1. Stable planetary orbits in multiple planetary systems

2. Extrasolar Planets • Systems with two or more Planets • Habitable Zones • HD 74156 • The System • Methods • Initial Conditions • Results • Eccentricities • Resonances • HD 38529, HD 169830, HD 168443

3. Extrasolar Planets • 102 planetary systems • 117 planets • 13 multiple planetary systems • 15 planetary systems in binaries

4. Systems with two or more Planets In collaboration with Prof. Erdi and his group from Budapest

5. Habitable Zones • Depends on the spectral type of the host star • Zones, where liquid water can exist • Planets have to stay on their orbits for • a very long time • The eccentricity should not be higher than 0.2

6. HD 74156 • The orbital parameters were taken from the • Geneva group of observers • Masses are Minimum Masses Mstar = 1.05 MSun HD 74156 b m = 1.6 Mjup sin i a = 0.28 AU e = 0.647 HD 74156 c m = 8.2 Mjup sin i a = 3.82 AU e = 0.354

7. The resonant region and the habitable zone

8. Region between the resonances (habitable zone) • Integration Methods • Numerical Integrations with the Lie-Series • Initial Conditions • Restricted 4-body problem • (star, 2 planets, massless test-planets) • Semimajor Axis: a = 0.6 to 1.4 AU • Eccentricities for the outer planet: • e = 0.3, 0.35, 0.4, 0.45 • Eccentricities for the inner planet: e = 0.5, ..., 0.7 • Integration Time: 105 years • Stability-Criterion • No close encounters within the Hill‘s sphere were allowed

9. e = 0.3 e = 0.35

10. e = 0.4 e = 0.45

11. Resonances • Integration Method • Numerical Integrations with the Lie-Series • The Fast Lyapunov Indicator (FLI) with a Bulirsch Stoer • integrator • Initial Conditions (Lie-Integrator) • Restricted 4-body problem • (star, 2 planets, massless test-planets) • Inner Resonances with the outer planet • Outer Resonances with the inner planet • Mean Anomaly: 0°, 30°,..., 330° • Different mean anomalies of the known planets • Inclinations: i = 0°, 10°,..., 50° • Integration Time: 105 years • Initial Conditions (FLI) • Different mean anomalies of the known planets • Semimajor axes: a = 0.7 and 1.3 AU

12. Stability-Criterion (Lie-Integrator) • No close encounters within the Hill‘s sphere were allowed • Stability-Criterion (FLI)

13. Inner Resonances with the outer planet: i = 0°

14. FLI: a = 1.3 AU

15. Outer Resonances with the inner planet: i = 0°

16. FLI: a = 0.7 AU

17. Inner Resonances with the outer planet: i = 20°

18. Outer Resonances with the inner planet: i = 20°

19. Inner Resonances with the outer planet: i = 50°

20. Outer Resonances with the inner planet: i = 50°

22. New Data HD 74156 b m = 1.86 MJup a = 0.294 AU e = 0.635 HD 74156 c m = 6.42 MJup a = 3.44 AU e = 0.561

23. HD 38529 HD 169830 HD 168443 Mstar = 1.39 MSun HD 38529 b m = 0.78 MJup a = 0.129 AU e = 0.29 HD 38529 c m = 12.7 MJup a = 3.68 AU e = 0.36 Mstar = 1.4 MSun HD 169830 b m = 3.03 MJup a = 0.82 AU e = 0.327 HD 169830 c m = 2.51 MJup a = 2.85 AU e = 0.0 Mstar = 1.01 MSun HD 168443 b m = 7.73 MJup a = 0.295 AU e = 0.53 HD 168443 c m = 17.23 MJup a = 2.9 AU e = 0.2