Dynamics of non natural bodies
Anne Lemaître
Facultés Universitaires Notre-Dame de la Paix (merge with UCL failed)
naXys (NAmur Center of Complex SYstems)
March 21, 2011
Dynamics of non natural bodies
Introduction
Namur’s specialities
Dynamics of natural bodies
I
I
I
I
I
I
I
Moon
Mercury
Asteroids
Galilean satellites
Saturnian satellites
Exoplanets
Galaxies and universe (cosmology)
Dynamics of non natural bodies
Introduction
Namur’s specialities
Dynamics of natural bodies
I
I
I
I
I
I
I
Moon
Mercury
Asteroids
Galilean satellites
Saturnian satellites
Exoplanets
Galaxies and universe (cosmology)
but also of artificial satellites
I Artificial satellite (H. Claes - 81)
I Critical inclination (F. Delhaise - 92)
I Lunar satellite (B. De Saedeleer - 06)
Dynamics of non natural bodies
Introduction
Present situation
I Influence of French Post-doc Florent Deleflie
I Stéphane Valk, Nicolas Delsate, Charles Hubaux : PhD on
artificial satellite dynamics
I ROMEO Team : BepiColombo mission : Julien Dufey
I
(Post-doc at ESA)
Reasons
I
I
I
I
Personal affinity
Softwares conception
Easier future careers ?
100 % male scientists
Dynamics of non natural bodies
Introduction
Present situation
I Influence of French Post-doc Florent Deleflie
I Stéphane Valk, Nicolas Delsate, Charles Hubaux : PhD on
artificial satellite dynamics
I ROMEO Team : BepiColombo mission : Julien Dufey
I
(Post-doc at ESA)
Reasons
I
I
I
I
Personal affinity
Softwares conception
Easier future careers ?
100 % male scientists
I Presentation of results, perspectives from 2008 to nowadays
I Space debris
I Artificial satellites
I Techniques used and improved
Dynamics of non natural bodies
Outline
Introduction
Space Debris
Symplectic integrators
Earth shadowing effects
Telluric planet satellite
Dynamics of non natural bodies
Space Debris
Space debris population
Catalogued objects (NASA)
I There are about 15 000 objects larger than 10 cm
TLE (Two Lines Elements) in a catalogue
I About 350 000 objects larger than 1 cm
I More than 3 × 108 objects larger than 1 mm
Composition of the catalogue
I
I
I
I
I
6 % Operational spacecrafts
24% Non-operational spacecrafts
17% Upper stages of rockets
13% Mission related debris
40% Debris mostly generated by explosions & collisions
Dynamics of non natural bodies
Space Debris
Optical observations in high-altitude orbits
Catalogue
Observations
[Schildknecht et al., 2005]
Dynamics of non natural bodies
Space Debris
Direct radiation pressure acceleration
The acceleration of a geostationary debris due to the direct
radiation pressure :
2
a
A r − r
arp = Cr Pr
,
r − r m r − r I Cr is the non-dimensional reflectivity coefficient (0 < Cr < 2),
I Pr = 4.56 10−6 N/m2 is the radiation pressure per unit of
I
I
I
mass for an object located at a distance of a = 1 AU,
r is the geocentric position of the space debris; r is the
geocentric position of the Sun,
A is the exposed area to the Sun of the space debris,
m is the mass of the space debris.
Non-gravitational force
Object
A/m (m2 /kg)
Object
A/m (m2 /kg)
Lageos 1 and 2
GPS (Block II)
0.0007
0.02
Moon
Space debris
1.3 10−10
???
