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The aim of this work is to explain how, starting from the orthogonality expression of two polynomials, we deduce the Schrödinger equation and the solution of the N-body problem including two-body correlations as well as the existence of shells. Generated by the behaviour of kinetic energy for a two-body interaction. The quantification of matters is obtained by the application of the weight function algorithm to the statement that two states are independents when their product integrated over the whole space is null leading to a two variables second order differential equation. The Nuclear Shell Model is a consequence of the kinetic energy behaviour for increasing number of nucleons in ground state. It leaves the mean field theory useless. [ABSTRACT FROM AUTHOR]

Reduced mass value and vector are well known for the two-body problem, but the many-body reduced vector problem is not solved yet. The study of many-body problems and their applications (such as vibrational spectroscopy) is one of the more important physical problems. Vibrational spectroscopy provides a powerful tool to perceive the molecular structures and atom motions of molecules. The water molecule is a threebody system stretching vibration that has been previously quantized; their frequencies were defined and showed the infrared (IR) absorption spectrum based on Morse potential. In this work, the reduced mass of the many-body problem is being solved and then used to study the intensity of the stretching vibration modes and show the ratio is in agreement with experiments. The molecule was studied in classical and quantum mechanics to determine its absorption intensity as an example of a reduced mass problem. The results show molecular atomic motions and changes in dipole and reduced mass vector. A Morse-like model for bending was predicted based on the spectroscopic vibration frequency and intensity, defining the bending potential depth of 93.5 kJ/mol. [ABSTRACT FROM AUTHOR]

One of the oldest problems in physics is that of calculating the motion of N particles under a specified mutual force: the N-body problem. Much is known about this problem if the specified force is non-relativistic gravity, and considerable progress has been made by considering the problem in one spatial dimension. Here, I review what is known about the relativistic gravitational N-body problem. Reduction to one spatial dimension has the feature of the absence of gravitational radiation, thereby allowing for a clear comparison between the physics of one-dimensional relativistic and non-relativistic self-gravitating systems. After describing how to obtain a relativistic theory of gravity coupled to N point particles, I discuss in turn the two-body, three-body, four-body, and N-body problems. Quite general exact solutions can be obtained for the two-body problem, unlike the situation in general relativity in three spatial dimensions for which only highly specified solutions exist. The three-body problem exhibits mild forms of chaos, and provides one of the first theoretical settings in which relativistic chaos can be studied. For N ≥ 4 , other interesting features emerge. Relativistic self-gravitating systems have a number of interesting problems awaiting further investigation, providing us with a new frontier for exploring relativistic many-body systems. [ABSTRACT FROM AUTHOR]

We investigate the existence of families of symmetric periodic solutions of second kind as continuation of the elliptical orbits of the two-dimensional Kepler problem for certain symmetric differentiable perturbations using Delaunay coordinates. More precisely, we characterize the sufficient conditions for its existence and its type of stability is studied. The estimate on the characteristic multipliers of the symmetric periodic solutions is the new contribution to the field of symmetric periodic solutions. In addition, we present some results about the relationship between our symmetric periodic solutions and those obtained by the averaging method for Hamiltonian systems. As applications of our main results, we get new families of periodic solutions for: the perturbed hydrogen atom with stark and quadratic Zeeman effect, for the anisotropic Seeligers two-body problem and to the planar generalized Størmer problem. [ABSTRACT FROM AUTHOR]

In this analytical study, we have presented a new type of solving procedure with the aim to obtain the coordinates of small mass m, which moves around primary MSun, referred to non-inertial frame of restricted two-body problem (R2BP) with a modified potential function (taking into account the variable velocity V → of central body MSun motion in a prescribed fixed direction) instead of a classical potential function for Kepler's formulation of R2BP. Meanwhile, system of equations of motion has been successfully explored with respect to the existence of an analytical way of presenting the solution in polar coordinates {r(t), φ(t)}. We have obtained an analytical formula for function t = t(r) via an appropriate elliptic integral. Having obtained the inversed dependence r = r(t), we can obtain the time dependence φ = φ(t). Also, we have pointed out how to express components of solution (including initial conditions) from cartesian to polar coordinates as well. [ABSTRACT FROM AUTHOR]

We consider the two- and n-body problems on the two-dimensional conformal sphere M R 2 , with a radius R > 0 . We employ an alternative potential free of singularities at antipodal points. We study the limit of relative equilibria under the SO(2) symmetry; we examine the specific conditions under which a pair of positive-mass particles, situated at antipodal points, can maintain a state of relative equilibrium as they traverse along a geodesic. It is identified that, under an appropriate radius–mass relationship, these particles experience an unrestricted and free movement in alignment with the geodesic of the canonical Killing vector field in M R 2 . An even number of bodies with pairwise conjugated positions, arranged in a regular n-gon, all with the same mass m, move freely on a geodesic with suitable velocities, where this geodesic motion behaves like a relative equilibrium. Also, a center of mass formula is included. A relation is found for the relative equilibrium in the two-body problem in the sphere similar to the Snell law. [ABSTRACT FROM AUTHOR]

