Institute for Astronomy University of Vienna Stable planetary

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

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

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

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

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

Institute for Astronomy University of Vienna HD 74156 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 • The orbital parameters were taken from the Geneva group of observers • Masses are Minimum Masses

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

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. 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

Institute for Astronomy University of Vienna e = 0. 35

Institute for Astronomy University of Vienna e = 0. 45

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: 105 years • Initial Conditions (FLI) • Different mean anomalies of the known planets • Semimajor axes: a = 0. 7 and 1. 3 AU

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

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

Institute for Astronomy University of Vienna FLI: a = 1. 3 AU

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

Institute for Astronomy University of Vienna FLI: a = 0. 7 AU

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

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

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

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

Institute for Astronomy University of Vienna

Institute for Astronomy University of Vienna 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

Institute for Astronomy University of Vienna HD 38529 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 HD 169830 HD 168443 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

Institute for Astronomy University of Vienna

Institute for Astronomy University of Vienna

Institute for Astronomy University of Vienna