Libration Amplitude Research Articles

Study of molecular dynamics in poly(n-alkyl methacrylates) by light induced absorption anisotropy.

Formation of absorbance anisotropy under linearly polarized irradiation at 405 nm in films of poly(ethyl methacrylate) (PEMA), poly(n-butyl methacrylate) (PnBMA), and poly(n-hexyl methacrylate) (PnHexMA) doped with azo compound has been studied in a wide temperature range below Tg. The anisotropy formation kinetics has been well described under the assumption that there are two types of environments in the polymer matrixes: mobile and immobile. The mobile environment allows the dopant to reorient in the course of cis-trans-cis isomerization cycle and the immobile environment does not. The fraction of mobile environments was estimated to be 0.12 in PEMA, 0.65 in PnBMA, and 1.0 in PnHexMA. An increase in the fraction of mobile environments within the series PEMA-PnBMA-PnHexMA have been associated with an increase in the size of alkyl nanodomains. The average angle of transition dipole rotation upon isomerization and the amplitude of molecules librations have been estimated in the temperature range studied.

TIDALLY MODULATED ERUPTIONS ON ENCELADUS:CASSINIISS OBSERVATIONS AND MODELS

We use images acquired by the Cassini Imaging Science Subsystem (ISS) to investigate the temporal variation of the brightness and height of the south polar plume of Enceladus. The plume’s brightness peaks around the moon’s apoapse, but with no systematic variation in scale height with either plume brightness or Enceladus’ orbital position. We compare our results, both alone and supplemented with Cassini near-infrared observations, with predictions obtained from models in which tidal stresses are the principal control of the eruptive behavior. There are three main ways of explaining the observations: (1) the activity is controlled by right-lateral strike slip motion; (2) the activity is driven by eccentricity tides with an apparent time delay of about 5 hr; (3) the activity is driven by eccentricity tides plus a 1:1 physical libration with an amplitude of about 0. ◦ 8 (3.5 km). The second hypothesis might imply either a delayed eruptive response, or a dissipative, viscoelastic interior. The third hypothesis requires a libration amplitude an order of magnitude larger than predicted for a solid Enceladus. While we cannot currently exclude any of these hypotheses, the third, which is plausible for an Enceladus with a subsurface ocean, is testable by using repeat imaging of the moon’s surface. A dissipative interior suggests that a regional background heat source should be detectable. The lack of a systematic variation in plume scale height, despite the large variations in plume brightness, is plausibly the result of supersonic flow; the details of the eruption process are yet to be understood.

The long-period forced librations of Titan

AbstractA moon in synchronous rotation has longitudinal librations because of its non-spherical mass distribution and its elliptical orbit around the planet. We study the librations of Titan with periods of 14.7y and 29.5y and include deformation effects and the existence of a subsurface ocean. We take into account the fact that the orbit is not Keplerian and has other periodicities than the main period of orbital motion around Saturn due to perturbations by the Sun, other planets and moons. An orbital theory is used to compute the orbital perturbations due to these other bodies.We numerically evaluate the amplitude of the long-period librations for many interior structure models of Titan constrained by the mass, radius and gravity field. Measurements of the librations may give constraints on the interior structure of the icy satellites.

Complements to the longitudinal librations of an elastic 3-layer Titan on a non-Keplerian orbit

AbstractTitan longitudinal librations are dependent on the satellite internal structure and the elastic behavior of the surface. The elastic deformation of the surface is related to the perturbing potential through the Love theory. In a previous paper, we described the deformation as a response to the tidal potential exerted by Saturn at orbital frequency. Here we improve the tidal deformation reponse by including the effect of the libration angle and the orbital perturbations. We then provide the libration amplitudes associated with the rotational model of a tidally deformed three-layer Titan evolving on a non-Keplerian orbit.

Inertial wave excitation and focusing in a liquid bounded by a frustum and a cylinder

