Dynamics of Asteroids an Overview Dr. Gerhard Hahn DLR PF - AK
Inventur im Sonnensystem 1967 2007 Monde - - 1 2 63 57 27 13 3
Sonnensystem 2007 1967 2007 Planeten 9 8 Monde 31 166 Zwergplaneten - 3 Kleinplaneten ~4000 ~160000 + ~200000 now ~220000 + ~240000 Trans-Neptune - >1000 Kentauren+SDOs 1 ~200 NEOs ~55 ~4700 now ~6350 Kometen ~600 ~2300 + ~1000 SOHO
Overview What are asteroids? How do they move? Orbits Astrometry Ephemerides Orbit determination Orbital elements Kepler Laws - Gravitation Law Perturbation Theory Where do they come from? What can we lean about the solar system history and evolution?
Inner Solar System
Hilda and Trojan asteroids
Near-Earth Asteroids
Asteroid Families
Outer Solar System
Outer Solar System
Orbital Motion 2-body motion N-body motion Osculating orbital elements - Epoch
Orbital Elements A Kepler orbit is specified by six orbital elements: Two define the shape and size of the ellipse: Eccentricity (e) Semimajor Axis (a) Two define the orientation of the orbital plane: Inclination (i) Longitude of the Ascending Node ( And finally: Argument of perihelion (ω ) defines the orientation of the ellipse in the orbital plane. Mean anomaly at epoch (M) defines the position of the orbiting body along the ellipse.
Orbit Determination Astrometric Positions (>= 3, as many as possible, long arc) Exact time Exact position of the observator(ies) Assumptions (orbit around the sun, assume some of the elements, etc.) Preliminary orbit ( and ephemeride) Orbit improvement Follow-up observations Linkage with previously known objects Detections of pre-discovery observations Definitive orbit (at standard epoch JD2455200.5 = 2010 Jan 4.0) Numbering and Naming of asteroids (after several apparitions 2-5) 221945 of which 15441 named (M.P.C. 2009 Oct. 4)
Minor Planet Center (MPC) Clearing house for all astrometry of Small Bodies Asteroids, comets, natural satellites Orbital Database Naming authority of the IAU (SBNC) Issues Minor Planet Circulars (monthly), Orbital Updates (daily), Discovery Circulars (when appropriate) Various web-based services (http://cfa-www.harvard.edu/iau/mpc.html ) Observations: K09B02G* C2009 01 17.08991 22 40 25.14 +26 17 25.2 21.3 V EB024G96 K09B02G C2009 01 17.09573 22 40 23.34 +26 17 00.8 21.2 V EB024G96 K09B02G C2009 01 17.10162 22 40 21.50 +26 16 36.7 21.2 V EB024G96 K09B02G C2009 01 17.10743 22 40 19.77 +26 16 12.7 21.1 V EB024G96 Orbital elements: 2009 BG2 Earth MOID = 0.1949 AU Epoch 2009 Jan. 9.0 TT = JDT 2454840.5 MPC M 348.91292 (2000.0) P Q n 0.25376329 Peri. 183.31104-0.99981929 +0.01711614 a 2.4708716 Node 357.62204-0.00929772-0.81981825 e 0.7712443 Incl. 11.49984-0.01658130-0.57236796 P 3.88 H 20.2 G 0.15 From 23 observations 2009 Jan. 17-18.
