The 3:2 spin-orbit resonant motion of Mercury

Our purpose is to build a model of rotation for a rigid Mercury, involving the planetary perturbations and the non-spherical shape of the planet. The approach is purely analytical, based on Hamiltonian formalism; we start with a first-order basic averaged resonant potential (including J ₂ and C ₂₂, and the first powers of the eccentricity and the inclination of Mercury). With this kernel model, we identify the present equilibrium of Mercury; we introduce local canonical variables, describing the motion around this 3:2 resonance. We perform a canonical untangling transformation, to generate three sets of action-angle variables, and identify the three basic frequencies associated to this motion. We show how to reintroduce the short-periodic terms, lost in the averaging process, thanks to the Lie generator; we also comment about the damping effects and the planetary perturbations. At any point of the development, we use the model SONYR to compare and check our calculations.     

The interior structure of Mercury and its core sulfur content

Depth-dependent interior structure models of Mercury are calculated for several plausible chemical compositions of the core and of the mantle. For those models, we compute the associated libration amplitude, obliquity, tidal deformation, and tidal changes in the external potential. In particular we study the relation between the interior structure parameters for five different mantle mineralogies and two different temperature profiles together with two extreme crust density values. We investigate the influence of the core light element concentration, temperature, and melting law on core state and inner and outer core size. We show that a sulfur concentration above 10 wt% is unlikely if the temperature at the core-mantle boundary is above 1850 K and the silicate shell at least 240 km thick. The interior models can only have an inner core if the sulfur weight fraction is below 5 wt% for core-mantle boundary temperature in the 1850-2200 K range. Within our modeling hypotheses, we show that with the expected precision on the moment of inertia the core size can be estimated to a precision of about 50 km and the core sulfur concentration with an error of about 2 wt%. This uncertainty can only be reduced when more information on the mantle mineralogy of Mercury becomes available. However, we show that the uncertainty on the core size estimation can be greatly reduced, to about 25 km, if tidal surface displacements and tidal variations in the external potential are considered.     

A Map for a Group of Resonant Cases in a quartic Galactic Hamiltonian

We present a map for the study of resonant motion in a potential made up of two harmonic oscillators with quartic perturbing terms. This potential can be considered to describe motion in the central parts of non-rotating elliptical galaxies. The map is based on the averaged Hamiltonian. Adding on a semi-empirical basis suitable terms in the unperturbed averaged Hamiltonian, corresponding to the 1:1 resonant case, we are able to construct a map describing motion in several resonant cases. The map is used in order to find thex − p _x Poincare phase plane for each resonance. Comparing the results of the map, with those obtained by numerical integration of the equation of motion, we observe, that the map describes satisfactorily the broad features of orbits in all studied cases for regular motion. There are cases where the map describes satisfactorily the properties of the chaotic orbits as well.     

Note on Mercury’s rotation: the four equilibria of the Hamiltonian model

Mercury is observed in a stable Cassini’s state, close to a 3:2 spin-orbit resonance, and a 1:1 node resonance. This present situation is not the only possible mathematical stable state, as it is shown here through a simple model limited to the second-order in harmonics and where Mercury is considered as a rigid body. In this framework, using a Hamiltonian formalism, four different sets of resonant angles are computed from the differential Hamiltonian equations, and each of them corresponds to four values of the obliquity; thanks to the calculation of the corresponding eigenvalues, their linear stability is analyzed. In this simplified model, two equilibria (one of which corresponding to the present state of Mercury) are stable, one is unstable, and the fourth one is degenerate. This degenerate status disappears with the introduction of the orbit (node and pericenter) precessions. The influence of these precession rates on the proper frequencies of the rotation is also analyzed and quantified, for different planetary models.     

