Resonant forcing of Mercury's libration in longitude

The period of free libration of Mercury's longitude about the position it would have had if it were rotating uniformly at 1.5 times its orbital mean motion is close to resonance with Jupiter's orbital period. The Jupiter perturbations of Mercury's orbit thereby lead to amplitudes of libration at the 11.86 year period that may exceed the amplitude of the 88 day forced libration determined by radar. Mercury's libration in longitude may be thus dominated by only two periods of 88 days and 11.86 years, where other periods from the planetary perturbations of the orbit have much smaller amplitudes.     

The free precession and libration of Mercury

An analysis based on the direct torque equations including tidal dissipation and a viscous core-mantle coupling is used to determine the damping time scales of O(¹⁰⁵) years for free precession of the spin about the Cassini state and free libration in longitude for Mercury. The core-mantle coupling dominates the damping over the tides by one to two orders of magnitude for the plausible parameters chosen. The short damping times compared with the age of the Solar System means we must find recent or on-going excitation mechanisms if such free motions are found by the current radar experiments or the future measurement by the MESSENGER and BepiColombo spacecraft that will orbit Mercury. We also show that the average precession rate is increased by about 30% over that obtained from the traditional precession constant because of a spin-orbit resonance induced contribution by the C₂₂ term in the expansion of the gravitational field. The C₂₂ contribution also causes the path of the spin during the precession to be slightly elliptical with a variation in the precession rate that is a maximum when the obliquity is a minimum. An observable free precession will compromise the determination of obliquity of the Cassini state and hence of C/MMR² for Mercury, but a detected free libration will not compromise the determination of the forced libration amplitude and thus the verification of a liquid core.     

Evolution of Mercury's obliquity

Mercury has a near-zero obliquity, i.e. its spin axis is nearly perpendicular to its orbital plane. The value of the obliquity must be known precisely in order to constrain the size of the planet's core within the framework suggested by Peale [Peale, S.J., 1976. Nature 262, 765-766]. Rambaux and Bois [Rambaux, N., Bois, E., 2004. Astron. Astrophys. 413, 381-393] have suggested that Mercury's obliquity varies on thousand-year timescales due to planetary perturbations, potentially ruining the feasibility of Peale's experiment. We use a Hamiltonian approach (free of energy dissipation) to study the spin-orbit evolution of Mercury subject to secular planetary perturbations. We can reproduce an obliquity evolution similar to that of Rambaux and Bois [Rambaux, N., Bois, E., 2004. Astron. Astrophys. 413, 381-393] if we integrate the system with a set of initial conditions that differs from the Cassini state. However the thousand-year oscillations in the obliquity disappear if we use initial conditions corresponding to the equilibrium position of the Cassini state. This result indicates that planetary perturbations do not force short-period, large amplitude oscillations in the obliquity of Mercury. In the absence of excitation processes on short timescales, Mercury's obliquity will remain quasi-constant, suggesting that one of the important conditions for the success of Peale's experiment is realized. We show that interpretation of data obtained in support of this experiment will require a precise knowledge of the spin-orbit configuration, and we provide estimates for two of the critical parameters, the instantaneous Laplace plane orientation and the orbital precession rate from numerical fits to ephemeris data. Finally we provide geometrical relationships and a scheme for identifying the correct initial conditions required in numerical integrations involving a Cassini state configuration subject to planetary perturbations.     

