Dynamics of Dynamics of AsteroidsAsteroids –– an an OverviewOverview Dr. Gerhard Hahn Dr. Gerhard Hahn –– DLR PF DLR PF -- AKAK
HGF-Ringvorlesung TU Berlin – 29. Oktober 2009
Inventur im Sonnensystem 1967 Inventur im Sonnensystem 1967 –– 20072007
Monde - - 1 2 63 57 27 13 3
HGF-Ringvorlesung TU Berlin – 29. Oktober 2009
Sonnensystem 2007
1967 2007Planeten
9
8
Monde 31
166Zwergplaneten
-
3
Kleinplaneten
~4000
~160000 + ~200000now
~220000 + ~240000
Trans-Neptune
-
>1000Kentauren+SDOs
1
~200
NEOs
~55
~4700now
~6350
Kometen
~600
~2300 + ~1000 SOHO
HGF-Ringvorlesung TU Berlin – 29. Oktober 2009
Overview
What
are
asteroids?How
do they
move?
Orbits –
Astrometry
–
Ephemerides
Orbit determinationOrbital elements
Kepler Laws
-
Gravitation LawPerturbation
Theory
Where
do they
come from?What
can
we
lean
about
the
solar system
history
and evolution?
HGF-Ringvorlesung TU Berlin – 29. Oktober 2009
Inner Solar System
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Hilda and Trojan asteroids
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Near-Earth Asteroids
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Asteroid Families
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Outer Solar System
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Outer Solar System
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Orbital Motion
2-body motionN-body
motion
Osculating
orbital elements
-
Epoch
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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.
HGF-Ringvorlesung TU Berlin – 29. Oktober 2009
Orbit DeterminationAstrometric Positions
(>= 3, as many
as possible, long
arc)
Exact
timeExact
position
of the
observator(ies)
Assumptions
(orbit
around
the
sun, assume
some
of the
elements, etc.)
Preliminary
orbit
( and ephemeride)Orbit improvement
Follow-up
observationsLinkage
with
previously
known
objects
Detections
of pre-discovery
observationsDefinitive 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)
HGF-Ringvorlesung TU Berlin – 29. Oktober 2009
Minor Planet Center (MPC)Clearing house
for
all astrometry
of Small BodiesAsteroids, comets, natural
satellitesOrbital DatabaseNaming
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 EB024G96K09B02G C2009 01 17.09573 22 40 23.34 +26 17 00.8 21.2 V EB024G96K09B02G C2009 01 17.10162 22 40 21.50 +26 16 36.7 21.2 V EB024G96K09B02G C2009 01 17.10743 22 40 19.77 +26 16 12.7 21.1 V EB024G96
Orbital elements:2009 BG2 Earth MOID = 0.1949 AUEpoch
2009 Jan. 9.0 TT = JDT 2454840.5 MPCM 348.91292 (2000.0) P Qn 0.25376329 Peri. 183.31104 -0.99981929 +0.01711614a 2.4708716 Node
357.62204 -0.00929772 -0.81981825e 0.7712443 Incl. 11.49984 -0.01658130 -0.57236796P 3.88 H 20.2 G 0.15From
23 observations
2009 Jan. 17-18.
HGF-Ringvorlesung TU Berlin – 29. Oktober 2009
Osculating Orbital Elements
Kepler elements
at a specified
Epoch
Because
these
elements
vary
with
timeDue
to Planetary
Perturbations
Short-period
VariationsSecular
Variations
HGF-Ringvorlesung TU Berlin – 29. Oktober 2009
HGF-Ringvorlesung TU Berlin – 29. Oktober 2009
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Proper ElementsProper 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)
HGF-Ringvorlesung TU Berlin – 29. Oktober 2009
Asteroid Families Bendjoya
& Zappala
(2004)
Discovered in 1918 by the Japanese researcher Kiyotsugu
Hirayama
Hierarchical Clustering Method (HCM)Wavelet Analysis Method (WAM)D Criterion
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Largest Families
from Zappala et al. (1995)
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KAM Theorem or the Search for Integrals of Motion
Much of the history of celestial mechanics has involved the search for integralsof 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 integralsTheir 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)
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Mean-Motion ResonancesKirkwood (1867)
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Short-period variations
Mean-motion
resonances(p) Pa
= q PJ
critical argument σ
= pλJ
– qλ
– (p – q) ϖ
5:2 resonance: p=5,q=2 => σ
= 5λJ – 2λ
– 3 ϖ
ϖ
= Longitude of perihelion
HGF-Ringvorlesung TU Berlin – 29. Oktober 2009
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 theStochasticity
Parameter, which measures the extent of the resonance
overlapwhen 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
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Secular Resonances From
Ch.Froeschle
& H.Scholl
(1988)
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Secular Resonances
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Special Groups of Asteroids
MainbeltFamilies
Kuiperbelt
(TNOs)
Transition
ObjectsNEOsCentaursSDOs
Trojans
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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 TNOs250 Centaurs
and SDOs
(scattered
disk
objects)
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Various
Subclasses
of TNOsClassicalResonantScatteredDetached
Centaurs
from David Jewitt‘s homepage
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TNO Dynamics and Simulations from Levison & Morbidelli (2003)
