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In a recent paper of Philcox, Goodman and Slepian, the solution of the elliptic Kepler's equation is given as a quotient of two contour integrals along a Jordan curve that contains in its interior the unique real solution of the elliptic Kepler's equation and does not include other complex zeroes. In this paper, we show that a similar explicit integral solution can be given for the hyperbolic Kepler's equation. With this purpose, we carry out a study of the complex zeros of the hyperbolic Kepler's equation in order to define suitable Jordan contours in the integrals. Even more, we show that appropriate elliptic Jordan contours can be defined for such integrals, which reduces the computing time. Moreover, using the ideas behind the fast Fourier transform (FFT) algorithm, these integrals can be approximated by the composite trapezoidal rule which gives an algorithm with spectral convergence as a function of the number of nodes. The results of some numerical experiments are presented to show that this implementation is a reliable and very accurate algorithm for solving the hyperbolic Kepler's equation. [ABSTRACT FROM AUTHOR]

This paper presents a new approach to approximate the solution of Kepler's equation. It is found that by means of a series approximation, an angle identity, the application of Sturm's theorem, and an iterative correction method, the need to evaluate transcendental functions or query lookup tables is eliminated. The final procedure builds upon Mikkola's approach. Initially, a fifteenth-order polynomial is derived through a series approximation of Kepler's equation. Sturm's theorem is used to prove that only one real root exists for this polynomial for the given range of mean anomaly and eccentricity. An initial approximation for this root is found using a third-order polynomial. Then, a single generalized Newton–Raphson correction is applied to obtain fourteenth-place accuracies in the elliptical case, which is near machine precision. This paper will focus on demonstrating the procedure for the elliptical case, though an application to hyperbolic orbits through a similar methodology may be similarly developed. [ABSTRACT FROM AUTHOR]

We study the perturbed-from-synchronous librational state of a double asteroid, modeled by the Full Two Rigid Body Problem (F2RBP), with primary emphasis on deriving analytical formulas which describe the system's evolution after deflection by a kinetic impactor. To this end, both linear and nonlinear (canonical) theories are developed. We make the simplifying approximations (to be relaxed in a forthcoming paper) of planar binary orbit and axisymmetric shape of the primary body. To study the effect of a DART-like hit on the secondary body, the momentum transfer enhancement parameter β is introduced and retained as a symbolic variable throughout all formulas derived, either by linear or nonlinear theory. Our approach can be of use in the context of the analysis of the post-impact data from kinetic impactor missions, by providing a precise modeling of the impactor's effect on the seconadry's librational state as a function of β . [ABSTRACT FROM AUTHOR]

This paper investigates the motion of a small particle moving near the triangular points of the Earth–Moon system. The dynamics are modeled in the Hill restricted 4-body problem (HR4BP), which includes the effect of the Earth and Moon as in the circular restricted 3-body problem (CR3BP), as well as the direct and indirect effect of the Sun as a periodic time-dependent perturbation of the CR3BP. Due to the periodic perturbation, the triangular points of the CR3BP are no longer equilibrium solutions; rather, the triangular points are replaced by periodic orbits with the same period as the perturbation. Additionally, there is a 2:1 resonant periodic orbit that persists from the CR3BP into the HR4BP. In this work, we investigate the dynamics around these invariant objects by performing a center manifold reduction and computing families of 2-dimensional invariant tori and their linear normal behavior. We identify bifurcations and relationships between families. Mechanisms for transport between the Earth, L 4 , and the Moon are discussed. Comparisons are made between the results presented here and in the bicircular problem (BCP). [ABSTRACT FROM AUTHOR]

In this paper, we address the problem of computing a preliminary orbit of a celestial body from one topocentric position vector P 1 and a very short arc (VSA) of optical observations A 2 . Using the conservation laws of the two-body dynamics, we write the problem as a system of 8 polynomial equations in 6 unknowns. We prove that this system is generically consistent, namely, for a generic choice of the data P 1 , A 2 , it always admits solutions in the complex field, even when P 1 , A 2 do not correspond to the same celestial body. The consistency of the system is shown by deriving a univariate polynomial v of degree 8 in the unknown topocentric distance at the mean epoch of the observations of the VSA. Through Gröbner bases theory, we also show that the degree of v is minimum among the degrees of all the univariate polynomials solving this problem. Even though we can find solutions to our problem for a generic choice of P 1 , A 2 , most of these solutions are meaningless. In fact, acceptable solutions must be real and have to fulfill other constraints, including compatibility with Keplerian dynamics. We also propose a way to select or discard solutions taking into account the uncertainty in the data, if present. The proposed orbit determination method is relevant for different purposes, e.g., the computation of a preliminary orbit of an Earth satellite with radar and optical observations, the detection of maneuvres of an Earth satellite, and the recovery of asteroids which are lost due to a planetary close encounter. We conclude by showing some numerical tests in the case of asteroids undergoing a close encounter with the Earth. [ABSTRACT FROM AUTHOR]