Dynamics of non natural bodies
Space Debris
Direct radiation pressure acceleration
Order of magnitude of the perturbations
10
-2
GM
10-4
J2
Acceleration [km/s2]
10-6
2
A/m 40 m /kg
A/m 10 m2/kg
10-8
2
A/m 1 m /kg
Sun
Moon
J22
10-10
2
A/m 0.01 m /kg
J3
10-12
10-14
Jupiter
-16
10
10000
15000
20000
25000
30000
35000
Distance ftom the Earth’s center [km]
40000
45000
50000
Dynamics of non natural bodies
Space Debris
Averaged Eccentricity and Inclination
Calculation of a potential :
2
a
2
a
A
A
W = Cr Pr
Cr Pr
m r − r m r
1
Pn (cos ψ)
n=0
r
r
n
Chao (2005) : averaging over the mean anomaly
< HSRP
2
A a
3
> = − Cr Pr
a e (C1 cos g1 + C2 cos g2
2
2
m r
+C3 cos g3 − C3 cos g4 + C4 cos g5 + C5 cos g6 ].
where Ck = Ck (i, ) with i the inclination and the obliquity
and gk are linear combinations of λ , and Ω.
Non singular cartesian variables
X1 =
X2 =
r 2P
r 2Q
L
L
sin e sin sin Ω sin
i
sin Ω
2
Y1
r 2P
=
cos e cos L
r 2Q
i
Y2 =
L
cos Ω sin
2
cos Ω
Dynamics of non natural bodies
Space Debris
Eccentricity and Inclination evolution
The long periodic motion of the eccentricity
Z
sin λ (t) + β0
X1 (t) = −
Ln
Z cos Y1 (t) =
cos λ (t) + α0
Ln
and consequently the very long periodic motion of the inclination
< X2 (t) >λ
−A0 sin (νt + θ0 )
< Y2 (t) >λ
sin − A0 cos (νt + θ0 )
Z 2 cos where ν =
and Z =
2
2n L
8<
Coupling the two degrees of freedom :
:
3
2
Cr Pr
X1 (t)
=
Y1 (t)
=
A
m
− LnZ
( ar )2 a.
Z cos Ln
sin λ (t) + B0 sin (νt + φ0 )
cos λ (t) + B0 cos (νt + φ0 )
Dynamics of non natural bodies
Space Debris
Eccentricity and Inclination evolution
0.4
50
45
0.35
40
0.3
35
Inclination [degree]
Eccentricity
0.25
0.2
0.15
30
25
20
15
0.1
10
0.05
5
0
0
0
200
400
600
800
Time [Days]
1000
1200
A/m = 10, 20 or 30 m2 /kg
Eccentricities
I Period = 1 year
I Values as high as 0.55
(A/m = 30 m2 /kg),
1400
1600
0
10
20
30
40
Time [Years]
50
60
70
80
A/m = 5, 10, 20 or 30 m2 /kg
Inclinations
I Very long periods reduced to
40 years for large A/m.
I Maximum value = 2 = 47◦
Dynamics of non natural bodies
Space Debris
Sensitivity to initial conditions
Geostationary space debris
+ gravity field
+ Luni-Solar perturbations
+ Solar Radiation pressure
I For A/m << 1 m2 /kg : apparition of chaos near the
separatrix of the pendulum (geostationary resonance)
Breiter et al (2005) and Wytrzyszczak et al (2007)
I For A/m > 1 m2 /kg : apparition and connection of chaos
with the value of A/m
Valk et al (2009)
By numerical integrations (20 000 orbits) and chaos indicators
Dynamics of non natural bodies
Space Debris
Sensitivity to initial conditions
Dynamics of non natural bodies
Space Debris
Numerical chaos indicators
d
x (t)
dt = f (x (t), α
),
x ∈ R2n ,
α
is a vector of parameters.
x0 , t0 = 0) is a solution of the flow.
φ(t,
We introduce the tangent vector δ(t) measuring the time evolution
of a small initial deviation δ0 with respect to φ(t).
The dynamics is given by the variational equations
δ˙ = d δ(t) = J(t) δ(t),
dt
∂f
with J(t) =
∂x |x =φ(t)
Dynamics of non natural bodies
Space Debris
LCN and MEGNO
LCN
The Lyapounov Characteristic Number λ = lim λ1 (t).
t→∞
t 1 δ(φ(t))
δ̇(φ(s))
1
λ1 (t) = ln
=
ds
t
t 0 δ(φ(s))
δ0 δ˙ · δ
where δ = δ and δ̇ =
δ
MEGNO
The Mean Exponential Growth factor of Nearby Orbits Y (φ(t))
is
based on a modified time-weighted version of the integral form of
the LCN (Cincotta et al 2000).