The proximity of Mercury's spin to Cassini state 1 from adiabatic invariance

In determining Mercury's core structure from its rotational properties, the value of the normalized moment of inertia, C/MR², from the location of Cassini 1 is crucial. If Mercury's spin axis occupies Cassini state 1, its position defines the location of the state, where the axis is fixed in the frame precessing with the orbit. Although tidal and core-mantle dissipation drive the spin to the Cassini state with a time scale O(¹⁰⁵) years, the spin might still be displaced from the Cassini state if the variations in the orbital elements induced by planetary perturbations, which change the position of the Cassini state, cause the spin to lag behind as it attempts to follow the state. After being brought to the state by dissipative processes, the spin axis is expected to follow the Cassini state for orbit variations with time scales long compared to the 1000 year precession period of the spin about the Cassini state because the solid angle swept out by the spin axis as it precesses is an adiabatic invariant. Short period variations in the orbital elements of small amplitude should cause displacements that are commensurate with the amplitudes of the short period terms. The exception would be if there are forcing terms in the perturbations that are nearly resonant with the 1000 year precession period. The precision of the radar and eventual spacecraft measurements of the position of Mercury's spin axis warrants a check on the likely proximity of the spin axis to the Cassini state. How confident should we be that the spin axis position defines the Cassini state sufficiently well for a precise determination of C/MR²? By following simultaneously the spin position and the Cassini state position during long time scale orbital variations over past 3 million years [Quinn, T.R., Tremaine, S., Duncan, M., 1991. Astron. J. 101, 2287-2305] and short time scale variations for 20,000 years [JPL Ephemeris DE 408; Standish, E.M., private communication, 2005], we show that the spin axis will remain within one arcsec of the Cassini state after it is brought there by dissipative torques. In this process the spin is located in the orbit frame of reference, which in turn is referenced to the inertial ecliptic plane of J2000. There are no perturbations with periods resonant with the precession period that could cause large separations. We thus expect Mercury's spin to occupy Cassini state 1 well within the uncertainties for both radar and spacecraft measurements, with correspondingly tight constraints on C/MR² and the extent of Mercury's molten core. Two unlikely caveats for this conclusion are: (1) an excitation of a free spin precession by an unknown mechanism or (2) a displacement by a dissipative core mantle interaction that exceeds the measurement uncertainties.     

Analytical model of the long-period forced longitude librations of Mercury

The shaking of Mercury’s orbit by the planets forces librations in longitude in addition to those at harmonics of the orbital period that have been used to detect Mercury’s molten core. We extend the analytical formulation of Peale et al. (Peale, S.J., Margot, J.L., Yseboodt, M. [2009]. Icarus 199, 1-8) in order to provide a convenient means of determining the amplitudes and phases of the forced librations without resorting to numerical calculations. We derive an explicit relation between the amplitude of each forced libration and the moment of inertia parameter (B-A)/C_m. Far from resonance with the free libration period, the libration amplitudes are directly proportional to (B-A)/C_m. Librations with periods close to the free libration period of ∼12 years may have measurable (∼arcsec) amplitudes. If the free libration period is sufficiently close to Jupiter’s orbital period of 11.86 years, the amplitude of the forced libration at Jupiter’s period could exceed the 35 arcsec amplitude of the 88-day forced libration. We also show that the planetary perturbations of the mean anomaly and the longitude of pericenter of Mercury’s orbit completely determine the libration amplitudes.While these signatures do not affect spin rate at a detectable level (as currently measured by Earth-based radar), they have a much larger impact on rotational phase (affecting imaging, altimetry, and gravity sensors). Therefore, it may be important to consider planetary perturbations when interpreting future spacecraft observations of the librations.     

Estimating Mercury's 88-day libration amplitude from orbit

In this paper it is derived that the libration of Mercury can be described by where Φ₀ is the unknown libration amplitude, M is Mercury's mean anomaly and K=−9.483. Φ₀ can be determined by comparing pairs of images of the same landmarks taken by an orbiter at different positions of Mercury. If the angle between the orbit plane of a polar orbiter and Mercury's line of periapsis is between −60° and 60° and if one landmark at the equator is imaged per day with a relative precision of , then the libration amplitude can be determined in two Mercury years (176 days) with an accuracy of or better, which is sufficient to answer the question whether Mercury has a solid or fluid core.     

