3BP model in mission design will result in a more smooth transition to full ephemeris model. PubDate: 2020-09-25

Abstract: Abstract 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 TNOs, which itself requires some dynamical production mechanism (Gladman et al. in Astron J Lett 697:L91–L94, 2009; Gomes et al. in Icarus 258:37–49, 2015; Batygin and Brown in Astrophys J 833(1):L3, 2016). A notable recent addition to the census of high-i minor bodies in the solar system is the retrograde asteroid 514107 Ka’epaoka’awela, which currently occupies a 1:-1 mean motion resonance with Jupiter at i = 163 \(^\circ \) (Wiegert et al. in Nature 543:687–689, 2017). In this work, we delineate a direct connection between retrograde Jupiter Trojans and high-i Centaurs. First, we backpropagate a large sample of clones of Ka’epaoka’awela for 100 Ma numerically and demonstrate that long-term stable clones tend to decrease their inclination steadily until it concentrates between 90 \(^\circ \) and 135 \(^\circ \) , while their eccentricity and semi-major axis increase, placing many of them firmly into the trans-Neptunian domain. Importantly, the clones show significant overlap with the synthetic high-i Centaurs generated in Planet 9 studies (Batygin et al. in Phys Rep 805:1–53, 2019), and hint at the existence of a relatively prominent, steady-state population of minor bodies occupying polar trans-Saturnian orbits. Second, through direct numerical forward modeling, we delineate the dynamical pathway through which conventional members of the Kuiper Belt’s scattered disk population can become retrograde Jovian Trojan resonators in the presence of Planet 9. PubDate: 2020-09-19

Abstract: Abstract 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: (1) comets that remain frozen during the entire time span and become observable only because of an increase in their orbital energy at the very end of their propagation; (2) comets whose perihelion distance performs an almost complete galactic cycle, while their galactic longitude of the ascending node and cosine of the galactic inclination remain almost constant; (3) comets whose perihelion distance and cosine of the galactic inclination perform a full galactic cycle, while their galactic longitude of the ascending node performs a half a cycle. This investigation allowed us to define four different zones for the previous perihelion distance, in which one or two of the above long-term behaviors dominate. Considering the distribution of the cosine of the ecliptic inclination and the galactic longitude of the ascending node at the previous perihelion distance, for the different zones, several fingerprints of the initial disk shape were highlighted. Such fingerprints appeared to be quite robust since they were still present considering the reconstructed orbital elements, i.e., the elements obtained from the original orbit after a backward propagation over one orbital period considering only the galactic tides. PubDate: 2020-09-02

Abstract: Abstract 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 samples during the simulation with respect to standard MC techniques. The two methods are first described and then applied to some test cases, providing evidence of the increased accuracy or the reduced computational burden with respect to a standard MC simulation. Finally, a sensitivity analysis is carried out to show how parameter setting affects the accuracy of the results and the numerical efficiency of the two methods. PubDate: 2020-08-27

Abstract: Abstract 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 are selected for the construction of transfers. The planar transfer orbits from each DRO to the LLO with zero inclination are firstly obtained and compared in the planar circular restricted three-body problem (PCR3BP) to reveal basic characteristics of the transfer solutions. The planar transfers are classified into several types based on characteristics. Each type is discussed in details, especially their transfer cost and time. Based on the planar transfers, nonplanar transfer orbits are constructed in the circular restricted three-body problem (CR3BP). Some nonplanar transfers are selected and compared to show effects of the LLO inclination. Then, the planar transfer orbits are refined in the planar bicircular restricted four-body problem (PBR4BP) with the gravity of the Sun. The comparison between results in the PCR3BP and PBR4BP shows that the gravity of the Sun can increase transfer options and reduce the transfer cost. Further analysis is carried out based on the realistic results in the PBR4BP, including the ballistic capture, departure and insertion locations, transfer cost and time, etc. The results are useful for selecting parking DROs and designing transport systems to the Moon. PubDate: 2020-08-25

Abstract: Abstract 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 describing the uncertainty of the solution itself. We study the behavior of the confidence region in the case of a simultaneous increase in the number of observations and the time span over which they are performed. More precisely, we describe the geometry of the confidence region for solutions in regular zones. We prove an estimate of the trend of the uncertainties in a set of positive measure of the phase space, made of invariant curve. Our result gives an analytical proof of some known numerical evidences. PubDate: 2020-08-13