2
Y (φ(t)) =
t
0
t
δ̇(φ(s))
s ds,
δ(φ(s))
1
Y (φ(t)) =
t
0
t
Y (φ(s))
ds
Dynamics of non natural bodies
Space Debris
MEGNO properties
I Stable, isochronous periodic
orbits :
Y (t) → 0.
I Quasi-periodic (regular)
Y (t) → 2
I Chaotic (irregular)
Y (t) λ/2 t
Breiter et al 2007
Dynamics of non natural bodies
Space Debris
High area-to-mass - A/m = 1 or 5 m2 /kg
180 × 70 initial conditions → 133 hours
MEGNO
4
4
yWAM1ALLmegno
x 10
yWAM5ALLmegno
x 10
12
6
4.219
4.219
10
5
4.218
4.218
8
4
4.217
4.217
3
4.216
2
6
4.216
4
1
4.215
4.215
2
0
4.214
4.214
0
−1
2
2.5
3
3.5
4
4.5
5
2
2.5
3
3.5
4
4.5
5
Dynamics of non natural bodies
Space Debris
High area-to-mass - A/m = 10 or 20 m2 /kg
MEGNO
4
yWAM10ALLmegno
x 10
yWAM20AllALLmegno
4.219
15
14
4.2190
4.218
12
4.2180
10
4.217
10
4.2170
4.216
6
Demi−Gd ax
8
4.2160
4.215
5
4
4.2150
4.214
2
4.2140
4.213
0
2
2.5
3
3.5
4
4.5
5
0
120
140
160
180
200
220
Anomal Moye
240
260
280
Dynamics of non natural bodies
Space Debris
FAM : Frequency Map Analysis
Frequency analysis (Laskar 1990, 1993 and 1995) :
∞
I A initial signal f (t) : f (t) =
pk e iνk t
k=1
I The approximation f
(t)
: f
(t)
=
N
iνk t
pk e
k=1
I The frequencies νk and the amplitudes pk for k = 1, . . . , N are
determined through an iterative scheme.
I Two indicators :
SDI = log |δδν| of the numerical Second Derivative
b
| where νa and νb represent the calculation of
DTI = log | νaν−ν
a
the frequency on two different time intervals Diffusion Time
Dynamics of non natural bodies
Space Debris
Identification of the two frequencies : secondary resonances
λ with period of 1 year and σ with period of 2.25 years
Dynamics of non natural bodies
Space Debris
Identification of the two frequencies : secondary resonances
112
14
80
12
Mean semi−major axis [km]
60
10
40
20
8
0
6
−20
4
−40
2
−60
0
−80
−2
−100
−120
−126.5 −90 −60 −30
0
30 60 90 120 150 180 210 235.5
Resonant angle [degree]
Dynamics of non natural bodies
Space Debris
Frequency analysis
Dynamics of non natural bodies
Space Debris
Perturbed pendulum
μ2
μ 2
H = − 2 −θ̇L+ 3 Re F200 (i) G200 (e) S2200 +F221 (i) G212 (e) S2212
2L
a
with F200 (i) = 43 (1 + cos i)2 3 and F221 (i) =
with G200 (e) = 1 − 52 e 2 + . . . 1 − 52 e 2 .
3
2
sin2 i 0
ψ2200 = 2ω + 2M + 2(Ω − θ) = 2(M + ω + Ω) − 2θ = 2 σ
S2200 = C22 cos ψ2200 + S22 sin ψ2200
= C22 cos 2σ + S22 sin 2σ = J22 cos 2(σ − σ0 ).
√
The variable σ is conjugated to L = μa .
