Libration Research Articles

The bright side of the Moon: The historical journey of our tidally locked satellite

Abstract The paper discusses a major shift in the (Western) science of our Moon. It reconsiders well-known developments, as the Aristotelian concept of distinct celestial and terrestrial worlds in which the Moon was one of seven perfect planetary spheres gave way to our modern view of a very Earth-like body, even endowed at one stage with oceans and continents. It examines the impact of pre-telescopic and telescopic observations in the period 1560 to 1660, notably those of Galileo Galilei and William Gilbert. Its new perspective is to present the strong evidence that supported the idea of the Moon as a planet. In large part this was because it is now known that aspects of the Moon’s orbit are unusual compared with the multiplicity of “moons” in the Solar System. It argues that a major reason was Aristotle’s influential argument for the theoretical impossibility of lunar libration. It concludes with the extraordinary work of William Gilbert (1544-1603) who, it is argued, was the true discoverer of libration because Galileo, who is often credited with it, failed to grasp the necessary dynamics of lunar motion. Notwithstanding, both played crucial roles in the emerging conception of our Moon as “another Earth”. Human activity is now degrading both bodies.

Irregular fixation – II. The orbits of irregular satellites

ABSTRACT Irregular satellites (ISs) are believed to have been captured during the Solar system’s dynamical history and provide clues for the Solar system’s formation and evolution. ISs occupy a large fraction of the Hill sphere of their host planet and their orbits are highly perturbed by the Sun. We use a novel formalism developed in Paper I to characterize their orbits in terms of an effective secular Hamiltonian (the Brown Hamiltonian) that accounts for their large orbital separations. We find that prograde satellites generally follow the Brown Hamiltonian, while retrograde satellites (which extend further) deviate more significantly. Nevertheless, the phase portrait is much better described by the Brown Hamiltonian for all satellites. We construct a semi-analytical criterion that predicts the librating orbit based on the effective energy due to the Brown Hamiltonian. We also check our results with highly accurate N-body integrations of satellite orbits, where initial conditions are loaded directly from the updated ephemeris from the NASA Horizons data base. Although the retrograde librating orbits occupy more area in the parameter space, the vast majority of librating ISs are prograde. Using our method, we find 13 librating satellites, 8 of them previously known to librate, and the rest shown to librate for the first time. Further observations of existing and new satellites could shed more light on the dynamical history of the Solar system and satellite formation and test our results.

Irregular fixation: I. Fixed points and librating orbits of the Brown Hamiltonian

ABSTRACT In hierarchical triple systems, the inner binary is slowly perturbed by a distant companion, giving rise to large-scale oscillations in eccentricity and inclination, known as von-Zeipel–Lidov–Kozai oscillations. Stable systems with a mild hierarchy, where the period ratio is not too small, require an additional corrective term, known as the Brown Hamiltonian, to adequately account for their long-term evolution. Although the Brown Hamiltonian has been used to accurately describe the highly eccentric systems on circulating orbits where the periapse completes a complete revolution, the analysis near its elliptical fixed points had been overlooked. We derive analytically the modified fixed points including the Brown Hamiltonian and analyse its librating orbits (where the periapse motion is limited in range). We compare our result to the direct three-body integrations of millions of orbits and discuss the regimes of validity. We numerically discover the regions of orbital instability, allowed and forbidden librating zones with a complex, fractal, structure. The retrograde orbits, where the mutual inclination is $\iota \ \gt\ 90\ \rm deg$, are more stable and allowed to librate for larger areas of the parameter space. We find numerical fits for the librating-circulating boundary. Finally, we discuss the astrophysical implications for systems of satellites, stars, and compact objects. In a companion paper (Paper II), we apply our formalism to the orbits of irregular satellites around giant planets.

