On the determination of the probability of collisons of NEAS

with the planets

MACE 2006

Rudolf DvorakADG, Institute of Astronomy

University of Vienna

Asteroids (Minor Planets) are present everywhere in our Solar System

Orbital elements of mercury

Eccentricity of Venus and Earth

0

50

100

150

200

1 2 3 4 5

num

ber

of a

ste

roid

s

sem im ajor axis [AU]

M ercury Venus Earth M ars Jupiter

3:1 8:3

5:2

1:1

3:2Hildas

Trojans

2:1

4:1

Hungarias

ATENs AM ORsAPOLLO s

N E As

Semimajor axis AU

Number of Asteroids

• Transport of Asteroids to become NEAsTransport of Asteroids to become NEAs• Fading of the NEAsFading of the NEAs

Transport from the mainbelt connected to the removal Transport from the mainbelt connected to the removal because of the resonances caused by Jupiterbecause of the resonances caused by Jupiter

‚‚Fading‘ caused by collisions with Fading‘ caused by collisions with terrestrial planets, by close encounter terrestrial planets, by close encounter and ejections and by falling into the sunand ejections and by falling into the sun

TWO MAIN QUESTIONS

• O‘Brien and Greenberg, Icarus 178 (2005)• The collisional and dynamical evolution of the main-

belt and NEA size distribution

• Collisions in the main-belt (Öpik, 1951)• Mean Motion Resonances (MMR)• Yarkovsky effect (radiation force on asteroids)• Resonance escape routes (overlapping of MMR)• Secular resonances• Resonant escape routes NEAs

• The removal from resonances is caused by CHAOS!The removal from resonances is caused by CHAOS!

An asteroid in a resonances (Mean motion resonance MMR An asteroid in a resonances (Mean motion resonance MMR suffers from larger perturbations from Jupiter than outside a suffers from larger perturbations from Jupiter than outside a resonance. The acting of small divisors because of the MMR resonance. The acting of small divisors because of the MMR causes large perturbations and shift the asteroid to larger and causes large perturbations and shift the asteroid to larger and larger eccentricities (still smaller than 0.1).larger eccentricities (still smaller than 0.1).

Then a sudden (thousands of years) increase in eccentricities Then a sudden (thousands of years) increase in eccentricities up to 0.3 and 0.4 leads to a region where the so called up to 0.3 and 0.4 leads to a region where the so called OVERLAPPING of resonances is acting.OVERLAPPING of resonances is acting.

This causes eventually an even larger eccentricity with later This causes eventually an even larger eccentricity with later encounters to Jupiter (also of Mars). encounters to Jupiter (also of Mars).

These encounters lead also to highly chaotic orbitsThese encounters lead also to highly chaotic orbits

PHASE SPACE OF THE PENDULUMx-axis: amplitude; y-axis: velocity

2 types of motion: inside libration, and outside

circulation ........

none

The eccentricity of the asteroid is proportional to the distance from the centeer

Libration – small ‚circles‘...e small

Circulation – large ‚banana‘ ... e large

Development of the eccentricity of a fictitious asteroid inside the 3:1 MMR for 1 million years

after J.Wisdom (1983)

NEAR EARTH ASTEROIDS

• Asteroids that have orbits that bring them within 1.3 AU (195 million kilometers) of the Sun are known as Earth-approaching or Near-Earth Asteroids (NEAs).

• NEAs are fragments which came from the main belt by a combination of asteroid collisions and the gravitational

influence of Jupiter.

• NEAs are grouped into three categories, named after 1221 Amor, 1862 Apollo, and 2062 Aten.

• Amors: Asteroids which cross Mars' orbit but do not quite reach the orbit of Earth. Eros.Eros.

• Apollos: Asteroids which cross Earth's orbit with a period greater than 1 year. GeographosGeographos

• Atens: Asteroids which cross Earth's orbit with a period less than 1 year. Ra-Shalom.Ra-Shalom.

The dynamical evolution of 2062 ATEN for 500000 years

• Billiard: Reflexion on the wall• For two ‚orbits‘ which deviate for a small angle this

angle doubles after each encounter• NEAs Reflexion whenever comes close to a planet

CHAOTIC MOTION CHAOTIC MOTION

Dvorak and Freistetter, Planetary and Space Science:Dynamical Evolution and collisions of asteroids with the

earth

• 1000 fictitious NEAs were investigated in a long term integrations

• Dynamical model: Venus to Saturn• Grid of initial conditions in• Semimajor axes 0.7 < a < 1.45• Eccentricity 0.1 < e < 0.8• RESULTS :• 1. Flow between these groups• 2. collision probabilities

Dynamical evolution of a fictious Aten for 500000 years

Collisions per 1 billion years:

Subatens 46

Atens 83

Apollos 52

Amors 3

Conclusions

• Collision probabilities:• ATEN every 500 million years• APOLLO every every 200 million years• Consequences: NEAs should disappear

within some 10 million years BUT• Because of a flux from the main belt

and outer parts a steady state of the NEA population is postulated (Wetherhill, Greenberg, O‘Brien...)