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
Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
1 Introduction
2 Satellites of asteroidsSystem Ida-DactylPrevious results
3 Stability testsNumerical simulationsClassical calculation of the potentialMacMillan potentialChaos indicator : MEGNO
4 Chaos MapsGravitational resonancesFrequency analysis
5 Analytical developmentMacMillan potentialApproximated formulation1:1 resonanceEquatorial resonant orbitsPolar resonant orbits
6 Conclusions
Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Motivation
Previous studies : rotation of the planets and natural satellitesand space debris on geostationary orbits
Collaboration Grasse - CNES : stability conditions for themotion of a probe around an asteroid
To test our methods on asteroid satellites (PhD - notpublished)
Stability : numerical tests and dynamical models
Several approaches of the potential of a non spherical body
Trace-free tensors in elliptical harmonicsGeometrical approachMacMillan potential : the only one presented here (Paolo)
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 asteroidEx : Antiope - Dynamics intensively studied in particular by Scheeres andcollaborators
2 A body is much smaller than the other one : asteroid and its satelliteEx : Ida-Dactyl
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
Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Dactyl :
Orbit data :
Semimajor axis (a) : 108 kmOrbital period (P) : 1.54 daysEccentricity (e) : ≥ 0.2?
Other data :
Mean radius : 0.7 kmPrincipal diameters : 1.6 × 1.4 × 1.2 kmShape : less irregular then IdaEllisoidal �t (radii) : 0,8 × 0,7 × 0,6 kmMass : ∼ 4.1012 kgSurface area : 6,3 km2
Volume : 1,4 km3
Spin period : > 8 hr
Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Ida-Dactyl simulations
J-M Petit et al : 1998, Belton, 1996Context :
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
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 integralsIntegrator : 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 di�erent sizes.
⇒ more precise bounds.
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 resonancesbetween the rotation of Ida and the orbital frequency of Dactyl.
Simulations results :
Most probable resonances 5:1 and 9:2
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, initialconditions of the satellite
Purpose : search for stable or resonant systems
Technique : chaos maps (MEGNO)
Software : NIMASTEP (N. Delsate) written for numericalintegration of an arti�cial satellite around a telluric planet
Di�erences : irregular shape and fast rotation of the primary,large eccentricity of the satellite, relative importance of theperturbations
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 planetsSmall deformations of a sphere
V (r , θ, λ) =GM
r
[1 +
∞∑n=2
n∑m=0
(Re
r
)n
Pnm(sin θ) (Cnm cos mλ+ Snm sin mλ)
]
(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 coe�cients
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) : Orbitalevolution around irregular bodies in Earth, Planets, Space.Four approaches of the potential :
Ivory's approach : direct calculation of the potential of anhomogeneous triaxial ellipsoid
Spherical harmonics approach (4th order)
Mascons approach : the body is approximated by a set ofpoint masses placed in a suitable place to reproduce the massdistribution
Polyhedral approach : the body is approximated by apolyhedron with a great number of faces
Axisymmetric ellipsoid (a = b = 10 km, c= 5 km) or triaxialellipsoid (a=30 km, b=10 km and c = 6.66 km).
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 ellipticorbits at a distance of 40km
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
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
2GM
Z +∞
λ1
„1− x2
s2− y2
s2 − h2− z2
s2 − k2
«ds√
s2 − h2√s2 − k2
where
h2 = a2 − b2 et k2 = a2 − c2 (a, b et c are the semi-majoraxes 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
Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
For each (x , y , z) :
x2
s2+
y2
s2 − h2+
z2
s2 − k2= 1 Equation of degree 3 in s2
Roots : λ21, λ22 et λ23 with 0 ≤ λ23 ≤ h2 ≤ λ22 ≤ k2 ≤ λ21.
Geometrically (x , y , z) is the intersection between
an ellipsoid with axes (√λ21,√λ21 − h2,
√λ21 − k2)
an hyperboloid of one sheet with axes
(√λ22,√λ22 − h2,
√k2 − λ22)
an hyperboloid of two sheets with axes
(√λ23,√h2 − λ23,
√k2 − λ23)
Ellipsoidal coordinates : (λ1, λ2, λ3)
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 (partialderivatives)
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
Case 2 -8.37 10−6 -1.25 10−5 -1.33 10−5 -1.27 10−5 -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
Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Orbits
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)
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 : : d
dtx(t) = f (x(t)), x ∈ IR2n.
φ(t) a solution function of time t
δφ(t) the tangent vector along φ(t) with δφ = ∂f∂x
(φ(t))δφ(t).
