The document discusses an international workshop on asteroids known as trojans, which orbit around the Lagrange points of planets, focusing on simulations involving Venus, Earth, and Mars. It highlights findings from N-body simulations that indicate captures into Trojan orbits and into mean motion resonances are not rare events, particularly for terrestrial planets. The aim is to gather statistics on captures related to varying inclinations and eccentricities across different regions.

1. International Workshop
on Paolo FARINELLA
Pisa, 14-16 June 2010
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Rudolf Dvorak and Richard Schwarz
AstroDynamicsGroup
Institute for Astronomy, University of Vienna

2. Trojans are asteroids known to orbit
around the Lagrange points of a
planet:
We observe Trojans around
Jupiter and Neptune and also Mars
JUPITER: L4 point = 2603 L5 point = 1473.
2001 QR322 L4 29.404 31.330
NEPTUNE: 2004 UP10 L4 29.354 31.209
2005 TN53 L4 28.252 32.236
2005 TO74 L4 28.727 31.782
2006 RJ103 L4 29.320 30.975
2007 VL305 L4 28.132 32.105
(5261) Eureka 1990 MB L5 1.425 1.622
MARS: 1998 VF31 L5 1.371 1.677
1999 UJ7 L4 1.465 1.584
2007 NS2 L5 1.441 1.606

3. The five Lagrange points SUN + PLANET

4. x

8. Eccentricity of the primaries: 0.05, 0.1, 0.2, 0.3

10. Thersites
is jumping from L_5
to L_4
I could not find any other
orbit having such interesting
dynamics

11. Can the terrestrial planets
Venus, Earth and Mars capture Trojans ?
N-body simulations in the Newtonian dynamical model
Sun, Venus – Saturn + 100 massless asteroids
4 different regions
Regions a (VeTe): 0.72 < a < 0.98
Regions b (TeMa): 1.02 < a < 1.50
Regions c (MaJu1): 1.54 < a < 2.20
Regions d (MaJu2): 2.2 < a < 2.5

12. Near Earth Asteroids and mainbelt asteroids
Mercury Venus Earth Mars Jupiter
200 3:1 8:3
150
a b c d
100 5:2 2:1
1:1
4:1
50
Hungarias Trojans
NEAs 3:2
ATENs APOLLOs AMORs
Hildas
0
1 2 3 4 5
semimajor axis [AU]

13. N-body simulations in the Newtonian dynamical model
Sun, Venus – Saturn + 100 massless fictitious asteroids
Regions a (VeTe): 0.72 < a < 0.98
For different inclinations 0<i<20 degrees
For some selected orbits for verifications – MeNe

14. 100 fictitious Trojans between Venus and the Earth

15. 100 fictitious asteroids between Venus and Earth for 10^7 years
Capture into 1:1 MMR ?

16. Capture into 1:1 MMR ?
100 fictitious asteroids between Venus and Earth for 10^7 years

17. Test for capture into the 1:1 MMR with Venus for the 100 asteroids

18. L_5
HORSESHOE
Venus captures into Trojans
L_4
360 deg
0
360 deg
0
360 deg
0

19. Captures by the EARTH
360 deg
0 degree
360 deg
0 degree

21. L5-Trojan->L4-Trojan->Horshoe-orbit->escape

22. Conclusions (Work in progress)
- Captures into the 1:1 MMR for Venus, Earth and Mars
are not rare events
- Captures into Trojan orbits also are possible but not so
often
Jumping orbits are to observe for all terrestrial planets
- For larger inclinations more captures
Results only for region between Venus and Earth
Goal:
statistics about captures for all 4 regions dependant on
inclination and eccentricities
Statistics of jumps and duration of a capture