AbstractThe mechanism of localized inertial wave excitation and its efficiency is investigated for an annular cavity rotating with $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\Omega _0$. Meridional symmetry is broken by replacing the inner cylinder with a truncated cone (frustum). Waves are excited by individual longitudinal libration of the walls. The geometry is non-separable and exhibits wave focusing and wave attractors. We investigated laboratory and numerical results for the Ekman number $E\approx 10^{-6}$, inclination $\alpha =5.71^\circ $ and libration amplitudes $\varepsilon \leq 0.2$ within the inertial wave band $0 < \omega < 2\Omega _0$. Under the assumption that the inertial waves do not essentially affect the boundary-layer structure, we use classical boundary-layer analysis to study oscillating Ekman layers over a librating wall that is at an angle $\alpha \neq 0$ to the axis of rotation. The Ekman layer erupts at frequency $\omega =f_{*}$, where $f_{*}\equiv 2 \Omega _0 \sin \alpha $ is the effective Coriolis parameter in a plane tangential to the wall. For the selected inclination this eruption occurs for the forcing frequency $\omega /\Omega _0=0.2$. For the librating lids eruption occurs at $\omega /\Omega _0=2$. The study reveals that the frequency dependence of the total kinetic energy $K_{\omega }$ of the excited wave field is strongly connected to the square of the Ekman pumping velocity $w_{{E}}(\omega )$ that, in the linear limit, becomes singular when the boundary layer erupts. This explains the frequency dependence of non-resonantly excited waves. By the localization of the forcing, the two configurations investigated, (i) frustum libration and (ii) lids together with outer cylinder in libration, can be clearly distinguished by their response spectra. Good agreement was found for the spatial structure of low-order wave attractors and periodic orbits (both characterized by a small number of reflections) in the frequency windows predicted by geometric ray tracing. For ‘resonant’ frequencies a significantly increased total bulk energy was found, while the energy in the boundary layer remained nearly constant. Inertial wave energy enters the bulk flow via corner beams, which are parallel to the characteristics of the underlying Poincaré problem. Numerical simulations revealed a mismatch between the wall-parallel mass fluxes near the corners. This leads to boundary-layer eruption and the generation of inertial waves in the corners.

Three body resonances in close orbiting planetary systems: tidal dissipation and orbital evolution

AbstractWe study the orbital evolution of a three-planet system with masses in the super-Earth regime resulting from the action of tides on the planets induced by the central star which cause orbital circularization. We consider systems either in or near to a three-body commensurability for which adjacent pairs of planets are in a first-order commensurability. We develop a simple analytic solution, derived from a time averaged set of equations, that describes the expansion of the system away from strict commensurability as a function of time, once a state where relevant resonant angles undergo small amplitude librations has been attained. We perform numerical simulations that show the attainment of such resonant states focusing on the Kepler 60 system. The results of the simulations confirm many of the scalings predicted by the appropriate analytic solution. We go on to indicate how the results can be applied to put constraints on the amount of tidal dissipation that has occurred in the system. For example, if the system has been in a librating state since its formation, we find that its present period ratios imply an upper limit on the time average of 1/Q′, With Q′ being the tidal dissipation parameter. On the other hand if a librating state has not been attained, a lower upper bound applies.

Phobos laser ranging: Numerical Geodesy experiments for Martian system science

Laser ranging is emerging as a technology for use over (inter)planetary distances, having the advantage of high (mm-cm) precision and accuracy and low mass and power consumption. We have performed numerical simulations to assess the science return in terms of geodetic observables of a hypothetical Phobos lander performing active two-way laser ranging with Earth-based stations. We focus our analysis on the estimation of Phobos and Mars gravitational, tidal and rotational parameters. We explicitly include systematic error sources in addition to uncorrelated random observation errors. This is achieved through the use of consider covariance parameters, specifically the ground station position and observation biases. Uncertainties for the consider parameters are set at 5mm and at 1mm for the Gaussian uncorrelated observation noise (for an observation integration time of 60s). We perform the analysis for a mission duration up to 5 years.It is shown that a Phobos Laser Ranging (PLR) can contribute to a better understanding of the Martian system, opening the possibility for improved determination of a variety of physical parameters of Mars and Phobos. The simulations show that the mission concept is especially suited for estimating Mars tidal deformation parameters, estimating degree 2 Love numbers with absolute uncertainties at the 10−2 to 10−4 level after 1 and 4 years, respectively and providing separate estimates for the Martian quality factors at Sun and Phobos-forced frequencies. The estimation of Phobos libration amplitudes and gravity field coefficients provides an estimate of Phobos׳ relative equatorial and polar moments of inertia with an absolute uncertainty of 10−4 and 10−7, respectively, after 1 year. The observation of Phobos tidal deformation will be able to differentiate between a rubble pile and monolithic interior within 2 years.For all parameters, systematic errors have a much stronger influence (per unit uncertainty) than the uncorrelated Gaussian observation noise. This indicates the need for the inclusion of systematic errors in simulation studies and special attention to the mitigation of these errors in mission and system design.

