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.     

Observations of metallic species in Mercury’s exosphere

From observations of the metallic species sodium (Na), potassium (K), and magnesium (Mg) in Mercury’s exosphere, we derive implications for source and loss processes. All metallic species observed exhibit a distribution and/or line width characteristic of high to extreme temperature - tens of thousands of degrees K. The temperatures of refractory species, including magnesium and calcium, indicate that the source process for the atoms observed in the tail and near-planet exosphere are consistent with ion sputtering and/or impact vaporization of a molecule with subsequent dissociation into the atomic form. The extended Mg tail is consistent with a surface abundance of 5-8% Mg by number, if 30% of impact-vaporized Mg remains as MgO and half of the impact vapor condenses. Globally, ion sputtering is not a major source of Mg, but locally the sputtered source can be larger than the impact vapor source. We conclude that the Na and K in Mercury’s exosphere can be derived from a regolith composition similar to that of Luna 16 soil (or Apollo 17 orange glass), in which the abundance by number is 0.0027 (0.0028) for Na and 0.0006 (0.0045) for K.     

Constraints on Mercury’s Na exosphere: Combined MESSENGER and ground-based data

We have used observations of sodium emission obtained with the McMath-Pierce solar telescope and MESSENGER’s Mercury Atmospheric and Surface Composition Spectrometer (MASCS) to constrain models of Mercury’s sodium exosphere. The distribution of sodium in Mercury’s exosphere during the period January 12-15, 2008, was mapped using the McMath-Pierce solar telescope with the 5″ × 5″ image slicer to observe the D-line emission. On January 14, 2008, the Ultraviolet and Visible Spectrometer (UVVS) channel on MASCS sampled the sodium in Mercury’s anti-sunward tail region. We find that the bound exosphere has an equivalent temperature of 900-1200 K, and that this temperature can be achieved if the sodium is ejected either by photon-stimulated desorption (PSD) with a 1200 K Maxwellian velocity distribution, or by thermal accommodation of a hotter source. We were not able to discriminate between the two assumed velocity distributions of the ejected particles for the PSD, but the velocity distributions require different values of the thermal accommodation coefficient and result in different upper limits on impact vaporization. We were able to place a strong constraint on the impact vaporization rate that results in the release of neutral Na atoms with an upper limit of 2.1 × 10⁶ cm⁻² s⁻¹. The variability of the week-long ground-based observations can be explained by variations in the sources, including both PSD and ion-enhanced PSD, as well as possible temporal enhancements in meteoroid vaporization. Knowledge of both dayside and anti-sunward tail morphologies and radiances are necessary to correctly deduce the exospheric source rates, processes, velocity distribution, and surface interaction.     

Impacts as sources of the exosphere on Mercury

Chemical processes associated with meteoroid bombardment of Mercury are considered. Meteoroid impacts lead to production of metal atoms as well as metal oxides and hydroxides in the planetary exosphere. By using quenching theory, the abundances of the main Na-, K-, Ca-, Fe-, Al-, Mg-, Si-, and Ti-containing species delivered to the exosphere during meteoroid impacts were estimated. Based on a correlation between the solar photo rates and the molecular constants of atmospheric diatomic molecules, photolysis lifetimes of metal oxides and SiO are estimated. Meteoroid impacts lead to the formation of hot metal atoms (0.2-0.4 eV) produced directly during impacts and of very hot metal atoms (1-2 eV) produced by the subsequent photolysis of oxides and hydroxides in the exosphere of Mercury. The concentrations of impact-produced atoms of the main elements in the exosphere are estimated relative to the observed concentrations of Ca, assumed to be produced mostly by ion sputtering. Condensation of dust grains can significantly reduce the concentrations of impact-produced atoms in the exosphere. Na, K, and Fe atoms are delivered to the exosphere directly by impacts while Ca, Al, Mg, Si, and Ti atoms are produced by the photolysis of their oxides and hydroxides. The chemistry of volatile elements such as H, S, C, and N during meteoroid bombardment is also considered. Our conclusions about the temperature and the concentrations of impact-produced atoms in the exosphere of Mercury may be checked by the Messenger spacecraft in the near future and by BepiColombo spacecraft some years later.     

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.     