μ2
3μ4 2
H = − 2 − θ̇L + 6 Re J22 cos 2(σ − σ0 )
2L
L
15μ4 2 2
− 6 Re e J22 cos 2(σ − σ0 ).
2L
Dynamics of non natural bodies
Space Debris
Pendulum - semi-major axis and resonant angle
μ2
F
H = − 2 − θ̇L + 6 cos 2(σ − σ0 )
2L
L
I a ageo
I Equilibria at different altitudes
I Period : 818.7 days = 2.24 years.
Breiter et al 2007
Dynamics of non natural bodies
Space Debris
Second frequency
We introduce through e 2 the second frequency, caused by the solar
radiation pressure ( = 0)
Z
e sin X1 (t) = −
sin λ (t) + β0
Ln
Z cos e cos Y1 (t) =
cos λ (t) + α0
Ln
For the obliquity equal to 0 :
e 2 = (e cos + e sin )2
Z2
2Z
2
2
=
+ α0 + β0 +
(α0 cos λS + −β0 sin λS )
2
2
LnS
L nS
e2 =
Z2
2Z
2
+γ +
γ cos (λS + δ).
2
2
LnS
L nS
with α0 = γ cos δ and β0 = γ sin δ
Dynamics of non natural bodies
Space Debris
Two frequencies expression
The final Hamiltonian is :
μ2
F
K (L, σ) = − 2 − θ̇L + 6 cos (2σ − 2σ0 )
2L
L
2G
− 6 cos (2σ − 2σ0 ) cos (λS + δ),
L
with
F
G
=
3μ4
=
15μ4 2
2 Re
Re2
J22 −
J22
15μ4 2
2 Re
Z
LnS
Z2
J22 ( L2 n2
S
+ γ2)
γ
2 cos (2σ − 2σ0 ) cos (λS + δ) = cos (2σ + λS − 2σ0 + δ)
+ cos (2σ − λS − 2σ0 − δ).
I σ librates or circulates
I σ̇ = constant.
I Only one secondary resonance ?
Dynamics of non natural bodies
Space Debris
Examples based on the second derivatives
Dynamics of non natural bodies
Space Debris
Examples based on the second derivatives
Dynamics of non natural bodies
Space Debris
Examples based on the dissipation
Dynamics of non natural bodies
Space Debris
Action - angle variables
I
I
I
I
Change of scale (new time)
Action-angle variables in the circulation or libration zone
Development in Bessel functions
New angle : η = λ + δ
For the circulation case (k ≡ the distance from the separatrix) :
J 2 1 b2
h(ψ, J) =
+
G
cos (k sin ψ) (2 cos ψ cos η)
+
2
2
2 J
−G sin (k sin ψ) (2 sin ψ cos η)
J 2 1 b2
+
G
J
(k)
cos(ψ
+
η)
+
G
J0 (k) cos(ψ − η)
+
0
2
2
2 J
+G J2 (k) cos (3ψ + η) + G J2 (k) cos (3ψ − η)
+G J2 (k) cos (ψ + η) + G J2 (k) cos (ψ − η)
+G J1 (k) cos (2ψ + η) − G J1 (k) cos (2ψ − η)
3 2
J0 (k) 1 − k ,
4
k
J1 (k) ,
2
k2
J2 (k) .
8
Dynamics of non natural bodies
Space Debris
Local resonant analysis
We isolate a resonance φ = jψ ± η and we build a canonical
transformation between (ψ, η, J, Λ) to (φ, η, J, Γ) defined by :
η
φ=ψ±
j
We replace ψ by φ ∓
<h>
=
or =
or =
η
j
J
Λ=Γ±
j
and we average.
J
J 2 1 b2
+
n
(Γ
∓
(J0 (k) + J2 (k)) cos φ
+
)
+
G
2
2
2 J
j
J 2 1 b2
J
+
n
(Γ
∓
J1 (k) cos 2φ
+
)
−
G
2
2
2 J
j
J 2 1 b2
J
+
n
(Γ
∓
J2 (k) cos 3φ
+
)
+
G
2
2
2 J
j
with calculation of a specific frequency (period) obtained by new
passages to action-angle variables.
