Rotation, libration, and gravitational field of Mercury

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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
dHoedt, Nicolas Rambaux, Julien Dufey Facultés
Universitaire Notre Dame de la Paix, Namur
We acknowledge PRODEX support/Belspo/ESA
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Rotation and libration of Mercury
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Rotation of the terrestrial planets
Mercury
Venus
Earth
Mars
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MERCURY Spin/orbit coupling 32 resonance
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What are librations?
Rotation (spin) 58,64 days
32 spin-orbit resonance
Solar torque
Revolution (orbit) 87,98 jours
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Notation

• Moment of inertia from gravitational harmonics

C
B
A
Mariner10 values C22 1?0.5 10-5 and
C/Mr20.325-0.380
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Torque (2)

• The z-component of the Liouville equations for a
  solid Mercury
• where r distance Mercury-Sun
• x angle between Sun and A

C
B
Sun
x
A
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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).

Liquid core
Solid core
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Impact of the core on the angle of libration in
longitude of 88 days from SONYR model
40 as
20 as
Cm/C 0.5 liquid core
Cm/C 1 solid core
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 Revisiting  a same place

• Only very high latitudes have a very frequent
   flyover  rate
• But lower latitude measurments contain more
  information
• -gt  Ideal  strategy?

30 km
10 km
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Track of the BC NAC (narrow angle camera) on
Mercury
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.
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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.
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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

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Peales 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?-1
Gravitational field and obliquity ( Cassini
state equation)
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Libration observation from Earth-based radar
measurements

• Radar echoes from solid planets are speckled.

Wavefront corrugations tied to Mercurys rotation
The time delay for the pattern to reproduce at
both stations is a direct measure of the rotation
rate.
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Earth-based libration observing strategy

• Illuminate Mercury with monochromatic radio
  signal from Goldstone radar (l3.5 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.

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Principle of Earth-based measurements of
libration.
.
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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 Cm/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 J2 and C22
? Mr2/C
amplitude of mean obliquity
from amplitudes of J2 and C22
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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
k can be determined from ephemerides orb can be
determined by ephemerides s can be determined
from radar observations (2.1).
orb
s
e2'obliquity
?8.6
also s can be determined from analytical approach
(1.6)
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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.

i0 7o I 8.6o
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Internal structure of Mercury
Parameters - Inner core radius - Sulfur
concentration.
Values for Mercury interior structure (MIS) model
a.
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Adjustment of the liquid core and solid inner
core densities
The growth of the inner core is modeled as
follows
- 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.
- 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
liquid
liquid
liquidus
liquidus
solidus
solidus
temperature
temperature
eutectic
eutectic
solid
solid
0FeS 100FeS 100Fe
0Fe
0FeS 100FeS 100Fe
0Fe
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Adjustment of the liquid core and solid inner
core densities
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Impact of the Sulfur concentration on the
librations
Liquid core cases
19 as
3.2 as
Solid core case
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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

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Remaining questions (1)

• Is the present obliquity (e) mean obliquity
  (e)? (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).

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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).

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Gravity of Mercury
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Different types of loading
Internal loading
Surface loading
Necessary if high gravity signal but small
topography
Good model if gravity and topography correlate
well
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Flexure model
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Global admittance analysis
Crustal density

• Admittance
• Cl depends on the rigidity of the lithosphere,
• Cl 1 for rigidity0, perfect compensation,
  isostasy
• 0 for an infinite flexural rigidity, no
  compensation
• Fit Cl to observations to extract global rigidity

topography
gravity anomaly internal mass load
degree of compensation
density jump rm-rc
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Gravity scientific performances noise level on
Doppler dataBepiColombo
Log10(velocity in mm/s)
Contributions to the spacecraft velocity from
given spherical harmonic of the gravity field for
the BepiColombo orbiter.
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Gravity scientific performances noise level on
Doppler dataMessenger
Log10(velocity in mm/s)
Contributions to the spacecraft velocity from
given spherical harmonic of the gravity field
for the Messenger orbiter.
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Gravity field determination from Doppler tracking
data
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Gravity

• Gravity field follows Kaula law c/l2 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/g2.
• In Milani et al. (2001) and in Garcia et al.
  (2004), one finds a scaling of 1/g.

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Article 1 gravity Garcia et al.
Planet Scaling 1/g/(GM/r) Scaling 1/g2 Scaling 1/g
Earth c10-5 (Kaula) c10-5 (Kaula) c10-5 (Kaula)
Mars c1.3 10-4 Correct value c0.7 10-4 c0.3 10-4
Mercury c1.8 10-4 Best estimation c0.7 10-4 c0.3 10-4 Garcia et al., Milani et al.
Mars c1.3 10-4 (Lemoine et al., 2001) c1.3 10-4 (Lemoine et al., 2001) c1.3 10-4 (Lemoine et al., 2001)
Mercury c1.8 10-4 Best estimation c1.3 10-4 Dehant et al c1.3 10-4
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Crossovers
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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.

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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).

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Rotation of Mercury contribution of altimetry
crossovers
BepiColombo inclination 90
BepiColombo inclination 91
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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 ?
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Conclusions of Veroniques part and introduction
for Tim Van Hoolsts part and Anne Lemaîtres
part!

• 88 day forced libration will be seen from the
  future space missionsradar it will provide us
  with information on the core state (Veros part)
  and possible core composition and dimension
  (Tims 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 (Annes part)
• 15 year free libration, ?at an observable
  level?
• Static gravity field for lithospheric and crustal
  properties
• Time variable part, tides, Love number k2, thus
  core state (Tims part).