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# N4.Lemaitre - "Stability of an asteroid satellite"

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Talk of the "International Workshop on Paolo Farinella (1953-2000): the Scientists, the man", Pisa, 14-16 June 2010

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### N4.Lemaitre - "Stability of an asteroid satellite"

1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 18. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions Orbits
19. 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. 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. 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)
22. 22. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions M=3.895551 106 kg, rotation rate = −3.76687 × 10−4 rad/s
23. 23. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions M=3.745722 106 kg, rotation rate = −3.76687 × 10−4 rad/s
24. 24. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions M=5.693498 106 kg, rotation rate = 3.76687 × 10−4 rad/s
25. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 41. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions
42. 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