On the oscillations in Mercury's obliquity

Mercury's capture into the 3:2 spin–orbit resonance can be explained as a result of its chaotic orbital dynamics. One major objective of MESSENGER and BepiColombo spatial missions is to accurately measure Mercury's rotation and its obliquity in order to obtain constraints on internal structure of the planet. Analytical approaches at the first-order level using the Cassini state assumptions give the obliquity constant or quasi-constant. Which is the obliquity's dynamical behavior deriving from a complete spin–orbit motion of Mercury simultaneously integrated with planetary interactions? We have used our SONYR model (acronym of Spin–Orbit N-bodY Relativistic model) integrating the spin–orbit N-body problem applied to the Solar System (Sun and planets). For lack of current accurate observations or ephemerides of Mercury's rotation, and therefore for lack of valid initial conditions for a numerical integration, we have built an original method for finding the libration center of the spin–orbit system and, as a consequence, for avoiding arbitrary amplitudes in librations of the spin–orbit motion as well as in Mercury's obliquity. The method has been carried out in two cases: (1) the spin–orbit motion of Mercury in the 2-body problem case (Sun–Mercury) where an uniform precession of the Keplerian orbital plane is kinematically added at a fixed inclination (S_2K case), (2) the spin–orbit motion of Mercury in the N-body problem case (Sun and planets) (S_n case). We find that the remaining amplitude of the oscillations in the S_n case is one order of magnitude larger than in the S_2K case, namely 4 versus 0.4 arcseconds (peak-to-peak). The mean obliquity is also larger, namely 1.98 versus 1.80 arcminutes, for a difference of 10.8 arcseconds. These theoretical results are in a good agreement with recent radar observations but it is not excluded that it should be possible to push farther the convergence process by drawing nearer still more precisely to the libration center. We note that the dynamically driven spin precession, which occurs when the planetary interactions are included, is more complex than the purely kinematic case. Nevertheless, in such a N-body problem, we find that the 3:2 spin–orbit resonance is really combined to a synchronism where the spin and orbit poles on average precess at the same rate while the orbit inclination and the spin axis orientation on average decrease at the same rate. As a consequence and whether it would turn out that there exists an irreducible minimum of the oscillation amplitude, quasi-periodic oscillations found in Mercury's obliquity should be to geometrically understood as librations related to these synchronisms that both follow a Cassini state. Whatever the open question on the minimal amplitude in the obliquity's oscillations and in spite of the planetary interactions indirectly acting by the solar torque on Mercury's rotation, Mercury remains therefore in a stable equilibrium state that proceeds from a 2-body Cassini state.     

Mercury's capture into the 3/2 spin-orbit resonance including the effect of core-mantle friction

The rotation of Mercury is presently captured in a 3/2 spin-orbit resonance with the orbital mean motion. The capture mechanism is well understood as the result of tidal interactions with the Sun combined with planetary perturbations [Goldreich, P., Peale, S., 1966. Astron. J. 71, 425-438; Correia, A.C.M., Laskar, J., 2004. Nature 429, 848-850]. However, it is now almost certain that Mercury has a liquid core [Margot, J.L., Peale, S.J., Jurgens, R.F., Slade, M.A., Holin, I.V., 2007. Science 316, 710-714] which should induce a contribution of viscous friction at the core-mantle boundary to the spin evolution. According to Peale and Boss [Peale, S.J., Boss, A.P., 1977. J. Geophys. Res. 82, 743-749] this last effect greatly increases the chances of capture in all spin-orbit resonances, being 100% for the 2/1 resonance, and thus preventing the planet from evolving to the presently observed configuration. Here we show that for a given resonance, as the chaotic evolution of Mercury's orbit can drive its eccentricity to very low values during the planet's history, any previous capture can be destabilized whenever the eccentricity becomes lower than a critical value. In our numerical integrations of 1000 orbits of Mercury over 4 Gyr, the spin ends 99.8% of the time captured in a spin-orbit resonance, in particular in one of the following three configurations: 5/2 (22%), 2/1 (32%) and 3/2 (26%). Although the present 3/2 spin-orbit resonance is not the most probable outcome, we also show that the capture probability in this resonance can be increased up to 55% or 73%, if the eccentricity of Mercury in the past has descended below the critical values 0.025 or 0.005, respectively.     