Evaluation of Nocturnal Aerosol Optical Depth Determining From a Lunar Photometry in Lanzhou of Northwest China

AbstractA Cimel Sun‐sky‐lunar photometer (CE318‐T) is designed to perform daytime and nighttime photometric measurements and calculate diurnal cycle of aerosol optical depth (AOD). Nevertheless, the determination of nocturnal AOD from CE318‐T requires a precise knowledge of extraterrestrial lunar irradiance, which significantly changes with moon's phase angle (MPA) and lunar libration in a single night. This study evaluated the 1‐year nocturnal AODs at Lanzhou by using three different methods, which were validated by collocated measurements of DIAL Lidar (as a reference) and Cimel software (as a proxy). The results indicated that three independent approaches could implement a similar performance to compute nocturnal AOD near full moon phase (i.e., MPA = 0°) under moderate aerosol loading. The spectral AOD values at nighttime calculated by ROLO Lunar Langley (Robotic Lunar Observatory model) and Sun‐moon Gain Factor (SMGF) methods are significantly underestimated under low moon's illumination or high MPA (MPA < −47° or MPA > 47°) and distinctly dependent on MPA. The RIMO correction factor (RCF) (ROLO Implementation for Moon photometry Observation correction factor) method could compensate the underestimated extraterrestrial lunar irradiances of ROLO model for about 6.76%–9.78%, and thus greatly improve the calculation accuracy of nighttime AOD. The day/night coherence transition test has demonstrated that we would obtain a good diurnal variation of AODs at Lanzhou after RCF correction. The overall averages of nocturnal AOD440nm differences between Cimel software and ROLO model and SMGF method are 0.064 ± 0.024 and 0.052 ± 0.022, respectively, while the corresponding difference of RCF is less than 0.021 ± 0.014 for all wavelengths, falling within uncertainty range of AERONET AOD products (∼±0.02). The diurnal variations of AODs determined from RCF method agree well with synchronous results of DIAL Lidar, with total mean AOD532nm differences of 0.038 ± 0.024 and 0.023 ± 0.017 in daytime and nighttime, respectively. The spectral AODs computed from RCF method are well consistent with Cimel software, although there are some discrepancies under low AOD cases (AOD440nm < 0.30), and the overall average of AOD440nm differences are less than −0.0053 ± 0.002 and −0.0185 ± 0.013 in daytime and nighttime, respectively. Our results confirmed that the CE318‐T photometer can be reasonably calculated nighttime AOD and Ångström exponent (AE440–870nm) at urban Lanzhou by using three independent methods, although the former two need to be greatly improved under low moon's illumination. The RCF method was proved to reliably calculate nocturnal AOD from moonlight irradiance, which didn't rely on MPA. A more accurate lunar irradiance model needs to be developed to improve the underestimation of current ROLO model. Long‐term climatological information of nocturnal AOD is crucial for comprehensively characterizing the diurnal variations of aerosol optical properties and atmospheric boundary layer structure during the winter at typical northern city of China, which deserves to be further investigated in the future.

Lunar laser ranging: The pointing error analysis to Lunokhod 1 and Lunokhod 2

The corner-cube reflectors (CCRs) are small optical instruments designed to accurately measure the distance between the Earth and the Moon. Signals from L1 (Lunokhod 1) and L2 (Lunokhod 2) can be used for higher precision analysis of lunar libration than Apollo reflector. Here, we describe the optical performance of L1 and L2 at near-infrared wavelength. Theoretically, because of the initial direction and lunar librations, the surfaces of these arrays do not, in general, point directly to the Earth. The laser incident angles for the CCRs and the far-field diffraction patterns (FFDP) of individual CCRs of possible incident angles are calculated. Finally, extract the optical cross-section as a function of incident angles. In the experiment, for the first time, superconducting nanowire single-photon detectors (SNSPDs) at a wavelength of 1064 nm are designed and fabricated aiming at lunar laser ranging (LLR), and the effective echo rates of L1 and L2 are measured. Using experimental results to check the orientation of the CCRs. The experimental results show that there is an error in the initial pointing of the CCRs, the initial pointing angles of L1 and L2 are about from 10o to 40o.