Monte Carlo modeling of sodium in Mercury’s exosphere during the first two MESSENGER flybys

We present a Monte Carlo model of the distribution of neutral sodium in Mercury’s exosphere and tail using data from the Mercury Atmospheric and Surface Composition Spectrometer (MASCS) on the MErcury Surface, Space ENvironment, GEochemistry, and Ranging (MESSENGER) spacecraft during the first two flybys of the planet in January and September 2008. We show that the dominant source mechanism for ejecting sodium from the surface is photon-stimulated desorption (PSD) and that the desorption rate is limited by the diffusion rate of sodium from the interior of grains in the regolith to the topmost few monolayers where PSD is effective. In the absence of ion precipitation, we find that the sodium source rate is limited to ∼10⁶-10⁷ cm⁻² s⁻¹, depending on the sticking efficiency of exospheric sodium that returns to the surface. The diffusion rate must be at least a factor of 5 higher in regions of ion precipitation to explain the MASCS observations during the second MESSENGER flyby. We estimate that impact vaporization of micrometeoroids may provide up to 15% of the total sodium source rate in the regions observed. Although sputtering by precipitating ions was found not to be a significant source of sodium during the MESSENGER flybys, ion precipitation is responsible for increasing the source rate at high latitudes through ion-enhanced diffusion.     

Spatial variations of the sodium/potassium ratio in Mercury’s exosphere uncovered by high-resolution spectroscopy

High-resolution spectroscopy of Mercury has been obtained with two different instruments in 2006: the EMMI instrument at the 3.6-m NTT telescope of ESO La Silla Chile and the ESPADON spectrograph at the 3.6-m CFHT telescope on top of Mauna Kea (Hawaii). The disk of the planet has been scanned for spatial variation of the exospheric species. The large spectral range and high resolution allow simultaneous measurements of the integrated column density of Na and K.We measure Na/K ratio between 80 and 400 with values between 60 and 90 when the telescope was pointed towards the subsolar region of Mercury’s disk and much larger value when we looked to other part of the exosphere. Moreover, we observed that the Na and K exospheres display very different spatial distributions. Even if these two species are probably ejected with very similar mechanisms from the surface, their differences in mass and sensitivity to solar pressure acceleration imply very different behavior in Mercury’s exosphere.     

Sodium winds on Mercury

Solar radiation acceleration imparts anti-sunward velocities to sodium atoms in the Mercury exosphere. The Earthward-directed vectors of the Sun-accelerated atom velocities can be observed from Earth as small Doppler shifts, either added to, or subtracted from the Earth-Mercury Doppler shifts. We measured these small Doppler shifts using high resolution spectrographs capable of detecting sodium velocity differences as small as 0.1 km/s. We report here four sets of observations performed at different Mercury true anomaly angles. For these measurements, the spectrograph slit was oriented first east-west, and then north-south on the planet so as to get east-west and north-south transects of the velocities. The velocity patterns in east-west transects could be explained in terms of sodium flows outwards from the subsolar point, except for unexpectedly large Earthward velocities observed above the dawn terminator, which we interpreted to be the result of evaporation of sodium as the cold surface is heated by the rising Sun. North-south transects also showed a general pattern consistent with sodium flows outwards from the subsolar point. However, in all cases, the velocities were higher in one hemisphere relative to the other. For two cases, excess sodium emission was observed in the same hemisphere as the velocity excess. We interpreted these results to mean that there existed sources of sodium at high latitudes, which could appear in either hemisphere.     

Orbital effects on Mercury’s escaping sodium exosphere

We present results from coronagraphic imaging of Mercury’s sodium tail over a 7° field of view. Several sets of observations made at the McDonald Observatory since May 2007 show a tail of neutral sodium atoms stretching more than 1000 Mercury radii (R_m) in length, or a full degree of sky. However, no tail was observed extending beyond 120 R_m during the January 2008 MESSENGER fly-by period, or during a similar orbital phase of Mercury in July 2008. Large changes in Mercury’s heliocentric radial velocity cause Doppler shifts about the Fraunhofer absorption features; the resultant change in solar flux and radiation pressure is the primary cause of the observed variation in tail brightness. Smaller fluctuations in brightness may exist due to changing source rates at the surface, but we have no explicit evidence for such changes in this data set. The effects of radiation pressure on Mercury’s escaping atmosphere are investigated using seven observations spanning different orbital phases. Total escape rates of atmospheric sodium are estimated to be between 5 and 13 × 10²³ atoms/s and show a correlation to radiation pressure. Candidate sources of Mercury’s sodium exosphere include desorption by UV sunlight, thermal desorption, solar wind channeled along Mercury’s magnetic field lines, and micro-meteor impacts. Wide-angle observations of the full extent of Mercury’s sodium tail offer opportunities to enhance our understanding of the time histories of these source rates.     

