Dynamical Evolution of Asteroids
Ovidiu C. Furdui
Astronomische Rechen-Institut (ARI)
Mönchhofstraβe 12-14, D-69120 Heidelberg, GERMANY
Characteristics of long-term dynamical evolution
Orbits of asteroids
™ Chaotic behaviour
Most asteroids travel in
fairly circular orbits, there
are some
notable
exceptions. One of the
most extreme of these is
3200 Phaethon. Another
asteroid, 944 Hidalgo, is
also thought by some to
be a defunct comet
because of its unusual
orbit.
Distribution and Kirkwood gaps
11
3.70
3.65
10
Semimajor axis (AU)
The solar system is a fascinating place in the
Universe. One century ago, it was thought that this
place is dominated by order, where several well defined
classes of celestial bodies are hierarchically disposed in
space. But this image has radically changed in our days.
Today, the advanced observational technique
transforms the mysterious dots of light from the sky in
well individualized bodies, with their own physical and
dynamical characteristics, history and origin. Some of
them have peculiar orbits, allowing a close encounter
with the Earth, sometimes at distances very
uncomfortable for us.
Many observational data, enriched by theoretical
results, reveal the «new» solar system as a tumultuous
region, where millions of interplanetary bodies of
different sizes follow their own orbital destinies, in a
chaotic manner, where gravitational captures and
collisional processes are current realities.
3.60
9
Semimajor axis
Introduction
8
7
3.55
3.50
3.45
6
3.40
5
3.35
0
200
400
600
800
1000
Years
Semi major axis variation of two
test particles, with almost identical
initial conditions, perturbed by Jupiter,
in the planar, circular, restricted threebody problem. Motion is dominated by
close encounters.
0
2000
4000
6000
8000
10000
Years
Semi major axis variation of two test
particles, with almost identical initial
conditions, perturbed by Jupiter, in the
planar, circular, restricted three-body
problem. Motion is free of close
encounters.
Dynamical classifications
Small bodies in the Solar System
- Asteroid belt
Within the main asteroid
belt are groups of asteroids
that cluster in certain orbital
elements. Such groups are
called families and assigned
the name of the lowest
numbered asteroid in the
family. Asteroid families are
thought to be formed when
an asteroid is disrupted in a
catastrophic
collision.
Theoretical studies indicate
that
such
catastrophic
collisions between asteroids
are common enough to
account for the number of
families observed.
About 95% of the known asteroids move in orbits between those
of Mars and Jupiter. These orbits, however, are not uniformly
distributed but exhibit "gaps" in the distribution of their semi major
axes. These so-called Kirkwood gaps are due to resonances with
Jupiter's orbital period. Gaps occur at 4:1, 7:2, 3:1, 5:2, 7:3, and 2:1
resonances, while concentrations occur at the 3:2 (Hilda group),
4:3 (Thule), and 1:1 (Trojan group) resonances.
a) Dynamical evolution of asteroid (3753) Cruithne on a horseshoe-like orbit
in respect to Earth; b). Dynamical evolutions of two Trojan asteroids, (1437)
Diomedes and (1208) Troilus, on tadpole-like orbits in respect to Jupiter. All these
motions are represented in corotational frames for 500 years. No close encounters
take place.
- Kuiper belt (old and scattered)
Source regions and dynamical transport mechanisms
- Comets (high e, unstable)
- Dust (debris disks)
Dynamics of close encounters
When the asteroid moves inside the gravitational
sphere of action of a planet, we consider that it
makes a close encounter with that planet.
¾The restricted three-body problem
Dynamical evolution of a fictitious asteroid placed in 3:1 mean motion resonance
with Jupiter, having the following keplerian elements at the initial epoch JD
2451545.0: a = 2:4893663 UA, e = 0:326086, I = 5±:34915, - = 0±, ! = 0± and M =
33±:2367. The dynamical system in which this integration was performed is Sun-the
four giant planets-asteroid. It becomes a NEA in about 100000 years.
r
2
⎧d r p
( 1 + mp ) r
rp
⎪ 2 = −G
rp3
⎪ dt
⎨ 2r
⎡1 r r
1 r
1 r ⎤
⎪d r
⎪ dt 2 = −G r 3 r + Gm p ⎢ d 3 ( r p − r ) − r 3 r p ⎥
p
⎣⎢
⎦⎥
⎩
¾Hill’s equations
⎧ ..
..
⎛ 3 mp ⎞
∂U H
⎪ξ − 2n p η = G ⎜⎜ 3 − 3 ⎟⎟ ξ =
ap d ⎠
∂ξ
⎪
⎝
⎨
.
η ∂U H
⎪ ..
⎪η + 2n p ξ = −Gm p d 3 = ∂η
⎩
¾Tisserand criterion
ap
a
+2
a
( 1 − e 2 ) ⋅ cos I = T ( const .)
ap
Resonant motions
¾ geometric and the dynamical interpretation of this
phenomenon, characterized by the libration of the
resonant argument (in standard notations)
σ p:q = pλp − qλ − ( p − q)ϖ
p/q resonance ratio
Variation of the resonant
argument for the asteroid
(4197) 1982 TA in respect
to Jupiter. The transition
between circulation and
libration shows that the
asteroid is captured in 10:3
mean motion resonance.
™Ovidiu Furdui, “Evolution of Celestial Bodies in the Vicinity of
Earth’s Orbit ”, Master Thesis, 2003
™ C.D. Murray, S.F. Dermott, Solar System Dynamics,
Cambridge University Press,1999
™ W.F. Bottke, R. Jedicke, A. Morbidelli, Understanding the
distribution of near-Earth asteroids, Science, 288, 2000.
™ A. Carusi, E. Dotto, Close Encounters of Minor Bodies with the
Earth, Icarus 124, 1996
™ Ch. Froeschl´e, A. Morbidelli, The secular resonances in the
solar system, in “Asteroids, Comets, Meteors 1993”, Kluwer
Academic Publishers,1994.
™ Near-Earth Object Dynamics Site (NEODyS).
™ Asteroid Dynamic Site (ASTDyS).
™ Minor Planet Center, The Smithsonian Astrophysical
Observatory
Acknowledgements
¾Öpik’s geometric formalism
Through a parametrization of the velocity vector u, we
obtain the relations between its orientation in space and the
corresponding heliocentric orbital elements of the body
References
Two types of dynamical behaviors of the MOID, computed for
one thousand years: a) periodic orbital approach - the asteroid
(1915) Quetzalcoatl; b) long-time orbital approach - the asteroid
1999 MN.
I would like to express gratitude to my supervisor
Prof. Dr. Rainer Spurzem, for his accurate guidance. I
am also pleased to acknowledge the support which
came from the staff of Astronomisches Rechen-Institut,
Heidelberg, to print the present poster.