Libration Amplitude Research Articles

Looking for Subsurface Oceans Within the Moons of Uranus Using Librations and Gravity

AbstractSeveral 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 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.

A perturbative treatment of the Yarkovsky-driven drifts in the 2:1 mean motion resonance

Aims. Our aim is to gain a qualitative understanding as well as to perform a quantitative analysis of the interplay between the Yarkovsky effect and the Jovian 2:1 mean motion resonance under the planar elliptic restricted three-body problem. Methods. We adopted the semi-analytical perturbation method valid for arbitrary eccentricity to obtain the resonance structures inside the Jovian 2:1 resonance. We averaged the Yarkovsky force so it could be applied to the integrable approximations for the 2:1 resonance and the ν5 secular resonance. The rates of Yarkovsky-driven drifts in the action space were derived from the quasi-integrable approximations perturbed by the averaged Yarkovsky force. Pseudo-proper elements of test particles inside the 2:1 resonance were computed using N-body simulations incorporated with the Yarkovsky effect to verify the semi-analytical results. Results. In the planar elliptic restricted model, we identified two main types of systematic drifts in the action space: (Type I) for orbits not trapped in the ν5 resonance, the footprints are parallel to the resonance curve of the stable center of the 2:1 resonance; (Type II) for orbits trapped in the ν5 resonance, the footprints are parallel to the resonance curve of the stable center of the ν5 resonance. Using the semi-analytical perturbation method, a vector field in the action space corresponding to the two types of systematic drifts was derived. The Type I drift with small eccentricities and small libration amplitudes of 2:1 resonance can be modeled by a harmonic oscillator with a slowly varying parameter, for which an analytical treatment using the adiabatic invariant theory was carried out.

A Hamiltonian for 1/1 rotational secondary resonances, and application to small satellites of Saturn and Jupiter

In this work, we study the dynamics of rotation of the small satellites Methone and Aegaeon and revisit previous works on the rotation of Prometheus, Metis, and Amalthea. In all cases, the surfaces of section computed with the standard spin–orbit model reveal that the synchronous regime with small amplitude of libration shares another large domain in the phase space. We reproduce and apply the hamiltonian theory given in Wisdom 2004 [1] to analytically characterize the detected structure as being a secondary resonance where the period of the physical libration is similar to the orbital period of the satellite. We also show that the amplitude of libration in the secondary resonance is always larger than in the case of the other mode. Since the current rotational states of these sorts of satellites should be synchronous, our results can be considered in evolutionary studies of their rotation.

The Global Shape, Gravity Field, and Libration of Enceladus

AbstractIn order to improve our understanding of the interior structure of Saturn's small moon Enceladus, we reanalyze radiometric tracking and onboard imaging data acquired by the Cassini spacecraft during close encounters with the moon. We compute the global shape, gravity field, and rotational parameters of Enceladus in a reference frame consistent with the International Astronomical Union's definition, where the center of the Salih crater is located at −5° East longitude. We recover a quadrupole gravity field with J3 and a forced libration amplitude of 0.091° ± 0.009° (3‐σ). We also compute a global shape model using a stereo‐photoclinometry technique with a global resolution of 500 m, although some local maps have higher resolutions ranging from 25 to 100 m. While our overall results are generally consistent with previous studies, we infer a thicker 27–33 km mean ice shell, a thinner 21–26 km mean ocean thickness, and a mean core density range of 2,270–2,330 kg/m3.

