Rotation, libration, and gravitational field of Mercury Véronique Dehant, Tim Van Hoolst, Pascal Rosenblatt, Mikael Beuthe, Nicolas Rambaux, Severine Rosat, Marie Yseboodt, Gregor Pfyffer Royal Observatory of Belgium, Brussels Anne Lemaître, Jacques Henrard, Sandrine d’Hoedt, Nicolas Rambaux, Julien Dufey Facultés Universitaire Notre Dame de la Paix, Namur We acknowledge PRODEX support/Belspo/ESA

Rotation and libration of Mercury

Rotation of the terrestrial planets MercuryVenusEarthMars

MERCURY: Spin/orbit coupling; 3:2 resonance

What are librations? Revolution (orbit) 87,98 jours Rotation (spin) 58,64 days Solar torque 3:2 spin-orbit resonance

Notation Moment of inertia from gravitational harmonics C B A Mariner10 values: C 22 = 1  and C/Mr 2 =

Torque (2) The z-component of the Liouville equations for a solid Mercury: where r = distance Mercury-Sun  = angle between Sun and A C B A Sun 

Effect of the core on the libration of Mercury Peale (1976): amplitude of the longitude 88-day libration is at least twice as large if the core is decoupled from the mantle (liquid). Solid core Liquid core

Impact of the core on the angle of libration in longitude of 88 days from SONYR model Cm/C ~1 ; solid core Cm/C ~0.5 ; liquid core 40 as 20 as

Earth Tracking Photographic measurements Orientation wrt the stars

13 « Revisiting » a same place Only very high latitudes have a very frequent « flyover » rate But lower latitude measurments contain more information -> « Ideal » strategy? 30 km 10 km

14 Track of the BC NAC (narrow angle camera) on Mercury 30km 10 km BepiColombo narrow angle camera groundtrack, in the case of the nominal orbit. At low altitudes two subsequent tracks do not cover the whole area between them.

15 Opposite side of the planet This represents the tracks on the opposite side of the planet of the preceding slide. At high altitudes two subsequent tracks do cover the whole area between them.

16 Possible observations of the surface Excentric polar orbit (alt. 400 – 1500 km) Periherm moving towards north pole (~16 ° in 200 days) Illumination conditions heavily constrain the possible observations Albedo features are best candidates for observation To correctly observe their patterns very low (less than 10°) or very high (more than 70°) phase angles are not permitted

17 Peale’s procedure We can determine the state of the core of Mercury through the measurement of the gravitational field, the obliquity and the libration. (Peale, 1976) ~ A γ -1 Gravitational field and obliquity (+ Cassini state equation)

Libration observation from Earth- based radar measurements Radar echoes from solid planets are speckled. Wavefront corrugations tied to Mercury’s rotation. 1sec telescope on Earth The time delay for the pattern to reproduce at both stations is a direct measure of the rotation rate.

Earth-based libration observing strategy Illuminate Mercury with monochromatic radio signal from Goldstone radar ( =3.5 cm) during ~10 minutes round-trip light time. Record echoes at Goldstone and at the Green Bank Telescopes for ~10 minutes. Perform cross-correlations between amplitude fluctuations recorded at both telescopes.

Principle of Earth-based measurements of libration..

Peale experiment Objective: obtain the ratio of the moment of inertia of the solid part of the planet to the moment of inertia of the whole planet: C m /C: And from a relation between the obliquity, the mass and moments of inertia of Mercury, either using a numerical integration or the mean Cassini state. amplitude of libration from amplitudes of J 2 and C 22 from amplitudes of J 2 and C 22 amplitude of mean obliquity  Mr 2 /C

Cassini State (Cassini 1693; Colombo 1966; Peale 1969) (i) Rotation rate is synchronous/commensurate with the orbital mean motion (ii) The angle between the spin axis and the normal to the orbital plane remains constant (iii) The spin axis, the normal to the orbital plane and the normal to the Laplace plane are always coplanar k orb s Ι= 8.6°  =2'=obliquity k can be determined from ephemerides; orb can be determined by ephemerides; s can be determined from radar observations (2.1’). also s can be determined from analytical approach (1.6’)

Laplace Plane Laplace plane: reference plane about which the axis the orbit is precessing due to the planetary perturbations We need a Laplace plane in order to compute the position of the Cassini equilibrium. i 0 = 7 o I’ = 8.6 o

Internal structure of Mercury Parameters: - Inner core radius; - Sulfur concentration. Values for Mercury interior structure (MIS) model a.

liquidus solidus Adjustment of the liquid core and solid inner core densities eutectic solidus 0%FeS 100%FeS 100%Fe 0%Fe eutectic liquidus 0%FeS 100%FeS 100%Fe 0%Fe - After the eutectic point is reached, the inner core grows by solidification (freezing) of the liquid outer core and thus the newly formed outer layers have the same concentration in light element as the remaining liquid core. The growth of the inner core is modeled as follows: - At the beginning, the inner core is created by precipitation of iron contained in the liquid core and thus has the density of pure solid iron; solid liquid temperature solid liquid temperature

Adjustment of the liquid core and solid inner core densities

Liquid core cases Solid core case Impact of the Sulfur concentration on the librations 19 as 3.2 as

