Dynamical Astronomy Research Articles

Advances In Celestial Mechanics And Dynamical Astronomy: A Literature Review

This research paper provides a comprehensive examination of recent developments within the realm of celestial mechanics and dynamical astronomy. By conducting an in-depth analysis of existing literature, this study aims to elucidate the pivotal advancements and contributions that have significantly enhanced our understanding of celestial body motion and dynamics. Through a meticulous evaluation of scholarly articles, books, and conference proceedings, this literature review delves into the fundamental concepts and methodologies that underpin celestial mechanics and dynamical astronomy. The paper explores a wide array of topics, encompassing orbital dynamics, dynamical systems, asterodynamics, planetary science, and explanatory systems. By synthesizing the findings of esteemed researchers and scientists, this study reveals the profound impact of novel numerical methods, analytical techniques, and observational discoveries on the field. Furthermore, it identifies areas ripe for future investigation, underscoring the vast potential for continued exploration and innovation within celestial mechanics and dynamical astronomy. Ultimately, this literature review seeks to provide a valuable resource for scholars, researchers, and students seeking insight into the latest advancements and emerging trends in celestial mechanics and dynamical astronomy, while also highlighting the boundless opportunities for future research and discovery.

Celestial Mechanics Leading up to the AAS Division on Dynamical Astronomy

This talk reviewed key developments leading to the establishment of the AAS Division on Dynamical Astronomy. In the 1890s, Simon Newcomb calculated astronomical constants and ephemerides for solar system bodies that were internationally adopted and used for 80 yr. The introduction of punch card equipment in the 1940s and computers in the 50s allowed much faster computations. Launches of Sputnik, Explorer, Vanguard, and Mercury satellites led to computing the orbits of satellites around the Earth and the field of astrodynamics. Dirk Brouwer's death in 1966 resulted in the celestial mechanics faculty at Yale spreading across the country. The space program, radar observations, and faster computers propelled advances in celestial mechanics. Teams at the Jet Propulsion Laboratory and Massachusetts Institute of Technology developed solar system ephemerides using numerical integrations fit to optical observations and radar ranging measurements. It was time for the AAS Division on Dynamical Astronomy.

Overview: Regularization and Numerical methods in Celestial Mechanics and Dynamical Astronomy

"The relevant study of the n-body problem (predicting individual motions of a group of celestial objects interacting with each other gravitational) is substantial in space dynamics. Further, the various perturbations effects, the collisions or the close encounters between celestial bodies are trajectory modifiers. In the digital age, the behaviour of non-integrable systems is usually and primarily studied with restricted models or coupled with the help of numerical integrators. Over time, these numerical integrators progressed, and the numerical challenge led to the introduction of many new techniques. The question is, which numerical algorithm to choose for the correct research? Is what we get through numerical integration close to the proper orbit? In other words, how can one be sure that the obtained trajectories reflect reality and that their prediction can be taken into account for a long time? This paper presents some analytical and numerical methods with highly accurate computations, such as regularization methods and symplectic integrators, which can be useful in obtaining the corresponding more accurate results."

Research in Dynamical Astronomy and the Quest for Planet Nine

Research in Dynamical Astronomy and the Quest for Planet Nine

MILESTONES IN THE DEVELOPMENT OF CELESTIAL MECHANICS

A brief outline of the development of ideas and a review of some achievements in modern celestial mechanics are given. The emphasis is on the fact that the classical definition of this science given by Laplace does not fully reflect the content of modern celestial mechanics, and the term dynamic astronomy is more capacious. Dynamic astronomy studies almost everything that moves and rotates in space: from dust particles to comets and asteroids, from satellites, planets and their satellites to stars and galaxies. This complex science includes not only the problems of classical, but also relativistic celestial mechanics, it includes the theory of equilibrium figures, various computational methods and computer simulation methods. Qualitative methods are of great importance, the culmination of which was the creation of the KAM theory. The development of celestial mechanics went through the practice of various applications, and the range of problems in it is exceptionally wide. A striking stimulus for the development of dynamical astronomy was the discovery of exoplanets around other stars. The article traces a chain of ideas from Keplerian orbits to osculating Lagrangian ellipses, from two-body problems to many-body problems, from Gaussian rings to models built on the basis of precessing analogues of these rings.