Libration- and Precession-driven Dissipation in the Fluid Cores of the TRAPPIST-1 Planets

The seven planets orbiting TRAPPIST-1 have sizes and masses similar to Earth and mean densities that suggest that their interior structures are comprised of a fluid iron core and rocky mantle. Here we use idealized analytical models to compute estimates of the viscous dissipation in the fluid cores of the TRAPPIST-1 planets induced by mantle libration and precession. The dissipation induced by the libration at orbital periods is largest for TRAPPIST-1b, of the order of 600 MW, and decreases with orbital distance, to values of 5–500 W for TRAPPIST-1h, depending on its triaxial shape. Extrapolating these results to the larger libration amplitudes expected at longer periods, dissipation may perhaps be as high as 1 TW in TRAPPIST-1b. Orbital precession induces a misalignment between the spin axes of the fluid core and mantle of a planet, the amplitude of which depends on the resonant amplification of its free precession and free core nutation. Assuming Cassini states, we show that the dissipation from this misalignment can reach a few TW for planets e and f. Our dissipation estimates are lower bounds, as we neglect ohmic dissipation, which may dominate if the fluid cores of the TRAPPIST-1 planets sustain magnetic fields. Our results suggest that dissipation induced by precession can be of the same order as tidal dissipation for the outermost planets, may perhaps be sufficient to supply the power to a generate a magnetic field in their liquid cores, and likely played an important role in the evolution of the TRAPPIST-1 system.

Effects of the Librationally Induced Flow in Mercury’s Fluid Core with an Outer Stably Stratified Layer

Observational constraints on Mercury’s thermal evolution and magnetic field indicate that the top part of the fluid core is stably stratified. Here we compute how a stable layer affects the core flow in response to Mercury’s main 88 day longitudinal libration, assuming various degrees of stratification, and study whether the core flow can modify the libration amplitude through viscous and electromagnetic torques acting on the core–mantle boundary (CMB). We show that the core flow strongly depends on the strength of the stratification near the CMB but that the influence of core motions on libration is negligible with or without a stably stratified layer. A stably stratified layer at the top of the core can, however, prevent resonant behavior with gravito-inertial modes by impeding radial motions and promote a strong horizontal flow near the CMB. The librationally driven flow is likely turbulent and might produce a nonaxisymmetric induced magnetic field with a strength of the order of 1% of Mercury’s dipolar field.

ExoMMR: A New Python Package to Confirm and Characterize Mean Motion Resonances

The study of orbital resonances allows for the constraint of planetary properties of compact systems. We can predict a system’s resonances by observing the orbital periods of the planets, as planets in or near mean motion resonance (MMR) have period ratios that reduce to a ratio of small numbers. However, a period ratio near commensurability does not guarantee a resonance; we must study the system’s dynamics and resonant angles to confirm resonance. Because resonances require in-depth study to confirm, and because two-body resonances require a measurement of the eccentricity vector which is quite challenging, very few resonant pairs or chains have been confirmed. We thus remain in the era of small-number statistics, not yet able to perform large population synthesis or informatics studies. To address this problem, we build a python package to find, confirm, and analyze MMRs, primarily through N-body simulations. We then analyze all near-resonant planets in the Kepler/K2 and TESS catalogs, confirming over 60 new resonant pairs and various new resonant chains. We additionally demonstrate the package’s functionality and potential by characterizing the mass–eccentricity degeneracy of Kepler-80g, exploring the likelihood of an exterior giant planet in Kepler-80, and constraining the masses of planets in Kepler-305. We find that our methods overestimate the libration amplitudes of the resonant angles and struggle to confirm resonances in systems with more than three planets. We identify various systems that are likely resonant chains but that we are unable to confirm, and highlight next steps for exoplanetary resonances.

The Mean Moment of Inertia for Irregularly Shaped Phobos and Its Application to the Constraint for the Two-Layer Interior Structure for the Martian Moon

The interior structure of Phobos has been the subject of debate in recent years, with the moment of inertia being a determining factor. To study this structure, we modeled Phobos with a two-layer structure and calculated its mean density and moment of inertia using updated gravity coefficients of degree-2 and forced libration amplitudes. By minimizing the misfit between modeled and derived moment of inertia, and observed and modeled mean density, we determined the frequency distribution for estimated parameters, including the core radius rc, core density ρc, and density ρm of the outer layer. Our results indicate that the optimized core radius is around 8.2 km for our models, along with a core density compromise of approximately 2500 kg·m−3, and an outer layer density of around 1400 kg·m−3. These values have remarkable sensitivity to the misfit function, implying a higher density likely inside Phobos compared to the outer layer. Given that the large core density was associated with ice content, it suggested that the fractional ice content in the outer layer is approximately 11% with a rock density of 2200 kg·m−3, while the content in the core is lower at 2.4% with a rock density of 3000 kg·m−3. The methodology introduced in this study can be further used to study the interior structure of irregularly shaped asteroids.