A Symplectic Mapping Model for the Study of 2:3 Resonant Trans-Neptunian Motion

A symplectic mapping is constructed for the study of the dynamical evolution of Edgeworth-Kuiper belt objects near the 2:3 mean motion resonance with Neptune. The mapping is six-dimensional and is a good model for the Poincaré map of the real system, that is, the spatial elliptic restricted three-body problem at the 2:3 resonance, with the Sun and Neptune as primaries. The mapping model is based on the averaged Hamiltonian, corrected by a semianalytic method so that it has the basic topological properties of the phase space of the real system both qualitatively and quantitatively. We start with two dimensional motion and then we extend it to three dimensions. Both chaotic and regular motion is observed, depending on the objects' initial inclination and phase. For zero inclination, objects that are phase-protected from close encounters with Neptune show ordered motion even at eccentricities as large as 0.4 and despite being Neptune-crossers. On the other hand, not-phase-protected objects with eccentricities greater than 0.15 follow chaotic motion that leads to sudden jumps in their eccentricity and are removed from the 2:3 resonance, thus becoming short period comets. As inclination increases, chaotic motion becomes more widespread, but phase-protection still exists and, as a result, stable motion appears for eccentricities up to e = 0.3 and inclinations as high as i = 15°, a region where plutinos exist.     

The sodium exosphere of Mercury: Comparison between observations during Mercury's transit and model results

In this study we compare the sodium exosphere observations made by Schleicher et al. [Schleicher, H., and 4 colleagues, 2004. Astron. Astrophys. 425, 1119-1124] with the result of a detailed numerical simulation. The observations, made during the transit of Mercury across the solar disk on 7 May 2003, show a maximum of sodium emission near the polar regions, with north prevalence, and the presence of a dawn-dusk asymmetry. We interpret this distribution as the resulting effect of two combined processes: the solar wind proton precipitation causing chemical alteration of the surface, freeing the sodium atoms from their bounds in the crystalline structure on the surface, and the subsequent photon-stimulated and thermal desorption of the sodium atoms. While we find that the velocity distribution of photon desorbed sodium can explain the observed exosphere population, thermal desorption seems to play a minor role only causing a smearing at the locations where Na atoms are released on the dayside. The observed and simulated distributions agree very well with this hypothesis and indicate that the combination of the proposed processes is able to explain the observed features.     

The 2/1 and 3/2 Resonant Asteroid Motion: A Symplectic Mapping Approach

A comparative study is made between the 2/1 and the 3/2 resonant asteroid motion, with the aim to understand their different behaviour (gap in the 2/1 resonance, group in the 3/2 resonance). A symplectic mapping model is used, for each of these two resonances, assuming the asteroid is moving in the three-dimensional space under the gravitational perturbation of Jupiter. It is found that these resonances differ in several points, and although there is, in general, more chaos in the phase space close to the 3/2 resonance, even in the model of circular orbit of Jupiter, there are regions, close to the secondary resonances, which are less chaotic in the 3/2 resonance compared to the 2/1 resonance, and consequently trapping can take place.     

The 0.7-5.3 μm IR spectra of Mercury and the Moon: Evidence for high-Ca clinopyroxene on Mercury

We present infrared spectra of Mercury and the Moon in the wavelength range 0.7-5.3 μm obtained with the SpeX spectrograph at the NASA Infrared Telescope Facility. The spectra were acquired from pole and terminator locations of Mercury's surface and of Mersenius C and the Copernicus central peak on the Moon. Spectra of both bodies were measured in close temporal succession and were reduced in the same manner with identical calibration stars to minimize differences in the reduction process. The Copernicus spectra display the expected absorption features due to mafic minerals in the near infrared and show spectral features in the SiO combination/overtone vibrational band region above 4 μm. The spectra of Mercury from longitude 170° and north and south mid-latitudes display a 1-μm absorption band indicative of high-Ca clinopyroxene, while a spectrum from longitude 260° and northern mid-latitudes does not. The Mercury spectra show a broad feature of low emittance over the full 3-5 μm thermal infrared region, but no narrow features in this spectral range. The longitude 260° spectrum shows excess thermal emission around 5 μm attributable to the existence of a thermal gradient in the insolated dayside regolith. The thermal-IR spectra suggest a significant difference in the compositional and/or structural properties of Mercury and the Moon that may be due to grain size, absorption coefficient, or the magnitude of near-surface thermal gradients. The results indicate that the composition of Mercury's surface is heterogeneous on regional scales, and that the near infrared wavelength range provides more discriminative information on the surface composition than the 2-4 μm region, where the solar reflected and thermally emitted radiation contribute approximately equally to the observed flux of these bodies.     