The Phobos geodetic control point network and rotation model

A new global control point network was derived for Phobos, based on SRC (Mars Express), Phobos-2, and Viking Orbiter image data. We derive 3-D Cartesian coordinates for 813 control points as well as improved pointing data for 202 SRC and Viking images in the Phobos-fixed coordinate system. The point accuracies vary from 4.5m on the Phobos nearside, to up to 67.0m on the farside, where we rely on Viking images (average point accuracy: 13.7m). From tracking of the control points we detect a librational motion synchronous to the Phobos orbital period and measure libration amplitude of 1.09°, in agreement with predictions from shape information assuming a uniform interior. This suggests that the interior of Phobos is homogeneous – but small local mass anomalies, e.g., associated with crater Stickney, cannot be ruled out. Our new control point network has a higher number of data points and higher point accuracy than previous data products and will be an important basis for accurate shape models and maps.

Librations of Probe Molecules in Polymeric Matrixes

Formation of optical anisotropy under linearly polarized light has been studied in films of poly(n-butyl methacrylate) and poly(n-hexyl methacrylate) doped with azo compound (1-naphthyl-p-azo-methoxybenzene). Time profiles of the absorption anisotropy and absorbance can not be fitted with a commonly used model of anisotropy formation, which includes the transition dipole turns upon cis-trans isomerization and the rotational diffusion: the experimental values of anisotropy happen to be much lower than the calculated ones. To explain the low anisotropy level, fast molecular librations, i.e., stochastic pendulum-like oscillations of azo molecules about their average orientations were introduced. Numerical simulation of anisotropy and absorbance time profiles with the modified model gives estimates for average libration amplitude of 45-60° at temperatures 25-65 K below T(g).

Librational response of a deformed 3-layer Titan perturbed by non-Keplerian orbit and atmospheric couplings

The analyses of Titan׳s gravity field obtained by Cassini space mission suggest the presence of an internal ocean beneath its icy surface. The characterization of the geophysical parameters of the icy shell and the ocean is important to constrain the evolution models of Titan. The knowledge of the librations, that are periodic oscillations around a uniform rotational motion, can bring a piece of information on the interior parameters.The objective of this paper is to study the librational response in longitude from an analytical approach for Titan composed of a deep atmosphere, an elastic icy shell, an internal ocean, and an elastic rocky core perturbed by the gravitational interactions with Saturn. We start from the librational equations developed for a rigid satellite in synchronous spin–orbit resonance. We introduce explicitly the atmospheric torque acting on the surface computed from the Titan IPSL GCM (Institut Pierre Simon Laplace General Circulation Model) and the periodic deformations of elastic solid layers due to the tides. We investigate the librational response for various interior models in order to compare and to identify the influence of the geophysical parameters and the impact of the elasticity.The main librations arise at two well-separated forcing frequency ranges: low forcing frequencies dominated by the Saturnian annual and semi-annual frequencies, and a high forcing frequency regime dominated by Titan׳s orbital frequency around Saturn. At low forcing frequency, the librational response is dominated by the Saturnian gravitational torque and the atmospheric torque has a small effect. In addition, the libration amplitude in that case is almost equal to the magnitude of the perturbation. The modulation of the gravitational torque amplitude at the orbital frequency with periodic deformation induces long-period terms in the librational response which contain information on the internal structure. At high forcing frequency the libration depends on the inertia of the layers and the elasticity can strongly reduce its amplitude at orbital frequency. For example, the amplitude of diurnal libration for oceanic models goes from about 320–390m if the icy shell is purely rigid to 60–85m when the elasticity is included, i.e. a reduction of about 80%. For models without ocean, diurnal libration goes from 52m in a rigid case to 50m for an elastic case, a very low reduction due to the weak deformation of an entirely solid satellite compared to the deformation of a thin icy shell. Oceanic models with elastic solid layers have the same order of libration amplitude than the oceanless models, which makes more challenging to differentiate them by the interpretation of librational motion.