We describe an approach to incorporating the physical effects of the absorption of energy by the event horizon of black holes in the scattering amplitudes based post-Minkowskian, point-particle effective description. Absorptive dynamics are incorporated in a model-independent way by coupling the usual point-particle description to an invisible sector of gapless internal degrees-of-freedom. The leading order dynamics of this sector are encoded in the low-energy expansion of a spectral density function obtained by matching an absorption cross section in the ultraviolet description. This information is then recycled using the scattering amplitudes based Kosower-Maybee-O'Connell in-in formalism to calculate the leading absorptive contribution to the impulse and change in rest mass of a Schwarzschild black hole scattering with a second compact body sourcing a massless scalar, electromagnetic or gravitational field. The results obtained are in complete agreement with previous worldline Schwinger-Keldysh calculations and provide an alternative on-shell scattering amplitudes approach to incorporating horizon absorption effects in the gravitational two-body problem. [ABSTRACT FROM AUTHOR]

In this analytical study, a novel solving method for determining the precise coordinates of a mass point in orbit around a significantly more massive primary body, operating within the confines of the restricted two-body problem (R2BP), has been introduced. Such an approach entails the utilization of a continued fraction potential diverging from the conventional potential function used in Kepler's formulation of the R2BP. Furthermore, a system of equations of motion has been successfully explored to identify an analytical means of representing the solution in polar coordinates. An analytical approach for obtaining the function t = t(r), incorporating an elliptic integral, is developed. Additionally, by establishing the inverse function r = r(t), further solutions can be extrapolated through quasi-periodic cycles. Consequently, the previously elusive restricted two-body problem (R2BP) with a continued fraction potential stands fully and analytically solved. [ABSTRACT FROM AUTHOR]

In the present paper, we efficiently solve the two-body problem for extreme cases such as those with high eccentricities. The use of numerical methods, with the usual variables, cannot maintain the perihelion passage accurately. In previous articles, we have verified that this problem is treated more adequately through temporal reparametrizations related to the mean anomaly through the partition function. The biparametric family of anomalies, with an appropriate partition function, allows a systematic study of these transformations. In the present work, we consider the elliptical orbit as a meridian section of the ellipsoid of revolution, and the partition function depends on two variables raised to specific parameters. One of the variables is the mean radius of the ellipsoid at the secondary, and the other is the distance to the primary. One parameter regulates the concentration of points in the apoapsis region, and the other produces a symmetrical displacement between the polar and equatorial regions. The three most used geodesy latitude variables are also studied, resulting in one not belonging to the biparametric family. However, it is in the one introduced now, which implies an extension of the biparametric method. The results obtained using the method presented here now allow a causal interpretation of the operation of numerous reparametrizations used in the study of orbital motion. [ABSTRACT FROM AUTHOR]

The theory of differential equations with deviating arguments is one of the important and significant branches of nonlinear analysis with numerous applications in most fields. Usually, equations of deviating arguments with deviation depend only on the time, however, when the deviation of the arguments depends upon both the state variable x and also the time t, is incredibly important theoretically and practically. Differential equations with state-dependent delays attract the interest of specialists since they widely arise from application models, such as the two-body problem of classical Electrodynamics, which also have many applications, especially in the class of problems that have past memories. In this paper, we study the existence of solutions of a self-reference (state dependence) quadratic integral equation of fractional orders of the form x(t) = x0 +I0,tαf1(t,x(x(ϕ(t)))).I0,tβ f2(t,x(x(ϕ)))) t∈[0,T], α, β ∈ (0, 1). The uniqueness of the solution will be studied. The continuous dependence of the unique solution on the initial data and the functions f1 and f2 will be proved. Some examples are included. The study establishes conditions for the solution's existence and uniqueness, according to Schauder's fixed point Theorem. [ABSTRACT FROM AUTHOR]

It is well known that exactly solvable models play an extremely important role in many fields of quantum physics. In this study, the Schrödinger equation is applied for a solution of a two-dimensional (2D) problem for two particles enclosed in a circle, confined in an oscillatory well, trapped in a magnetic field, interacting via the Coulomb, Kratzer, and modified Kratzer potentials. In the framework of the Nikiforov–Uvarov method, we transform 2D Schrödinger equations with potentials for which the three-dimensional Schrödinger equation is exactly solvable, into a second-order differential equation of a hypergeometric-type via transformations of coordinates and particular substitutions. Within this unified approach which also has pedagogical merit, we obtain exact analytical solutions for wave functions in terms of special functions such as a hypergeometric function, confluent hypergeometric function, and solutions of Kummer's, Laguerre's, and Bessel's differential equations. We present the energy spectrum for any arbitrary state with the azimuthal number m. Interesting aspects of the solutions unique to the 2D case are discussed. [ABSTRACT FROM AUTHOR]

