1. Institute for Astronomy University of Vienna Stable planetary orbits in multiple planetary systems

2. Institute for Astronomy University of Vienna 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. Institute for Astronomy University of Vienna Extrasolar Planets 102 planetary systems 117 planets 13 multiple planetary systems 15 planetary systems in binaries

4. Institute for Astronomy University of Vienna Systems with two or more Planets In collaboration with Prof. Erdi and his group from Budapest

5. Institute for Astronomy University of Vienna 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. Institute for Astronomy University of Vienna M star = 1.05 M Sun HD 74156 b m = 1.6 M jup sin i a = 0.28 AU e = 0.647 HD 74156 c m = 8.2 M jup sin i a = 3.82 AU e = 0.354 HD 74156 The orbital parameters were taken from the Geneva group of observers Masses are Minimum Masses

7. Institute for Astronomy University of Vienna The resonant region and the habitable zone

8. Institute for Astronomy University of Vienna 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: 10 5 years Stability-Criterion No close encounters within the Hill‘s sphere were allowed

9. Institute for Astronomy University of Vienna e = 0.3e = 0.35

10. Institute for Astronomy University of Vienna e = 0.4e = 0.45

11. Institute for Astronomy University of Vienna 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: 10 5 years Initial Conditions (FLI) Different mean anomalies of the known planets Semimajor axes: a = 0.7 and 1.3 AU

12. Institute for Astronomy University of Vienna Stability-Criterion (Lie-Integrator) No close encounters within the Hill‘s sphere were allowed Stability-Criterion (FLI)

13. Institute for Astronomy University of Vienna Inner Resonances with the outer planet: i = 0°

14. Institute for Astronomy University of Vienna FLI: a = 1.3 AU

15. Institute for Astronomy University of Vienna Outer Resonances with the inner planet: i = 0°

16. Institute for Astronomy University of Vienna FLI: a = 0.7 AU

17. Institute for Astronomy University of Vienna Inner Resonances with the outer planet: i = 20°

18. Institute for Astronomy University of Vienna Outer Resonances with the inner planet: i = 20°

19. Institute for Astronomy University of Vienna Inner Resonances with the outer planet: i = 50°

20. Institute for Astronomy University of Vienna Outer Resonances with the inner planet: i = 50°

21. Institute for Astronomy University of Vienna

22. Institute for Astronomy University of Vienna HD 74156 b m = 1.86 M Jup a = 0.294 AU e = 0.635 HD 74156 c m = 6.42 M Jup a = 3.44 AU e = 0.561 New Data

23. Institute for Astronomy University of Vienna HD 38529 HD 169830 HD 168443 M star = 1.39 M Sun HD 38529 b m = 0.78 M Jup a = 0.129 AU e = 0.29 HD 38529 c m = 12.7 M Jup a = 3.68 AU e = 0.36 M star = 1.4 M Sun HD 169830 b m = 3.03 M Jup a = 0.82 AU e = 0.327 HD 169830 c m = 2.51 M Jup a = 2.85 AU e = 0.0 M star = 1.01 M Sun HD 168443 b m = 7.73 M Jup a = 0.295 AU e = 0.53 HD 168443 c m = 17.23 M Jup a = 2.9 AU e = 0.2

24. Institute for Astronomy University of Vienna

25. Institute for Astronomy University of Vienna

26. Institute for Astronomy University of Vienna