The planet Mercury: Synthesis of resolved images of unknown part in the longitude range 250-290°W

Plans to send orbiter missions to Mercury (e.g., NASA's Messenger and ESA's BepiColombo) have prompted renewed efforts to investigate the surface of Mercury using ground-based remote sensing. While the highest resolution instrumentation optical telescopes (e.g. HST) cannot be used at small angular distances (<45°) from the Sun (Mercury's elongation never exceeds 28° seen from Earth), advanced ground-based astronomical techniques and modern processing software can be used to construct resolved images of the poorly known part of Mercury. Our observations of the planet presented here were carried out mainly in April and May, 2002, at evening elongation of the planet, at the Skinakas astrophysical observatory of Heraklion University (Crete, Greece). A synthesis of the acquired images of the hemisphere of Mercury, which was not observed by the Mariner 10 mission (1974-1975), is presented. A double rim basin with an internal diameter of about 1000 km and an external rim about 2000 km is suggested by the data. We present the observational method, the data analysis approach, and the resulting images.     

The optical effects of small iron particles that darken but do not redden: Evidence of intense space weathering on Mercury

Submicroscopic iron particles larger than about 50 nm, infused throughout mineral grains or glasses, are abundant in planetary materials altered by their environment such as shocked meteorites and lunar agglutinate glasses. Such particles darken their host material but do not redden their spectra but to date there has been no theoretical treatment of their optical effects. Using Mie theory, we modify the Hapke (2001) radiative transfer model of the effects of space weathering to include these effects. Comparison with laboratory measurements shows that the new treatment reproduces the relationship between submicroscopic iron size, abundance and reflectance. We apply this new model to near-IR spectra of Mercury recently obtained by the MESSENGER spacecraft and find that submicroscopic iron is much more abundant on Mercury than in lunar soils, with typical total submicroscopic iron abundances near 3.5 wt.% compared to about 0.5 wt.% for lunar soils We also find that the ratio of iron particles that darken but do not redden to the abundance of very small iron particles that impart the red slope to space weathered material is much larger than lunar (6 vs. 2). Both the total submicroscopic iron abundance and ratio of particle size fractions are consistent with the higher production of melt and vapor in micrometeorite impact on Mercury relative to the Moon (Cintala, 1992) that enables more accumulation of space weathering products before sequestration by regolith overturn. The radiative transfer model cannot directly constrain the abundance of opaque minerals on Mercury because of ambiguities between the darkening effects of opaques and submicroscopic iron particles larger than 50 nm, but assuming the opaques are the ultimate source of the submicroscopic iron, our results place a lower limit of 4-20 wt.% on opaque abundance on Mercury depending on the composition of the opaque phase and whether titanium metal also contributes to the space weathering effect.     

A Symplectic Mapping Model as a Tool to Understand the Dynamics of 2/1 Resonant Asteroid Motion

We present a 3-D symplectic mapping model that is valid at the 2:1 mean motion resonance in the asteroid motion, in the Sun-Jupiter-asteroid model. This model is used to study the dynamics inside this resonance and several features of the system have been made clear. The introduction of the third dimension, through the inclination of the asteroid orbit, plays an important role in the evolution of the asteroid and the appearance of chaotic motion. Also, the existence of the secondary resonances is clearly shown and their role in the appearance of chaotic motion and the slow diffusion of the elements of the orbit is demonstrated. This revised version was published online in July 2006 with corrections to the Cover Date.     

The effects of the target material properties and layering on the crater chronology: The case of Raditladi and Rachmaninoff basins on Mercury

In this paper we present a crater age determination of several terrains associated with the Raditladi and Rachmaninoff basins. These basins were discovered during the first and third MESSENGER flybys of Mercury, respectively. One of the most interesting features of both basins is their relatively fresh appearance. The young age of both basins is confirmed by our analysis on the basis of age determination via crater chronology. The derived Rachmaninoff and Raditladi basin model ages are about 3.6 Ga and 1.1 Ga, respectively. Moreover, we also constrain the age of the smooth plains within the basins' floors. This analysis shows that Mercury had volcanic activity until recent time, possibly to about 1 Ga or less. We find that some of the crater size-frequency distributions investigated suggest the presence of a layered target. Therefore, within this work we address the importance of considering terrain parameters, as geo-mechanical properties and layering, into the process of age determination. We also comment on the likelihood of the availability of impactors able to form basins with the sizes of Rachmaninoff and Raditladi in relatively recent times.     