The influence of Mercury’s inner core on its physical libration

As Mercury orbits the Sun, gravitational torques on its equatorial elliptical shape give rise to a planetary libration. The amplitude of Mercury’s libration, as determined from Earth-based radar speckle pattern observations, suggests that only the mantle participates in the motion. This indicates a decoupling between the core and the mantle, and therefore that the outermost part of the core must be fluid. If a solid inner core is present at the center of Mercury, the equatorial elliptical shape of the latter may become misaligned with that of Mercury’s mantle, leading to an internal gravitational torque between the two. If this torque is large, it may participate in the dynamics of Mercury’s libration. The goal of this work is to determine whether Mercury’s observed librations can be used to place constraints on the properties of its inner core. We present a comparison between predicted and observed librations for a range of interior models of Mercury, with various inner core sizes and fluid core densities. We show that a marginally better fit to observations can be achieved for interior models that have an inner core radius larger than 400 km. However, the improvement in fit is small, and it is not possible to draw robust conclusions on the size of Mercury’s inner core on the basis of existing libration data. Nevertheless, our study demonstrates that the influence of the inner core on the libration of Mercury could be detected with a decade worth of accurate observations.     

Potential relations between Caloris basin and Mercury's sodium exosphere

We have developed a 1D thermal model of Mercury's regolith, in order to simulate the heat diffusion in the upper subsurface (first 10 m). We assume in our model that the thermophysical properties of the Hermean regolith are similar to those of the lunar regolith. We apply our thermal model to the Caloris basin which slopes induce distortions of the surface temperature compared to results obtained for a perfect spherical planet. This thermal model is then coupled with a 3D Monte Carlo model of Mercury's sodium exosphere [Leblanc, F., Johnson, R.E., 2003. Icarus 164, 261-281; Leblanc, F., Delcourt, D., Johnson, R.E., 2003b. J. Geophys. Res. 108 (E12), doi:10.1029/2003JE002151/.5136], in order to describe the signatures of Caloris basin on Mercury's sodium exosphere in term of temporal and spatial variabilities. In particular, we find a motion of the maxima of sodium density in the exosphere towards the Northern hemisphere similar to the one observed by Potter et al. [Potter, A.E., Morgan, T.H., Killen, R.M., 1999. Planet. Space Sci., 47, 1441-1449] but did not reproduce the observed change of the emission brightness. The main conclusion of this study is that the Caloris basin-exosphere relations might be observable from the Earth which we hope will motivate new observations of Mercury's exosphere.     

Physical libration of the moon in longitude

The application of modern computing techniques in the study of the physical libration of the Moon in longitude brings into a new perspective this problem that has been debated so much in the past.     

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.     

A nonlinear analysis of the moon's physical libration in longitude

The Euler equations for the forced physical librations of the Moon have already been solved by using a digital computer to perform the semi-literal mathematical manipulations. Very near resonance, the computer solution for the physical libration in longitude is complemented by the solution of the appropriate Duffing equation with a dissipation term. Because of its apparent proximity to a resonant frequency, the term whose argument is 2 - twice the mean angular distance of the Moon's perigee from the ascending node of its orbit - is especially important. Its phase, which soon should be measurable, is related to the Moon's anelasticity. The term's frequency, in units of the sidereal month, increases as the semi-major axis of the Moon's orbit about the Earth increases. Using the Moon's mechanical ellipticity of Koziel and the rate of increase of the semi-major axis of MacDonald, it is estimated that the 2 term will cross the resonant frequency in 130 million years and, if the rate of energy dissipation is sufficiently low, a transient libration will be induced.     