Predicting the Stability of Hierarchical Triple Systems with Convolutional Neural Networks

The dynamical stability of hierarchical triple systems is a long-standing question in celestial mechanics and dynamical astronomy. Assessing the long-term stability of triples is challenging because it requires computationally expensive simulations. Here we propose a convolutional neural network model to predict the stability of equal-mass hierarchical triples by looking at their evolution during the first 5 × 105 inner binary orbits. We employ the regularized few-body code tsunami to simulate 5 × 106 hierarchical triples, from which we generate a large training and test data set. We develop 12 different network configurations that use different combinations of the triples’ orbital elements and compare their performances. Our best model uses six time series, namely, the semimajor axes ratio, the inner and outer eccentricities, the mutual inclination, and the arguments of pericenter. This model achieves excellent performance, with an area under the ROC curve score of over 95% and informs of the relevant parameters to study triple systems stability. All trained models are made publicly available, which allows predicting the stability of hierarchical triple systems 200 times faster than pure N-body methods.

Norm of the Position Shift of a Celestial Body in a Dynamical Astronomy Problem

The averaging method is widely used in celestial mechanics, in which a mean orbit is introduced and slightly deviates from an osculating one, as long as disturbing forces are small. The difference $$\delta {\mathbf{r}}$$ in the celestial body positions in the mean and osculating orbits is a quasi-periodic function of time. Estimating the norm $$\left\| {\delta {\mathbf{r}}} \right\|$$ for deviation is interesting to note. Earlier, the exact expression of the mean-square norm for one problem of celestial mechanics was obtained: a zero-mass point moves under the gravitation of a central body and a small perturbing acceleration $${\mathbf{F}}$$. The vector $${\mathbf{F}}$$ is taken to be constant in a co-moving coordinate system with axes directed along the radius vector, the transversal, and the angular momentum vector. Here, we solved a similar problem, assuming the vector $${\mathbf{F}}$$ to be constant in the reference frame with axes directed along the tangent, the principal normal, and the angular momentum vector. It turned out that $${{\left\| {\delta {\mathbf{r}}} \right\|}^{2}}$$ is proportional to $${{a}^{6}}$$, where $$a$$ is the semi-major axis. The value $${{\left\| {\delta {\mathbf{r}}} \right\|}^{2}}{{a}^{{ - 6}}}$$ is the weighted sum of the component squares of $${\mathbf{F}}$$. The quadratic form coefficients depend only on the eccentricity and are represented by the Maclaurin series in even powers of $$e$$ that converge, at least for $$e < 1$$. The series coefficients are calculated up to $${{e}^{4}}$$ inclusive, so that the correction terms are of order $${{e}^{6}}$$.

The use of particle distributions in Solar system small body dynamics

ABSTRACT The extraterrestrial material, called meteoroids, that constantly enters the Earth’s atmosphere gives us a unique opportunity to examine the motion and population of small bodies in the Solar system. This exploration requires simulating the motion of these particles. Currently, only the timing of meteoroids encountering the Earth is well predicted by such simulations, while other parameters are uncertain. This can be remedied by proper stochastic representation and estimation using a sufficient number of samples. We propose methods to both represent simulations in a stochastic manner and to improve sampling using Importance Sampling. We also demonstrate these methods practically with a test model. Using the test model resulted in an error reduction by a factor of 3 without increase in computation time. Thus, we validated that these techniques can be implemented on and are compatible with Solar system small body dynamics models. Based on these results we predict that when properly implemented on a larger and more complex model, Importance Sampling can improve sampling numbers by several orders of magnitude without increasing computation time, depending on the simulation in question. The methods presented here bring advantages such as; greatly reduced estimation errors, fitting models without re-running simulations, model comparisons without sample variations, circumventing unknown properties using invariant measures, representing large particle numbers without additional errors. This methodology has wide application possibility and will enable larger, more reliable and reusable simulations of dynamical astronomy.