Dynamics of non natural bodies
Space Debris
Conclusions
I Same analysis for the libration zone
I Good agreement of periods and amplitudes of the secondary
resonances web
I Obliquity : cos = 1 → combinations jψ + 2λ
I Objects with high area-to-mass ratios (A/m) are good
candidates to the recently discovered debris population,
I Such objects present highly irregular (chaotic) dynamics near
the geosynchronous region
I The numerical investigations (MEGNO and Frequency
analysis & Numerical analysis) showed an intricate web of
secondary resonances with stable islands
I Stability for centuries (hazardous situation or parking orbits)
Dynamics of non natural bodies
Space Debris
Conclusions
Publications
I
I
I
Valk, S., Lemaître, A., Anselmo, L., Analytical and semi-analytical investigations of
geosynchronous space debris with high area-to-mass ratios influenced by solar radiation pressure,
Advances in Space Research, 41, 1077–1090, 2008.
Valk, S., Lemaître, A., Deleflie, F., Semi-analytical theory of mean orbital motion for
geosynchronous space debris under gravitational influence, Advances in Space Research, 43,
1070–1082, 2009.
Lemaître, A., Delsate, N., Valk, S., A web of secondary resonances for large A/m geostationary
debris. CM & DA, 104, 383–402, 209.
Softwares
I
I
I
Valk : " resonant " numerical integration of geostationary motions
Delsate : NIMASTEP Numerical Integraiton of the Motion of an Artificial Satellite orbiting a
TElluric Planet
MEGNO and FAM
Dynamics of non natural bodies
Symplectic integrators
Introduction
I
I
I
I
I
Space missions : too short for symplectic integrators
Space debris problematic : stability for centuries
Excellent symplectic integrators existing
Application and adaptation to a new field of applications
Four classes of symmetric symplectic integrators (Laskar et
Robutel, 2001) have been used.
SABA2n (τ )
SABA2n+1 (τ )
SBAB2n (τ )
of order O(τ 2n ε + τ 2 ε2 ) where ε |B|/|A|
SBAB2n+1 (τ )
I The Hamiltonian system is separated into two integrable parts :
H(v , Λ,r , θ) = A(v ,r , Λ) + B(r , θ)
Dynamics of non natural bodies
Symplectic integrators
Hamiltonian Formalism
H(p , q ) = H(v , Λ,r , θ)
= Hkepl (v ,r ) + Hrot (Λ) + Hgeo (r , θ) + H3b (r ) + Hsrp (r )
I Hkepl (v ,r ) = v22 − μr
I Hrot (Λ) = θ̇ Λ
I Hgeo =
I H3b
− Rμe
∞ n
Cnm Vnm (r , θ) + Snm Wnm (r , θ)
n=2 m=0
1
= − μi r −ri − rr·3ri
i
i
I Hsrp =
2
a
A
Pr m r −r (conservative form, without Earth shadow)
Dynamics of non natural bodies
Symplectic integrators
Results
42164.140 km
e0
=
0.1
i0
=
0.1 rad
Ω0
=
0 rad
ω0
=
0 rad
M0
=
0 rad
JD
=
2448281.5
Perturbation : J2
Time span : 100 years
Time step : 10 hours
CPU time : 3 seconds
Integrator : SABA4
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
Time [Years]
60
70
80
90
100
0.100001
0.1
0.0999995
0.099999
0.0999985
0.099998
0.0999975
0.099997
0.0999965
0.099996
0.0999955
Argument of perigee [rad] Longitude of asc. node [rad]
=
Semi-major axis [km]
a0
0.1
0.09999
0.09998
0.09997
0.09996
0.09995
0.09994
0.09993
0.09992
Resonant angle [rad]
I
I
I
I
I
Initial conditions :
Inclination [rad]
I
Eccentricity
Keplerian elements
42164.2
42164.1
42164
42163.9
42163.8
42163.7
42163.6
42163.5
42163.4
7
6
5
4
3
2
1
0
7
6
5
4
3
2
1
0
7
6
5
4
3
2
1
0
Dynamics of non natural bodies
Symplectic integrators
Conclusions
I
I
I
I
I
I
I
Symplectic integrator : precise, fast, robust (Ch. Hubaux)
Long time integrations : parking orbits, stability islands
TLE : 15 000 objects of size > 10 cm
Synthetic population of debris of any size (power law) : naXys
