Stable planetary orbits in multiple planetary systems

Institute for Astronomy University of Vienna

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

•102 planetary systems•117 planets•13 multiple planetary systems•15 planetary systems in binaries


Systems with two or more Planets

Star apl [AU] epl Mpl sin i [MJup] Ups And 0.059 0.012 0.69

0.829 0.28 1.19 2.53 0.27 3.75

HD 38529 0.129 0.29 0.78 3.68 0.36 12.7

55 Cnc 0.11 0.02 0.84 0.24? 0.34? 0.21? 5.9 0.16 4.05

Gliese 876 0.13 0.12 0.56 0.21 0.27 1.98

HD 74156 0.28 0.647 1.6 3.82 0.354 8.2

HD 168443 0.295 0.53 7.73 2.9 0.2 17.23

HD 37124 0.54 0.1 0.75 2.5 0.69 1.2

HD 82943 0.73 0.54 0.88 1.16 0.41 1.63

HD 169830 0.82 0.327 3.03 2.85 0.0 2.51

HD 12661 0.83 0.096 2.3 2.56 <0.1 1.57

HD 160691 1.5 0.31 1.7 2.3? 0.8? 1?

47 Uma 2.1 0.096 2.41 3.73 <0.1 0.76

Epsilon Eridani 3.3 0.608 0.86 40?? 0.3?? 0.1??

In collaboration with Prof. Erdi and his group from Budapest


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


Mstar = 1.05 MSun

HD 74156 bm = 1.6 Mjup sin ia = 0.28 AUe = 0.647

HD 74156 cm = 8.2 Mjup sin ia = 3.82 AUe = 0.354

HD 74156• The orbital parameters were taken from the Geneva group of observers• Masses are Minimum Masses

Institute for Astronomy University of ViennaThe resonant region and the habitable zone


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


e = 0.3 e = 0.35


e = 0.4 e = 0.45


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


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

•Stability-Criterion (FLI)


Inner Resonances with the outer planet: i = 0°


FLI:a = 1.3 AU


Outer Resonances with the inner planet: i = 0°


FLI:a = 0.7 AU




Outer Resonances with the inner planet:i = 20°




Outer Resonances with the inner planet:i = 50°


stable

0

5

10

15

20

25

30

0 20 40 60

Inclination

Inner Resonances

Outer Resonances


HD 74156 bm = 1.86 MJup

a = 0.294 AUe = 0.635

HD 74156 cm = 6.42 MJup

a = 3.44 AUe = 0.561

New Data

Institute for Astronomy University of ViennaHD 38529 HD 169830 HD

168443

Mstar = 1.39 MSun

HD 38529 bm = 0.78 MJup

a = 0.129 AUe = 0.29HD 38529 cm = 12.7 MJup

a = 3.68 AUe = 0.36

Mstar = 1.4 MSun

HD 169830 bm = 3.03 MJup

a = 0.82 AUe = 0.327HD 169830 cm = 2.51 MJup

a = 2.85 AUe = 0.0

Mstar = 1.01 MSun

HD 168443 bm = 7.73 MJup

a = 0.295 AUe = 0.53HD 168443 cm = 17.23 MJup

a = 2.9 AUe = 0.2




Stable planetary orbits in multiple planetary systems

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