Correction to: Effect of the rotation and tidal dissipation history of stars on the evolution of close-in planets

This is an erratum for the publication Bolmont & Mathis 2016 (Celestial Mechanics and Dynamical Astronomy, 126, 275-296, https://doi.org/10.1007/s10569-016-9690-3). There was a small mistake for the spin integration of our code which we corrected and we take advantage of this erratum to investigate a bit further the influence of a planet on the spin of its host star.

Measuring the mass distribution in stellar systems

One of the fundamental tasks of dynamical astronomy is to infer the distribution of mass in a stellar system from a snapshot of the positions and velocities of its stars. The usual approach to this task (e.g., Schwarzschild's method) involves fitting parametrized forms of the gravitational potential and the phase-space distribution to the data. We review the practical and conceptual difficulties with this approach and describe a novel statistical method for determining the mass distribution that does not require determining the phase-space distribution of the stars. We show that this new estimator out-performs other distribution-free estimators for the harmonic and Kepler potentials.

Final state predictions for J2 gravity perturbed motion of the Earth’s artificial satellites using Bispherical coordinates

In this paper, initial value problem for dynamical astronomy will be established using Bispherical coordinates. A computational algorithm is developed for the final state predictions for J2 gravity perturbed motion of the Earth’s artificial satellites. This algorithm is important in targeting, rendezvous maneuvers as well for scientific researches. The applications of the algorithm are illustrated by numerical examples of some test orbits of different eccentricities. The numerical results are extremely accurate and efficient.

New families of symplectic splitting methods for numerical integration in dynamical astronomy

We present new splitting methods designed for the numerical integration of near-integrable Hamiltonian systems, and in particular for planetary N-body problems, when one is interested in very accurate results over a large time span. We derive in a systematic way an independent set of necessary and sufficient conditions to be satisfied by the coefficients of splitting methods to achieve a prescribed order of accuracy. Splitting methods satisfying such (generalized) order conditions are appropriate in particular for the numerical simulation of the Solar System described in Jacobi coordinates. We show that, when using Poincaré Heliocentric coordinates, the same order of accuracy may be obtained by imposing an additional polynomial equation on the coefficients of the splitting method. We construct several splitting methods appropriate for each of the two sets of coordinates by solving the corresponding systems of polynomial equations and finding the optimal solutions. The experiments reported here indicate that the efficiency of our new schemes is clearly superior to previous integrators when high accuracy is required.

Improving spectral bounds for clustering problems by Lagrangian relaxation

Clustering is one of the most important issues in data mining, image segmentation, VLSI design, parallel computing and many other areas. We consider the general problem of partitioning n points into k clusters by maximizing the affinity measure of the points into the clusters. This objective function, referred to as Ratio Association, generalizes the classical (Minimum) Sum-of-Squares clustering problem, where the affinity is measured as closeness in the Euclidean space. This generalized version has emerged in the context of the approximation of chemical conformations for molecules, and in explaining transportation phenomena in dynamical systems, especially in dynamical astronomy. In particular, we refer to the dynamical systems application in the paper. Although successful heuristics have been developed to approximately solve the problem, the conventional spectral bounds proposed in the literature are not tight enough for “large” instances to assert the quality of those heuristics or to allow solving the problem exactly. In this paper, we investigate how to tighten the spectral bounds by using Lagrangian relaxation and Subgradient optimization methods.

Application of Symplectic Algebraic Dynamics Algorithm to Circular Restricted Three-Body Problem

Symplectic algebraic dynamics algorithm (SADA) for ordinary differential equations is applied to solve numerically the circular restricted three-body problem (CR3BP) in dynamical astronomy for both stable motion and chaotic motion. The result is compared with those of Runge–Kutta algorithm and symplectic algorithm under the fourth order, which shows that SADA has higher accuracy than the others in the long-term calculations of the CR3BP.