Mercury's sodium exosphere

Mercury's neutral sodium exosphere is simulated using a comprehensive 3D Monte Carlo model following sodium atoms ejected from Mercury's surface by thermal desorption, photon stimulated desorption, micro-meteoroid vaporization and solar wind sputtering. The evolution of the sodium surface density with respect to Mercury's rotation and its motion around the Sun is taken into account by considering enrichment processes due to surface trapping of neutrals and ions and depletion of the sodium available for ejection from the surfaces of grains. The change in the sodium exosphere is calculated during one Mercury year taking into account the variations in the solar radiation pressure, the photo-ionization frequency, the solar wind density, the photon and meteoroid flux intensities, and the surface temperature. Line-of-sight column densities at different phase angles, the supply rate of new sodium, average neutral and ion losses over a Mercury year, surface density distribution and the importance of the different processes of ejection are discussed in this paper. The sodium surface density distribution is found to become significantly nonuniform from day to night sides, from low to high latitudes and from morning to afternoon because of rapid depletion of sodium atoms in the surfaces of grains mainly driven by thermal depletion. The shape of the exosphere, as it would be seen from the Earth, changes drastically with respect to Mercury's heliocentric position. High latitude column density maxima are related to maxima in the sodium surface concentration at high latitudes in Mercury's surface and are not necessarily due to solar wind sputtering. The ratio between the sodium column density on the morning side of Mercury's exosphere and the sodium column density on the afternoon side is consistent with the conclusions of Sprague et al. (1997, Icarus 129, 506-527). The model, which has no fitting parameters, shows surprisingly good agreement with recent observations of Potter et al. (2002, Meteor. Planet. Sci. 8, 3357-3374) successfully explaining their velocity and column density profiles vs. heliocentric distance. Comparison with this data allows us to constrain the supply rate of new sodium atoms to the surface. We also discuss the possible origins of the strong high latitude emissions (Potter and Morgan, 1990, Science 248, 835-838; 1997a, Adv. Space Res. 19, 1571-1576; 1997b, Planet. Space Sci. 45, 95-100; Sprague et al., 1998, Icarus 135, 60-68) and the strong variations of the total content of the sodium exosphere on short (Potter et al., 1999, Planet. Space Sci. 47, 1441-1449) and long time scales (Sprague et al., 1997, Icarus 129, 506-527).     

Monte-Carlo simulation of Mercury's exosphere

Because of its proximity to the Sun and its small size, Mercury has not been able to retain its atmosphere and only a thin exosphere surrounds the planet. The exospheric pressure at the planetary surface is approximately 10⁻¹⁰ mbar, set by the Mariner 10 occultation experiment. The existence of gaseous species H, He, and O has been established by Mariner 10. In addition Na, K, and Ca have been observed by ground based instrumentation. Other elements are expected to be found in Mercury's exosphere since the total pressure of the known species is almost two orders of magnitude less than the exospheric pressure.It is intended to measure these exospheric particle densities in situ with an instrument on board of ESA's BepiColombo Mercury Planetary Orbiter (MPO) spacecraft. Since the expected exospheric densities are very small we developed a Monte-Carlo computer model to investigate if such a measurement is feasible along the MPO spacecraft orbit. We model energy and ejection angle distributions of the particles at the surface, with the emission process determining the actual distribution functions. Our model follows the trajectory of each particle by numerical integration until the particle hits Mercury's surface again or escapes from the calculation domain. Using a large set of these trajectories bulk parameters of the exospheric gas are derived, e.g., particle densities for various atomic and molecular species. Our study suggests that a mass spectrometric measurement is feasible and, at least at MPO's periherm, all species that are released from the surface will be observed.     

