Pr

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University of Namur - Belgium Department of
Mathematics Dynamical systems
2 professors 10 researchers
Celestial mechanics - resonances - spin orbit
Mercury S. DHoedt (Phd in 2007)
N. Rambaux (Post-doc) J. Dufey (Phd
student)
MORE 9 months for Rambaux in 2006
9 6 4 months in 2007
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• Objectives
• to build a complete model of rigid rotation for
  Mercury including the gravitational
  field and the planetary perturbations
• to identify the relevant frequencies and
  amplitudes as functions of the parameters
• to isolate any contribution of the rotation on
  the orbiter

• Objectives of this meeting
• to understand exactly which precision and which
  variables are important
• to identify our direct partners for the software
  
• to make collective decisions about standards and
  constants

Mathematical point of view
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Planet Mercury Resonance spin-orbit 32
Why ? Capture in resonance (Correia Laskar
2004)
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3 1 reference frames
Spin-orbit resonances reference frames
Ecliptic at some epoch
Basic frame (X0, Y0, Z0) Spin frame (X2, Y2, Z2)
Body frame (X3, Y3, Z3)
These frames are linked together by two sets of
Eulers angles (h, K, /) and (g, J, l).
K basic obliquity
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4th frame (X1, Y1, Z1) linked to the Orbit
wo argument of the pericenter io
inclination Wo longitude of the ascending node
?o true anomaly
All these frames are centered at the center of
mass of Mercury
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d i / dt 0 d ?/ dt C
Laplace plane
?

• Dependent on the model
• Mean or instantaneous scales ?
• Calibrated by data or calculated

(Peale 2005 - Yseboodt Margot 2005)
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General Hamiltonian
Convention lowercase letters variables
uppercase letters momenta
H - ?2 / 2 Lo2 (two body problem) T (L,
G, H, l, g, h) (rotational kinetic
energy) VG (L, G, H, l, g, h, Lo, Go, Ho,
lo, go, ho) (gravitational
potential) VP (Lo, Go, Ho, lo, go, ho, t)
(planetary perturbations)
Rotation l, g, h L, G, H Orbit lo, go, ho Lo,
Go, Ho
H H ( L , G , H , l , g , h , Lo , Go , Ho , lo
, go , ho, t)
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• Delaunays elements associated to the elements of
  Mercurys
• center of mass
• Lo, Go, Ho, lo, go, ho (Orbit)
• with lo the mean anomaly, go wo , ho ?o
• (orbit of the Sun around Mercury)
• Andoyers variables L, G, H, l, g, h
  (Rotation)
• with G the norm of the angular momentum (in the
  direction of Z2)
• L G cos J (projection of the
  angular momentum on Z3)
• H G cos K (projection of the
  angular momentum on Z0)

(ao, eo, io)
T T (l, g, h, L, G, H) L2 / 2 I3
(G2 - L2) (sin2 l / 2I1 cos2 l / 2I2)
T (?l , ?, ?, L, G, ???) (Andoyer - Deprit)
g spin FAST l precession body frame on spin
frame SLOW h precession spin frame on basic
frame SLOW
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H H ( L , G , H , l , g , h , Lo , Go , Ho , lo
, go , ho, t)
dl / dt ?H / ?L dg / dt ?H / ?G dh / dt ?H
/ ?H dlo / dt ?H / ?Lo dgo / dt ?H / ?Go dho
/ dt ?H / ?Ho
dL / dt - ?H / ?l dG / dt - ?H / ?g dH / dt
- ?H / ?h dLo / dt - ?H / ?lo dGo / dt -
?H / ?go dHo / dt - ?H / ?ho
Rotation
Orbite
12 variables for the planet
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• To keep the ROTATION variables 6
• To introduce an ORBITAL theory
• (Keplerian, ephemerides, secular)