In this paper, we present a way of combining the computation of invariant tori and their stable and unstable manifolds with the multiple shooting technique. We start by showing some of the results of Jorba (Nonlinearity 14(5):943–976, 2001) that should be modified in order to introduce the multiple shooting technique in these computations. After that, by a direct application in the planar elliptic restricted three-body problem (PERTBP), how to modify the equations and methods to compute the above-mentioned objects is introduced. In particular, the structure of the (systems of) equations and matrices involved in these computations is shown. An application of these computations can be found in Duarte and Jorba (Invariant manifolds of tori near L 1 and L 2 in the planar elliptic restricted three-body problem II. The Dynamics of Comet Oterma, Preprint 2023), where the dynamics of comet 39P/Oterma is modelled as a PERTBP. [ABSTRACT FROM AUTHOR]

In this paper, the Lidov–Kozai mechanism was studied in the region of the Hilda group and Jupiter Trojans. Asteroids of these populations move in 3:2 and 1:1 orbital resonances with Jupiter. The study was carried out using numerical integration of real asteroids’ equations of motion. A simplified dynamical model was adopted. Perturbations from only Jupiter moving in a fixed elliptical orbit were taken into account. Classical secular perturbations were excluded from osculating elements at every print step, and derived orbital inclinations and eccentricities were plotted versus a perihelion argument ω . As a result, it was found that usual positions of a maximum of the eccentricity and, accordingly, a minimum of the inclination ( ω = 90 ∘ , 270 ∘ ) are shifted in these resonant regions. For Hildas, the maximum of the eccentricity is achieved with perihelion argument values ω = 0 ∘ , 180 ∘ . For L4 Trojans, it is achieved with ω = 30 ∘ , 210 ∘ , and for L5 Trojans—with ω = 150 ∘ , 330 ∘ . [ABSTRACT FROM AUTHOR]

Two-dimensional Poincaré maps are widely used for discovering planar periodic and quasi-periodic orbits. These maps have well-defined and easily identifiable chains of islands that are indicative of quasi-periodic orbits. These chains of islands surround fixed points, which indicate periodic orbits. When expanding the Poincaré maps to four dimensions the problem becomes more complex, not only must the four dimensions be displayed in a manner that can be easily interpreted visually, but the structures within the Poincaré maps that indicate quasi-periodic motion are much more complex. An additional factor making these structures more complex is that spatial orbits can experience short-term, quasi-periodic behavior similar to that experienced by a planar orbit, but these spatial orbits can also undergo long-term, quasi-periodic behavior that causes periodic change in the shape of the orbit. In addition to providing indications of quasi-periodic behavior, the shape of the 4D Poincaré structures can provide insight into the transforming nature of the orbit. This paper uses the “space-plus-color” method for displaying the 4D Poincaré maps. Various structures within a 4D Poincaré map relating to spatial, quasi-periodic orbits in the Earth-Moon, circular restricted three-body problem are identified and analyzed. Trends among the structures are identified so that inferences can be made about the orbit relating to structures that have not yet been documented. Structures that clearly highlight a fixed point are identified, and a multiple-shooter method is used to converge on the periodic orbit. Similar structures are identified within 4D Poincaré maps with different Jacobi constants to demonstrate that these structures are not unique to a particular Jacobi constant. [ABSTRACT FROM AUTHOR]

Numerous missions leverage the weak stability boundary in the Earth–Moon–Sun system to achieve a safe and cost-effective access to the lunar environment. These transfers are envisaged to play a significant role in upcoming missions. This paper proposes a novel method to design low-energy transfers by combining the recent Theory of Functional Connections with a homotopic continuation approach. Planar patched transfer legs within the Earth–Moon and Sun–Earth systems are continued into higher-fidelity models. Eventually, the full Earth–Moon transfer is adjusted to conform to the dynamics of the planar Earth–Moon Sun-perturbed, bi-circular restricted four-body problem. The novelty lies in the avoidance of any propagation during the continuation process and final convergence. This formulation is beneficial when an extensive grid search is performed, automatically generating over 2000 low-energy transfers. Subsequently, these are optimized through a standard direct transcription and multiple shooting algorithm. This work illustrates that two-impulse low-energy transfers modeled in chaotic dynamic environments can be effectively formulated in Theory of Functional Connections, hence simplifying their overall design process. Moreover, its synergy with a homotopic continuation approach is demonstrated. [ABSTRACT FROM AUTHOR]