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.

Characterising the interior of five inner Saturnian moons using Cassini ISS data

Context. The Cassini-Huygens space mission provided a huge amount of data for the Saturnian system. While the mission ended in September 2017, there is still much information to be retrieved from the data. Aims. Here, from their orbital motion, we infer the interior of the five inner Saturnian moons, Atlas, Prometheus, Pandora, Janus, and Epimetheus. Our results benefit from a recent study of the shape of these moons, which provides a much better estimation of their 3D shape and volume. Methods. Using Cassini Imaging Science Subsystem (ISS) astrometric data, we again reduced the Cassini ISS images with an improved shape modelling. We then integrated the orbital motion of the inner Saturnian moons and solved for a large set of physical parameters for this system, including the masses and physical libration amplitudes of the moons. Results. We obtain a density value (in g cm−3 with a 1σ error bar based on the digital terrain models and spherical harmonics available today) of 0.3935 ± 0.0033, 0.4873 ± 0.0026, 0.5045 ± 0.0017, 0.6233 ± 0.0015, and 0.6413 ± 0.0002 for Atlas, Prometheus, Pandora, Epimetheus, and Janus, respectively. We were able to determine the physical libration amplitudes of Prometheus, Pandora, Epimetheus, and Janus to be 0.98 ± 0.08, −5.2 ± 0.7, −6.0 ± 0.6, and −0.39 ± 0.16 (degrees and 1σ error bar), respectively. Our solutions for Epimetheus and Janus are in full agreement with a former estimation performed directly from the measurement of the rotation of these two moons. Conclusions. We confirm the large porosity of these five moons and their increase in density as a function of their size. Our estimations of the physical librations of Prometheus, Pandora, Janus, and Epimetheus are consistent with a homogeneous interior, but for Pandora a heterogeneous interior is also plausible.

Dynamics of two liquids in a non-uniformly rotating horizontal cylinder

Controlling the shape of the interfacial boundary is an important technological task. The effect of rotation velocity modulation on the interface between light viscous and denser low viscosity fluid in a rotating horizontal cylinder is studied experimentally. Liquids are characterized by a high contrast of viscosities, these are glycerin and a low-viscosity (denser) fluorinert FC-40. Glycerin is stained with rhodamine, which makes it possible to study the shape of the interface by photo registration with high accuracy when illuminating liquids with a green laser, which causes rhodamine fluorescence. Experiments are carried out in the centrifuged state of liquids when the gravity field has no effect while varying the rotation speed, as well as the amplitude and frequency of librations. It is found that librations qualitatively change the shape of the interface near the ends of the cavity without affecting the cylindrical interface in the middle cavity part. It is shown that this is due to different "viscous" interactions of liquids with the ends of the oscillating cavity, as a result of which the interface near the ends performs radial oscillations with a frequency of librations, and the latter leads to an averaged radial displacement of the wetting boundary of the ends, which increases with the modulation amplitude. At certain amplitudes, the contact line reaches the outer boundary of the cavity. It is shown that the dynamic equilibrium of the interface between liquids of different densities is determined by the averaged interaction of liquids of different viscosity with the oscillating ends.

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.