Source dependency of exospheric sodium on Mercury

Due to a large solar radiation effect, the sodium exosphere exhibits many interesting effects, including the formation of an extended corona and a tail-like structure. The current suite of observations allows us to study some physical properties of the sodium exosphere, such as the source rates and the interaction with the surface, both experimentally and theoretically. In order to quantify the complex variations in the sodium exosphere in more detail, we use an exospheric model with the Monte-Carlo method to examine the surface interactions of a sodium atom, including the surface thermal accommodation rate and the sticking coefficient. The source rates from different components, such as the photon stimulated desorption (PSD), the meteoroid impact vaporization (MIV), and the solar wind ion sputtering (IS), can be constrained by comparing our exospheric model calculations with the published observational data. The detected terminator to limb (TL) ratio on the disk and the tail production rate can be explained with no sticking effect and small thermal accommodation rates. We also examine the best fit of the MIV source evolution, through comparison with the disk-averaged emission. The resultant discrepancy between the observations and the model fit may reflect the surface variation in the sodium abundance. A comprehensive mapping of the surface geochemical composition of the surface by the MESSENGER and Bepi-Colombo missions should give us more information about the nature of this surface-bound exosphere.     

Mercury and Moon He exospheres: Analysis and modeling

Helium is one of the first elements clearly identified in the lunar exosphere (Hoffman, J.H., Hodges, R.R., Johnson, F.S., Evans, D.E. [1973]. Proc. Lunar Sci. Conf. 3, 2865–2875). Apollo 17 measured the He density at the surface during four lunations. It confirmed the expected day to night asymmetry of the He exosphere with a maximum density near the dawn terminator on the nightside. Few years later, the first detection of Mercury’s He exosphere was successfully obtained by Mariner 10 (Broadfoot, A.L., Shemansky, D.E., Kumar, S. [1976]. Geophys. Res. Lett. 3, 577–580). These observations highlighted similar global distribution of the He exosphere at Mercury and at the Moon, but also significant differences that have never been convincingly explained.In this paper, we model the He exosphere at the Moon and Mercury with the same approach. The energy accommodation of the exospheric He particles interacting with the surface can be roughly constrained using Apollo 17 and Mariner 10 measurements. Neither a low energy accommodation, as suggested by Shemansky and Broadfoot (Shemansky, D.E., Broadfoot, A.L. [1977]. Rev. Geophys. 15, 491–499), nor a full energy accommodation, as suggested by Hodges (Hodges Jr., R.R. [1975]. The Moon, 14, 139–157), can fit all the observations. These observations and their modeling suggest a diurnal variation of the energy distribution of the He ejected from the surface that cannot be explained satisfactorily by any of the present theories on the gas–surface interaction in surface-bounded exospheres.     

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).     

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.     

First observations of the Na exosphere of Mercury with the high-resolution spectrograph of the 3.5M Telescopio Nazionale Galileo

Analysis of three spectra of the exosphere of Mercury in the Na-D lines are presented. Spectra were secured with the high-resolution spectrograph (SARG) of the 3.5 M Telescopio Nazionale Galileo (TNG, located on the Roque de los Muchachos, Canaries) on the evenings of August 23 and 24, 2002. Spectra have resolution ; the slit length was 26.7”. The Na column abundances range from 4.3 to with the highest abundances being close to the illuminated limb. Our observations at true anomaly angles (TAA) from 171°to 174°show the traces of the emission lines to be strongly peaked at the illuminated limb, supportive of recent modeling that shows thermal desorption to be a strong factor in determining the distribution of Na about the planet.     

The variability of Mercury's exosphere by particle and radiation induced surface release processes