Water scaffolding in collagen: Implications on protein dynamics as revealed by solid‐state NMR

Solid-state NMR studies of collagen samples of various origins confirm that the amplitude of collagen backbone and sidechain motions increases significantly on increasing the water content. This conclusion is supported by the changes observed in three different NMR observables: (i) the linewidth dependence on the 1H decoupling frequency; (ii) 13C CSA changes for the peptide carbonyl groups, and (iii) dephasing rates of 1H-13C dipolar couplings. In particular, a nearly threefold increase in motional amplitudes of the backbone librations about C-Cα or N-Cα bonds was found on increasing the added water content up to 47 wt%D2 O. On the basis of the frequencies of NMR observables involved, the timescale of the protein motions dependent on the added water content is estimated to be of the order of microseconds. This estimate agrees with that from wideline T2(1)H NMR measurements. Also, our wideline 1H NMR measurements revealed that the timescale of the microsecond motions in proteins reduces significantly on increasing the added water content, i.e., an ∼15-fold increase in protein motional frequencies is observed on increasing the added water content to 45 wt% D2 O. The observed changes in collagen dynamics is attributed to the increase in water translational diffusion on increasing the amount of added water, which leads to more frequent "bound water/free water" exchange on the protein surface, accompanied by the breakage and formation of new hydrogen bonds with polar functionalities of protein.

Phobos' shape and topography models

The global shape and the dynamic environment are fundamental properties of a body. Other properties such as volume, bulk density, and models for the dynamic environment can subsequently be computed based on such models. Stereo-photogrammetric methods were applied to derive a global digital terrain model (DTM) with 100m/pixel resolution using High Resolution Stereo Camera images of the Mars Express mission and Viking Orbiter images. In a subsequent least-squares fit, coefficients of the spherical harmonic function to degree and order 45 are computed. The dynamic models for Phobos were derived from a polyhedron representation of the DTM. The DTM, spherical harmonic function model, and dynamic models, have been refined and represent Phobos' dynamic and geometric topography with much more detail when compared to Shi et al. (2012) and Willner et al. (2010) models, respectively. The volume of Phobos has been re-determined to be in the order of 5741km3 with an uncertainty of only 0.6% of the total volume. This reduces the bulk density to 1.86±0.013g/cm3 in comparison to previous results. Assuming a homogeneous mass distribution a forced libration amplitude for Phobos of 1.14° is computed that is in better agreement with observations by Willner et al. (2010) than previous estimates.

Insights on Hydrogen-Bond Lifetimes in Liquid and Supercooled Water

We study the temperature dependence of the lifetime of geometric and geometric/energetic water hydrogen-bonds (H-bonds), down to supercooled water, through molecular dynamics. The probability and lifetime of H-bonds that break either by translational or librational motions and those of energetic broken H-bonds, along with the effects of transient broken H-bonds and transient H-bonds, are considered. We show that the fraction of transiently broken energetic H-bonds increases at low temperatures and that this energetic breakdown is caused by oxygen-oxygen electrostatic repulsions upon too small amplitude librations to disrupt geometric H-bonds. Hence, differences between geometric and energetic continuous H-bond lifetimes are associated with large H-bond energy fluctuations, in opposition to moderate geometric fluctuations, within common energetic and geometric H-bond definition thresholds. Exclusion of transient broken H-bonds and transient H-bonds leads to H-bond definition-independent mean lifetimes and activation energies, ~11 kJ/mol, consistent with the reactive flux method and experimental scattering results. Further, we show that power law decay of specific temporal H-bond lifetime probability distributions is associated with librational and translational motions that occur on the time scale (~0.1 ps) of H-bond breaking /re-forming dynamics. While our analysis is diffusion-free, the effect of diffusion on H-bond probability distributions where H-bonds are allowed to break and re-form, switching acceptors in between, is shown to result in neither exponential nor power law decay, similar to the reactive flux correlation function.

The forced libration of Europa’s deformable shell and its dependence on interior parameters

One of the most important goals of future missions to jovian moon Europa will be to unambiguously determine and characterize the putative subsurface ocean, as well as characterizing the overlying ice shell. In addition to magnetic, altimetry and gravity measurements, observations of Europa’s librations are expected to contribute to the realization of such an important goal. The longitudinal libration of Europa’s shell in the presence of a subsurface ocean has been previously studied assuming that Europa’s internal solid layers behave rigidly. However, at the frequency of the acting diurnal tides and external gravitational torques, the response of Europa’s interior is not rigid but rather viscoelastic. In this paper, we develop a differential libration model that takes into account the effect of diurnal deformation on the forced longitudinal libration of Europa’s internal solid layers. We apply our libration model to a rather large range of possible interior models of Europa to investigate the dependence of the shell libration amplitude on the geophysical parameters that characterize the interior of Europa, in particular the shell. From all analyzed interior parameters, we find that the poorly constrained rigidity of the shell has the largest effect on the libration amplitude. This results suggests thus that future libration observations could be very useful to constrain the value of this parameter for Europa. However, we also notice that the effect of the viscosity of a dissipating ice sublayer at the bottom of the shell (if present) on the libration amplitude of the shell would not allow for an unambiguous determination of the shell rigidity. Furthermore, the dependence of the shell libration amplitude on the shell thickness and density is rigidity-dependent and weak in comparison to the dependence on the rigidity. As a result, libration observations would not be able to provide any information on the thickness and/or density of the shell without previous determination of the rigidity.

