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

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.

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

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.

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.

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.

The paper focuses on the study and development of a mission to use the L1 libration point for deployment of a tether system in the direction of a moon in the planar elliptic restricted three-body problem on the example of the Mars-Phobos system. An orbiting spacecraft, which deploys the tether system, is located at the L1 libration point and is held at this point by the low thrust of its engines. The classical equations in the Nechville variables are converted into equations in polar coordinates. For the particular case when the tether is inextensible and two primaries moving in circular orbits about their mass center, these equations are integrated in quadrature, and the equilibrium positions and the oscillation period of the tether are found. As a result of eccentricity, the primaries move on ellipses around the barycenter, which rotates with angular velocity equal to that of the primaries, the position of the two-body pulsates along the axis connecting them. A new mission architecture is proposed, which includes three successive stages: initial deployment to the upper pulsation point (perigee), angular stabilization of the tether relative to the lower stable position, and maintaining a constant distance to a moon’s surface. An end mass with measuring equipment of this tether system can be set directly on the moon’s surface. Numerical simulations have shown the effectiveness of the proposed control laws of the tether system at all stages of the mission for the Mars-Phobos system, in which the L1 libration point is located quite close to the Phobos’ surface (~ 3.4 km). This paper is the first effort, using to justify publicly the possibility of implementing a mission with a tether system “attached” at L1 libration point to study surface of moons based on the proposed control laws and the sequence of their application. The results of this study can be used to enable many future missions throughout the solar system. If in the future similar missions will be approved, then undoubtedly more advanced control methods of this kind of systems will be developed.

Given the interest in future space missions devoted to the exploration of key moons in the solar system and that may involve libration point orbits, an efficient design strategy for transfers between moons is introduced that leverages the dynamics in these multi-body systems. The moon-to-moon analytical transfer (MMAT) method is introduced, comprised of a general methodology for transfer design between the vicinities of the moons in any given system within the context of the circular restricted three-body problem, useful regardless of the orbital planes in which the moons reside. A simplified model enables analytical constraints to efficiently determine the feasibility of a transfer between two different moons moving in the vicinity of a common planet. In particular, connections between the periodic orbits of such two different moons are achieved. The strategy is applicable for any type of direct transfers that satisfy the analytical constraints. Case studies are presented for the Jovian and Uranian systems. The transition of the transfers into higher-fidelity ephemeris models confirms the validity of the MMAT method as a fast tool to provide possible transfer options between two consecutive moons.

This paper presents an analytical framework for in-plane two-impulse transfers between two Lissajous orbits around a single collinear libration point in the linearized circular restricted three-body problem. Based on the categorization of orbits near a collinear libration point in the linearized dynamics, we formulate two-impulse transfers as a two-point boundary value problem with a fixed time of flight (ToF). We explicitly derive and numerically verify the solution, along with its subsets corresponding to transfers involving asymptotic invariant manifolds. Additionally, we derive some general properties of solutions, including the inherent constraints arising from the limited distribution of invariant manifolds are identified and compared to their single-impulse counterparts. We further investigate the relation between the performance and the constraints of two-impulse transfers by numerically evaluating the cost of maneuvers for amplitude-decreasing transfers. Our main results demonstrate improved feasibility compared to the single-impulse transfers with minimum additional cost for a relatively large ToF, regardless of the size of the amplitude size, as well as the existence of a lower bound on the cost similar to the one found in the single-impulse case.