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Solar System Formation Models (Nice model)
Planetary
MigrationInteraction with
TNOs
GasdynamicsMB EvolutionInner Planet FormationSpecial Events
LHBTrojans
Published in 3 papers in Nature 435 (2005)Tsiganis et al. 459-461.Morbidelli et al. 462-465.Gomes et al. 466-469
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Nice Model Simulations
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Trojans – Asteroids in the 1:1 MMR with a Planet
from Marzari et al. (2004)
L4 L5Mars 3 1Jupiter 2048 1421Neptune 6 -
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Co-Orbitals with Venus and Earth
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Near-Earth Asteroids (NEA)
OriginAsteroid BeltMars-Crosser
Population
(extinct) Short-period
CometsPopulation
Atens
(IEO)ApollosAmors
Orbital EvolutionFast or
Slow
„Track“
Dynamical
and Physical
„Endstates“
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Apollo – Amor – Apollo
from Hahn & Lagerkvist (1988)
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Resonance Jumping and Encounters
from Hahn & Lagerkvist (1988)
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from Hahn & Lagerkvist (1988)
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Resonance Jumping and Orbital Evolution
from Milani et al. (1989)
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Resonance Jumping and Orbital Evolution
from Milani et al. (1989)
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Orbital Classes in the NEA Population and their Interaction
from Milani et al. (1989)
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Tisserand Parameter
The Tisserand parameter, which is a quantity based onthe circular restricted three-body problem, relative to a planetwith semi-major axis ap , as defined by
can be used to characterize the orbital behavior of a bodyencountering that planet. Relative to Jupiter, it also might be used to distinguish between cometary (TJ <3) and asteroidal orbits (TJ >3)
HGF-Ringvorlesung TU Berlin – 29. Oktober 2009
Mainbelt Evolution
CollisionsFamily
Formation
Insertion into
Resonant
RegionsMMR: 3:1, 5:2, 2:1SR: ν5
, ν6
, ν16
Yarkovsky
Drift
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Yarkovsky/YORP Effect From
Broz
et al. (2005)
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Yarkovsky Effect From
Broz
et al. (2005)
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Planet-Crossing Evolution and Endstates
MC – EC – VC – SunRandom
Walk
Close EncountersResonances
with
Terrestrial
Planets
Protection
–
Ejection
–
Collision
(dynamical) Lifetime
~10 –
20 Myr„refill“-time
of similar
order
from Farinella et al. (1994)
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Orbital Integration over 60 Myr – Endstates from
Gladman
et al. (2000)
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NEODyS – General Information
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NEODyS – 1866 Sisyphus
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NEODyS – 99942 Apophis
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NEODyS – 99942 Apophis – Impactor Table
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99942 Apophis – 2004 MN4
Was passiert nach diesem Vorbeiflug?
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Bahn von Apophis vor und nach dem Vorbeiflug
HGF-Ringvorlesung TU Berlin – 29. Oktober 2009
Nur wenn Apophis durch‘s Schlüsselloch fliegt wird es gefährlich !
after Giorgini et al. (2008)
Durchmesser = 270 mUmlaufszeit = 323.5 Tage
Einschlagwahrscheinlichkeit heute für 2036 ist 1:250000
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Websites relevant for Asteroid Dynamics
AstDyS
–
Asteroids
Dynamic
Sitehttp://hamilton.dm.unipi.it/astdys/
NEODyS
–
Near-Earth
Objects
Dynamic
Sitehttp://newton.dm.unipi.it/neodys/
OrbFit
Software Packagehttp://adams.dm.unipi.it/~orbmaint/orbfit/
JPL –
Solar System Dynamicshttp://ssd.jpl.nasa.gov/
MPC –
Minor
Planet Centerhttp://www.cfa.harvard.edu/iau/mpc.html
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References and SourcesFigures
and Plots
Asteroids
and Comets
Groups
(Petr Scheirich)http://sajri.astronomy.cz/asteroidgroups/groups.htmhttp://sajri.astronomy.cz/asteroidgroups/groups.htm
BooksModern Celestial Mechanics; Aspects of Solar System Dynamics (Alessandro Morbidelli) can
be
downloaded
for
free
from
Alessandro‘s
webpage
http://www.oca.eu/morbyhttp://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
HGF-Ringvorlesung TU Berlin – 29. Oktober 2009
References and SourcesArticles
and PapersBendjoya, 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)
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Numerical Integration Methods
Predictor-Corrector
MethodsBurlisch-StoerRADAU (E.Everhart)
Symplectic Integrator MethodsSWIFT -
Regularized Mixed Variable Symplectic (RMVS) (H. Levison)
Mercury (J.E.Chambers)
HGF-Ringvorlesung TU Berlin – 29. Oktober 2009
Solar System Model -
DE405 (JPL)Number
of Perturbers
(planets)
8, 4 (+4-> sun)8 + 4 asteroidsE+M or
separated
Asteroids
as massless
bodiesCheck for
close
encounters
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