Weak
Influence of the rotation on the orbit
Influence of the orbit on the rotation
Potential rotation singularities
Important
?1 l g h ?1 G ?2 - l ?2 G - L
G (1 - cos J) ?3 - h ??3 G - H G (1 -
cos K)
?1 FAST ?2 SLOW ?3 SLOW
T T (?1, ?2, ?3, ?1, ?2, ?3) (?1 - ?2)2
/ 2 I3 (?12 - (?1 - ?2)2) (sin2 ?2 / 2I1
cos2 ?2 / 2I2) T (?? , ?2, ?, ?1, ?2,
???) (Andoyer - Deprit)
I1 lt I2 lt I3 3 principal moments of inertia
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Introduction of the resonant angle s
s ??????/2 ?l? ?? ??l3 L1 ( G) ?? ??L3
lo go , ho Lo Lo 3/2 L1 Go , H?
H H2B (L1, L0) T (?? , ?2, ?, ?1, ?2, ???)
VG (L1, L2, L3, s, l2, l3, Lo, Go, Ho, lo, go,
ho) VP (Lo, Go, Ho, lo, go, ho, t)
ANGLES 4 SLOW (?? ??l3, go , ho)
2 FAST (???, l?) --gt 1 FAST (l?) 1 SLOW (s)
Averaging over the fast angle lo
VG VG (L1, L2, L3, s, l2, l3, Lo, Go, Ho, - ,
go, ho) --gt 5 angles
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K kernel H2B T part of VG with keplerian
motion for Mercury in K
s1 s - ho - go ??????? s3 l3 ho L1, ??, L3
go , ho Go Go L1 Ho Ho L1- L3
H K (s1, ??, s3, L1, ??, L3) VG(s1, ??, s3,
go, ho, L1, ??, L3) short periodic terms VP
?C20 and C 22
Calculation of the equilibria of K selection of
the Cassinis equilibrium 1
(s1, s2, s3) (0, 0, 0) J 0 and K io ?1 G
s1 0 the spin axis points out to the Sun at the
pericenter s2 0 the spin axis coincides with the
third axis of inertia s3 0 the two nodes
coincide (orbital and rotational)
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The 4 values of the ecliptic obliquity in
Cassinis equilibria
MERCURY
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Rambaux et al. 2007
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Precession of the node Modification of the
equilibrium
(s1, s2, s3) (0, 0, 0), J 0, L? G and K
i0 1.?
Usual obliquity ? (angle between spin axis and
normal to orbit) can be calculated by cos ??
cos io cos K sin io sin K cos (?3)
(s1, s2, s3) (0, 0, 0) J 0 and K i0 G
norm of the rotational angular momentum
?
Obliquity q 0
(s1, s2, s3) (0, 0, 0) J 0 and K i0 e, e
1.6 G norm of the new rotational angular
momentum
Obliquity q e
?
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Mercury is not blocked at the exact 32
resonance small oscillations about this
equilibrium.
s1 h1 s2 h2 s3 h3 L1 L10 x1 L2 x2 L3
L30 x3
translation to the equilibrium
K a h12 b h22 2 c h1h3 d h32 e x12 f
x22 2 g x1x3 h x32
canonical transformation (untangling
transformation) to introduce the angle-action
coordinates (Ji, yi)
K A (u2 U2) B (v2 V2) C (w2 W2) n1
J1 n2 J2 n3 J3
u v2J1 sin y1 U v2J1 cos y1 v v2J2
sin y2 V v2J2 cos y2 w v2J3 sin y3
W v2J3 cos y3
T1 15.86 years T2 583.74 years (wobble) T3
1065.08 years
Proper frequencies or periods
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Elimination of the short periodic terms
H H2B T (?2) VG22 (s, l2, l3, lo, go, ho)
VGRC VP (lo, go, ho, t)
average over lo (88 days)
H K (s1, s2, s3) R (s1, s2, s3, go, ho,t )
V
Cassinis equilibrium
q Q ?W / ?P et p P - ?W / ?Q (first
order) higher orders (convergence problems)
Keplerian orbit Complete orbit (IMCCE) lo and
lJ, lV
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Motion of the pole (wobble) at the surface of
Mercury
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Short periodic motion of ?1
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Coupling of motions 1 and 3 close to Mercury

• K a h12 b h22 2 c h1h3 d h32
• e x12 f x22 2 g x1x3 h x32 - ?1 ?1 - ?2
  ?1 - ?1 ?3

L1 G L2 G (1 - cos J) L3 G (1 - cos K)
d h??/ dt 2 e ?1 2 g ?3 - ?1 - ?2 d ???/
dt - 2 a ?1 - 2 c ?3 d h??/ dt 2 f ?2 d
???/ dt - 2 b ?2 d h??/ dt 2 g ?1 2 h ?3
- ?1 d ???/ dt - 2 c ?1 - 2 d ?3
Motion of the obliquity d2 ???/ dt2 - 2 c
dh?/ dt - 2 d dh?/ dt ? d2
???/ dt2 A ?3 B ?1 C
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Motion 1 resonance 32 15 years
Motion 3 resonance 11 1000 years
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K a h12 2 c h1h3 d h32 e x12 2 g x1x3
h x32 - ?1 ?1 - ?2 ?1 - ?1 ?3
Indice 1 15 years Indice 3 1065 years
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Adiabatic effect on motion 3 on motion 1 Rambaux,
Lemaitre, DHoedt (2007)
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• Summary
• Averaged model of rotation 2007
• First short-periodic terms 2007
• Complete model of rotation 2008
• Checks and comparisons 2007 and 2008

• Equations of the orbiter
• Role of our team
• Précisions on the frequencies, amplitudes
• Numerical integration, semi-analytical
  expansions