Huge number of initial conditions
Present collaboration with A. Rossi : first publication
Indirect application : Carletti T., Hubaux Ch. and Libert
A.-S., MNRAS, Global symplectic integrators (application to
exoplanetary systems)
Dynamics of non natural bodies
Earth shadowing effects
Cylindric geometry
Solar Radiation Pressure (SPR) is not a continuous function
Dynamics of non natural bodies
Earth shadowing effects
Cylindrical Earth shadow (non-singular orbital elements)
I Cylindrical Earth shadow is crossed when (Escobal, 1976) :
r 2 cos2 ψ = r 2 − RE2
with
r · r = r r cos ψ
I After some algebra :
p 2 (β(r ) cos V +ξ(r ) sin V )2 +RE2 (1+ke cos V +he sin V )2 −p 2 = 0
with p = a(1 − e 2 ), V the eccentric longitude, he = e sin and ke = e cos .
I An analytical solution of this equation can be found and the
boundaries V1 and V2 of the Earth shadow are known as
function of a, e, r .
I Problem : no Hamiltonian formalism.
Dynamics of non natural bodies
Earth shadowing effects
Geocentric cartesian coordinates
Cylindrical Earth shadow :
r · r
+
r
r 2 − RE2 < 0
This condition is introduced in the computation of the SRP
perturbation:
∇r Hsrp ⇒ ∇r [Hsrp S(r )]
where S is a switch function :
⎧
r · r 2
⎨
+ r − RE2 < 0
1 if
S(r ) =
r
⎩
0 otherwise
Dynamics of non natural bodies
Earth shadowing effects
Averaged Earth shadowing effects
Ideas : Kozai (1961) - Aksnes (1976)
I
I
I
I
Geostationary orbit : resonant averaging process
Classical averaging method : integration from 0 to 2π
Interruption in the effect of the solar radiation pressure
Replaced by an integration from 0 to the "entrance anomaly"
and from the "exit anomaly" to 2π.
I Calculation of increments (in particular Δa)
I Special adjustment of the initial conditions
I Pubication
Valk S. and Lemaître A., Semi-analytical investigations of high area-to-mass ratio
geosynchronous space debris including Earth’s shadowing effects, Advances in Space Research
42, 1429–1443, 2008.
Dynamics of non natural bodies
Earth shadowing effects
Numerical integration: 25 years
Dynamics of non natural bodies
Earth shadowing effects
Umbra and penumbra
The shape of the cone is introduced in the SRP perturbation:
∇r Hsrp ⇒ ∇r [Hsrp S(r , α, β)]
Dynamics of non natural bodies
Earth shadowing effects
Continuous formulation
I Numerous hypotheses
I An hyperbolic function
1
formulation
0.9
I Adaptation of the umbra and
0.8
0.7
penumbra behaviors
S
I Inserted in the symplectic
0.6
0.5
integrator for large A/m
0.4
I Checked by NIMASTEP
I Discovery of a very long
0.3
periodic effect, depending on
A/m
0.2
0.1
0
0
5
10
15
20
25
Time [1.14 min]
30
35
40
45
Dynamics of non natural bodies
Earth shadowing effects
Long term evolution of semi-major axis period
Period [years]
15000
10000
5000
0
3.8
4
4.2
4
x 10
4.4
a0 [km]
20
10
15
5
Am [m2/kg]
Initial conditions :
e0 = 0, i0 = Ω0 = ω0 = M0 = 0, JD0 = 2448281.5
Dynamics of non natural bodies
Telluric planet satellite
Motivation
I Sabbatic stay in 2008 at Grasse (F. Deleflie)
I CNES : B. Meyssignac - BepiColombo mission (ROMEO
contract)
I
I
I
I
I
I
I
I
Fast numerical stability analysis for the probe : FAM (Laskar)
Interpretation of results : OK but surprising structures
Collaboration with N. Delsate in January 2009
Collaboration with Ph. Robutel a few months after
Completely different map and analysis (NIMASTEP)
Interesting region of frozen orbits revisited
Context : any telluric planet and even asteroid satellite
Paper published in CM&DA in 2010
by Delsate, Robutel, Lemaitre and Carletti.