Inferring the Mean Thickness of the Outer Ice Shell of Enceladus From Diurnal Crustal Deformation

AbstractThe thickness of the outer ice shell plays an important role in several geodynamical processes at ocean worlds. Here, we show that observations of tidally driven diurnal surface displacements can constrain the mean ice shell thickness, . Such estimates are sensitive to any significant structural features that break spherical symmetry such as faults and lateral variation in ice shell thickness and structure. We develop a finite‐element model of Enceladus to calculate diurnal tidal displacements for a range of values in the presence of such structural heterogeneities. Consistent with results from prior studies, we find that the presence of variations in ice shell thickness can significantly amplify deformation in thinned regions. If major faults are also activated by tidal forcing—such as Tiger Stripes on Enceladus—their characteristic surface displacement patterns could easily be measured using modern geodetic methods. Within the family of Enceladus models explored, estimates of that assume spherical symmetry a priori can deviate from the true value by as much as ∼41% when structural heterogeneities are present. Additionally, we show that crustal heterogeneities near the South Pole produce differences of up to 35% between Love numbers evaluated at different spherical harmonic orders. A ∼41% range in estimates of from Love numbers is smaller than that found with approaches relying on static gravity and topography (∼250%) or analyzing diurnal libration amplitudes (∼85%) to infer at Enceladus. As such, we find that analysis of diurnal tidal deformation is a relatively robust approach to inferring mean crustal thickness.

Earth observations from the Moon's surface: dependence on lunar libration

Abstract. Observing the Earth from the Moon's surface has important scientific advantages. The angular diameter of the Earth as seen from the Moon's surface is 1.8–2.0∘ (the angular size varies due to the change in the Earth–Moon distance). The libration of the Moon in latitude reaches an amplitude of 6.68∘ and has a main period of 27.21 d (or 653.1 h). The libration of the Moon in longitude, reaching an amplitude of 7.9∘, has a period of 27.55 d (or 661.3 h). This causes the center of the Earth to move in the Moon's sky in a rectangle measuring 13.4∘ × 15.8∘. The trajectory of the Earth's motion in this rectangle changes its shape within a period of 6 years. This apparent librational movement of the Earth in the Moon's sky complicates observations of the Earth. This paper proposes that this disadvantage be turned into an advantage and that a multi-slit spectrometer be placed on the Moon's surface on a fixed platform. The libration motion and the daily rotation of the Earth will act as a natural replacement for the scanning mechanism.

Spherical harmonic decomposition and interpretation of the shapes of the small Saturnian inner moons

Context. The Cassini-Huygens space mission made a series of observations of Saturn’s small satellites during its grand finale stage. These measurements were performed in order to study the shape, geology, and surface composition of the small satellites as well as to study the impact of the environment, in particular the rings, on these small bodies. Aims. The purpose of this study is to focus on the shape analysis of the small Saturnian satellites in order to describe their global figure and large-scale topography, as well as to deduce fundamental quantities, gravity field, and amplitude of the diurnal libration by assuming that the bodies are homogeneous. Methods. We used two approaches in this study. On the one hand, we directly exploited the Cassini images of the small satellites by performing limb measurements and deducing a confidence interval on the shape measurements. On the other hand, we used previously published shape models which combine limb measurements and control points. These shape models were then decomposed and described in spherical harmonics. Results. We found that the shape of the small satellites can be described with a confidence interval between 50 and 150 m. The low degree in spherical harmonics (degree 2) indicated that Telesto, Pandora, Pan, Janus, and Helene have a degree 2 shape close to the Omega sequence, which was defined recently, where the potential is constant along a meridian perpendicular to the longest axis. The degree 2 shape of Epimetheus, on the other hand, is close to the Roche sequence. In contrast, Prometheus, Calypso, and Atlas are in the Low-Brown region. The root mean square spectrum and spherical harmonic maps then allowed us to describe the topography of the satellites, and in particular to highlight equatorial ridges for some satellites including Daphnis. Finally, our estimates of the libration amplitude in the homogeneous case provide values in agreement with previously published librational measurements for Epimetheus while highlighting the proximity of the resonance for Epimetheus, Pandora, and Prometheus. Conclusions. The high resolution images of the internal satellites have allowed us to describe the geology and the geophysics of these bodies. Future comparison of these amplitudes with new librational measurements deduced, for example, by the astrometric method, will allow us to obtain information on the internal structure of these bodies. Similar studies could be carried out on the internal satellites of Jupiter using images from the Europa Clipper (NASA) or JUICE (ESA) missions.

