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In Book 1, Proposition 7, Problem 2 of his 1687 Philosophiae Naturalis Principia Mathematica, Isaac Newton poses and answers the following question: Let the orbit of a particle moving in a central force field be an off-center circle. How does the magnitude of the force depend on the position of the particle onthat circle? In this article, we identify a potential that can produce such a force, only at zero energy. We further map the zero-energy orbits in this potential to finite-energy free motion orbits on a sphere; such a duality is a particular instance of a general result by Goursat, from 1887. The map itself is an inverse stereographic projection, and this fact explains the circularity of the zero-energy orbits in the system of interest. Finally, we identify an additional integral of motion—an analogue of the Runge-Lenz vector in the Coulomb problem—that is responsible for the closeness of the zero-energy orbits in our problem.

The many problems faced by the theory of general relativity (GR) have always motivated us to explore the modified theory of GR. Considering the importance of studying the black hole (BH) entropy and its correction in gravity physics, we study the correction of thermodynamic entropy for a kind of spherically symmetric black hole under the generalized Brans–Dicke (GBD) theory of modified gravity. We derive and calculate the entropy and heat capacity. It is found that when the value of event horizon radius r+ is small, the effect of the entropy-correction term on the entropy is very obvious, while for larger values r+, the contribution of the correction term on entropy can be almost ignored. In addition, we can observe that as the radius of the event horizon increases, the heat capacity of BH in GBD theory will change from a negative value to a positive value, indicating that there is a phase transition in black holes. Given that studying the structure of geodesic lines is important for exploring the physical characteristics of a strong gravitational field, we also investigate the stability of particles’ circular orbits in static spherically symmetric BHs within the framework of GBD theory. Concretely, we analyze the dependence of the innermost stable circular orbit on model parameters. In addition, the geodesic deviation equation is also applied to investigate the stable circular orbit of particles in GBD theory. The conditions for the stability of the BH solution and the limited range of radial coordinates required to achieve stable circular orbit motion are given. Finally, we show the locations of stable circular orbits, and obtain the angular velocity, specific energy, and angular momentum of the particles which move in circular orbits.

The properties of modified Hayward black hole space-time can be investigated through analyzing the particle geodesics. By means of a detailed analysis of the corresponding effective potentials for a massive particle, we find all possible orbits which are allowed by the energy levels. The trajectories of orbits are plotted by solving the equation of orbital motion numerically. We conclude that whether there is an escape orbit is associated with b\\(b\\) (angular momentum). The properties of orbital motion are related to b\\(b\\), α\\(\\alpha \\) (α\\(\\alpha \\) is associated with the time delay) and β\\(\\beta \\) (β\\(\\beta \\) is related to 1-loop quantum corrections). There are no escape orbits when b<4.016M\\(b<4.016M\\), α=0.50\\(\\alpha = 0.50\\) and β=1.00\\(\\beta = 1.00\\). For fixed α=0.50\\(\\alpha = 0.50\\) and β=1.00\\(\\beta = 1.00\\), if b<3.493M\\(b < 3.493M\\), there only exist unstable orbits. Comparing with the regular Hayward black hole, we go for a reasonable speculation by mean of the existing calculating results that the introduction of the modified term makes the radius of the innermost circular orbit (ISCO) and the corresponding angular momentum larger.

Computer algebra methods are used to determine the equilibrium orientations of a system of two bodies connected by a spherical hinge that moves in a central Newtonian force field on a circular orbit under the action of gravitational torque. Primary attention is given to the study of equilibrium orientations of the two-body system in the plane of the circular orbit. By applying symbolic differentiation, differential equations of motion are derived in the form of Lagrange equations of the second kind. A method is proposed for transforming the system of trigonometric equations determining the equilibria into a system of algebraic equations, which in turn are reduced by calculating the resultant to a single algebraic equation of degree 12 in one unknown. The roots of the resulting algebraic equation determine the equilibrium orientations of the two-body system in the circular orbit plane. By applying symbolic factorization, the algebraic equation is decomposed into three polynomial factors, each specifying a certain class of equilibrium configurations. The domains with an identical number of equilibrium positions are classified using algebraic methods for constructing a discriminant hypersurface. The equations for the discriminant hypersurface determining the boundaries of domains with an identical number of equilibrium positions in the parameter space of the problem are obtained via symbolic computations of the determinant of the resultant matrix. By numerical analysis of the real roots of the resulting algebraic equations, the number of equilibrium positions of the two-body system is determined depending on the parameters.

In the paper, the problem of forming and maintaining the small satellites formation in the near-earth projected circular orbits is considered. The satellite formation reconfiguration and formation-keeping control laws are proposed by employing the passivity-based output feedback control. For the complete nonlinear and time-dependent dynamics of the relative motion of a pair of satellites in elliptical orbits, new combined control algorithms, including a consensus protocol, are proposed and analyzed. A comparison of the control modes using the passivity-based output feedback control and the proportional-differential controller with and without the consensus algorithm is given. On the basis of the passification method, the algorithm is obtained ensuring the stable motion of the slave satellite relative to the orbit of the master satellite. To improve the accuracy of the satellites’ positioning, a consensus protocol based on measurements of the relative positions of the satellites is proposed and studied. Computer simulations of the proposed algorithms for options to construct formations are provided for two projected circular orbits of 8 satellites, demonstrating the efficiency of the proposed control schemes. It is shown that the resulting passivity-based output feedback control provides better accuracy than the PD controller. It is also shown that the use of the consensus protocol further increases the positioning accuracy of the satellite constellation.