Osculating Orbital Elements Kepler elements at a specified Epoch Because these elements vary with time Due to Planetary Perturbations Short-period Variations Secular Variations
Proper Elements Proper elements are obtained as a result of the elimination of short and longperiodic perturbations from their instantaneous, osculating counterparts, and thus represent a kind of average characteristics of motion. Knezevic et al. (2004) Carpino et al. (1999)
Asteroid Families Bendjoya & Zappala (2004) Discovered in 1918 by the Japanese researcher Kiyotsugu Hirayama Hierarchical Clustering Method (HCM) Wavelet Analysis Method (WAM) D Criterion
Largest Families from Zappala et al. (1995)
KAM Theorem or the Search for Integrals of Motion Much of the history of celestial mechanics has involved the search for integrals of motion. The search was doomed to fail; eventually Poincaré proved that there was no analytic integral for the problem of the Sun and two planets. However, there do exist non-analytic integrals Their discovery culminated in the Kolmogorov-Arnold-Moser (KAM) theorem, a fundamental result in the mathematics of chaos. The theorem guarantees the existence of invariant curves (i.e., other integrals) as long as the perturbations are not too large and the coupling is not too near any resonance. Understanding the exact meaning of the word near was crucial, because resonances (like the rational numbers) are dense Although the KAM theorem is of fundamental importance for the mathematical structure of chaos, the strict conditions of the theorem are rarely satisfied in the solar system. from Lecar et al. (2001)
Mean-Motion Resonances Kirkwood (1867)
Short-period variations Mean-motion resonances (p) P a = q P J critical argument σ = pλ J qλ (p q) ϖ 5:2 resonance: p=5,q=2 => σ = 5λ J 2λ 3 ϖ ϖ = Longitude of perihelion
Chaos in the Solar System Lecar et al. (2001) Overlapping resonances account for its Kirkwood gaps and were used to predict and find evidence for very narrow gaps in the outer belt. In many cases we can estimate the Lyapunov Time and even the Crossing Time (the time for a small body to develop enough eccentricity to cross the orbit of the perturber). Both times depend on the Stochasticity Parameter, which measures the extent of the resonance overlap when numerically integrating a clone, initially differing infinitesimally from the original trajectory, the two would have separated exponentially. In fact, that is the definition of a chaotic orbit: exponential dependence on initial conditions
Secular Resonances From Ch.Froeschle & H.Scholl (1988)
Secular Resonances
Special Groups of Asteroids Mainbelt Families Kuiperbelt (TNOs) Transition Objects NEOs Centaurs SDOs Trojans
TNO - Trans-Neptunian Objects Suggested by various people in the 1940 s and 1950 s therefore sometimes called the Edgeworth-Kuiper Belt First object (1992 QB1) by David Jewitt & Jane Luu See http://www2.ess.ucla.edu/~jewitt/kb.html 1100 TNOs 250 Centaurs and SDOs (scattered disk objects)
Various Subclasses of TNOs Classical Resonant Scattered Detached Centaurs from David Jewitt s homepage
TNO Dynamics and Simulations from Levison & Morbidelli (2003)
Solar System Formation Models (Nice model) Planetary Migration Interaction with TNOs Gasdynamics MB Evolution Inner Planet Formation Special Events LHB Trojans Published in 3 papers in Nature 435 (2005) Tsiganis et al. 459-461. Morbidelli et al. 462-465. Gomes et al. 466-469
Nice Model Simulations
Trojans Asteroids in the 1:1 MMR with a Planet from Marzari et al. (2004) L4 L5 Mars 3 1 Jupiter 2048 1421 Neptune 6 -
Co-Orbitals with Venus and Earth
Near-Earth Asteroids (NEA) Origin Asteroid Belt Mars-Crosser Population (extinct) Short-period Comets Population Atens (IEO) Apollos Amors Orbital Evolution Fast or Slow Track Dynamical and Physical Endstates
Apollo Amor Apollo from Hahn & Lagerkvist (1988)
Resonance Jumping and Encounters from Hahn & Lagerkvist (1988)
from Hahn & Lagerkvist (1988)
Resonance Jumping and Orbital Evolution from Milani et al. (1989)
Resonance Jumping and Orbital Evolution from Milani et al. (1989)
Orbital Classes in the NEA Population and their Interaction from Milani et al. (1989)
Tisserand Parameter The Tisserand parameter, which is a quantity based on the circular restricted three-body problem, relative to a planet with semi-major axis a p, as defined by can be used to characterize the orbital behavior of a body encountering that planet. Relative to Jupiter, it also might be used to distinguish between cometary (T J <3) and asteroidal orbits (T J >3)
Mainbelt Evolution Collisions Family Formation Insertion into Resonant Regions MMR: 3:1, 5:2, 2:1 SR: ν 5, ν 6, ν 16 Yarkovsky Drift
Yarkovsky/YORP Effect From Broz et al. (2005)
Yarkovsky Effect From Broz et al. (2005)
Planet-Crossing Evolution and Endstates MC EC VC Sun Random Walk Close Encounters Resonances with Terrestrial Planets Protection Ejection Collision (dynamical) Lifetime ~10 20 Myr refill -time of similar order from Farinella et al. (1994)
Orbital Integration over 60 Myr Endstates from Gladman et al. (2000)
