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Stable planetary orbits in multiple planetary systems

Stable planetary orbits in multiple planetary systems. 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. Extrasolar Planets

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Stable planetary orbits in multiple planetary systems

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  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°

  21. 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

  22. 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

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