Librations and obliquity of Mercury from the BepiColombo radio-science and camera experiments

A major goal of the BepiColombo mission to Mercury is the determination of the structure and state of Mercury's interior. Here the BepiColombo rotation experiment has been simulated in order to assess the ability to attain the mission goals and to help lay out a series of constraints on the experiment's possible progress. In the rotation experiment pairs of images of identical surface regions taken at different epochs are used to retrieve information on Mercury's rotation and orientation. The idea is that from observations of the same patch of Mercury's surface at two different solar longitudes of Mercury the orientation of Mercury can be determined, and therefore also the obliquity and rotation variations with respect to the uniform rotation.The estimation of the libration amplitude and obliquity through pattern matching of observed surface landmarks is challenging. The main problem arises from the difficulty to observe the same landmark on the planetary surface repeatedly over the MPO mission lifetime, due to the combination of Mercury's 3:2 spin-orbit resonance, the absence of a drift of the MPO polar orbital plane and the need to combine data from different instruments with their own measurement restrictions.By assuming that Mercury occupies a Cassini state and that the spacecraft operates nominally we show that under worst case assumptions the annual libration amplitude and obliquity can be measured with a precision of, respectively, 1.4 arcseconds (as) and 1.0 as over the nominal BepiColombo MPO lifetime with about 25 landmarks for rather stringent illumination restrictions. The outcome of the experiment cannot be easily improved by simply relaxing the observational constraints, or increasing the data volume.     

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 MESSENGER mission to Mercury: scientific objectives and implementation

Mercury holds answers to several critical questions regarding the formation and evolution of the terrestrial planets. These questions include the origin of Mercury's anomalously high ratio of metal to silicate and its implications for planetary accretion processes, the nature of Mercury's geological evolution and interior cooling history, the mechanism of global magnetic field generation, the state of Mercury's core, and the processes controlling volatile species in Mercury's polar deposits, exosphere, and magnetosphere. The MErcury Surface, Space ENvironment, GEochemistry, and Ranging (MESSENGER) mission has been designed to fly by and orbit Mercury to address all of these key questions. After launch by a Delta 2925H-9.5, two flybys of Venus, and two flybys of Mercury, orbit insertion is accomplished at the third Mercury encounter. The instrument payload includes a dual imaging system for wide and narrow fields-of-view, monochrome and color imaging, and stereo; X-ray and combined gamma-ray and neutron spectrometers for surface chemical mapping; a magnetometer; a laser altimeter; a combined ultraviolet–visible and visible-near-infrared spectrometer to survey both exospheric species and surface mineralogy; and an energetic particle and plasma spectrometer to sample charged species in the magnetosphere. During the flybys of Mercury, regions unexplored by Mariner 10 will be seen for the first time, and new data will be gathered on Mercury's exosphere, magnetosphere, and surface composition. During the orbital phase of the mission, one Earth year in duration, MESSENGER will complete global mapping and the detailed characterization of the exosphere, magnetosphere, surface, and interior.     

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.     

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.     

Analytical Development of Rigid Mercury Nutation Series

In this paper, series of a rigid model of Mercury nutations are computed. The method used is based on the calculation of the forces produced by the Sun on Mercury as considered as a rigid body. In order to take into account the indirect effects coming from the orbit perturbations of Mercury, we used the ephemerides VSOP87 (Bretagnon and Francou, 1988). Due to non-negligible difference between the principal moment of inertia A and B in the case of Mercury, we compute also terms due to the triaxiality in addition to the general terms coming from J ₂. With a truncation level of 10 ^–3 mas (milliarcsecond), related to the present-day precision of the Mercury precession constant, 173 terms in longitude ( sin ) and 166 terms in obliquity () are computed. The value of the dynamical flattening used is H _D = (C – A)/C = 2.3 × 10^–4 (Anderson, 1987).     