The effect of inclined third-body perturbation on the perturbed body's inclination evolution has been investigated extensively. However, under various initial orbital elements, the exact effect of the J 2 perturbation and the inclined third-body perturbation on perturbed body's inclination is still an open problem. In this paper, we use a simplified model for long-term orbital variations to analyze the free and forced inclinations for almost circular orbits with low inclinations. We analyze the influence of the J 2 perturbation on the inclination evolution under various initial orbital elements. The results show that the J 2 perturbation consistently decreases the maximum value of the inclination. Besides, the J 2 perturbation either decreases the amplitude of the inclination's oscillation, or enables the perturbed body to orbit closer to the equator of the central body. These effects enable the objects with initially low inclinations to maintain their orbits near the equator of the central body for a long time. The results of the study are applied to circum-Jovian orbits and the triple asteroid system 87 Sylvia. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the existence of dust clouds moving near the triangular points of the Earth–Moon system, the so-called Kordylewski clouds. The study is based on using some simplified planar models to find possible locations for these clouds. The validity of these predictions is tested by means of numerical simulations on a realistic model. The simplified models are based on the Earth–Moon restricted three-body problem plus the direct gravitational effect of the Sun on the particles (this is the so-called bicircular model), the solar radiation pressure and the Poynting–Robertson effect. The analysis of these models shows that there are some stability regions in the Earth–Moon plane, at some distance of the triangular points. The stability of these regions has been tested numerically in realistic (nonplanar) models. The results show that particles in these regions persist for some time (about a century), but it is very remarkable that many of these particles also escape the Earth–Moon system. If we perform backwards in time numerical simulations we obtain a similar result: particles also escape the Earth–Moon system after a similar time. From this point of view, the clouds are not a stable region in the classical sense of the term, but a region with "slow diffusion" where interplanetary particles stay for some years. [ABSTRACT FROM AUTHOR]

Coupling between the rotational and translational motion of a rigid body can have a profound effect on spacecraft motion in complex dynamical environments. While there is a substantial amount of study of rigid-body coupling in a non-uniform gravitational field, the spacecraft is often considered as a point-mass vehicle. By contrast, the full-N body problem (FNBP) evaluates the mutual gravitational potential of the rigid-body celestial objects and any other body, such as a spacecraft, under their influence and treats all bodies, including the spacecraft, as a rigid body. Furthermore, the perturbing effects of the FNBP become more pronounced as the celestial bodies become smaller and/or more significantly aspherical. Utilizing the comprehensive framework of dynamics and gravitational influences within the FNBP, this research investigates the dynamics of spacecraft modeled as rigid bodies in binary systems characterized by nearly circular mutual orbits. The paper presents an examination of the perturbation effects that arise in this circular restricted full three-body problem (CRF3BP), aiming to assess and validate the extent of these effects on the spacecraft's overall motion. Numerical results provided for spacecraft motion in the CRF3BP in a binary asteroid system demonstrate non-negligible trajectory divergence when utilizing rigid-body versus point mass spacecraft models. These results also investigate the effects of shape and inertia tensors of the bodies and solar radiation pressure in those models. [ABSTRACT FROM AUTHOR]

This paper presents an improved alpha shape-based linear interpolation method, and an improved binning method within the continuum method framework for accurate and efficient planar phase space long-term density propagation. The density evolution equation is formulated for the continuum density propagation under the influence of the solar radiation pressure and Earth’s oblateness using semi-analytical equations. The concept of the alpha shape is included to get accurate interpolated density within the non-convex hull enclosing all the samples for the highly deformed and elongated density distribution. The improved binning method increases the density accuracy by considering the variant nonlinearity of the density within each alpha shape triangulation, which calculates the joint and marginal density as the weighted sum of density weights per bin area and per bin width, respectively. The suitable sample number for the continuum method and the suitable grid number for performing the linear interpolation are selected by trading off the density accuracy and the computational effort. The superiority of the improved alpha shape-based continuum method is demonstrated for accurate and efficient density propagation in the context of the high-altitude and high area-to-mass ratio satellite long-term propagation. [ABSTRACT FROM AUTHOR]