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The research on highly inclined mean motion resonances (MMRs), even retrograde resonances, has drawn more attention in recent years. However, the dynamics of polar resonance with inclination i ≈ 90 ∘ have received much less attention. This paper systematically studies the dynamics of polar resonance and their effects on the Kozai–Lidov mechanism in the circular restricted three-body problem (CRTBP). The maps of dynamics are obtained through the numerical method and semi-analytical method, by mutual authenticating. We investigate the secular dynamics inside polar resonance. The phase-space portraits on the e − ω plane are plotted under exact polar resonance and considering libration amplitude of critical angle σ. Simultaneously, we investigate the evolution of 5000 particles in polar resonance by numerical integrations. We confirm that the e − ω portraits can entirely explain the results of numerical experiments, which demonstrate that the phase-space portraits on the e − ω plane obtained through the semi-analytical method can represent the real Kozai–Lidov dynamics inside polar resonance. The resonant secular dynamical maps can provide meaningful guidance for predicting the long-term evolution of polar resonant particles. As a supplement, in the polar 2 / 1 case, we analyze the pure secular dynamics outside resonance, and confirm that the effect of polar resonance on secular dynamics is pronounced and cannot be ignored. Our work is a meaningful supplement to the general inclined cases and can help us understand the evolution of asteroids in polar resonance with the planet. [ABSTRACT FROM AUTHOR]

Many asteroids move in the belt between the orbits of Mars and Jupiter under the gravitational attraction of the Sun and planets in the Solar system. If one of these asteroids does not leave the belt during a period, it is considered to be temporarily stable on that time-scale. This paper aims to study the time-scales on which asteroids could stay in the main belt. A simplified situation is studied in which the initial orbital inclinations and the longitudes of the ascending nodes of the asteroids are set to zero. Numerical integration is used to study the temporal stability of the main-belt asteroids. In the integration, the distribution of the instability time for randomly generated particles can be fitted with a function. Thus a reasonable method is presented to choose an integration time-scale based on the percentage of the already unstable particles in relation to all unstable particles. A total of 151 000 particles are generated and then integrated for 8.09 × 106 yr. The integration data are used to train the probabilistic neural networks to predict the stability of particles. A temporal stability map in the a - e plane is obtained from the prediction results of the neural networks. [ABSTRACT FROM AUTHOR]

In the preliminary trajectory design of the multi-target rendezvous problem, a model that can quickly estimate the cost of the orbital transfer is essential. The estimation of the transfer time using solar sails between two arbitrary orbits is difficult and usually requires to solve an optimal control problem. Inspired by the successful applications of the deep neural networks in nonlinear regression, this work explores the possibility and effectiveness of mapping the transfer time for solar sails from the orbital characteristics using the deep neural networks. Furthermore, the Monte Carlo Tree Search method is investigated and used to search the optimal sequence considering a multi-asteroid exploration problem. The obtained sequences from preliminary design will be solved and verified by sequentially solving the optimal control problem. Two examples of different application backgrounds validate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

With the discoveries of more minor bodies in retrograde resonances with giant planets, such as 2015 BZ509 and 2006 RJ2, our curiosity about the Kozai-Lidov dynamics inside the retrograde resonance has been sparked. In this study, we focus on the 3D retrograde resonance problem and investigate how the resonant dynamics of a minor body impacts on its own Kozai-Lidov cycle. First we deduce the action-angle variables and canonical transformations that deal with the retrograde orbit specifically. After obtaining the dominant Hamiltonian of this problem, we then carry out the numerical averaging process in closed form to generate phase-space portraits on an e − ω space. The retrograde 1:1 resonance is particularly scrutinized in detail, and numerical results from a CRTBP model show a great agreement with our semi-analytical portraits. On this basis, we inspect two real minor bodies currently trapped in retrograde 1:1 mean motion resonance. It is shown that they have different Kozai-Lidov states, which can be used to analyse the stability of their unique resonances. In the end, we further inspect the Kozai-Lidov dynamics inside the 2:1 and 2:5 retrograde resonance, and find distinct dynamical bifurcations of equilibrium points on phase-space portraits. [ABSTRACT FROM AUTHOR]