On the basis of a general action principle, we revisit the scale invariant field equation using the cotensor relations by Dirac (1973). This action principle also leads to an expression for the scale factor λ , which corresponds to the one derived from the gauging condition, which assumes that a macroscopic empty space is scale-invariant, homogeneous, and isotropic. These results strengthen the basis of the scale-invariant vacuum (SIV) paradigm. From the field and geodesic equations, we derive, in current time units (years, seconds), the Newton-like equation, the equations of the two-body problem, and its secular variations. In a two-body system, orbits very slightly expand, while the orbital velocity keeps constant during expansion. Interestingly enough, Kepler's third law is a remarkable scale-invariant property. [ABSTRACT FROM AUTHOR]

We introduce a novel point-particle effective description of ANO vortex solitons in the critical Abelian Higgs Model (AHM) in d = 2 + 1 based on the small winding expansion. Identifying the effective vortices with the elementary quanta of a complex scalar field, relativistic vortex-vortex scattering amplitudes are calculated as a diagrammatic, perturbative expansion in the winding number N. Making use of powerful techniques recently developed for analyzing the post-Minkowskian two-body problem in general relativity, we efficiently extract the contribution to the loop integrals from the classical potential region, with the resulting velocity expansion subsequently resummed to all orders. The main result of this paper is an analytic expression for the classical, vortex-vortex potential at O (N2), or one-loop, with exact velocity dependence. By truncating the resulting effective Hamiltonian at O (p2) we derive an analytic, perturbative expression for the metric on the 2-vortex moduli space. Finally, the emergence of the critical AHM from the classical limit of the N = 2 supersymmetric AHM, and the resulting constraints on the point-particle EFT is described in detail using an on-shell superspace construction for BPS states in d = 2 + 1. [ABSTRACT FROM AUTHOR]

In contact Hamiltonian systems, the so‐called dissipated quantities are akin to conserved quantities in classical Hamiltonian systems. In this article, a Noether's theorem for non‐autonomous contact Hamiltonian systems is proved, characterizing a class of symmetries which are in bijection with dissipated quantities. Other classes of symmetries which preserve (up to a conformal factor) additional structures, such as the contact form or the Hamiltonian function, are also studied. Furthermore, making use of the geometric structures of the extended tangent bundle, additional classes of symmetries for time‐dependent contact Lagrangian systems are introduced. The results are illustrated with several examples. In particular, the two‐body problem with time‐dependent friction is presented, which could be interesting in celestial mechanics. [ABSTRACT FROM AUTHOR]

Explicit methods to generate new canonical sets of elements are presented, together with an analysis on the structure of existing canonical elements that are used in the study of the perturbed two-body problem. New non-singular sets of canonical elements are introduced. Among these sets, the resonant canonical variables are non-singular for circular orbits and display an action-angle-like behaviour for the unperturbed Kepler problem, if their associated generalized frequency is tuned to match the conserved energy. They also prove to be an efficient tool for both qualitative and quantitative studies of perturbations. [ABSTRACT FROM AUTHOR]

We describe the formalism to compute gravitational-wave observables for compact binaries via the effective field theory framework in combination with modern tools from collider physics. We put particular emphasis on solving the 'multi-loop' integration problem via the methodology of differential equations and expansion by regions. This allows us to bootstrap the two-body relativistic dynamics in the Post-Minkowskian (PM) expansion from boundary data evaluated in the near-static (soft) limit. We illustrate the procedure with the derivation of the total spacetime impulse in the scattering of non-spinning bodies to 4PM (three-loop) order, i.e. O (G4), including conservative and dissipative effects. [ABSTRACT FROM AUTHOR]

A brief outline of the development of ideas and a review of some achievements in modern celestial mechanics are presented. 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 artificial satellites, planets, and their satellites to stars and galaxies. This complex science includes problems of both classical and relativistic celestial mechanics; it includes the theory of equilibrium figures and various numerical 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, and from Gaussian rings to models built on the basis of precessing analogues of these rings. [ABSTRACT FROM AUTHOR]

In this paper, a universal computational algorithm is constructed by using F and G series, which can be applied for any of conic orbit. In particular, to find the solution of the two–body problem. In this context, the solution of geocentric system motion of the Mercury planet in the solar system is found using the obtained computational algorithm. [ABSTRACT FROM AUTHOR]

The contribution to the correction of the perihelion precession of binary systems due to energy loss from gravitational radiation emissions, as a follow-up to our previous papers, is obtained for the OJ287 system, which has a large discrepancy not satisfied by the Schwarzschild metric. Using our formulation of the perihelion advance correction from the rate of change of the orbital energy, as well as our alternate correction and spin contributions from previous papers, we obtain a theoretical value of 39.76 degrees for the total perihelion precession angle correction of the OJ287 system, within 1.7% of the observed value of 39.1 ± 0.1. [ABSTRACT FROM AUTHOR]