The interior structure of Mercury constrained by the low-degree gravity field and the rotation of Mercury

Among all the planets of the solar system Mercury stands out, because it has a relatively high average density compared to its size. To account for this high density, Mercuryʼs core radius is thought to be larger than 34 of its radius. Here, we use recent data about the second-degree gravity field coefficients and measurements about Mercuryʼs rotation – obliquity and 88-day libration amplitude – to obtain constraints on Mercuryʼs interior structure. By combining the gravity field data and the obliquity measurements, the mean moment of inertia of Mercury can be determined. If the coupling between the core and the mantle is neglected then the gravity field data together with the libration amplitude provide an estimate of Mercuryʼs silicate shell moment of inertia. However, since the effect of core–mantle coupling on the 88-day libration amplitude can be about as large as the libration amplitudeʼs uncertainty (Van Hoolst et al., 2012) we use as data the mean moment of inertia and the 88-day libration amplitude to infer knowledge about Mercuryʼs interior structure.We use two different interior modeling approaches. The first one is based on Rivoldini et al. (2009) and uses a set of 5 mantle mineralogies, a crust with a given thickness and density, and two mantle temperatures. In the second setting the density of the mantle, the thickness, and density of the crust, and the temperature of the core mantle boundary are parameters of the model. In both cases, we assume that the core is made of iron and the light element sulfur and use the temperature, pressure, and the concentration of sulfur in the core together with the melting temperature of iron–sulfur to determine the radius of the inner core.Our results show that the data provide a strong constraint on the radius of the core and on its average density or equivalently on the fraction of sulfur in the core if sulfur is the only light element in the core. We find that Mercury has a core radius of 2004±39 km, an average core density of 7233±267 kg/m3, and a sulfur fraction of 4.5±1.8 wt%. The other parameters of the model, in particular the density of the mantle are however, only weakly constrained by the data. The geodesy data can also not distinguish between a fully liquid core and the existence of an inner core.

Phobos interior from librations determination using Doppler and star tracker measurements

Numerical simulations have been performed to assess the precision that can be obtained on Phobos libration angles by using two types of data acquired by a probe landed on the Martian moon: (1) Direct-To-Earth (DTE) Doppler data and (2) Star-Tracker (ST) data. Compared to independent estimates, combination of DTE and ST data provides the more precise estimates of Phobos libration angles at the 10−3−10−5 degree level. Short period libration amplitudes are functions of the relative moments of inertia. Thus their determination would provide constraints on mass distributions inside Phobos. We show that the longitudinal libration amplitude at the orbital period, while being clearly distinct from the resonance period, is the most interesting signal for that purpose, because it leads to the best precision on (B−A)/C (10−5 after only 10 weeks of operation). Nevertheless, we showed that inferring the individual moments of inertia A, B and C with a precision sufficient (<1%) to distinguish the rotational behavior of an homogenous from that of an heterogenous body requires the determination of one of the degree-two gravity field coefficient at the percent level.In addition, we stress that ST and DTE data combination allows de-correlating librational motion from orbital motion since ST measurements are sensitive to the moon's rotation and do not depend on its ephemeris, whereas DTE Doppler data are sensitive to both motions as well as to Phobos' surface displacement due to the tides raised by Mars. Furthermore, the ephemeris of Phobos has to be known at the centimeter level to allow measuring the tidal surface displacement with enough precision to conclude on the nature of Phobos' interior (rubble pile versus monolithic) as one step toward constraining its origin and evolution.