Energy dissipation in synchronous binary asteroids

Synchronous binary asteroids can experience libration about their tidally-locked equilibrium, which will result in energy dissipation. This is an important topic to the Asteroid Impact and Deflection Assessment, where excitation caused by the DART kinetic impact in the Didymos binary asteroid system may be reduced through dissipation before Hera arrives to survey the effects of the impact. We develop a numeric model for energy dissipation in binary asteroids to explore how different system configurations affect the rate of energy dissipation. We find tumbling within the synchronous state eliminates a systematic trend in libration damping on short timescales (several years), but not over long times (hundreds of years) depending on the material conditions. Furthermore, damping of libration, eccentricity, and fluctuations in the semimajor axis are primarily dependent on the stiffness of the secondary, whereas the semimajor axis secular expansion rate is dictated by the stiffness of the primary, as expected. Systems experiencing stable planar libration in the secondary can see a noticeable reduction in libration amplitude after only a few years depending on the stiffness of the secondary, and thus dissipation should be considered during Hera’s survey of Didymos. For a very dissipative secondary undergoing stable libration, Hera may be able to calculate the rate of libration damping in Dimorphos and therefore constrain its tidal parameters.

Inferred Properties of Planets in Mean-motion Resonances are Biased by Measurement Noise

Planetary systems with mean-motion resonances hold special value in terms of their dynamical complexity and their capacity to constrain planet formation and migration histories. The key toward making these connections, however, is to have a reliable characterization of the resonant dynamics, especially the so-called “libration amplitude,” which qualitatively measures how deep the system is into the resonance. In this work, we identify an important complication with the interpretation of libration amplitude estimates from observational data of resonant systems. Specifically, we show that measurement noise causes inferences of the libration amplitude to be systematically biased to larger values, with noisier data yielding a larger bias. We demonstrated this through multiple approaches, including using dynamical fits of synthetic radial velocity data to explore how the libration amplitude distribution inferred from the posterior parameter distribution varies with the degree of measurement noise. We find that even modest levels of noise still result in a slight bias. The origin of the bias stems from the topology of the resonant phase space and the fact that the available phase-space volume increases nonuniformly with increasing libration amplitude. We highlight strategies for mitigating the bias through the usage of particular priors. Our results imply that many known resonant systems are likely deeper in resonance than previously appreciated.

Dynamics of Polar Resonances and Their Effects on Kozai–Lidov Mechanism

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.

On the semi-analytical construction of halo orbits and halo tubes in the elliptic restricted three-body problem

The halo orbits of the spatial circular restricted three-body problem are largely considered in space-flight dynamics to design low-energy transfers between celestial bodies. A very efficient analytical method for the computation of halo orbits, and the related transfers, has been obtained from the high-order resonant Birkhoff normal forms defined at the Lagrangian points L1−L2. In this paper, by implementing a non-linear Floquet–Birkhoff resonant normal form, we provide the definition of orbits, as well as their manifold tubes, which exist in a large order approximation of the elliptic three-body problem and generalize the halo orbits of the circular problem. Since the libration amplitude of such halo orbits is large (comparable to the distance of L1−L2 from the secondary body), and the Birkhoff normal forms are obtained through series expansions at the Lagrangian points, we provide also an error analysis of the method with respect to the orbits of the genuine elliptic restricted three-body problem.

The Effects of Disk-induced Apsidal Precession on Planets Captured into Mean Motion Resonance

The process of migration into resonance capture has been well studied for planetary systems where the gravitational potential is generated exclusively by the star and planets. However, massive protoplanetary disks add a significant perturbation to these models. In this paper we consider two limiting cases of disk-induced precession on migrating planets and find that small amounts of precession significantly affect the equilibrium reached by migrating planets. We investigate these effects with a combination of semianalytic models of the resonance and numerical integrations. We also consider the case of the disk’s dispersal, which can excite significant libration amplitude and can cause ejection from resonance for large enough precession rates. Both of these effects have implications for interpreting the known exoplanet population and may prove to be important considerations as the population of well-characterized exoplanet systems continues to grow.