Dynamics of non natural bodies
Telluric planet satellite
First Results concerning BepiColombo
Dynamics of non natural bodies
Telluric planet satellite
Problems of the analysis
I Nothing similar obtained by NIMASTEP
I Exchange of emails, numerical simulations, Ph. Robutel’s
expertise
I Different time scales (presence of escapes after 6 weeks)
I No consideration of "crashes" on Mercury’s surface (Mercury
is a point mass and the planet can be crossed)
Dynamics of non natural bodies
Telluric planet satellite
Revisited map
Dynamics of non natural bodies
Telluric planet satellite
Dynamics over 30 years
The system of differential equations describing the probe motion is
given by
¨ +r
¨ rp
¨ pot +r
¨r =r
Upot (r , λ, φ)
U
¨ rp
r
=
=
=
μ
μ
− +
r
r
μ
C r Pr
XX ∞
n
Rp
r
n=2 m=0
r · r
1
−
r − r r 3
a
r − r 2
n
Pnm (sin φ)(Cnm cos mλ + Snm sin mλ)
A r − r
m r − r with
A/m < 0.01
Dynamics of non natural bodies
Telluric planet satellite
Dynamics of the eccentricity vector
Eccentricity amplitude
SDI
Dynamics of non natural bodies
Telluric planet satellite
Simplified Hamiltonian
I Successive tests to keep the minimum to reproduce this map
I No radiation pressure, J2 and C22 , i = 0, quadripolar
I Average over the short periodic effects (longitudes of the
I
satellite and Sun)
Final averaged Hamiltonian :
! 3ε "
ε
H2
K =
I εJ2 =
J2
4G 3
J2 RE2
a5
1−3
G2
and
+
3b
8
ε3b =
2
5 (1 − G )
1−
H2
G2
M
3 (1−e 2 )3/2
M a
!
2
2
sin ω − H − 2 + 2G
⇒
I K ≡ KJ2 + KKozai−Lidov
I K (ω, Ω, G , H) = K (ω, −, G , H) : H is constant
I Heureuse coïncidence
γ=
2
#
εJ2
ε3b
Dynamics of non natural bodies
Telluric planet satellite
Equilibria
Curves obtained by
∂K
∂ω
=0=
∂K
∂G
Dynamics of non natural bodies
Telluric planet satellite
Phase spaces
Dynamics of non natural bodies
Telluric planet satellite
Phase spaces
Dynamics of non natural bodies
Telluric planet satellite
Phase spaces
Dynamics of non natural bodies
Telluric planet satellite
Conclusions
I J2 acts as a protector for the Kozai-Lidov mechanism for the
eccentricity
I slowing down their apparition
I slowing down the motion of e to 1 when H tends to 0
I A small region is missing here it mathematical curiosity
I J3 changes the positions of the previous equilibria
I Frozen orbits : already known (numerically, partially
analytically)
I Our contribution : general analytical frame and analysis
I Already used for satellites of asteroids and Mercury (by us), for
Europa, Mars, Venus, etc (by others)
I Fast answer in a recent review of paper (eccentricity changing
of behavior)
I Publication :
Delsate, N.,Robutel Ph., Lemaître, A. and Carletti, T.: Frozen orbits at
high eccentricity and inclination, CM&Da 108, 275-300.
Dynamics of non natural bodies
Telluric planet satellite
Conclusions
Thank you for your attention !