International Workshop on Paolo FARINELLA

Pisa, 14-16 June 2010

Rudolf Dvorak and Richard Schwarz Rudolf Dvorak and Richard Schwarz AstroDynamicsGroupAstroDynamicsGroup

Institute for Astronomy, University of ViennaInstitute for Astronomy, University of Vienna

C A P T U R E D

T R O J A N S

Trojans are asteroids known to orbit around the Lagrange points of a planet:

We observe Trojans around

Jupiter and Neptune and also Mars

(5261) Eureka 1990 MB L5 1.425 1.622 1998 VF31 L5 1.371 1.677 1999 UJ7 L4 1.465 1.584 2007 NS2 L5 1.441 1.606

2001 QR322 L4 29.404 31.330 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

L4 point = 2603 L5 point = 1473. JUPITER:

NEPTUNE:

MARS:

The five Lagrange points SUN + PLANET

x

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

Thersites is jumping from L_5

to L_4

I could not find any other orbit having such interesting

dynamics

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

0

50

100

150

200

1 2 3 4 5semimajor axis [AU]

Mercury Venus Earth Mars Jupiter

3:1 8:3

5:2

1:1

3:2Hildas

Trojans

2:1

4:1

Hungarias

ATENs AMORsAPOLLOs

NEAs

Near Earth Asteroids and mainbelt asteroids

a b dc

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

100 fictitious Trojans between Venus and the Earth

100 fictitious asteroids between Venus and Earth for 10^7 years

Capture into 1:1 MMR ?

100 fictitious asteroids between Venus and Earth for 10^7 years

Capture into 1:1 MMR ?

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

Venus captures into Trojans

L_4

L_5 HORSESHOE

360 deg

0

360 deg

0

360 deg

0

360 deg

0 degree

Captures by the EARTH360 deg

0 degree

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

Conclusions (Work in progress)

- Captures into the 1:1 MMR for Venus, Earth and Marsare 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