A note on the instantaneous rotational velocity of Mercury

The fluctuation in the angular velocity of the present rotation of Mercury is investigated. The instantaneous rotational rate in terms of orbital mean motion at different positions along Mercury's orbit is given. At aphelion the rotational velocity decreases substantially because the solar gravitational torque on the two thermal bulges on Mercury's surface tends to retard the rotation of Mercury before aphelion passage. It is found that the difference between the rotational periods derived from the motions at perihelion and aphelion is 4.68 min and that the maximum rate of rotation occurs atf=/4 andf=7/4, wheref is the true anomaly.     

Obliquity evolution of extrasolar terrestrial planets

We have investigated the obliquity evolution of terrestrial planets in habitable zones (at ∼1 AU) in extrasolar planetary systems, due to tidal interactions with their satellite and host star with wide varieties of satellite-to-planet mass ratio (m/Mp) and initial obliquity (γ₀), through numerical calculations and analytical arguments. The obliquity, the angle between planetary spin axis and its orbit normal, of a terrestrial planet is one of the key factors in determining the planetary surface environments. A recent scenario of terrestrial planet accretion implies that giant impacts of Mars-sized or larger bodies determine the planetary spin and form satellites. Since the giant impacts would be isotropic, tilted spins (sinγ₀∼1) are more likely to be produced than straight ones (sinγ₀∼0). The ratio m/Mp is dependent on the impact parameters and impactors' mass. However, most of previous studies on tidal evolution of the planet-satellite systems have focused on a particular case of the Earth-Moon systems in which m/Mp?0.0125 and γ₀∼10° or the two-body planar problem in which γ₀=0° and stellar torque is neglected. We numerically integrated the evolution of planetary spin and a satellite orbit with various m/Mp (from 0.0025 to 0.05) and γ₀ (from 0° to 180°), taking into account the stellar torques and precessional motions of the spin and the orbit. We start with the spin axis that almost coincides with the satellite orbit normal, assuming that the spin and the satellite are formed by one dominant impact. With initially straight spins, the evolution is similar to that of the Earth-Moon system. The satellite monotonically recedes from the planet until synchronous state between the spin period and the satellite orbital period is realized. The obliquity gradually increases initially but it starts decreasing down to zero as approaching the synchronous state. However, we have found that the evolution with initially tiled spins is completely different. The satellite's orbit migrates outward with almost constant obliquity until the orbit reaches the critical radius ∼10-20 planetary radii, but then the migration is reversed to inward one. At the reversal, the obliquity starts oscillation with large amplitude. The oscillation gradually ceases and the obliquity is reduced to ∼0° during the inward migration. The satellite eventually falls onto the planetary surface or it is captured at the synchronous state at several planetary radii. We found that the character change of precession about total angular momentum vector into that about the planetary orbit normal is responsible for the oscillation with large amplitude and the reversal of migration. With the results of numerical integration and analytical arguments, we divided the m/Mp-γ₀ space into the regions of the qualitatively different evolution. The peculiar tidal evolution with initially tiled spins give deep insights into dynamics of extrasolar planet-satellite systems and discussions of surface environments of the planets.     

On the figure of the planet Mercury

The figure of Mercury is estimated in terms of an isostatic form of equilibrium which tends to be controlled by the situation near perihelion passage at the 32 resonance spin rate. The ratios of the principal moments of inertia for Mercury are: (1)(C–A)/C7×10^–5; (2)(C–B)/C5×10^–5 and (3)(B–A)/C2×10^–5. The thermal effect on Mercury's figure during solidification forces Mercury's rotation to be trapped in the 32 resonance lock as its spin rate is being slowed by tidal effects. It is shown that the process of trapping of Mercury has been naturally affected by the instantaneous solidification of Mercury into a shape with two thermal bulges, and that the two permanent thermal bulges stabilize the planet's rotation.     