Aims. The asteroid 2015 BZ509 is the first asteroid confirmed to be in retrograde co-orbit resonance (or 1/−1 resonance) with the giant planets in the solar system. While Saturn is the only giant planet whose trojans have not yet been discovered, we identify some small bodies among centaurs and damocloids that are potentially in 1/−1 resonance with Saturn in the present study. Methods. We integrate numerically the motion of the 1000 clones (including the nominal orbit) of each centaur whose orbit has a semi-major axis between 9.3 au and 9.8 au and an inclination i > 90°. To confirm and evaluate the 1/−1 resonant configurations mentioned above, we introduce a useful one degree integrable approximation for planar 1/−1 resonance. Results. We identify four candidates potentially in 1/−1 resonance with Saturn. The capture of candidates in this particular resonant state during the 40 000 yr integration time span is very common for 2006 RJ2 (906/1000 clones), 2006 BZ8 (878/1000 clones), and 2017 SV13 (998/1000 clones), and it is less likely for 2012 YE8 (426/1000 clones). According to our statistical results, 2006 RJ2 is the best candidate to be currently in a 1/−1 mean motion resonance with Saturn, and 2017 SV13 is another important potential candidate. Moreover, 2012 YE8 and 2006 BZ8 are also centaurs of interest but their current and long-term 1/−1 resonant state with Saturn is less likely. The proportions of the clones captured in the relative long-term stable co-orbit resonance (over 10 000 yr) are also given. The motions of the 2006 RJ2, 2015 BZ509, and 2006 BZ8 in the solar system are just around the ideal equilibrium points of the 1/−1 resonance given by the planar semi-analytical model. Conclusions. Small bodies in retrograde co-orbit resonance with giant planets may be more common than previously expected. Identification of these potential mysterious minor bodies encourages the search for such objects on a larger scale in our solar system. The findings of this paper are also useful for understanding the origin and dynamical evolution of centaurs and damocloids on retrograde orbits. [ABSTRACT FROM AUTHOR]

The Hill stability of the nine known triple asteroid systems in the solar system has been investigated in a framework of the three body system. In this paper, the Sun and triple-asteroid system are treated as a four body system to analyze the influence of the Sun on the Hill stability of the triple subsystem. First, the relationship of the total energy and the angular momentum between the four body system and the triple subsystem is derived. It is found that the total energy of this 1-3 configuration four body system is the sum of the energy of the triple subsystem and the energy of a two-body system composed of the Sun and the mass center of the subsystem; so is the angular momentum. Then, the Hill stability of the triple subsystem is reinvestigated using a previous criterion in the four body problem (Gong and Liu in Mon. Not. R. Astron. Soc. 462:547-553, 2016) and the results are compared to those in the three body problem. Among the nine known triple-asteroid systems, 1995 CC and 1999 TC are Hill stable for both models; the others are stable in the three body model while not stable in the four body model. In addition, the exploration of Pluto by New Horizons has attracted great attention in recent years, the Sun-Pluto-Charon-Hydra four body system is investigated in the paper, and it is found that the system is Hill stable. [ABSTRACT FROM AUTHOR]

A falling meteor that blazed and exploded over the Russian Ural Mountains on February 15, 2013, has been attracting attention from all over the world. It was estimated by the Russian Academy of Science and the U.S. National Aeronautics and Space Administration (NASA) that the meteor entered Earth's atmosphere at a speed of approximately 33?40 km/s. The energy released by the meteor is on the level of hundreds of kilotons of TNT. The threat posed by such falling meteors and Earth-crossing asteroids (ECAs) cannot be excluded as one of the most unpredictable risks to Earth and humans. [ABSTRACT FROM PUBLISHER]

We consider an elliptic restricted four-body system including three primaries and a massless particle. The orbits of the primaries are elliptic, and the massless particle moves under the mutual gravitational attraction. From the dynamic equations, a quasi-integral is obtained, which is similar to the Jacobi integral in the circular restricted three-body problem (CRTBP). The energy constant C determines the topology of zero velocity surfaces, which bifurcate at the equilibrium point. We define the concept of Hill stability in this problem, and a criterion for stability is deduced. If the actual energy constant Cac(>0) is bigger than or equal to the critical energy constant Ccr, the particle will be Hill stable. The critical energy constant is determined by the mass and orbits of the primaries. The criterion provides a way to capture an asteroid into the Earth-Moon system. [ABSTRACT FROM AUTHOR]

A gravity tractor spacecraft propelled by a diffractive sail tracking a displaced orbit offers a potential option for deflecting near-Earth asteroids. This study delves into the preliminary dynamics, evaluates the deflection capabilities, and investigates the stability and control of the diffractive sail on the displaced orbit near the asteroid. Unlike reflective sails, diffractive solar sails generate tangential radiation pressure force, offering multiple advantages. These include the elimination of tilt attitude requirements, no offset angle restrictions from the target asteroid's flight direction, and superior deflection performance with an equivalent area-to-mass ratio when the diffraction angle surpasses 50 deg. To maintain position, a controllable diffractive sail model using spatial light modulators, coupled with a feedback control law, is proposed. This control design stabilizes the periodic displaced orbit without necessitating tilt attitude adjustments towards the sun, as validated by our numerical simulations. [ABSTRACT FROM AUTHOR]