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The Earth–Moon libration points no longer exhibit the dynamical characteristics of “equilibrium points” due to perturbation effects when applying the ephemeris model. By decoupling the forced motions within the ephemeris model and computing the dynamical substitute trajectories, we can reconstruct a dynamical system that recovers the “equilibrium points” feature. Diverging from the conventional analytical approach rooted in the framework of Newtonian mechanics, this paper presents a novel method for calculating dynamical substitute based on the Hamiltonian mechanics framework. First, the Hamiltonian equations for the ephemeris model are formulated. Subsequently, the problem of decoupling forced motions is reformulated as solving a nonautonomous differential equation through canonical transformations. Then, an iterative method based on frequency analysis is employed for the computation. Eventually, approximate analytical solutions for five libration points over a 360 yr period are provided. Simulation results demonstrate that the computed approximate analytical solutions are in excellent agreement with the numerical integration results derived from the ephemeris model, thereby validating the efficacy of the proposed method. The Hamiltonian dynamical system derived herein enables the analysis of nonlinear central manifold motions via canonical transformations, facilitating the construction of higher-order analytical solutions for libration point orbits. This framework also provides a robust foundation for exploring characterization parameters of libration point orbits within the real Earth–Moon system.

In this paper I analyze the context in which Cassini 1st described lunar libration and proposed its interpretation.

This paper studies the dynamics and libration suppression of a tethered system with a moving climber in circular orbits. The tethered system is modeled by a two-piece dumbbell model that consists of one main satellite, one climber and one end-body connected by two straight, massless and inextensible tethers. A new tension control strategy to suppress the libration of the tethered system due to the moving climber is proposed by reeling in-out tether at the end-body without thrust. The control strategy is implemented with the sliding mode control to suppress the libration angle of the climber to zero by the end of climber’s transfer phase. The numerical results show that the proposed control strategy is very effective in suppressing the libration of the climber in the three-body tethered system with tension control only. Furthermore, cases with limited tension control are examined. It reveals that a longer tether between the climber and the end-body is required to supplement the limited tension in suppressing the libration of the climber.

Several of the icy moons in the Jupiter and Saturn systems appear to possess internal liquid water oceans. Our knowledge of the Uranian moons is more limited but a future tour of the system has the potential to detect subsurface oceans. Planning for this requires an understanding of how the moons' internal structures—with and without oceans—relate to observable quantities. Here, we show that the amplitude of forced physical librations could be diagnostic of the presence or absence of subsurface oceans within the Uranian moons. In the presence of a decoupling global ocean, ice shell libration amplitudes at Miranda, Ariel, and Umbriel will exceed 100 m if the shells are <30${< } 30$km$\\mathrm{k}\\mathrm{m}$thick. The presence of oceans could also imply significant tidal heating within the last few hundred million years. Combining librations with the quadrupole gravity field could provide comprehensive constraints on the internal structures and histories of the Uranian moons. Plain Language Summary Several of the icy moons in the Jupiter and Saturn systems appear to possess internal liquid water oceans. Our knowledge of the Uranian moons is more limited but a future tour of the Uranus system has the potential to detect subsurface oceans. Planning for this requires an understanding of how the moons' internal structures—with and without oceans—relate to observable quantities. Here, we show that certain aspects of their rotational states could be diagnostic of the presence or absence of internal liquid water oceans within several of the Uranian moons and that combining this with measurements of the gravity field could provide comprehensive constraints on the internal structures and histories of the Uranian moons. Key Points Measuring physical libration amplitudes can be used to detect subsurface liquid water oceans within the Uranian moons Combining librations with gravity measurements can yield comprehensive constraints on the interiors of the Uranian moons Thick oceans are easier to detect but finding thin oceans may require libration amplitudes to be measured to within better than 10 m

The paper considers the possibility of constructing of a space elevator that is attached to a moon and deployed towards its planet. The space elevator consists of a climber, end mass, and a tether connecting an attachment point on the moon and the end mass. In the static state of the climber, the space elevator is a double pendulum. The stability of the zero equilibrium position of the double pendulum fixed on the lunar surface under the L1 libration point and deployed towards the planet is investigated in the frame of the planar circular restricted three-body problem. The nonlinear equations of motion of the double pendulum are derived and then linearized. The condition for the stability of the double pendulum motion is obtained in analytical form. The performed analysis indicates that, for the Mars-Phobos system, the minimum required length for a stable space elevator range from 3.5 to 40 km, depending on the mass ratio between the end mass and the climber, which is from 40 to 1/4. In the Earth-Moon system the minimum required length is between 284·103 and 375·103 km. Obviously, constructing a space elevator anchored on the Moon's surface and extending towards the Earth poses significant challenges. In contrast, building a space elevator anchored on Phobos’ surface holds considerable promise for future missions, such as establishing a transportation corridor between Phobos’ surface and orbit of Mars, or as a means of maintaining a long-duration orbital station in a stable position between Mars and Phobos.