Next-generation Laser Ranging at Lunar Geophysical Network and Commercial Lander Payload Service Sites

Five retroreflector arrays currently on the Moon reflect short laser pulses back to Earth, allowing range to be measured. Each array has multiple small corner cubes. Due to variable lunar optical librations of the direction to Earth, the tilted arrays spread return times of single photons in the returned laser pulse, degrading the synthesized multiphoton normal point range accuracy. The Next Generation Lunar Retroreflectors (NGLRs) and MoonLIGHT reflectors currently being fabricated are larger 10 cm single corner cubes that do not spread the pulse. The Lunar Geophysical Network (LGN) mission will place NGLRs at three separated sites on the lunar nearside. The Commercial Lander Payload Service (CLPS) and early Artemis missions will precede the LGN mission. Solutions that include 6 yr of simulated Lunar Laser Ranging (LLR) data to two sites in the north and two in the south show improvement in the uncertainties of many science parameters. Lunar solution parameters include displacement Love numbers h 2 and l 2, tidal dissipation at several frequencies, fluid-core/solid-mantle boundary (CMB) dissipation, and moment of inertia combinations (C–A)/B and (B–A)/C, with principal moments of inertia A < B < C. Submeter-accuracy coordinates of the new reflectors will result from the first month of well-distributed data. There are benefits other than lunar science: gravitational physics includes the equivalence principle; Earth science includes terrestrial tidal dissipation and ranging station positions and motions; and astronomical constants with GM(Earth+Moon) for the gravitational constant times the mass of the Earth–Moon system. Improvements are illustrated for h 2, l 2, (C–A)/B, (B–A)/C, equivalence principle, and GM(Earth+Moon).

A Radial Limit on Polar Circumbinary Orbits from General Relativity

Abstract A particle orbiting a misaligned eccentric orbit binary undergoes nodal precession either around the binary angular momentum vector (a circulating orbit) or around a stationary inclination (a librating orbit). In the absence of general relativity (GR), the stationary inclination is inclined by 90° to the binary angular momentum vector (aligned with the binary eccentricity vector) and does not depend on the particle semimajor axis. GR causes apsidal precession of the binary orbit. Close to the binary, the behavior of the particle is not significantly affected, and a librating particle precesses with the binary. However, we find that the stationary inclination and the minimum inclination required for libration both increase with the particle semimajor axis. There is a critical radius beyond which there are no librating orbits, only circulating orbits, and therefore there is a maximum orbital radius for a stationary polar-orbiting body. The critical radius is within planet-forming regions around binaries with a semimajor axis ≲1 au. This has implications for the search for misaligned circumbinary planets and the radial extent of polar circumbinary disks.

The irradiance instrument subsystem (IRIS) on the airborne-lunar spectral irradiance (air-LUSI) instrument

The objective of the airborne lunar spectral irradiance (air-LUSI) project is to make low uncertainty, SI-traceable measurements of the LUSI in the visible to near-infrared region from an aircraft above most of the optically absorbing components of the atmosphere. The measurements are made from a NASA ER-2 aircraft, which can fly at altitudes of approximately 20 km above sea level. Air-LUSI measurements, corrected for residual atmospheric attenuation, are designed to provide a matrix of low uncertainty top-of-the-atmosphere lunar irradiances at known lunar phase and libration angles to be compared and combined with other lunar irradiance data sets to constrain the uncertainties in models of lunar irradiance and reflectance. The measurements are also expected to provide insight into the differences between models and satellite sensor measurements of lunar irradiance. This paper describes the development and characterization of the air-LUSI subsystem for acquiring lunar measurements, called the irradiance instrument subsystem, prior to flight.

Earth-Based Repeat-Pass SAR Interferometry of the Moon: Spatial–Temporal Baseline Analysis