A cornerstone result in Newtonian mechanics is Bertrand's Theorem concerning the behavior of the solutions of the classical two-body problem. It states that among all possible gravitational laws there are only two exhibiting the property that all bounded orbits are closed. One of these is Newtonian gravitation, the other being Hookean gravitation. We present a comprehensive proof of Bertrand's Theorem that is accessible to undergraduate students who are familiar with basic notions from advanced calculus and differential equations. [ABSTRACT FROM AUTHOR]

Recently, Hamiltonian neural networks (HNNs) have been introduced to incorporate prior physical knowledge when learning the dynamical equations of Hamiltonian systems. Hereby, the symplectic system structure is preserved despite the data-driven modeling approach. However, preserving symmetries requires additional attention. In this research, we enhance HNN with a Lie algebra framework to detect and embed symmetries in the neural network. This approach allows us to simultaneously learn the symmetry group action and the total energy of the system. As illustrating examples, a pendulum on a cart and a two-body problem from astrodynamics are considered. [ABSTRACT FROM AUTHOR]

Efficient trigonometrically-fitted explicit two-derivative improved Runge–Kutta-Nystr o ¨ m methods with three stage fifth-order, denoted as TFTDIRKN5 method is derived for direct solving special type of second-order ordinary differential equation in the form y ′ (t) = f (t , y (t)) with oscillatory solution. Order conditions of proposed method that includes previous estimated slopes, k - i are presented through Taylor series expansion and comparison of coefficients with power of h. Second-order initial value problems (IVPs) are integrated exactly with numerical solution in linear composition of set functions e i ω t and e - i ω t with ω ∈ R. Certain coefficients of proposed methods are depend on the principle frequency of the numerical problems for deriving trigonometrically-fitted improved Runge–Kutta-Nystr o ¨ m direct methods with two-derivative term. The proposed method is analysed numerically to prove that it is zero stable, consistent and convergent, which are critical for solving problems effectively. Stability region and error analysis of proposed method are investigated. The numerical tests show that the proposed method performs better in comparison with other existing Runge–Kutta-Nystr o ¨ m methods with similar algebraic order. [ABSTRACT FROM AUTHOR]

During the interception of a high-speed gliding vehicle (HGV), the predicted intercept point (PIP) is updated many times, so the interceptor must continuously adjust its trajectory. In this paper, we propose an online guidance correction algorithm for the interceptor, which can effectively meet the position constraints and energy management of the guidance process. Firstly, we simplify the interceptor's maneuvering problem to a two-body orbital transfer problem. The Lambert orbit transfer method is used to enable the interceptor to maneuver towards the target's PIP. Secondly, by comparing the target's high probability existence area (THPEA) with the interceptor-interceptable area (IIA), the engine start threshold is set in stages to avoid frequent redundancy maneuvers caused by frequent engine starting during missile correction. This approach achieves energy management. Simulation results show that our midcourse guidance strategy is effective. [ABSTRACT FROM AUTHOR]

We consider the Kepler two-body problem in the presence of a cosmological constant Λ. Several dimensionless parameters characterizing the possible orbit typologies are used to identify open and closed trajectories. The qualitative picture of the two-body motion is described and critical parameters of the problem are found. [ABSTRACT FROM AUTHOR]

General relativity provides a correction of the perihelion precession which provided the correct value for inertial masses like Mercury. But for binary systems with high eccentricities and large gravitational masses, such as the OJ287 system, there is a large discrepancy not satisfied by this general relativistic correction. The first order correction obtained from the Schwarzschild metric gives a theoretic value of 33.06 degrees, in contrast to the observed value of 39.1 ± 0.1 . In a previous paper, while adopting the perturbation method, they use the Schwarzschild metric and complex integration to obtain higher order expansion terms to make up for this discrepancy. We will be providing an alternate method of correction to the perihelion precession of binary systems, such as that of the OJ287 system, using an alternate formulation of the relativistic potential which provides us an extra term dependent of the radial velocity of the inertial mass. We also obtain values for two other systems, such as the Sagittarius A*-S2, which also has a significant discrepancy, to test for consistency. [ABSTRACT FROM AUTHOR]

This paper presents a finite element method to search for equilibrium points around a heterogeneous small body, which degenerates from the full two-body problem in finite element form. The gravitational potential, acceleration, gravitational gradient matrix, and the linearized perturbation equation, which are key formulas for solving the equilibrium points and discriminating their stabilities, are interpolated by the tetrahedral model nodes of the small body. The finite element method could capture the complex internal structures of small bodies and provide a uniform and simple formula for various configurations. The method is applied to the Kuiper Belt Object (486958) Arrokoth, which is a contact binary asteroid. Three types of heterogeneous structures are hypothesized to investigate the evolutions of equilibrium points, namely density disparity of two sub-lobes, hardcore structure, and cavity structure. Under the condition of constant Arrokoth mass, the density disparity of the two sub-lobes greatly influences the local gravitational field. It thus has a significant influence on the positions of equilibrium points. The hardcore and cavity structures have less impact on the equilibrium points. These three simulation groups verified that the equilibrium points of a heterogeneous small body are quite different from those of a homogeneous small body. Such investigation of the equilibrium points may give an in-depth understanding of the dynamical environment around the heterogeneous small body, which is significant for future deep-space missions. [ABSTRACT FROM AUTHOR]

