N4.Lemaitre - "Stability of an asteroid satellite"
1. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Stability of an asteroid satellite
Anne Lemaître, Audrey Compère, Nicolas Delsate
Department of Mathematics FUNDP Namur
14 June 2010
2. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
1 Introduction
2 Satellites of asteroids
System Ida-Dactyl
Previous results
3 Stability tests
Numerical simulations
Classical calculation of the potential
MacMillan potential
Chaos indicator : MEGNO
4 Chaos Maps
Gravitational resonances
Frequency analysis
5 Analytical development
MacMillan potential
Approximated formulation
1:1 resonance
Equatorial resonant orbits
Polar resonant orbits
6 Conclusions
3. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Motivation
Previous studies : rotation of the planets and natural satellites
and space debris on geostationary orbits
Collaboration Grasse - CNES : stability conditions for the
motion of a probe around an asteroid
To test our methods on asteroid satellites (PhD - not
published)
Stability : numerical tests and dynamical models
Several approaches of the potential of a non spherical body
Trace-free tensors in elliptical harmonics
Geometrical approach
MacMillan potential : the only one presented here (Paolo)
4. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Vocabulary
Binary asteroid : system of two asteroids
Two categories :
1 The two bodies have the same size : double asteroid
Ex : Antiope - Dynamics intensively studied in particular by Scheeres and
collaborators
2 A body is much smaller than the other one : asteroid and its satellite
Ex : Ida-Dactyl
5. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Natural system Ida-Dactyl
Ida : main belt asteroid (Koronis family), very irregular shape and fast spin
Ida Dactyl
Mass (4.2 ± 0.6) × 1016 kg ∼ 4.1012 kg
Diameter 59.8 × 25.4 × 18.6 km 1.6 × 1.4 × 1.2 km
6. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Dactyl :
Orbit data :
Semimajor axis (a) : 108 km
Orbital period (P) : 1.54 days
Eccentricity (e) : ≥ 0.2?
Other data :
Mean radius : 0.7 km
Principal diameters : 1.6 × 1.4 × 1.2 km
Shape : less irregular then Ida
Ellisoidal t (radii) : 0,8 × 0,7 × 0,6 km
Mass : ∼ 4.1012 kg
Surface area : 6,3 km2
Volume : 1,4 km3
Spin period : 8 hr
7. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Ida-Dactyl simulations
J-M Petit et al : 1998, Belton, 1996
Context :
Ida mass is not known precisely.
Each value of the mass corresponds to a Keplerian orbit for Dactyl
To constraint the mass of Ida by Dactyl's orbit
Belton,1996
8. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Petit et al
1. Stability bounds on Ida mass
First model
Ida is represented by an ellipsoid.
Gravitational potential : elliptic integrals
Integrator : Bulirsch and Stoer with a precision of 10−10
Masses : between 3.65 × 1016 and 5.7 × 1016 kg
Results :
Orbits with M 4.93 × 1016 kg (q 63 km) are very unstable.
→ crash or escape after several hours or days
The other orbits are stable for hundreds of years.
Second model
Approximation of Ida by a collection of 44 spheres of dierent sizes.
⇒ more precise bounds.
9. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Resonant stable orbits
The Ida-Dactyl system should be stable for long time ⇒ search for resonances
between the rotation of Ida and the orbital frequency of Dactyl.
Simulations results :
Most probable resonances 5:1 and 9:2
10. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Numerical simulations
Model : a point mass orbiting an ellipsoid
Parameters : shape, mass and spin of the primary, initial
conditions of the satellite
Purpose : search for stable or resonant systems
Technique : chaos maps (MEGNO)
Software : NIMASTEP (N. Delsate) written for numerical
integration of an articial satellite around a telluric planet
Dierences : irregular shape and fast rotation of the primary,
large eccentricity of the satellite, relative importance of the
perturbations
11. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
First calculation of the potential
Based on the spherical harmonics as for the telluric planets
Small deformations of a sphere
n
GM ∞
Re n
V (r , θ, λ) = 1+ Pnm (sin θ) (Cnm cos mλ + Snm sin mλ)
r n=2 m=0
r
(r , θ, λ) are the spherical coordinates
Re is the equatorial radius
Pnm are the Legendre's polynomials
Cnm et Snm are the spherical harmonics coecients
12. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Check of the integrations
Paper of A. Rossi, F. Marzari and P. Farinella (1999) : Orbital
evolution around irregular bodies in Earth, Planets, Space.
Four approaches of the potential :
Ivory's approach : direct calculation of the potential of an
homogeneous triaxial ellipsoid
Spherical harmonics approach (4th order)
Mascons approach : the body is approximated by a set of
point masses placed in a suitable place to reproduce the mass
distribution
Polyhedral approach : the body is approximated by a
polyhedron with a great number of faces
Axisymmetric ellipsoid (a = b = 10 km, c= 5 km) or triaxial
ellipsoid (a=30 km, b=10 km and c = 6.66 km).
13. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Tests of Rossi, Marzari and Farinella
Four cases :
Case 0 : Sphere (not considered here)
Case 1 : Axisymmetric ellipsoid with inclined circular orbits
(i = 10◦ ) at a distance of 20km - 5835 mascons - 1521 faces -
Mass = 2.0831015 kg - ρ = 1g cm3 .
Case 2 : Axisymmetric ellipsoid with inclined elliptic orbits
(e = 0.2)
Case 3 : Axisymmetric ellipsoid with distant inclined elliptic
orbits at a distance of 40km
14. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Comparisons
First tests : Variation of the ascending node (in radians s
−1 ) :
Secular Theory (J2 ) polygones mascons spherical harmonics
Case 1
circular inclined -7.7 10−6 -1.09 10−5 -1.11 10−5 -1.07 10−5
Case 2
elliptic inclined -8.37 10−6 -1.25 10−5 -1.33 10−5 -1.27 10−5
Case 3
elliptic, inclined and distant -7.10 10−7 -7.76 10−7 -7.92 10−7 -7.85 10−7
15. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
MacMillan potential
New potential : Potential for an ellipsoid : MacMillan (1958)
V (x , y , z ) = 3 GM
+∞
x2 y2 z2 ds
Z „ «
1− − − √ √
2 λ1 s2 s 2 − h2 s2 − k2 s 2 − h2 s 2 − k 2
where
h
2
= a2 − b2 et k 2 = a2 − c 2 (a, b et c are the semi-major
axes of the ellipsoid with a ≥ b ≥ c )
(x , y , z ) are the cartesian coordinates of the point
λ1 is the rst ellipsoidal coordinate of the point
16. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
For each (x , y , z ) :
x
2 y
2 z
2
+ 2 + 2 =1 Equation of degree 3 in s 2
s2 s − h2 s − k2
Roots : λ2 , λ2 et λ2 with 0 ≤ λ2 ≤ h2 ≤ λ2 ≤ k 2 ≤ λ2 .
1 2 3 3 2 1
Geometrically (x , y , z ) is the intersection between
an ellipsoid with axes ( λ2 , λ2 − h2 ,
1 1 λ2 − k 2 )
1
an hyperboloid of one sheet with axes
( λ2 ,
2 λ2 − h2 ,
2 k2 − λ2 )
2
an hyperboloid of two sheets with axes
( λ2 ,
3 h2 − λ2 ,
3 k2 − λ2 )
3
Ellipsoidal coordinates : (λ1 , λ2 , λ3 )
17. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
New tests and comparisons with Rossi et al
Calculation of the force components explicitly (partial
derivatives)
Gauss-Legendre quadrature for the integrals
Introduction in NIMASTEP
New tests : Variation of the ascending node (in radians s
−1 ) :
Secular Theory (J2 ) polygones mascons spherical harmonics Mac Millan
Case 1 -7.7 10−6 -1.09 10−5 -1.11 10−5 -1.07 10−5 -1.11 10−5
−6 −5 −5 −5
Case 2 -8.37 10 -1.25 10 -1.33 10 -1.27 10 -1.33 10−5
Case 3 -7.10 10−7 -7.76 10−7 -7.92 10−7 -7.85 10−7 -7.86 10−7
18. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Orbits
19. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Tests on the system Ida-Dactyl
Test on eccentric Dactyl orbits :
Resultats :
Crash or escapes for M ≥ 5 × 1016 kg
Regular orbits for M ≤ 5 × 1016 kg
⇒ same results as Petit et al. (1998)
20. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Chaos indicator : MEGNO
MEGNO = Mean Exponential Growth factor of Nearby Orbits
(Cincotta et Simo, 2000)
Dynamical system : : dt x (t ) = f (x (t )), x ∈ IR2n .
d
φ(t ) a solution function of time t
δφ (t ) the tangent vector along φ(t ) with δ˙φ = ∂ x (φ(t ))δφ (t ).
∂f
The MEGNO is :
t t
2 ˙ 1
Z Z
δφ · δφ
Yφ (t ) = t δφ · δφ
s ds and Yφ = t Yφ (s ) ds
0 0
= measure of the divergence rate between two close orbits.
Periodic orbit : Yφ → 0
Quasi-periodic orbit : Yφ → 2
Chaotic orbit : Yφ is increasing with time
21. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Chaos Maps
We set :
the mass and the rotation rate of the asteroid (ellipsoid)
the initial conditions of the satellite (a=148.8km, i = 3 rad)
a the largest semi-axis of the ellipsoid
Variations of the primary shape (through the semi-axes b and c ).
Integrator : Runge-Kutta-Fehlberg with variable step
Precision : 10−12
Results of the chaos indicator MEGNO are given in the plane (b/a,c/a)
25. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
However let us remind that the mass is constant in these graphics, some of
these cases correspond to impossible values of the densities (chosen between 1
and 3 gr/cm3 ) - The mass M and the axis a are xed.
26. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Evolution of the MEGNO with time
After 0.1 year after 1 year
after 5 years after 10 years
27. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Spin
v = −2.5 10−4 rad/s v = −4.0 10−4 rad/s
v = −3.76687 10−4 rad/s Inuence of the spin v
28. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Semi major-axis
a =130 km a =170 km
a =148.8 km Evolution with semi-major axis
29. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Reference case
M=3.895551 106 kg, initial orbit i 3 rad)
rotation rate v = −3.76687 × 10−4 rad/s
30. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Gravitational resonance
A resonance between
the rotation of the primary (P = 4, 63 hours)
the orbital period of the satellite (specic to each point)
Tests on a few points
Q1 : b=18.6 km, c=8.9 km and Y → 2 - period of 2.50 days
Q2 : b=18.9 km, c=8.9 km and Y → +∞ - period of 2.48 days
Q3 : b=20.1 km, c=8.9 km and Y → 2 - period of 2.48 days
Gravitational resonance 1:13
31. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Frequency analysis (J. Laskar)
c= 8.9 km is constant and b varies
Analysis (a ∗ cos (M ), a ∗ sin(M )) :
32. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Second case
M=3.745722 106 kg, i 2.99), v = −3.76687 × 10−4 rad/s
33. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Choosing again c = 8.9 km and b varies with time
Analysis of (a ∗ cos (M ) , a ∗ sin(M ))
34. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Analytical development
MacMillan Potential for an ellipsoid (1958) :
V (x , y , z ) = 3 GM
+∞
x2 y2 z2 ds
Z „ «
1− − − √ √
2 λ1 s2 s 2 − h2 s2 − k2 s 2 − h2 s 2 − k 2
with h2 = a2 − b2 and k 2 = a2 − c 2
a, b et c are the semi- axes of the ellipsoid with a ≥ b ≥ c .
Chauvineau, B., Farinella, P. and Mignard, F. (Icarus, 1993)
Planar orbits about a triaxial body - Application to asteroidal
satellites
Scheeres, D. (Icarus, 1994) Dynamics about uniformly rotating
triaxial ellipsoids : applications to asteroids
35. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Expansion of the potential
Expansion of MacMillan potential in powers of h/R and k /R
where R 2 = x 2 + y 2 + z 2
Keplerian orbit about a rotating body (about its vertical axis)
perturbed by MacMillan potential
√
Delaunay's Hamiltonian momentum : L = µ¯ a
µ µ 2 2 3µ 2 2 2 2
H = −
2L2 − 10R 3 (h + k ) + 10R 5 (y h + z k )
36. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
1:1 resonance, circular and equatorial
The curve corresponds to an curve : k
2 − 2h 2 0
37. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
1:1 resonance model
= L sin(M +
2
Simplications : z = 0 and y
µ − φ)
φ=v t
Resonant variable : σ = M + − φ
Same equilibria as Scheeres or others
µ µ4 2 2 3µ4 2 (1 − cos 2σ).
H =− −v L − 6 (h + k ) + h
2L2 10L 20L6
The exact 1:1 resonance : v =n: k
2 − 2h2 = 0
38. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Other resonances in the equatorial cases
= L sin(f +
2
z = 0 and y
µ − φ)
The eccentricity is used to develop f in multiples of M
Extraction of the resonant angle σ
σ is now conjugated to P = L − G .
Introduction of the pericentre motion (second degree of
freedom) responsible for the multipliers of the exact resonance
Higher orders of resonances require higher powers of the
eccentricity
Case Ida - Dactyl : potential 5:1 or 9:2 resonance (eccentricity
of Dactyl high)
39. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Non-equatorial cases : polar case
Map of the resonances between the rotation of Vesta and the
orbital motion of a polar satellite : numerical work
LAMO HAMO
14
1000 13
Paper of Tricarico and Sykes
The dynamical environment of Vesta
12
900 11
800
10 submitted to Planetary and Space Science
1:2 9
Distance Range [km]
Orbital Period [hour]
8
700
7
2:3
600 6
5
500
1:1 4
400
3
4:3
3:2
300 2
300 400 500 600 700 800 900 1000
Initial Radius [km]
Figure 4: Distance range as a function of the initial radius of a circular orbit, computed over a
period of 50 days. The central mark in each bar represents the median of the range. The rotation
period used for Vesta is of 5.3421288 hours (Harris et al., 2008). Five spin-orbit resonances have
been identified and marked in the plot. The 1:1 resonance affects the largest interval in initial
radius, but the strongest perturbations come from the 2:3 resonance. The leftmost data point,
40. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Our results
Numerical integration with NIMASTEP (especially drawn for
polar orbits)
Resonance map : position and importance of each resonance
Complete agreement with Tricarico and Sykes
Discovery of smaller structures ignored by Tricarico and Sykes
Analysis of each resonance to compare their width and shape
41. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
42. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Conclusions
MEGNO is very ecient for the detection of gravitational
resonances
Use of the frequency map for the identication of the
resonances
Eciency and precision of MacMillan potential for ellipsoidal
bodies
Explicit approximated formulation in h and k
Specic i : j resonance models : strength, width, equilibria
Equatorial and polar cases (Ida and Vesta)
Paolo's contribution : pioneer and omnipresent in the
literature about asteroid dynamics