Earth-based repeat-pass synthetic aperture radar (SAR) interferometry (InSAR) is a powerful tool to obtain the high-precision lunar terrain and deformation information, due to the short revisit period as well as the all-weather, long-time, and large-coverage characteristics of radar lunar observation. In this article, the baseline formation mechanism is investigated. Theoretical analyses show that the interferometric baselines mainly come from the lunar libration; however, the lunar libration may also result in a significant change of baselines and further lead to serious geometric decorrelation. To address this problem and ensure the Earth-based repeat-pass InSAR of the Moon, the precise relative motion model between the radar and the lunar target, considering lunar libration, is established, and the subradar point (SRP) wagging phenomenon is analyzed. Then, to evaluate the effects of SRP wagging on InSAR coherence, a geometric decorrelation model based on azimuth angle and incident angle is proposed. Based on that, the spatial–temporal baseline for Earth-based repeat-pass InSAR of the Moon is analyzed based on numerical simulation. Results show that the temporal baseline has obvious periodicity with a cycle of about 27 days. When the temporal baseline is an integral multiple of 27, the spatial baseline usually has the minimum value and the geometric decorrelation effect is relatively slight. Moreover, using the geometric decorrelation criteria, we examine the effects of radar frequency, position, and target location on the quantity of viable interferometric pairs, which has guiding significance for the selection of radar system parameters, radar position, and lunar target area.

Near-Earth Asteroid Surveillance Constellation in the Sun-Venus Three-Body System

The threat of potential hazardous near-Earth asteroid (PHA) impact on Earth is increasingly attracting public attention. Monitoring and early warning of those PHAs are the premise of planetary defense. In this paper, we proposed a novel concept of surveillance constellation of heterogeneous wide-field near-Earth asteroid (NEA) surveyors (CROWN), in which six space-based surveyors are loosely deployed in Venus-like orbits to detect the NEAs along the direction of the sunlight. First, the concept and overall design of the NEA surveillance constellation are discussed. Second, the transfer and deployment trajectory of the surveyors are investigated based on the Sun-Venus three-body system. The Sun-Venus libration orbit is taken as the parking orbit, and its stable invariant manifolds are used to reduce the deployment fuel consumption. Next, the detection performance of the CROWN was evaluated considering constraints of apparent visual magnitude and field of view. The NEA orbit determination (OD) using the CROWN was studied and verified. Simulation results show that the CROWN can be deployed with a total velocity increment of approximately 300 m/s. During the 5 years of observation, 99.8% of PHAs can be detected and the OD precision is better than a single-surveyor system. This paper can provide a reference for the construction of future asteroid defense system.

Testing the Motion of Lunar Retroreflectors

AbstractThe Moon lacks recognizable plates moving with respect to each other, unlike the Earth. Analysis of Lunar Laser Ranging data does not find large Earth‐like (&gt;1 cm/year) relative motions for the five lunar retroreflectors. The distances between reflectors show few mm/year rates that are not statistically significant. Despite this lack of individual significance, curiously, all of the well determined distances have negative rates. There is an extra non‐synchronous lunar rotation rate of 1.0 ± 0.4 mas/yr that is suspected to arise from unmodeled very long period effects of physical libration in longitude.

Constraints on the interior structure of Phobos from tidal deformation modeling

Phobos is one of the two moons of Mars, the only other terrestrial planet to have companions. In spite of certain common orbital characteristics between Earth’s Moon and Phobos, such as almost circular, low-inclination, and synchronous orbits, the origin of the Martian moons remains to be understood. This unsettled state of affairs reflects a scarcity of data pertinent to its interior structure and bulk composition, both of which hold the key to solving the question of the provenance of the satellites. Drawing on the importance of interior structure as a means of providing further constraints on this problem, we construct a series of models of the interior of Phobos in accordance with current observations and determine their tidal deformation. The models include a homogeneous body, ice-rock mixtures, models with positive and negative density gradients, and layered models. To compute the deformation of Phobos precisely, we solve the three-dimensional elastostatic problem to obtain the full displacement field. For this, we rely on a higher-order spectral-element method and to account for a correct representation of shape and resulting displacement field, we accurately mesh the figure of Phobos by employing the digital terrain model of Willner et al. (2014). To enable us to further distinguish between models, we also rely on currently available geophysical data (e.g., magnitude of libration in longitude, mean density, gravity field, and moments of inertia). Our results show that the homogeneous, ice-rock mixture, and negative density gradient models are largely degenerate and therefore difficult to separate, but that models with a density increase with depth can be differentiated from the former model families. From a purely deformational point of view, we find that for positive gradient and layered models, the largest deformation occurs on the rim of the Stickney crater, rather than immediately at the sub-Mars point on Phobos, as in the case of the other models, where tidal forces are largest. These observations will be of considerable interest and should provide important anchoring points for the future exploration of Phobos as envisaged with the Martian Moons Exploration mission.