Eccentricity and spin precession are key observables in gravitational-wave astronomy, encoding precious information about the astrophysical formation of compact binaries together with fine details of the relativistic two-body problem. However, the two effects can mimic each other in the emitted signals, raising issues around their distinguishability. Since inferring the existence of both eccentricity and spin precession simultaneously is – at present – not possible, current state-of-the-art analyses assume that either one of the effects may be present in the data. In such a setup, what are the conditions required for a confident identification of either effect? We present simulated parameter inference studies in realistic LIGO/Virgo noise, studying events consistent with either spin precessing or eccentric binary black hole coalescences and recovering under the assumption that either of the two effects may be at play. We quantify how the distinguishability of eccentricity and spin precession increases with the number of visible orbital cycles, confirming that the signal must be sufficiently long for the two effects to be separable. The threshold depends on the injected source, with inclination, eccentricity, and effective spin playing crucial roles. In particular, for injections similar to GW190521, we find that it is impossible to confidently distinguish eccentricity from spin precession. [ABSTRACT FROM AUTHOR]

Consider the dynamics of two point masses on a surface of constant curvature subject to an attractive force analogue of Newton's inverse square law, that is under a 'cotangent' potential. When the distance between the bodies is sufficiently small, the reduced equations of motion may be seen as a perturbation of an integrable system. We take suitable action-angle coordinates to average these perturbing terms and describe dynamical effects of the curvature on the motion of the two-bodies. [ABSTRACT FROM AUTHOR]

In his 1842 lectures on dynamics C.G. Jacobi summarized difficulties with differential equations by saying that the main problem in the integration of differential equations appears in the choice of right variables. Since there is no general rule for finding the right choice, it is better to introduce special variables first, and then investigate the problems that naturally lend themselves to these variables. This paper follows Jacobi's prophetic observations by introducing certain "meta" variational problems on semi-simple reductive groups G having a compact subgroup K. We then use the Maximum Principle of optimal control to generate the Hamiltonians whose solutions project onto the extremal curves of these problems. We show that there is a particular sub-class of these Hamiltonians that admit a spectral representation on the Lie algebra of G. As a consequence, the spectral invariants associated with the spectral curve produce a large number of integrals of motion, all in involution with each other, that often meet the Liouville complete integrability criteria. We then show that the classical integrals of motion associated, with the Kowalewski top, the two-body problem of Kepler, and Jacobi's geodesic problem on the ellipsoid can be all derived from the aforementioned Hamiltonian systems. We also introduce a rolling geodesic problem that admits a spectral representation on symmetric Riemannian spaces and we then show the relevance of the corresponding integrals on the nature of the curves whose elastic energy is minimal. [ABSTRACT FROM AUTHOR]

Gaussian Mixture Models and State Transition Tensors are combined to propagate state uncertainty to demonstrate an analytical and accurate future uncertainty and conjunction assessment. The initial uncertainty distribution is split into a Gaussian Mixture Model to reduce the size of covariance associated with each mixture component. This reduction in size allows for a significantly more accurate nonlinear mapping of each individual component. For higher accuracy in the mixture component propagation, State Transition Tensors are used to capture the nonlinear effects due to higher-order dynamics, as shown already in work by Fujimoto and Scheeres. In this paper, this GMM-STT method is used to analyze collisions using different dynamical models. The first case study demonstrates the application of this method in two-body problem dynamics; the second also adds J 2 secular effects; and the third brings in full J 2 and Solar Radiation Pressure dynamics through a Simplified Dynamical System. This paper also includes a two-body problem repeating conjunction example to establish a relationship between the propagation time and the number of mixture components needed for an accurate probability of collision computation. In each example, as the number of mixture components increases, the probability of collision converges with the Monte Carlo truth, proving the feasibility of this method. • Orbit Determination models make accurate uncertainty propagation essential. • Propagation with nonlinear dynamics leads to non-Gaussian future uncertainty. • Conjunction models require accurately propagated future state and uncertainty. • Gaussian Mixture Model - State Transition Tensor model gives accurate uncertainty. • Model accuracy is confirmed with comparison to Monte Carlo Probability of Collision. [ABSTRACT FROM AUTHOR]

We have applied an absorbing boundary condition (ABC) method to calculations of the level density (LD) and the continuum level density (CLD). The results of LD and CLD in the ABC calculation are compared with those calculated by the complex scaling method (CSM). The energy dependences of LD in ABC show similar behavior to those in CSM around the resonance energy but they are prominently different, especially at the lower-energy region, due to the different sequence of the energy eigenvalues in the complex plane. The phase shift for two-body scattering problems is also derived from CLD. The phase shift obtained by ABC is almost the same as that calculated from CSM although the ABC method seems to be slightly inferior to the CSM calculation in the numerical precision. The improvement of the numerical precision in ABC is also discussed. [ABSTRACT FROM AUTHOR]