Limits to Mercury's magnesium exosphere from MESSENGER second flyby observations

The discovery measurements of Mercury's exospheric magnesium, obtained by the MErcury Surface, Space ENvironment, GEochemistry, and Ranging (MESSENGER) probe during its second Mercury flyby, are modeled to constrain the source and loss processes for this neutral species. Fits to a Chamberlain exosphere reveal that at least two source temperatures are required to reconcile the distribution of magnesium measured far from and near the planet: a hot ejection process at the equivalent temperature of several tens of thousands of degrees K, and a competing, cooler source at temperatures as low as 400 K. For the energetic component, our models indicate that the column abundance that can be attributed to sputtering under constant southward interplanetary magnetic field conditions is at least a factor of five less than the rate dictated by the measurements. Although highly uncertain, this result suggests that another energetic process, such as the rapid dissociation of exospheric MgO, may be the main source of the distant neutral component. If meteoroid and micrometeoroid impacts eject mainly molecules, the total amount of magnesium at altitudes exceeding ∼100 km is found to be consistent with predictions by impact vaporization models for molecule lifetimes of no more than two minutes. Though a sharp increase in emission observed near the dawn terminator region can be reproduced if a single meteoroid enhanced the impact vapor at equatorial dawn, it is much more likely that observations in this region, which probe heights increasingly near the surface, indicate a reservoir of volatile Mg being acted upon by lower-energy source processes.     

Global contraction of planetary bodies due to despinning: Application to Mercury and Iapetus

We extend previous work on the global tectonic patterns generated by despinning with a self-consistent treatment of the isotropic despinning contraction that has been ignored. We provide simple analytic approximations that quantify the effect of the isotropic despinning contraction on the global shape and tectonic pattern. The isotropic despinning contraction of Mercury is ∼93 m (T/1 day)⁻², where T is the initial rotation period. If we take into account both the isotropic contraction and the degree-2 deformations associated with despinning, the preponderance of compressional tectonic features on Mercury’s surface requires an additional isotropic contraction ?1 km (T/1 day)⁻², presumably due to cooling of the interior and growth of the solid inner core. The isotropic despinning contraction of Iapetus is ∼9 m (T/16 h)⁻², and it is not sensitive to the presence of a core or the thickness of the elastic lithosphere. The tectonic pattern expected for despinning, including the isotropic contraction, does not explain Iapetus’ ridge. Furthermore, the ridge remains unexplained with the addition of any isotropic compressional stresses, including those generating by cooling.     

Photometric correction of Mercury's global color mosaic

During the third flyby of Mercury by the MESSENGER spacecraft, a dedicated disk-integrated photometric sequence was acquired with the wide-angle multispectral camera to observe Mercury's global photometric behavior in 11 spectral filters over as broad a range of phase angle as possible within the geometric constraints of the flyby. Extraction of disk-integrated measurements from images acquired during this sequence required careful accounting for scattered light and residual background effects. The photometric model fit to these measurements is shown to fit observed radiances at phase angles below 110°, possibly except where both solar incidence and emission angles are high (>70°). The complexity of the scattered light at wavelengths greater than 828 nm contributes to a less accurate photometric correction at these wavelengths. The model is used to correct the global imaging data set acquired at a variety of geometries to a common geometry of incidence angle=30°, emission angle=0°, and phase angle=30°, yielding a relatively seamless mosaic. The results here will be used to correct image mosaics of Mercury acquired in orbit.     

Mercury radar speckle dynamics

Current data reveal that Mercury is a dynamic system with a core which has not yet solidified completely and is at least partially decoupled from the mantle. Radar speckle displacement experiments have demonstrated that the accuracy in spin-dynamics determination for Earth-like planets can approach 10⁻⁵. The extended analysis of space-time correlation properties of radar echoes shows that the behavior of speckles does not prevent estimation of Mercury’s instantaneous spin-vector components to accuracy of a few parts in 10⁷. This limit can be reached with more powerful radar facilities and leads to constraining the interior in more detail from effects of spin dynamics, e.g., from observation of the core-mantle interplay through high precision monitoring of the 88-day spin-variation of Mercury’s crust.