The apparent lack of lunar-like swirls on Mercury: Implications for the formation of lunar swirls and for the agent of space weathering

Images returned by the MESSENGER spacecraft from the Mercury flybys have been examined to search for anomalous high-albedo markings similar to lunar swirls. Several features suggested to be swirls on the basis of Mariner 10 imaging (in the craters Handel and Lermontov) are seen in higher-resolution MESSENGER images to lack the characteristic morphology of lunar swirls. Although antipodes of large impact basins on the Moon are correlated with swirls, the antipodes of the large impact basins on Mercury appear to lack unusual albedo markings. The antipodes of Mercury’s Rembrandt, Beethoven, and Tolstoj basins do not have surface textures similar to the “hilly and lineated” terrain found at the Caloris antipode, possibly because these three impacts were too small to produce obvious surface disturbances at their antipodes. Mercury does have a class of unusual high-reflectance features, the bright crater-floor deposits (BCFDs). However, the BCFDs are spectral outliers, not simply optically immature material, which implies the presence of material with an unusual composition or physical state. The BCFDs are thus not analogs to the lunar swirls. We suggest that the lack of lunar-type swirls on Mercury supports models for the formation of lunar swirls that invoke interaction between the solar wind and crustal magnetic anomalies (i.e., the solar-wind standoff model and the electrostatic dust-transport model) rather than those models of swirl formation that relate to cometary impact phenomena. If the solar-wind standoff hypothesis for lunar swirls is correct, it implies that the primary agent responsible for the optical effects of space weathering on the Moon is solar-wind ion bombardment rather than micrometeoroid impact.     

The Homoclinic Tangle of A 1:2 Resonance in a 2-D Hamiltonian System

We study in great detail the geometry of the homoclinic tangle, with respect to the energy, corresponding to an unstable periodic orbit of type 1:2, on a surface of section representing a 2-D Hamiltonian system. The tangle consists of two resonance areas, in contrast with the tangles of type-l or -{l, m, k, x = 0} considered in previous studies, that consist of only one resonance area. We study the intersections of the inner and outer lobes of the same resonance area and of the two resonance areas. The intersections of the lobes follow certain rules. The detailed study of these rules allows us to derive quantitative relations about the number of intersections and to understand the complex behavior of the higher order lobes by studying the lower order lobes. We find 1st, 2nd, 3rd, etc. order intersections formed by lobes making 1, 2, 3, etc. turns around an island. After a sufficiently high order of iterations a lobe may intersect its image and thus produce a Poincaré recurrence. Numerical results for a wide interval of energies are presented. The number of intersections changes through tangencies. In any finite interval of the energy between two tangencies of 1st order, an infinite number of higher order tangencies occur and thus, according to the Newhouse theorem, there exist nearby islands of stability.     

视超光速射电节点的频谱演化:持续供能的喷流—激波模型

本文提出一个持续供能的喷流—激波模型来解释视超光速射电节点频谱演化的普遍特性．理论计算指出，射电爆发频谱的反转频率νm和频谱极大流量Sm之间的关系具有典型的3阶段演化方式，并且与类星体3C345中观测到的视超光速节点C4的演化行为相当好地符合．令Sm∝ν，则在上升阶段ζ≤3；在平坦变化阶段ζ≈0，而在衰减阶段ζ≈1．爆发幅度和流量极大的迟后时间对频率的关系△Smax（ν）和△t（ν），都具有Blazar天体中观测到的射电爆发的普遍形式．本文提出的激波模型，考虑到驱动气体的持续注入和激波后等离子体的绝热膨胀．因此提供了对相对论喷流中激波形成和演化的物理原因更深入的理解．并对Blazar天体中视超光速节点频谱演化提出了新的解释．