In the present work, we define a new anomaly, Ψ$$ \Psi $$, termed semifocal anomaly. It is determined by the mean between the true anomaly, f$$ f $$, and the antifocal anomaly, f′$$ {f}^{\prime } $$; Fukushima defined f′$$ {f}^{\prime } $$ as the angle between the periapsis and the secondary around the empty focus. In this first part of the paper, we take an approach to the study of the semifocal anomaly in the hyperbolic motion and in the limit case corresponding to the parabolic movement. From here, we find a relation between the semifocal anomaly and the true anomaly that holds independently of the movement type. We focus on the study of the two‐body problem when this new anomaly is used as the temporal variable. In the second part, we show the use of this anomaly—combined with numerical integration methods—to improve integration errors in one revolution. Finally, we analyze the errors committed in the integration process—depending on several values of the eccentricity—for the elliptic, parabolic, and hyperbolic cases in the apsidal region. [ABSTRACT FROM AUTHOR]

In this manuscript, we review the motion of a two-body celestial system (planet–sun) for a Yukawa-type correction on Newton's gravitational potential using Hamilton's formulation. We reexamine the stability using the corresponding linearization Jacobian matrix, and verify that the conditions of Bertrand's theorem are met for radii ≪ 10 15 m, meaning that bound closed orbits are expected. Applied to the solar system, we present the equation of motion of the planet, then solve it both analytically and numerically. Making use of the analytical expression of the orbit, we estimate the Yukawa strength α and find it to be larger than the nominal value ( 10 − 8 ) adopted in previous studies, in that it is of order ( α = 10 − 4 − 10 − 5 ) for the terrestrial planets (Mercury, Venus, earth, Mars, and Pluto) and even larger ( α = 10 − 3 ) for the giant planets (Jupiter, Saturn, Uranus, and Neptune). Taking the inputs ( r m i n , v m a s , e ) observed by NASA, we analyse the orbits analytically and numerically for both the estimated and nominal values of α and determine the corresponding trajectories. For each obtained orbit, we recalculate the characterizing parameters ( r m i n , r m a x , a , b , e ) and compare their values according to the potential (Newton with/without Yukawa correction) and method (analytical and/or numerical) used. When compared to the observational data, we conclude that the path correction due to Yukawa correction is on the order of up to 80 million km (20 million km) as the maximum deviation occurring for Neptune (Pluto) for a nominal (estimated) value of α. [ABSTRACT FROM AUTHOR]

The post-Minkowskian expansion of Einstein's general theory of relativity has received much attention in recent years due to the possibility of harnessing the computational power of modern amplitude calculations in such a classical context. In this brief review, we focus on the post-Minkowskian expansion as applied to the two-body problem in general relativity without spin, and we describe how relativistic quantum field theory can be used to greatly simplify analytical calculations based on the Einstein–Hilbert action. Subtleties related to the extraction of classical physics from such quantum mechanical calculations highlight the care which must be taken when both positive and negative powers of Planck's constant are at play. In the process of obtaining classical results in both Einstein gravity and supergravity, one learns new aspects of quantum field theory that are obscured when using units in which Planck's constant is set to unity. The scattering amplitude approach provides a self-contained framework for deriving the two-body scattering valid in all regimes of energy. There is hope that the full impact of amplitude computations in this field may significantly alter the way in which gravitational wave predictions will advance in the coming years. [ABSTRACT FROM AUTHOR]

The problem of the two‐body gravitational interaction has been solved numerically based on the classical mechanics principles. One of the bodies is a deformable three‐axis ellipsoid (central body) and the other is a material point (satellite). The relationship of the angular discrepancy between the calculated and actual positions of the satellite pericenter with central body's gravity anomaly has been established. [ABSTRACT FROM AUTHOR]

Gravitational waves, despite predicted within the Theory of General Relativity about a century ago, were observed experimentally only in the last few years, thanks to the recent advances of the LIGO and Virgo detectors. In this work, we introduce a simple theoretical model to study the gravitational irradiation of binary systems and derive the characteristics of the emitted gravitational waves (e.g., in terms of frequency). The simplicity of the model relies first on the consideration of the two-body problem and circular orbits. Second, we follow an energetic approach and make use of only one result of the Theory of General Relativity, to make the procedure clear and suitable for undergraduate education. The numerical solution is finally compared with the experimental data available for a few black hole mergers. [ABSTRACT FROM AUTHOR]