The MEGNO is :
Yφ(t) =2
t
Zt
0
δφ · δφδφ · δφ
s ds and Yφ =1
t
Zt
0
Yφ(s) ds
= 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
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 stepPrecision : 10−12
Results of the chaos indicator MEGNO are given in the plane (b/a,c/a)
Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
M=3.895551 106 kg, rotation rate = −3.76687× 10−4 rad/s
Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
M=3.745722 106 kg, rotation rate = −3.76687× 10−4 rad/s
Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
M=5.693498 106 kg, rotation rate = 3.76687× 10−4 rad/s
Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
However let us remind that the mass is constant in these graphics, some ofthese cases correspond to impossible values of the densities (chosen between 1and 3 gr/cm3) - The mass M and the axis a are �xed.
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
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 In�uence of the spin v
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
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
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 (speci�c to each point)
Tests on a few points
Q1 : b=18.6 km, c=8.9 km and Y → 2 - period of 2.50 daysQ2 : b=18.9 km, c=8.9 km and Y → +∞ - period of 2.48 daysQ3 : b=20.1 km, c=8.9 km and Y → 2 - period of 2.48 days
Gravitational resonance 1:13
Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Frequency analysis (J. Laskar)
c = 8.9 km is constant and b variesAnalysis (a ∗ cos(M), a ∗ sin(M)) :
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
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))
Several high order resonances (1:21 or 1:20 or 2:43)
Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Analytical development
MacMillan Potential for an ellipsoid (1958) :
V (x , y , z) =3
2GM
Z +∞
λ1
„1− x2
s2− y2
s2 − h2− z2
s2 − k2
«ds√
s2 − h2√s2 − k2
with h2 = a2 − b2 and k2 = a2 − c2
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
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/Rwhere R2 = x2 + y2 + z2
Keplerian orbit about a rotating body (about its vertical axis)perturbed by MacMillan potential
Delaunay's Hamiltonian momentum : L =õa
H = − µ
2L2− µ
10R3(h2 + k2) +
3µ
10R5(y2h2 + z2k2)
Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
1:1 resonance, circular and equatorial
The curve corresponds to an curve : k2 − 2h2 ' 0
Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
1:1 resonance model
Simpli�cations : z = 0 and y = L2
µ sin(M +$ − φ)
φ = v t
Resonant variable : σ = M +$ − φSame equilibria as Scheeres or others
H = − µ
2L2− v L− µ4
10L6(h2 + k2) +
3µ4
20L6h2(1− cos 2σ).
The exact 1:1 resonance : v = n : k2 − 2h2 = 0
Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Other resonances in the equatorial cases
z = 0 and y = L2
µ sin(f +$ − 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 offreedom) responsible for the multipliers of the exact resonance
Higher orders of resonances require higher powers of theeccentricity
Case Ida - Dactyl : potential 5:1 or 9:2 resonance (eccentricityof Dactyl high)
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 theorbital motion of a polar satellite : numerical work
300 400 500 600 700 800 900 1000Initial Radius [km]
300
400
500
600
700
800
900
1000
Dis
tance
Ran
ge
[km
]
1:1
3:2
4:3
1:2
2:3
HAMOLAMO
2
3
4
5
6
7
8
9
10
11
12
13
14
Orb
ital
Per
iod [
hour]
2
3
4
5
6
7
Figure 4: Distance range as a function of the initial radius of a circular orbit, computed over aperiod of 50 days. The central mark in each bar represents the median of the range. The rotationperiod used for Vesta is of 5.3421288 hours (Harris et al., 2008). Five spin-orbit resonances havebeen identified and marked in the plot. The 1:1 resonance a!ects the largest interval in initialradius, but the strongest perturbations come from the 2:3 resonance. The leftmost data point,with initial radius of 370 km, reaches a lowest distance just below 300 km, only a few km awayfrom Vesta’s surface. The orbital radius of HAMO and LAMO is also marked.
19
Paper of Tricarico and SykesThe dynamical environment of Vesta
submitted to Planetary and Space Science
Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Our results
Numerical integration with NIMASTEP (especially drawn forpolar 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
Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
Conclusions
MEGNO is very e�cient for the detection of gravitationalresonances
Use of the frequency map for the identi�cation of theresonances
E�ciency and precision of MacMillan potential for ellipsoidalbodies
Explicit approximated formulation in h and k
Speci�c i : j resonance models : strength, width, equilibria
Equatorial and polar cases (Ida and Vesta)
Paolo's contribution : pioneer and omnipresent in theliterature about asteroid dynamics