1. On the determination of the probability of collisons of NEAS with the planets MACE 2006 Rudolf Dvorak ADG, Institute of Astronomy University of Vienna

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

3. Orbital elements of mercury

4. Eccentricity of Venus and Earth

6. Semimajor axis AU NumberofAsteroidsNumberofAsteroids

7. 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 because of the resonances caused by Jupiter ‚Fading‘ caused by collisions with terrestrial planets, by close encounter and ejections and by falling into the sun TWO MAIN QUESTIONS

8. 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

10. 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 suffers from larger perturbations from Jupiter than outside a resonance. The acting of small divisors because of the MMR causes large perturbations and shift the asteroid to larger and larger eccentricities (still smaller than 0.1). An asteroid in a resonances (Mean motion resonance MMR suffers from larger perturbations from Jupiter than outside a resonance. The acting of small divisors because of the MMR causes large perturbations and shift the asteroid to larger and larger eccentricities (still smaller than 0.1). Then a sudden (thousands of years) increase in eccentricities up to 0.3 and 0.4 leads to a region where the so called OVERLAPPING of resonances is acting. Then a sudden (thousands of years) increase in eccentricities up to 0.3 and 0.4 leads to a region where the so called OVERLAPPING of resonances is acting. This causes eventually an even larger eccentricity with later encounters to Jupiter (also of Mars). This causes eventually an even larger eccentricity with later encounters to Jupiter (also of Mars). These encounters lead also to highly chaotic orbits

11. PHASE SPACE OF THE PENDULUM x-axis: amplitude; y-axis: velocity 2 types of motion: inside libration, and outside circulation........ none

12. 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)

13. 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.

14. NEAs are grouped into three categories, named after 1221 Amor, 1862 Apollo, and 2062 Aten. Eros.Amors: Asteroids which cross Mars' orbit but do not quite reach the orbit of Earth. Eros. GeographosApollos: Asteroids which cross Earth's orbit with a period greater than 1 year. Geographos Ra-Shalom.Atens: Asteroids which cross Earth's orbit with a period less than 1 year. Ra-Shalom.

15. The dynamical evolution of 2062 ATEN for 500000 years

16. 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

17. 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

19. Dynamical evolution of a fictious Aten for 500000 years

21. Collisions per 1 billion years: Subatens 46 Atens 83 Apollos 52 Amors 3

22. 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...)