The strategy of capturing near-Earth asteroids to Sun-Earth L 1 /L 2 Lyapunov orbit using a lunar flyby is investigated in this paper. Families of Lyapunov orbits are calculated in the Sun-Earth circular restricted three-body problem (CR3BP), and Lyapunov orbits with small amplitudes serve as the final parking orbits for the captured asteroids. Then the Sun-Earth-Moon bicircular model (BCM) is used to describe the dynamical behavior of the captured asteroid flying close to the Moon. With the assumption that the lunar flyby is simplified as an orbit in the Moon-centered two-body problem, analytical expressions are derived to describe the energy changes of the stable manifolds of the Lyapunov orbits with a lunar flyby. According to the analytical results of the energy changes, optimizations of capturing asteroids to Sun-Earth L 1 /L 2 Lyapunov orbits using a lunar flyby are then carried out. Numerical results indicate that the asteroid capture method using a lunar flyby can potentially save energy compared to the direct asteroid capture strategy without a lunar flyby. Moreover, the asteroid capture with a lunar flyby also has the potential to reduce the total flight time compared to the direct capture strategy without a lunar flyby. • Stable manifolds with a lunar flyby are investigated to capture asteroids. • Analytical expressions of energy changes with a lunar flyby are derived. • Analytical results of energy changes are used for optimizations. • The lunar flyby can help to reduce the energy of capturing asteroids. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to search for orbits that can be used in the Brazilian proposed Aster mission. This mission is under study and its objective is to use a spacecraft to observe the system 2001SN263, which is a triple asteroid system. With respect to the two-body problem (spacecraft and the main asteroid), the symmetries of the orbits are broken by the oblateness of the main body of the system, the solar radiation pressure, and the gravitational attraction of the two moons of the main body. Additionally, the masses of these two moons have errors associated with their predicted values, which reinforce the asymmetry and require extra effort to maintain the observational objectives of the mission. The idea is to find orbits that remain for some time observing the three bodies of that system, even if the physical parameters of the bodies are not the ones expected from observations made from the Earth. This is accomplished by studying the effects of errors in all the physical properties of the three asteroids in the trajectories described by a spacecraft that is orbiting this system. Several important and useful trajectories are found, which are the ones that can observe the desired bodies, even if the physical parameters are not the expected ones. To express our results, we built time histories of the relative distances between each of the asteroids and the spacecraft. They are used to select the trajectories according to the amount of time that we need to observe each body of the system. In this way, the first objective of this research is to search for trajectories to keep the spacecraft close to the three bodies of the system as long as possible, without requiring orbital maneuvers. The errors for the masses of the two smaller and lesser known bodies are taken into consideration, while the mass of the most massive one is assumed to be known, because it was determined with higher precision by observations. [ABSTRACT FROM AUTHOR]

The quasi-homogeneous two-body problem aims at studying the interaction between two point particles under a prescribed potential in the form of W (r) = − A r a − B r b , where A, B > 0 are constants and r is the mutual distance between two particles. Important examples include the Manev potential (a = 1, b = 2) and the Schwarzschild potential (a = 1, b = 3). It is well known that power two serves as a threshold value for the homogeneous potential: One is able to observe significant differences regarding the solution dynamics as the power of the homogeneous potential exceeds two from below. This phenomenon remains observable for quasi-homogeneous potentials. In this paper, we shall provide a complete characterization of the whole phase space of the quasi-homogeneous two-body problem in terms of global existence and singularity for all the possible b > a > 0. In particular, one is able to generalize the result of the Manev and Schwarzschild two-body problem to all the quasi-homogeneous potentials. Two techniques are presented in this paper: One is the variational method based on the energy, and the other is a direct computation of collision time based on the integrability of two-body systems. [ABSTRACT FROM AUTHOR]

Recent higher-order explicit Runge–Kutta methods are compared with the classic fourth-order (RK4) method in long-term integration of both energy-conserving and lossy systems. By comparing quantity of function evaluations against accuracy for systems with and without known solutions, optimal methods are proposed. For a conservative system, we consider positional accuracy for Newtonian systems of two or three bodies and total angular momentum for a simplified Solar System model, over moderate astronomical timescales (tens of millions of years). For a nonconservative system, we investigate a relativistic two-body problem with gravitational wave emission. We find that methods of tenth and twelfth order consistently outperform lower-order methods for the systems considered here. [ABSTRACT FROM AUTHOR]

The Einstein–Hilbert action of the general theory of relativity (GR) is the integral of the scalar curvature R. It is a theory that is drawn from the Equivalence principle and has predictions that come out as a consequence of the principle, in observables. Testing such observables to find confirmation/infirmation of the principle has formed a significant chunk of tests of GR itself. It is expected that quantum corrections to GR may add additional higher powers of R to the Einstein–Hilbert action, or more generally, modifying the action into a generic class of functions of the Ricci scalar. Testing the fate of the prized equivalence principle, in such modified theories of gravity, hence become important in order to obtain a more generic theory of gravitation, and consequently, of gravitating objects. In this study, it is shown that a Post-Newtonian (PN) expansion of a class of f R theories lead to a sequence of solutions to the two-body problem, which follows the equivalence principle (EP) at the Newtonian order, and generalizes to the 'effacing principle' at a higher PN order. [ABSTRACT FROM AUTHOR]