NEODyS General Information
NEODyS 1866 Sisyphus
NEODyS 99942 Apophis
NEODyS 99942 Apophis Impactor Table
99942 Apophis 2004 MN 4 Was passiert nach diesem Vorbeiflug?
Bahn von Apophis vor und nach dem Vorbeiflug
Nur wenn Apophis durch s Schlüsselloch fliegt wird es gefährlich! Durchmesser = 270 m Umlaufszeit = 323.5 Tage Einschlagwahrscheinlichkeit heute für 2036 ist 1:250000 after Giorgini et al. (2008)
Websites relevant for Asteroid Dynamics AstDyS Asteroids Dynamic Site http://hamilton.dm.unipi.it/astdys/ NEODyS Near-Earth Objects Dynamic Site http://newton.dm.unipi.it/neodys/ OrbFit Software Package http://adams.dm.unipi.it/~orbmaint/orbfit/ JPL Solar System Dynamics http://ssd.jpl.nasa.gov/ MPC Minor Planet Center http://www.cfa.harvard.edu/iau/mpc.html
References and Sources Figures and Plots Asteroids and Comets Groups (Petr Scheirich) Books http://sajri.astronomy.cz/asteroidgroups/groups.htm Modern Celestial Mechanics; Aspects of Solar System Dynamics (Alessandro Morbidelli) can be downloaded for free from Alessandro s webpage http://www.oca.eu/morby/) A.E. Roy, "Orbital Motion, Third Edition," Adam Hilger, 1988 S.W. McCuskey, "Introduction to Celestial Mechanics," Addison-Wesley, 1963. J.M.A. Danby, "Fundamentals of Celestial Mechanics," Second Edition, Willmann-Bell, 1988. V. Szebehely, "Adventures in Celestial Mechanics," University of Texas Press, 1989
References and Sources Articles and Papers Bendjoya, Ph. & V. Zappala Asteroid Family Identification in Asteroids III, pp. 613-666 (2004) Broz, M. et al. Non-gravitational forces acting on small bodies IAU Symp 229, 351 365 (2006) Carpino, M. et al. Long-term numerical integrations and synthetic theories for the motion of the outer planets Astron. & Astrophys. 181, 182-194 (1999) Chambers, J.E. A hybrid symplectic integrator that permits close encounters between massive bodies MNRAS 304, 793-799 (1999) Farinella, P. et al. Asteroids falling into the Sun Nature 371, 314-317 (1994) Froeschle, Ch. & H. Scholl Secular Resonances: new results Celest. Mech. Dyn. Astron. 43, 113-117 (1988) Giorgini, J.D et al. Predicting the Earth encounters of (99942) Apophis Icarus 193, 1-19 (2008) Gladman, B. et al. The Near-Earth Object Population Icarus 46, 176-189 (2000) Hahn, G. & C.-I. Lagerkvist Orbital Evolution Studies of Planetcrossing Asteroids Celest. Mech. Dyn. Astron. 43, 285-302 (1988) Kirkwood D. In Meteoric Astronomy: A Treatise on Shooting-Stars, Fireballs, and Aerolites. Philadelphia: Lippincott. (1867) Knezevic, Z. et al. The Determination of Asteroid Proper Elements in Asteroids III, pp.603-612 (2004) Lecar, M. et al. Chaos in the Solar System Annu. Rev. Astron. Astrophys. 39, 581 631 (2001) Levison, H.F. & A. Morbidelli The formation of the Kuiper belt by the outward transport of bodies during Neptune s migration Nature 426, 419-421 (2003) Marzari, F. et al. Origin and Evolution of Trojan Asteroids in Asteroids III, pp. 725 738 (2004) Milani, A. et al. Dynamics of Planetcrossing Asteroids: Classes of Orbital Behaviour Project SPACEGUARD Icarus 78, 212 269 (1989) Wisdom, J. Chaotic Behavior and the Origin of the 3/1 Kirkwood Gap Icarus 56, 51-74 (1983) Zappala, V. et al. Asteroid Families: Search of a 12,487-Asteroid Sample Using Two Different Clustering Techniques Icarus 116, 291-314 (1995)
Numerical Integration Methods Predictor-Corrector Methods Burlisch-Stoer RADAU (E.Everhart) Symplectic Integrator Methods SWIFT - Regularized Mixed Variable Symplectic (RMVS) (H. Levison) Mercury (J.E.Chambers)
Solar System Model - DE405 (JPL) Number of Perturbers (planets) 8, 4 (+4-> sun) 8 + 4 asteroids E+M or separated Asteroids as massless bodies Check for close encounters