Transfer between libration orbits through the outer branches of manifolds for Phobos exploration

The transfer between libration orbits in the Mars–Phobos system is challenging under the strong gravitational perturbation of Mars and Phobos. In this paper, the outer branches of manifolds of libration orbits around Phobos are employed for impulse and continuous thrust transfer between orbits, respectively. The 1:1 resonant orbits are found to be appropriate intermediate orbits for impulse transfer, which can connect the manifolds of various kinds of libration orbits. For the continuous thrust transfer, the convex optimization method is adopted to design the low-thrust trajectory in the three-body problem. The Poincaré section is established to find the best matching points on the outer manifolds. The L1/L2 homoclinic and L1−L2 heteroclinic transfer trajectories are designed and discussed. The simulations show the feasibility of the transfer by outer manifolds with both impulse and continuous thrust. The proposed method can avoid the potential hazard caused by the complicated dynamics near Phobos and provide a reliable transfer method, which can be used in future Phobos exploration missions.

Analysis of Photoelectric Occultations and Development of a Digital Model of the Lunar Libration Zone

Analysis of Photoelectric Occultations and Development of a Digital Model of the Lunar Libration Zone

Global analysis of direct transfers from Lunar orbits to Sun-Earth libration point regimes

In this paper we look forward to obtain a global picture of simple transfer opportunities from Lunar to Sun-Earth libration orbits, useful for a preliminary design of these mission scenarios. Considering the trajectory of the Chinese Change’2 spacecraft as a reference case, the main transfer families are characterized and classified. In a second step, the results are analyzed and extended to departures from more general families of orbits about the Moon. Finally, we also include a preliminary sensitivity analysis of the first transfer correction maneuver (TCM1) to cancel injection errors. The methodology is of general applicability to the transfer analysis involving libration point final orbits in other general multi-body restricted systems.

A new telescope with three fields of view to measure the orientation parameters of the Moon and terrestrial planets

For lots of scientific questions about lunar physics deep inside the Moon, in-situ observation on lunar physical libration is one of the most potential ways. In this paper, we propose a brand new optical telescope functioned with simultaneously observing multiple (here there are three) fields of view (FOVs) for in-situ observation of lunar physical libration. The telescope can be placed at any place with any attitude on the Moon and do not require manned install, control or operation. It passively, continuously and simultaneously observe stars in three FOVs along with rotation of the Moon. Libration is to be measured and studied from celestial motion of the directions of three FOVs from image processing. The concept and design of this telescope are firstly introduced in this paper. The principle and feasibility of the method of in-situ observation are also demonstrated. From simulation, precision of the determined lunar physical libration is expected to be several milliarcsecs, about two orders of magnitude better than the current precision of libration by lunar laser ranging observation. Libration data with milliarcsec precision level can play a valuable role in the study of the physics and dynamics of the interior of the Moon. This telescope can also be applied to observe the rotation of other terrestrial planets like Mars.

Physical Libration of the Moon: An Extended Problem

The work is devoted to the study of the physical libration of the Moon. Interest in the traditional topic related to the rotation of the Moon is stirred up by the activity of many countries regarding the development of circumlunar space. Scientifically, the main agenda is to consider the viscosity of the core. At this stage of the theory development, such effects as indirect and direct perturbations from the planets were considered, the 4th order harmonic was included in the gravitational potential of the Moon, and the mean tidal potential was also considered. The inclusion of the described effects in the equations of the Moon’s rotation led to a significant improvement in the solution when compared with the corresponding data from the DE421 theory, although the residual differences still remain greater than the 1 ms accuracy required by the theory. The influence of the direct effect from the planets was milliseconds. The influence of the 4th harmonic manifested itself as a systematic shift of the order of 0.85 $$'' $$ in the residual differences in libration in longitude. Considering the tide made it possible to reduce the residual differences in latitude by almost an order of magnitude. In this case, the main factor that reduces the residual differences is changes in the second-order Stokes coefficients. The calculations were carried out using the DE421 ephemeris built at NASA Jet Propulsion Laboratory.