G. Contopoulos: Order and chaos in dynamical astronomy, springer, physics and astronomy library, 2002

G. Contopoulos: Order and chaos in dynamical astronomy, springer, physics and astronomy library, 2002

Straight-line oscillations generating three-dimensional motions in the photogravitational restricted three-body problem

The Sitnikov configuration is a special case of the restricted three-body problem where the two primaries are of equal masses and the third body of a negligible mass moves along a straight line perpendicular to the orbital plane of the primaries and passes through their center of mass. It may serve as a toy model in dynamical astronomy, and can be used to study the three-dimensional orbits in more applicable cases of the classical three-body problem. The present paper concerns the straight-line oscillations of the Sitnikov family of the photogravitational circular restricted three-body problem as well as the associated families of three-dimensional periodic orbits. From the stability analysis of the Sitnikov family and by using appropriate correctors we have computed accurately 49 critical orbits at which families of 3D periodic orbits of the same period bifurcate. All these families have been computed in both cases of equal and non-equal primaries, and consist entirely of unstable orbits. They all terminate with coplanar periodic orbits. We have also found 35 critical orbits at which period doubling bifurcations occur. Several families of 3D periodic orbits bifurcating at these critical Sitnikov orbits have also been given. These families contain stable parts and close upon themselves containing no coplanar orbits.

Chaos and Complexity in Astrophysics

Methods and techniques of the theory of nonlinear dynamical systems and patterns can be useful in astrophysical applications. Some works on the subjects of dynamical astronomy, stellar pulsation and variability, as well as spatial complexity in extended systems, in which such approaches have already been utilized, are reviewed. Prospects for future directions in applications of this kind are outlined.

MOTION IN PARABOLIC CYLINDRICAL COORDINATES: APPLICATION TO J2GRAVITY PERTURBED TRAJECTORIES

In this paper, initial value problem for dynamical astronomy will be established using parabolic cylindrical coordinates. Computation algorithm is developed for the initial value problem of gravity perturbed trajectories. Applications of the algorithm for the problem of final state predication are illustrated by numerical examples of seven test orbits of different eccentricities. The numerical results are extremely accurate and efficient in predicating final state for gravity perturbed trajectories which is of extreme importance for scientific researches as well as for military purposes. Moreover, an additional efficiency of the algorithm is that, for each of the test orbits, the step size used for solving the differential equations of motion is larger than 70% of the step size used for obtaining its reference final state solution.

Gravitational Effects of Minor Planets and Their Mass Determination

There are two groups of methods for asteroid mass determination: astrophysical and dynamical. In dynamical methods, the mass of an asteroid is estimated from its perturbation of the motion of some other body. In our work, we tried to find out whether it was possible to determine the correction of perturbing mass separately from the corrections of the six osculating elements of the perturbed asteroid. As a consequence, we introduce a modified method of asteroid mass determination. The procedure for the mass determination is based on a calculation of the perturbed orbit using observations that cover the preor post-encounter part of the orbit. The preand post-encounter parts of the perturbed orbit are linked by some chosen value for the mass of a perturbing body. The procedure is an iterative one. The resulting mass of the perturbing asteroid is found to provide the minimum O C residuals in the rms sense. We calculated the masses of the eight largest asteroids independently for all found perturbed asteroids (62 bodies), using the standard and modified method. Some of these perturbed asteroids were never used before for this purpose (11 bodies). This thesis calculates the mass of (4) Vesta based on newly found encounters (A. Kovacevic, 2005, A&A, 430, 319), as well as the mass of (16) Psyche, which is the first successful attempt—based on a dynamical method—that leads to the conclusion that the composition of an asteroid is metallic (M. Kuzmanoski & A. Kovacevic, 2002, A&A, 395, L17). As can be seen from numerical tests, the modified method provides results that are in good agreement with the standard method and adopted values of the asteroid’s masses. This method can be used as a tool for asteroid mass determination. This work has been realized within the project “Dynamical Astronomy and Planetology” (No. 146004) supported by the Ministry of Science and Environmental Protection of the Republic of Serbia.

Simulation of the full two rigid body problem using polyhedral mutual potential and potential derivatives approach

Herein we investigate the coupled orbital and rotational dynamics of two rigid bodies modelled as polyhedra, under the influence of their mutual gravitational potential. The bodies may possess any arbitrary shape and mass distribution. A method of calculating the mutual potential’s derivatives with respect to relative position and attitude is derived. Relative equations of motion for the two body system are presented and an implementation of the equations of motion with the potential gradients approach is described. Results obtained with this dynamic simulation software package are presented for multiple cases to validate the approach and illustrate its utility. This simulation capability is useful both for addressing questions in dynamical astronomy and for enabling spacecraft missions to binary asteroid systems.