Dynamical Evolution of Planets in Disks

We study the evolution of a system consisting of two protoplanets still embedded in a protoplanetary disk. Results of two different numerical approaches are presented. In the first kind of model the motion of the disk material is followed by fully viscous hydrodynamical simulations, and the planetary motion is determined by N-body calculations including exactly the gravitational potential from the disk material. In the second kind we only solve the N-body part and add additional analytically given forces which model the effect of the torques of the disk. This type of modeling is of course orders of magnitudes faster than the full hydro-model. Another advantage of this two-fold approach is the possibility of adjusting the otherwise unknown parameters of the simplified model.The results give very good agreement between the methods. Using two different initial setups for the planets and disk, we obtain in the first case a resonant trapping into the 3:1 resonance, and in the second case a trapping into the 2:1 resonance. Resonant capture leads to a rise in the eccentricity and to an alignment of the spatial orientation of orbits. The characteristics of the numerical results agree very favorably with those of three observed planetary systems (GJ 876, HD 82943, and 55 Cnc) known to be in mean motion resonances.     

Planetary long periodic terms in Mercury’s rotation: a two dimensional adiabatic approach

The averaged spin-orbit resonant motion of Mercury is considered, with e the orbital eccentricity, and i _o the orbital inclination introduced as very slow functions of time, given by any secular planetary theory. The basis is our Hamiltonian approach (D’Hoedt, S., Lemaître, A.: Celest. Mech. Dyn. Astron. 89:267–283, 2004) in which Mercury is considered as a rigid body. The model is based on two degrees of freedom; the first one is linked to the 3:2 resonant spin-orbit motion, and the second one to the commensurability of the rotational and orbital nodes. Mercury is assumed to be very close to the Cassini equilibrium of the model. To follow the motion of rotation close to this equilibrium, which varies with respect to time through e and i _o , we use the adiabatic invariant theory, extended to two degrees of freedom. We calculate the corrections (remaining functions) introduced by the time dependence of e and i _o in the three steps necessary to characterize the frequencies at the equilibrium. The conclusion is that Mercury follows the Cassini equilibrium (stays in the Cassini forced state), in an adiabatic behavior: the area around the equilibrium does not change by more than ${\varepsilon}$ for times smaller than ${\frac{1}{\varepsilon}}$ . The role of the inclination and the eccentricity can be dissociated and measured in each step of the canonical transformation.     

A heat flow and physical properties package for the surface of Mercury

European Space Agencies fifth cornerstone mission BepiColombo includes a ‘Surface Element’ to land a scientific payload on the surface of Mercury. The current strawman payload includes a heat flow and physical properties package (HP³), focussing on key thermal and mechanical properties of the near-surface material (down to a depth of 2–5 m) and the measurement of heat flow from Mercury's interior, an important constraining parameter for models of the planet's interior and evolution. We present here an overview of the HP³ experiment package and its possible accommodation in a self-inserting ‘mole’ device. A mole is considered to be the most appropriate deployment method for HP³, at least in the currently-assumed case of an airbag-assisted soft landing architecture for the Mercury Surface Element.     

The existence of families of periodic orbits in theN-body problem

It is proved that monoparametric families of periodic orbits of theN-body problem in the plane, for fixed values of all masses, exist in a rotating frame of reference whosex axis contains always two of the bodiesP ₁ andP ₂. A periodic motion of theN-body problem is obtained as a continuation ofN–2 symmetric periodic orbits of the circular restricted three-body problem whose periods are in integer dependence, by increasing the masses of the originallyN–2 massless bodiesP ₃, ...,P _k. The analytic continuation, for sufficiently small values of theN–2 bodiesP ₃ ...P _k and finite values for the masses ofP ₁ andP ₂ has been proved by the continuation method and the solution itself has been found explicitly to a linear approximation in the small masses. Also, the possible application of the above periodic orbits to the study of the Solar system and of stellar systems is mentioned.     