The paper is devoted to the investigation of the possibility of constructing a double pendulum fixed at the L1 libration point in the framework of the planar circular restricted three-body problem. Possible configurations of pendulum equilibrium positions depending on the ratios of masses and lengths of single pendulums composing the double pendulum are shown. The stability of two equilibrium positions is proved using Sylvester’s criterion. In the first position, the pendulum is oriented toward a Moon, and it is oriented toward a Planet in the second position. Small motions near these stable equilibrium configurations are studied. The natural frequencies and mode ratios are obtained analytically, and their dependence on the mass and length ratios of the pendulums is analyzed. The conducted studies demonstrate the possibility of building a space elevator in the Mars–Phobos system from the L1 libration point to a Moon (distance from the L1 point to the surface of Phobos ~ 3.4 km) or to a Planet (distance from the L1 point to the surface of Mars ~ 7800 km). This also opens up the opportunity of building a two-part space elevator from Mars to Phobos. The obtained natural frequencies and mode ratios allow us to predict in advance the possible motions of a space elevator under small perturbations relative to the stable equilibrium position.

Enceladus exhibits some remarkable phenomena, including water geysers spraying through surface cracks, a global ice shell that is librating atop an ocean, a large luminosity, and rapid outward orbital migration. Here, we model the coupled evolution of Enceladus’s orbit and interior structure. We find that Enceladus is driven into a periodic state—a limit cycle. Many of Enceladus’s observed phenomena emerge from the model, and the predicted values for the orbital eccentricity, libration amplitude, shell thickness, and luminosity agree with observations. A single limit cycle lasts around 10 million years, and has three distinct stages: (1) freezing, (2) melting, and (3) resonant libration. In our model, Enceladus is currently in the freezing stage, meaning that its ice shell is getting thicker. That pressurizes the ocean, which in turn cracks the shell and pushes water up through the cracks. In this stage, the orbital eccentricity increases, as Saturn pushes Enceladus deeper into resonance with Dione. Once the eccentricity is sufficiently high, tidal heating begins to melt the shell, which is the second stage of the cycle. In the third stage, the shell remains close to 3 km thick. At that thickness, the shell’s natural libration frequency is resonant with the orbital frequency. The shell’s librations are consequently driven to large amplitude, for millions of years. Most of the tidal heating of Enceladus occurs during this stage, and the observed luminosity is a relic from the last episode of resonant libration, while the present-day heat production is small (∼1 GW).

Observing and controlling macroscopic quantum systems has long been a driving force in quantum physics research. In particular, strong coupling between individual quantum systems and mechanical oscillators is being actively studied 1 – 3 . Whereas both read-out of mechanical motion using coherent control of spin systems 4 – 9 and single-spin read-out using pristine oscillators have been demonstrated 10 , 11 , temperature control of the motion of a macroscopic object using long-lived electronic spins has not been reported. Here we observe a spin-dependent torque and spin-cooling of the motion of a trapped microdiamond. Using a combination of microwave and laser excitation enables the spins of nitrogen–vacancy centres to act on the diamond orientation and to cool the diamond libration via a dynamical back-action. Furthermore, by driving the system in the nonlinear regime, we demonstrate bistability and self-sustained coherent oscillations stimulated by spin–mechanical coupling, which offers the prospect of spin-driven generation of non-classical states of motion. Such a levitating diamond—held in position by electric field gradients   under vacuum—can operate as a ‘compass’ with controlled dissipation and has potential use in high-precision torque sensing 12 – 14 , emulation of the spin-boson problem 15 and probing of quantum phase transitions 16 . In the single-spin limit 17 and using ultrapure nanoscale diamonds, it could allow quantum non-demolition read-out of the spin of nitrogen–vacancy centres at ambient conditions, deterministic entanglement between distant individual spins 18 and matter-wave interferometry 16 , 19 , 20 . Coupling the spins of many nitrogen–vacancy centres in a trapped diamond to its orientation produces a spin-dependent torque and spin-cooling of the motion of the diamond.

This paper studies the libration-free cargo transfer control of a partial space elevator where the main satellite may change its orbital state in the transfer period. The orbital motion of the main satellite in the climber transfer period is first studied. Then, a reduced-order libration-free dynamics of the partial space elevator is derived. Accordingly, a novel libration-free switching control strategy is proposed to stabilize cargo transportation with two alternating controllers. The Controller I controls the cargo speed in the libration-free mode by a shrinking horizon model predictive control based on the reduced-order libration-free dynamic mode of the partial space elevator. It leads to high computational efficiency in control. Once the libration is induced by the cargo transfer, the control turns off the Controller I and activates the Controller II to suppress the libration to zero within one time step of the Controller I by a novel prescribed-time control law based on the fixed-time control scheme. The stability of the control is proved in the Lyapunov framework. The validity and effectiveness of the proposed control strategy are demonstrated by computation simulation. Simulation results reveal that the proposed control strategy is effective in keeping stable cargo transportation while ensuring the equilibrium state at the end of transportation.

The paper investigates the feasibility of designing and deploying a space elevator fixed at the L1 libration point in the Mars–Phobos system in the framework of the planar circular restricted three-body problem. Two configurations of the space elevator are discussed. One is directed towards Phobos and the other towards Mars. In the first case, the length of the elevator is limited by the distance to the surface of Phobos (about 3.4 km), and in the second by the distance to the surface of Mars (about 7800 km). The law of motion of the climber is proposed, including the acceleration part, the braking part and the main part of the climbing (or descending) of the climber at constant velocity. The influence of the mass ratio of the climber and the end body is analyzed. It is also shown that it is possible to turn the elevator 180 degrees from the direction of Phobos to the direction of Mars and back when the climber is at the end point of the elevator. This is achieved using the well-known control law of the elevator length. This is the first preliminary study on the design of the Mars–Phobos space elevator using the L1 libration point, based on theoretical statements and numerical simulation results.