The workshop gathered theorists working on different though connected areas concerning the recent discovery of gravitational waves. It fostered new collaborations between the quantum gravitational scattering amplitude and the general relativity community, leading to the calculation of new, high-order, terms in the post-Newtonian and post-Minkowskian perturbative approaches to the physics of binary systems, at both analytical and numerical level, in order to construct the waveform templates necessary for the analysis of LIGO/Virgo data. The use of recent progress on gravitational scattering and radiation in ultra-relativistic collisions of elementary particles or strings improved the determination of parameters appearing in the effective-one-body approach to the relativistic two-body problem. Various consequences of modified gravity theories for the LIGO/Virgo discoveries were also explored. [ABSTRACT FROM AUTHOR]

We show that the Zwanzig first-order perturbation theory can be obtained directly from a truncated Taylor series expansion of a two-body perturbation theory and that such truncation provides a more accurate prediction of thermodynamic properties than the full two-body perturbation theory. This unexpected result is explained by the quality of the resulting approximation for the fluid radial distribution function. We prove that the first-order and the two-body perturbation theories are based on different approximations for the fluid radial distribution function. To illustrate the calculations, the square-well fluid is adopted. We develop an analytical expression for the two-body perturbed Helmholtz free energy for the square-well fluid. The equation of state obtained using such an expression is compared to the equation of state obtained from the first-order approximation. The vapor-liquid coexistence curve and the supercritical compressibility factor of a square-well fluid are calculated using both equations of state and compared to Monte Carlo simulation data. Finally, we show that the approximation for the fluid radial distribution function given by the first-order perturbation theory provides closer values to the ones calculated via Monte Carlo simulations. This explains why such theory gives a better description of the fluid thermodynamic behavior. [ABSTRACT FROM AUTHOR]

We present the theory and algorithms for computing analytical energy gradients for explicitly correlated second-order Møller-Plesset perturbation theory (MP2-F12). The main difficulty in F12 gradient theory arises from the large number of two-electron integrals for which effective two-body density matrices and integral derivatives need to be calculated. For efficiency, the density fitting approximation is used for evaluating all two-electron integrals and their derivatives. The accuracies of various previously proposed MP2-F12 approximations [3C, 3C(HY1), 3*C(HY1), and 3*A] are demonstrated by computing equilibrium geometries for a set of molecules containing first- and second-row elements, using double-ζ to quintuple-ζ basis sets. Generally, the convergence of the bond lengths and angles with respect to the basis set size is strongly improved by the F12 treatment, and augmented triple-ζ basis sets are sufficient to closely approach the basis set limit. The results obtained with the different approximations differ only very slightly. This paper is the first step towards analytical gradients for coupled-cluster singles and doubles with perturbative treatment of triple excitations, which will be presented in the second part of this series. [ABSTRACT FROM AUTHOR]

We revisit the two-body problem, where one body can be deformed under the action of tides raised by the companion. Tidal deformation and consequent dissipation result in spin and orbital evolution of the system. In general, the equations of motion are derived from the tidal potential developed in Fourier series expressed in terms of Keplerian elliptical elements, so that the variation of dissipation with amplitude and frequency can be examined. However, this method introduces multiple index summations and some orbital elements depend on the chosen frame, which is prone to confusion and errors. Here, we develop the quadrupole tidal potential solely in a series of Hansen coefficients, which are widely used in celestial mechanics and depend just on the eccentricity. We derive the secular equations of motion in a vectorial formalism, which is frame independent and valid for any rheological model. We provide expressions for a single average over the mean anomaly and for an additional average over the argument of the pericentre. These equations are suitable to model the long-term evolution of a large variety of systems and configurations, from planet–satellite to stellar binaries. We also compute the tidal energy released inside the body for an arbitrary configuration of the system. [ABSTRACT FROM AUTHOR]

Two groups of astronomers used the large telescopes Keck and VLT for decades to observe trajectories of bright stars near the Galactic Centre. Based on results of their observations, the astronomers concluded that trajectories of the stars are roughly elliptical and foci of the orbits are approximately coincide with the Galactic Centre position. In a last few years, a self-gravitating dark matter core–halo distribution was suggested by Ruffini, Argüelles, Rueda (RAR) and this model was actively used in consequent studies. In particular, recently it has been claimed that the RAR-model provides a better fit of trajectories of bright stars in comparison to the conventional model with a supermassive black hole. The dark matter distribution with a dense core having a constant density as it was suggested in the RAR-model leaves trajectories of stars elliptical like in Kepler's two-body problem. However, in this case not the foci of the ellipses coincide with the Galactic Centre but their centres while the orbital periods do not depend on semi-major axes. These properties are not consistent with the observational data for trajectories of bright stars. [ABSTRACT FROM AUTHOR]