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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<ul><li>1.Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development ConclusionsStability of an asteroid satelliteAnne Lematre, Audrey Compre, Nicolas DelsateDepartment of Mathematics FUNDP Namur 14 June 2010 </li></ul><p>2. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions 1 Introduction2 Satellites of asteroids System Ida-Dactyl Previous results3Stability tests Numerical simulations Classical calculation of the potential MacMillan potential Chaos indicator : MEGNO4Chaos Maps Gravitational resonances Frequency analysis5Analytical development MacMillan potential Approximated formulation 1:1 resonance Equatorial resonant orbits Polar resonant orbits6Conclusions 3. IntroductionSatellites of asteroids Stability tests Chaos Maps Analytical development ConclusionsMotivation Previous studies : rotation of the planets and natural satellitesand space debris on geostationary orbitsCollaboration Grasse - CNES : stability conditions for themotion of a probe around an asteroidTo test our methods on asteroid satellites (PhD - notpublished)Stability : numerical tests and dynamical modelsSeveral approaches of the potential of a non spherical bodyTrace-free tensors in elliptical harmonicsGeometrical approachMacMillan potential : the only one presented here (Paolo) 4. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development ConclusionsVocabularyBinary asteroid :system of two asteroids Two categories : 1The two bodies have the same size : double asteroidEx : Antiope - Dynamics intensively studied in particular by Scheeres andcollaborators 2A body is much smaller than the other one : asteroid and its satelliteEx : Ida-Dactyl 5. Introduction Satellites of asteroids Stability tests Chaos MapsAnalytical development ConclusionsNatural system Ida-DactylIda : main belt asteroid (Koronis family), very irregular shape and fast spin Ida DactylMass(4.2 0.6) 1016 kg 4.1012 kgDiameter59.8 25.4 18.6 km1.6 1.4 1.2 km 6. Introduction Satellites of asteroids Stability tests Chaos MapsAnalytical 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 km2Volume: 1,4 km3Spin period :8 hr 7. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development ConclusionsIda-Dactyl simulationsJ-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 DactylTo constraint the mass of Ida by Dactyl's orbit Belton,1996 8. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development ConclusionsPetit 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 1010 Masses : between 3.65 1016 and 5.7 1016 kgResults :Orbits with M4.93 1016 kg (q63 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 ConclusionsResonant stable orbitsThe 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. IntroductionSatellites of asteroids Stability tests Chaos Maps Analytical development ConclusionsNumerical simulationsModel : a point mass orbiting an ellipsoidParameters : shape, mass and spin of the primary, initialconditions of the satellitePurpose : search for stable or resonant systemsTechnique : chaos maps (MEGNO)Software : NIMASTEP (N. Delsate) written for numericalintegration of an articial satellite around a telluric planetDierences : irregular shape and fast rotation of the primary,large eccentricity of the satellite, relative importance of theperturbations 11. IntroductionSatellites of asteroids Stability tests Chaos Maps Analytical development ConclusionsFirst calculation of the potentialBased on the spherical harmonics as for the telluric planets Small deformations of a spherenGM Ren V (r , , ) =1+Pnm (sin ) (Cnm cos m + Snmsin m) r n=2 m=0 r (r , , ) are the spherical coordinatesRe is the equatorial radiusPnm are the Legendre's polynomials Cnm et Snm are the spherical harmonics coecients 12. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development ConclusionsCheck of the integrationsPaper 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 anhomogeneous triaxial ellipsoidSpherical harmonics approach (4th order)Mascons approach : the body is approximated by a set ofpoint masses placed in a suitable place to reproduce the massdistributionPolyhedral approach : the body is approximated by apolyhedron with a great number of facesAxisymmetric 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 ConclusionsTests 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 - = 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. IntroductionSatellites of asteroidsStability tests Chaos MapsAnalytical development ConclusionsComparisons First tests : Variation of the ascending node (in radianss1 ) :Secular Theory (J2 )polygones mascons spherical harmonicsCase 1circular inclined-7.7 106-1.09 105 -1.11 105 -1.07 105Case 2elliptic inclined-8.37 106 -1.25 105 -1.33 105 -1.27 105Case 3elliptic, inclined and distant -7.10 107 -7.76 107 -7.92 107 -7.85 107 15. Introduction Satellites of asteroidsStability testsChaos Maps Analytical development ConclusionsMacMillan potential New potential : Potential for an ellipsoid : MacMillan (1958) V (x , y , z ) = 3 GM+x2 y2 z2ds Z 1 2 1 s2 s 2 h2 s2 k2s 2 h2 s 2 k 2whereh2 = a2 b2 et k 2 = a2 c 2 (a, b et c are the semi-majoraxes of the ellipsoid with a b c )(x , y , z ) are the cartesian coordinates of the point1 is the rst ellipsoidal coordinate of the point 16. IntroductionSatellites of asteroidsStability tests Chaos Maps Analytical development Conclusions For each (x , y , z ) :x2y 2z2+ 2 + 2 =1 Equation of degree 3 in s 2s2 s h2s k2 Roots : 2 , 2 et 2 with 0 2 h2 2 k 2 2 .1 23 3 21Geometrically (x , y , z ) is the intersection betweenan ellipsoid with axes ( 2 , 2 h2 ,1 1 2 k 2 ) 1an hyperboloid of one sheet with axes( 2 , 2 2 h2 ,2k2 2 ) 2an hyperboloid of two sheets with axes( 2 , 3 h2 2 ,3k2 2 ) 3 Ellipsoidal coordinates : (1 , 2 , 3 ) 17. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development ConclusionsNew tests and comparisons with Rossi et al Calculation of the force components explicitly (partialderivatives)Gauss-Legendre quadrature for the integralsIntroduction in NIMASTEPNew tests : Variation of the ascending node (in radians s 1 ) :Secular Theory (J2 ) polygones masconsspherical harmonicsMac MillanCase 1 -7.7 106 -1.09 105-1.11 105 -1.07 105 -1.11 105 6 55 5 Case 2-8.37 10 -1.25 10-1.33 10 -1.27 10 -1.33 105Case 3-7.10 107 -7.76 107-7.92 107 -7.85 107 -7.86 107 18. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development ConclusionsOrbits 19. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development ConclusionsTests on the system Ida-DactylTest on eccentric Dactyl orbits : Resultats :Crash or escapes for M 5 1016 kgRegular orbits for M 5 1016 kg same results as Petit et al. (1998) 20. Introduction Satellites of asteroidsStability tests Chaos Maps Analytical development ConclusionsChaos 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 (t ) a solution function of time t (t ) the tangent vector along (t ) with = x ((t )) (t ).fThe MEGNO is :t t2 1Z Z Y (t ) = t s ds and Y = t Y (s ) ds0 0 = measure of the divergence rate between two close orbits.Periodic orbit : Y 0 Quasi-periodic orbit : Y 2 Chaotic orbit : Y is increasing with time 21. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development ConclusionsChaos 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 : 1012Results of the chaos indicator MEGNO are given in the plane (b/a,c/a) 22. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions M=3.895551 106 kg, rotation rate = 3.76687 104 rad/s 23. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development Conclusions M=3.745722 106 kg, rotation rate = 3.76687 104 rad/s 24. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development ConclusionsM=5.693498 106 kg, rotation rate = 3.76687 104 rad/s 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 ConclusionsEvolution of the MEGNO with timeAfter 0.1 yearafter 1 yearafter 5 years after 10 years 27. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development ConclusionsSpinv = 2.5 104 rad/sv = 4.0 104 rad/s v = 3.76687 104 rad/s Inuence of the spin v 28. IntroductionSatellites of asteroids Stability tests Chaos Maps Analytical development ConclusionsSemi major-axis a =130 km a=170 kma =148.8 kmEvolution with semi-major axis 29. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development ConclusionsReference case M=3.895551 106 kg, initial orbit i 3 rad)rotation rate v = 3.76687 104 rad/s 30. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development ConclusionsGravitational resonance A resonance betweenthe 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 ConclusionsFrequency 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 ConclusionsSecond case M=3.745722 106 kg,i2.99), v = 3.76687 104 rad/s 33. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical development ConclusionsChoosing again c = 8.9 km and b varies with timeAnalysis of (a cos (M ) , a sin(M )) 34. Introduction Satellites of asteroidsStability testsChaos Maps Analytical development ConclusionsAnalytical developmentMacMillan Potential for an ellipsoid (1958) :V (x , y , z ) = 3 GM+x2 y2 z2ds Z 1 2 1 s2 s 2 h2 s2 k2s 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 asteroidalsatellites Scheeres, D. (Icarus, 1994) Dynamics about uniformly rotatingtriaxial ellipsoids : applications to asteroids 35. IntroductionSatellites of asteroids Stability tests Chaos Maps Analytical development ConclusionsExpansion of the potentialExpansion of MacMillan potential in powers of h/R and k /Rwhere R 2 = x 2 + y 2 + z 2Keplerian orbit about a rotating body (about its vertical axis)perturbed by MacMillan potentialDelaunay's Hamiltonian momentum : L = a 223 2 2 2 2 H = 2L2 10R 3 (h + k ) + 10R 5 (y h + z k ) 36. Introduction Satellites of asteroids Stability tests Chaos MapsAnalytical development Conclusions1:1 resonance, circular and equatorial The curve corresponds to an curve : k 2 2h 20 37. Introduction Satellites of asteroids Stability tests Chaos Maps Analytical developmentConclusions1:1 resonance model= L sin(M + 2Simplications :z = 0 and y )=v t Resonant variable : = M + Same equilibria as Scheeres or others 4 2 2342 (1 cos 2).H =vL 6 (h + k ) +h2L2 10L 20L6 The exact 1:1 resonance : v =n: k 2 2h2 = 0 38. Introduction Satellites of asteroids Stability testsChaos Maps Analytical development ConclusionsOther resonances in the equatorial cases = L sin(f +2z = 0 andy )The eccentricity is used to develop f in multiples of MExtraction of the resonant angle is now conjugated to P = L G .Introduction of the pericentre motion (second degree offreedom) responsible for the multipliers of the exact resonanceHigher orders of resonances require higher powers of theeccentricityCase Ida - Dactyl : potential 5:1 or 9:2 resonance (eccentricityof Dactyl high) 39. Introduction Satellites of asteroidsStability tests Chaos Maps Analytical development ConclusionsNon-equatorial cases : polar case Map of the resonances between the rotation of Vesta and the orbital motion of a polar satellite : numerical workLAMOHAMO14 1000 13Paper of Tricarico and SykesThe dynamical environment of Vesta12 90011 80010 submitted to Planetary and Space Science 1:29 Distance Range [km] Orbital Period [hour]8 7007 2:3 6006 5 5001:1 4 40034:33:2 3002 300 400 500600700800 9001000Initial 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 identied and marked in the plot. The 1:1 resonance aects the largest interval in initial radius, but the strongest perturbations come from the 2:3 resonance. The left...</p>