Impact of the Sulfur concentration on the librations 19 as 3.2 as % light element increases  Radius  of the core increases  Core moment of inertia increases  Mantle moment of inertia decreases  libration amplitude increases

Remaining questions (1) Is the present obliquity (  ) = mean obliquity (  )? (What is the contamination of the free precession to the obliquity?)  Theoretical value for mean obliquity (importance of theory), observation for present obliquity (observation by radar and camera experiment with BC [SIMBIO-SYS, MORE, startracker]). What is the value of the obliquity?  Ephemerides value for orbital plane position (importance of ephemerides), observation for spin axis position (observation by radar and camera experiment with BC [SIMBIO-SYS, MORE, startracker]). Is Mercury in the Cassini state/equilibrium?  Ephemerides value for invariant plane position (importance of ephemerides), Ephemerides value for orbital plane position (importance of ephemerides), observation for spin axis position (observation by radar and camera experiment with BC [SIMBIO-SYS, MORE, startracker]).

Remaining questions (2) What is the value of the libration amplitude?  observation of libration angle (observation by radar and camera experiment with BC [SIMBIO-SYS, MORE, startracker]). What is the contamination of the 88-day libration from free libration?  observation of libration angle (observation by radar and camera experiment with BC [SIMBIO-SYS, MORE, startracker]). What are the value of the gravity coefficients?  Gravity observation with BC [MORE]).

Gravity of Mercury

Different types of loading Surface loading Internal loading Necessary if high gravity signal but small topography Good model if gravity and topography correlate well

Flexure model

Global admittance analysis Admittance: C l depends on the rigidity of the lithosphere, C l = 1 for rigidity=0, perfect compensation, isostasy = 0 for an infinite flexural rigidity, no compensation Fit C l to observations to extract global rigidity gravity anomaly ~ internal mass load Crustal density density jump  m -  c topography degree of compensation

Gravity scientific performances; noise level on Doppler data BepiColombo Contributions to the spacecraft velocity from given spherical harmonic of the gravity field for the BepiColombo orbiter. Log 10 (velocity in mm/s)

Gravity scientific performances; noise level on Doppler data Messenger Contributions to the spacecraft velocity from given spherical harmonic of the gravity field for the Messenger orbiter. Log 10 (velocity in mm/s)

Gravity field determination from Doppler tracking data

Gravity Gravity field follows Kaula law: c/l 2 where l is the degree of the gravity coefficient and c is a constant. c is a scaling which depends on the planet. If one considers that terrestrial planets support stresses scaled by a factor g, the gravity anomalies are scaled by 1/g, and as the gravity coefficient are scaled by GM/r, one has a general scaling of 1/g/(GM/r). In the literature (Kaula, 1993, Vincent & Bender, 1990, Wu et al., 1995…), one finds a scaling of 1/g 2. In Milani et al. (2001) and in Garcia et al. (2004), one finds a scaling of 1/g.

PlanetScaling 1/g/(GM/r) Scaling 1/g 2 Scaling 1/g Earthc=10 -5 (Kaula) Marsc= Correct value c= c= Mercuryc= Best estimation c= c= Garcia et al., Milani et al. Marsc= (Lemoine et al., 2001) Mercuryc= Best estimation c= Dehant et al c= Article 1: gravity Garcia et al.

Crossovers

7-day crossover network on the planet surface rotation matrices (model) Displacement of the network in the inertial frame Least-squares fit a posteriori uncertainties on the rotation parameters Rotation of Mars: contribution of altimetry crossovers MGS orbit repeatability: 7 days (error: 0.08 %) – 88 revolutions Objective: Detection of the nutation in longitude and in obliquity; Better determination of the LOD.

Rotation of Mars: contribution of altimetry crossovers Observed values: Nutation: never observed; LOD: formal error of ~ 4 mas (Konopliv et al. 2001). Simulation results: Nutation: precision as low as 18 mas for longitude and 7 mas for obliquity; LOD: precision 27 mas. Conclusion: Liability of the least-squares estimator (stability and decrease of the uncertainties on the rotation parameters); Possibility to detect the nutation; In order to improve the LOD determination, we need more crossovers; Simulation based on less than 1 million of crossovers while actual number of crossovers: 24 millions (Neumann et al. 2001).

Rotation of Mercury: contribution of altimetry crossovers BepiColombo inclination 90° BepiColombo inclination 91°

Rotation of Mercury: contribution of altimetry crossovers Uncertainties of the rotation parameters estimated from the altimetry crossover depend strongly on the precisions of BELA and of the orbit determination. The number of crossovers depends strongly on the orbit inclination. MGS: precision for crossovers is 100 m BP: ?

Conclusions of Veronique’s part and introduction for Tim Van Hoolst’s part and Anne Lemaître’s part! 88 day ‘forced’ libration will be seen from the future space missions+radar; it will provide us with information on the core state (Vero’s part) and possible core composition and dimension (Tim’s part). Peale experiment uses libration angle, gravity coefficients in order to get solid moment of inertia, thus core state, and core moment of inertia. It is important to have support from the theory (Anne’s part) ~15 year ‘free’ libration, ?at an observable level? Static gravity field for lithospheric and crustal properties Time variable part, tides, Love number k 2, thus core state (Tim’s part).