Mercury's close orbit around the Sun, its weak intrinsic magnetic field and the absence of an atmosphere (P_surface<1×10⁻⁸ Pa) results in a strong direct exposure of the surface to energetic ions, electrons and UV radiation. Thermal processes and particle-surface-collisions dominate the surface interaction processes leading to surface chemistry and physics, including the formation of an exosphere (N?10¹⁴ cm⁻²) in which gravity is the dominant force affecting the trajectories of exospheric atoms. NASA's Mariner 10 spacecraft observed the existence of H, He, and O in Mercury's exosphere. In addition, the volatile components Na, K, and Ca have been observed by ground based instrumentation in the exosphere. We study the efficiency of several particle surface release processes by calculating stopping cross-sections, sputter yields and exospheric source rates. Our study indicates surface sputter yields for Na between values of about 0.27 and 0.35 in an energy range from 500 eV up to 2 keV if Na⁺ ions are the sputter agents, and about 0.037 and 0.082 at an energy range between 500 eV up to 2 keV when H⁺ are the sputter agents and a surface binding energy of about 2 eV to 2.65 eV. The sputter yields for Ca are about 0.032 to 0.06 and for K atoms between 0.054 to 0.1 in the same energy range. We found a sputter yield for O atoms between 0.025 and 0.04 for a particle energy range between 500 eV up to 2 keV protons. By taking the average solar wind proton surface flux at the open magnetic field line area of about 4×10⁸ cm⁻² s⁻¹ calculated by Massetti et al. (2003, Icarus, in press) the resulting average sputtering flux for O is about 0.8-1.0×10⁷ cm⁻² s⁻¹ and for Na approximately 1.3-1.6×10⁵ cm⁻² s⁻¹ depending on the assumed Na binding energies, regolith content, sputtering agents and solar activity. By using lunar regolith values for K we obtain a sputtering flux of about 1.0-1.4×10⁴ cm⁻² s⁻¹. By taking an average open magnetic field line area of about 2.8×10¹⁶ cm² modelled by Massetti et al. (2003, Icarus, in press) we derive an average surface sputter rate for Na of about 4.2×10²¹ s⁻¹ and for O of about 2.5×10²³ s⁻¹. The particle sputter rate for K atoms is about 3.0×10²⁰ s⁻¹ assuming lunar regolith composition for K. The sputter rates depend on the particle content in the regolith and the open magnetic field line area on Mercury's surface. Further, the surface layer could be depleted in alkali. A UV model has been developed to yield the surface UV irradiance at any time and latitude over a Mercury year. Seasonal and diurnal variations are calculated, and Photon Stimulated Desorption (PSD) fluxes along Mercury's orbit are evaluated. A solar UV hotspot is created towards perihelion, with significant average PSD particle release rates and Na fluxes of about 3.0×10⁶ cm⁻² s⁻¹. The average source rates for Na particles released by PSD are about 1×10²⁴ s⁻¹. By using the laboratory obtained data of Madey et al. (1998, J. Geophys. Res. 103, 5873-5887) for the calculation of the PSD flux of K atoms we get fluxes in the order of about 10⁴ cm⁻² s⁻¹ along Mercury's orbit. However, these values may be to high since they are based on idealized smooth surface conditions in the laboratory and do not include the roughness and porosity of Mercury's regolith. Further, the lack of an ionosphere and Mercury's small, temporally and spatially highly variable magnetosphere can result in a large and rapid increase of exospheric particles, especially Na in Mercury's exosphere. Our study suggests that the average total source rates for the exosphere from solar particle and radiation induced surface processes during quiet solar conditions may be of the same order as particles produced by micrometeoroid vaporization. We also discuss the capability of in situ measurements of Mercury's highly variable particle environment by the proposed NPA-SERENA instrument package on board ESA's BepiColombo Mercury Planetary Orbiter (MPO).     

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.     

Rapid changes in the sodium exosphere of Mercury

We imaged Mercury in sodium D₁ and D₂ emission for 6 days during the period 13–20 November 1997 using a 10×10-arc s aperture image slicer coupled to a high-resolution spectrograph. We corrected the sodium images for smearing by the terrestrial atmosphere by computing the actual seeing function from surface reflection images, and used this function to correct the sodium images. During the period of observation, large daily changes took place in both the total amount of sodium and its distribution over the planet. Total sodium increased by a factor of about 3 during this period. The sodium emission was brightest at longitudes near the subsolar longitude in the range 130–150°, with excess sodium at northern latitudes on some days, and excess sodium at southern latitudes on other days. There are no obviously outstanding geologic features at this longitude. The rapid changes observed during this period suggest a connection with solar activity, since the planet itself is apparently geologically inactive. The F10.7 cm solar flux during this period varied only slightly, with an increase of about 15%, probably insufficient to account for the observed changes. However, there were a number of coronal mass ejection (CME) events, some of which were directed towards the general area of Mercury. We suggest that the changes in the visible neutral sodium atmosphere might be a result of the effect of CMEs on Mercury.     

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.