Mothership-Cubesat Radioscience for Phobos Geodesy and Autonomous Navigation

The knowledge of the interior structure (e.g., homogeneous, porous, or fractured) of Martian moons will lead to a better understanding of their formation as well as the early solar system. One approach to inferring the interior structure is via geodetic characteristics, such as gravity field and libration. Geodetic parameters can be derived from radiometric tracking measurements. A feasible mothership-CubeSat mission is proposed in this study with following purposes, (1) performing inter-sat Doppler measurements, (2) improving the understanding of Phobos as well as the dynamic model, (3) securing the mothership as well as the primary mission, and (4) supporting autonomous navigation, given the long distance between the Earth and Mars. This study analyzes budgets of volume, mass, power, deployment Δv, and link, and the Doppler measurement noise of the system, and gives a feasible design for the CubeSat. The accuracy of orbit determination and geodesy is revealed via the Monte-Carlo simulation of estimation considering all uncertainties. Under an ephemeris error of the Mars-Phobos system ranging from 0 to 2 km, the autonomous orbit determination delivers an accuracy ranging from 0.2 m to 21 m and 0.05 mm/s to 0.4 cm/s. The geodesy can return 2nd-degree gravity coefficients at an accuracy of 1‰, even in the presence of an ephemeris error of 2 km. The achieved covariance of gravity coefficients and libration amplitude indicates an excellent possibility to distinguish families of interior structures.

Low-temperature librations and dynamical transition in proteins at differing hydration levels.

Hydration of water affects the dynamics and in turn the activity of biomacromolecules. We investigated the dependence of the librational oscillations and the dynamical transition on the hydrating conditions of two globular proteins with different structure and size, namely β-lactoglobulin (βLG) and human serum albumin (HSA), by spin-label electron paramagnetic resonance (EPR) in the temperature range of 120-270 K. The proteins were spin-labeled with 5-maleimide spin-label on free cysteins and prepared in the lyophilized state, at low (h = 0.12) and full (h = 2) hydration levels in buffer. The angular amplitudes of librations are small and almost temperature independent for both lyophilized proteins. Therefore, in these samples, the librational dynamics is restricted and the dynamical transition is absent. In the small and compact beta-structured βLG, the angular librational amplitudes increase with temperature and hydrating conditions, whereas hydration-independent librational oscillations whose amplitudes rise with temperature are recorded in the large and flexible alpha-structured HSA. Both βLG and HSA at low and fully hydration levels undergo the dynamical transition at about 230 K. The overall results indicate that protein librational dynamics is activated at the low hydration level h = 0.12 and highlight biophysical properties that are common to other biosamples at cryogenic temperatures.

Butane isomers mobility and framework dynamics in UiO-66 (Zr) MOF: Impact of the hydroxyl groups in zirconia cluster

UiO-66 (Zr) is a metal-organic framework (MOF) known for its thermal and chemical stability and wide range of adsorption-based applications. This MOF exhibits high separation selectivity for butane isomers. It has been earlier inferred that the separation performance of the material depends on the hydroxylation state of the zirconia cluster. In this contribution, we apply 2H solid-state NMR to characterize the dynamics of both the MOF organic framework itself and butane isomers in hydroxylated and dehydroxylated forms of UiO-66. It is established that the rate of π-flipping and the amplitude of the phenylene ring plane librations in the framework are higher for the dehydroxylated form. Self-diffusion coefficients of butane isomers have been estimated for both forms of UiO-66. The diffusivity is higher for n-butane in the dehydroxylated form, whereas the diffusion of isobutane is not affected by the presence of OH groups in the zirconia cluster of the MOF. Higher diffusivity of n-butane in dehydroxylated form is accounted for by the larger effective diameter of the window between the adjacent cages in this form, which arises from faster rotation and larger amplitude of framework linker libration. This rationalizes the higher efficiency of the dehydroxylated form of UiO-66(Zr) material for butane isomers separation.