Global MHD modeling of Mercury's magnetosphere with applications to the MESSENGER mission and dynamo theory

We use a global magnetohydrodynamic (MHD) model to simulate Mercury's space environment for several solar wind and interplanetary magnetic field (IMF) conditions in anticipation of the magnetic field measurements by the MESSENGER spacecraft. The main goal of our study is to assess what characteristics of the internally generated field of Mercury can be inferred from the MESSENGER observations, and to what extent they will be able to constrain various models of Mercury's magnetic field generation. Based on the results of our simulations, we argue that it should be possible to infer not only the dipole component, but also the quadrupole and possibly even higher harmonics of the Mercury's planetary magnetic field. We furthermore expect that some of the crucial measurements for specifying the Hermean internal field will be acquired during the initial fly-bys of the planet, before MESSENGER goes into orbit around Mercury.     

Regularization of Motion Equations with L-Transformation and Numerical Integration of the Regular Equations

The sets of L-matrices of the second, fourth and eighth orders are constructed axiomatically. The defining relations are taken from the regularization of motion equations for Keplerian problem. In particular, the Levi-Civita matrix and KS-matrix are L-matrices of second and fourth order, respectively. A theorem on the ranks of L-transformations of different orders is proved. The notion of L-similarity transformation is introduced, certain sets of L-matrices are constructed, and their classification is given. An application of fourth order L-matrices for N-body problem regularization is given. A method of correction for regular coordinates in the Runge–Kutta–Fehlberg integration method for regular motion equations of a perturbed two-body problem is suggested. Comparison is given for the results of numerical integration in the problem of defining the orbit of a satellite, with and without the above correction method. The comparison is carried out with respect to the number of calls to the subroutine evaluating the perturbational accelerations vector. The results of integration using the correction turn out to be in a favorable position.     

Properties and Peculiarities of Mercury's Regolith. Disk-integrated Polarimetry in 2000–2002

On the basis of the data from ground-based polarimetric, photometric, and other observations, as well as from space measurements (Mariner 10), we survey the investigations of the properties and peculiarities of Mercury's regolith in detail. We also present the results of our own observations performed during three apparitions of the planet in 2000–2002. An analysis of the published data points to essentially more intensive maturation processes in the Hermean surface regolith compared to that on the lunar surface. In addition, the orbital characteristics of Mercury allow us to suppose that the intensity of its regolith maturation and, therefore, the optical properties of its surface can noticeably depend on the planetocentric longitude. Polarimetric observations of Mercury's surface (the planetocentric longitude range was 265°–330°) carried out in 2000–2002 with a 70-cm reflector actually detected a polarization degree varying with an amplitude of about 1.5%. To ascertain the nature of these variations, additional observations of Mercury in a maximally wide range of planetocentric longitudes of the viewed surface are required.     

The Spin-Orbit Resonant Rotation of Mercury: A Two Degree of Freedom Hamiltonian Model

The paper develops a hamiltonian formulation describing the coupled orbital and spin motions of a rigid Mercury rotation about its axis of maximum moment of inertia in the frame of a 3:2 spin orbit resonance; the (ecliptic) obliquity is not constant, the gravitational potential of mercury is developed up to the second degree terms (the only ones for which an approximate numerical value can be given) and is reduced to a two degree of freedom model in the absence of planetary perturbations. Four equilibria can be calculated, corresponding to four different values of the (ecliptic) obliquity. The present situation of Mercury corresponds to one of them, which is proved to be stable. We introduce action-angle variables in the neighborhood of this stable equilibrium, by several successive canonical transformations, so to get two constant frequencies, the first one for the free spin-orbit libration, the other one for the 1:1 resonant precession of both nodes (orbital and rotational) on the ecliptic plane. The numerical values obtained by this simplified model are in perfect agreement with those obtained by Rambaux and Bois [Astron. Astrophys. 413, 381–393]. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

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.     

N-body simulation of giant molecular clouds in a galaxy

We madeN-body simulation of Giant Molecular Clouds in a model galaxy and confirmed that their random motion is generated and accelerated by purely gravitational interaction among them in some 10^7–8 yr. The results of the present simulation agree well with those of the previous theoretical calculation by Fukunaga (1984a, b).Paper presented at the IAU Third Asian-Pacific Regional Meeting, held in Kyoto, Japan, between 30 September–6 October, 1984.