Dynamics of Asteroids Asteroids – – an Overview Overview Dr. … · 2009-11-05 · Solar System...

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


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


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?


Inner Solar System


Hilda and Trojan asteroids


Near-Earth Asteroids


Asteroid Families


Outer Solar System


Orbital Motion

2-body motionN-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 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)


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.


Osculating Orbital Elements

Kepler elements

at a specified

Epoch

Because

these

elements

vary

with

timeDue

to Planetary

Perturbations

Short-period

VariationsSecular

Variations


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)


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 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)


Mean-Motion ResonancesKirkwood (1867)


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


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


Secular Resonances From

Ch.Froeschle

& H.Scholl

(1988)


Secular Resonances


Special Groups of Asteroids

MainbeltFamilies

Kuiperbelt

(TNOs)

Transition

ObjectsNEOsCentaursSDOs

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 TNOs250 Centaurs

and SDOs

(scattered

disk

objects)


Various

Subclasses

of TNOsClassicalResonantScatteredDetached

Centaurs

from David Jewitt‘s homepage


TNO Dynamics and Simulations from Levison & Morbidelli (2003)


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


Nice Model Simulations


Trojans – Asteroids in the 1:1 MMR with a Planet

from Marzari et al. (2004)

L4 L5Mars 3 1Jupiter 2048 1421Neptune 6 -


Co-Orbitals with Venus and Earth


Near-Earth Asteroids (NEA)

OriginAsteroid BeltMars-Crosser

Population

(extinct) Short-period

CometsPopulation

Atens

(IEO)ApollosAmors

Orbital EvolutionFast or

Slow

„Track“

Dynamical

and Physical

„Endstates“


Apollo – Amor – Apollo

from Hahn & Lagerkvist (1988)


Resonance Jumping and Encounters



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 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)


Mainbelt Evolution

CollisionsFamily

Formation

Insertion into

Resonant

RegionsMMR: 3:1, 5:2, 2:1SR: ν5

, ν6

, ν16

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 – 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)


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 MN4

Was passiert nach diesem Vorbeiflug?


Bahn von Apophis vor und nach dem Vorbeiflug


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


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


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

http://sajri.astronomy.cz/asteroidgroups/groups.htm

http://www.oca.eu/morby/


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)


Numerical Integration Methods

Predictor-Corrector

MethodsBurlisch-StoerRADAU (E.Everhart)

Symplectic Integrator MethodsSWIFT -

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 asteroidsE+M or

separated

Asteroids

as massless

bodiesCheck for

close

encounters

Dynamics of Asteroids Asteroids – – an Overview Overview Dr. … · 2009-11-05 · Solar System...

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