The paper describes the method for the development of programs for two-parameter multiple impulse correction of altitude and inclination of circular orbits. The considered corrections feature the simultaneous adjustment of altitude and inclination of the orbit under limited thrust of the propulsion system of the spacecraft. Low thrust-to-weight ratio of the spacecraft leads to the need for a correction program consisting of several burns of the propulsion system. Thrust vector orientation, burn time, and its operation duration are determined as propulsion system parameters.

Recently, a novel 4 D Einstein–Gauss–Bonnet gravity was formulated by Glavan and Lin (Phys Rev Lett 124(8):081301, 2020). Although whether the theory is well defined is currently debatable, the spherically symmetric black hole solution is still meaningful and worthy of study. In this paper, we study the geodesic motions in the spacetime of the spherically symmetric black hole solution. First of all, we find that a negative GB coupling constant is allowable, as in which case the singular behavior of the black hole can be hidden inside the event horizon. Then we calculate the innermost stable circular orbits for massive particles, which turn out to be monotonic decreasing functions of the GB coupling constant. Furthermore, we study the unstable photon sphere and shadow of the black hole. It is interesting to find that the proposed universal bounds on black hole size in Lu and Lyu (Phys Rev D 101(4):044059, 2020) recently can be broken when the GB coupling constant takes a negative value.

A bstract Building upon the worldline effective field theory (EFT) formalism for spinning bodies developed for the Post-Newtonian regime, we generalize the EFT approach to Post-Minkowskian (PM) dynamics to include rotational degrees of freedom in a manifestly covariant framework. We introduce a systematic procedure to compute the total change in momentum and spin in the gravitational scattering of compact objects. For the special case of spins aligned with the orbital angular momentum, we show how to construct the radial action for elliptic-like orbits using the Boundary-to-Bound correspondence. As a paradigmatic example, we solve the scattering problem to next-to-leading PM order with linear and bilinear spin effects and arbitrary initial conditions, incorporating for the first time finite-size corrections. We obtain the aligned-spin radial action from the resulting scattering data, and derive the periastron advance and binding energy for circular orbits. We also provide the (square of the) center-of-mass momentum to O G 2 , which may be used to reconstruct a Hamiltonian. Our results are in perfect agreement with the existent literature, while at the same time extend the knowledge of the PM dynamics of compact binaries at quadratic order in spins.

A bstract We extend the boundary-to-bound (B2B) correspondence to incorporate radiative as well as conservative radiation-reaction effects. We start by deriving a map between the total change in observables due to gravitational wave emission during hyperbolic-like motion and in one period of an elliptic-like orbit, which is valid in the adiabatic expansion for non-spinning as well as aligned-spin configurations. We also discuss the inverse problem of extracting the associated fluxes from scattering data. Afterwards we demonstrate, to all orders in the Post-Minkowskian expansion, the link between the radiated energy and the ultraviolet pole in the radial action in dimensional regularization due to tail effects. This implies, as expected, that the B2B correspondence for the conservative sector remains unchanged for local-in-time radiation-reaction tail effects with generic orbits. As a side product, this allows us to read off the energy flux from the associated pole in the tail Hamiltonian. We show that the B2B map also holds for non-local-in-time terms, but only in the large-eccentricity limit. Remarkably, we find that all of the trademark logarithmic contributions to the radial action map unscathed between generic unbound and bound motion. However, unlike logarithms, other terms due to non-local effects do not transition smoothly to quasi-circular orbits. We conclude with a discussion on these non-local pieces. Several checks of the B2B dictionary are displayed using state-of-the-art knowledge in Post-Newtonian/Minkowskian theory.

To be observed and analyzed by the network of current gravitational-wave detectors (LIGO, Virgo, KAGRA), and in anticipation of future third generation ground-based (Einstein Telescope, Cosmic Explorer) and space-borne (LISA) detectors, inspiralling compact binaries—binary star systems composed of neutron stars and/or black holes in their late stage of evolution prior the final coalescence—require high-accuracy predictions from general relativity. The orbital dynamics and emitted gravitational waves of these very relativistic systems can be accurately modelled using state-of-the-art post-Newtonian theory. In this article we review the multipolar-post-Minkowskian approximation scheme, merged to the standard post-Newtonian expansion into a single formalism valid for general isolated matter system. This cocktail of approximation methods (called MPM-PN) has been successfully applied to compact binary systems, producing equations of motion up to the fourth-post-Newtonian (4PN) level, and gravitational waveform and flux to 4.5PN order beyond the Einstein quadrupole formula. We describe the dimensional regularization at work in such high post-Newtonian calculations, for curing both ultra-violet and infra-red divergences. Several landmark results are detailed: the definition of multipole moments, the gravitational radiation reaction, the conservative dynamics of circular orbits, the first law of compact binary mechanics, and the non-linear effects in the gravitational-wave propagation (tails, iterated tails and non-linear memory). We also discuss the case of compact binaries moving on eccentric orbits, and the effects of spins (both spin-orbit and spin–spin) on the equations of motion and gravitational-wave energy flux and waveform.