On the librations and tides of large icy satellites

The gravitational force exerted by a central planet deforms its orbiting moons (tides) and periodically changes their rotation (librations). Tides and librations of large icy satellites rotating synchronously with their orbital motion are usually studied separately but we here show that tides strongly reduce the libration amplitude of large icy satellites with a subsurface ocean. Moreover, libration also slightly changes the tides. Periodic tides affect libration in three ways: first the periodic tidal bulges modify the gravitational torque exerted by the central planet on the satellite, secondly torques between the periodic tidal bulges of a solid layer and the static shape of another solid layer couple the libration of the ice shell and solid interior, and thirdly, the zonal tides periodically change the polar moment of the satellite, which acts as the inertia for rotational motion. We develop a formalism to include these elastic effects in the study of librations and calculate the amplitudes of the longitudinal librations at orbital period for the largest icy satellites in the Solar System: Europa, Ganymede, Callisto, and Titan. Without a subsurface ocean, elastic effects on libration are of the order of a few percent. However, if a subsurface ocean exists as indicated by several observations for the satellites considered, elastic tides have a crucial effect. If the solid layers of the satellites are considered rigid, the libration amplitude is at least one order of magnitude larger than for an entirely solid satellite. Periodic elastic tidal deformations of the layers, in particular of the ice shell, counterbalance this large increase and keep the libration amplitude at the level of the libration of the satellite without subsurface ocean. Besides gravitationally forced librations, seasonal variations in the atmosphere of Titan also cause the rotation of Titan to change on a seasonal timescale. We show that tidal elastic effects can strongly increase the amplitude of these long-term variations in the rotation rate by a factor of about four with respect to a rigid shell. We predict that these variations can have an amplitude above the detection limit of the Cassini radar.

Inertial waves in a spherical shell induced by librations of the inner sphere: experimental and numerical results

We begin with an experimental investigation of the flow induced in a rotating spherical shell. The shell globally rotates with angular velocity Ω. A further periodic oscillation with angular velocity 0 ⩽ ω ⩽ 2Ω, a so-called longitudinal libration, is added on the inner sphere's rotation. The primary response is inertial waves spawned at the critical latitudes on the inner sphere, and propagating throughout the shell along inclined characteristics. For sufficiently large libration amplitudes, the higher harmonics also become important. Those harmonics whose frequencies are still less than 2Ω behave as inertial waves themselves, propagating along their own characteristics. The steady component of the flow consists of a prograde zonal jet on the cylinder tangent to the inner sphere and parallel to the axis of rotation, and increases with decreasing Ekman number. The jet becomes unstable for larger forcing amplitudes as can be deduced from the preliminary particle image velocimetry observations. Finally, a wave attractor is experimentally detected in the spherical shell as the pattern of largest variance. These findings are reproduced in a two-dimensional numerical investigation of the flow, and certain aspects can be studied numerically in greater detail. One aspect is the scaling of the width of the inertial shear layers and the width of the steady jet. Another is the partitioning of the kinetic energy between the forced wave, its harmonics and the mean flow. Finally, the numerical simulations allow for an investigation of instabilities, too local to be found experimentally. For strong libration amplitudes, the boundary layer on the inner sphere becomes unstable, triggering localized Görtler vortices during the prograde phase of the forcing. This instability is important for the transition to turbulence of the spherical shell flow.

Librational fluctuations in protein glasses

Librational motions in the region of the protein “glass” (or dynamic) transition are analysed for spin-labelled haemoglobin, serum albumin and β-lactoglobulin by EPR spectroscopy. A discontinuity in the temperature dependence of the mean-square librational amplitude, <α2>, occurs in the region of 200K as found for the mean-square atomic displacement, <r2>, at the protein dynamic transition by Mössbauer spectroscopy and neutron scattering. The discontinuity in <α2> vs. T can be described by the Vogel–Tammann–Fulcher equation, implying a finite glass transition temperature. Above the dynamic transition, <α2> vs. 1/T can be approximated by the Arrhenius law with activation energies similar to those usually found for <r2>, and relaxation processes in glass-forming media and the hydration shells of proteins. Similar results are found for librational fluctuations of membranous Na,K-ATPase spin-labelled either on superficial SH groups or on those essential to activity.

Long term stability of Earth Trojans

We explore the long-term stability of Earth Trojans by using a chaos indicator, the Frequency Map Analysis. We find that there is an extended stability region at low eccentricity and for inclinations lower than about $$50^{\circ }$$ even if the most stable orbits are found at $$i \le 40^{\circ }$$ . This region is not limited in libration amplitude, contrary to what found for Trojan orbits around outer planets. We also investigate how the stability properties are affected by the tidal force of the Earth–Moon system and by the Yarkovsky force. The tidal field of the Earth–Moon system reduces the stability of the Earth Trojans at high inclinations while the Yarkovsky force, at least for bodies larger than 10 m in diameter, does not seem to strongly influence the long-term stability. Earth Trojan orbits with the lowest diffusion rate survive on timescales of the order of $$10^9$$ years but their evolution is chaotic. Their behaviour is similar to that of Mars Trojans even if Earth Trojans appear to have shorter lifetimes.