ClassificationClassification ofof thethe knownknown multimulti-- planetaryplanetary systemssystems (S.Ferraz(S.Ferraz--Mello,Mello, 2005)2005)
ClassClass IaIa ––>> PlanetsPlanets inin meanmean motionmotion resonanceresonance (HD82943, Gliese876,HD128311,55Cnc,HD202206) ClassClass IbIb ÆÆ LowLow--eccentricityeccentricity nearnear--resonantresonant planetplanet pairspairs (47Uma) ClassClass IIIIÆÆ NonNon--resonantresonant planetsplanets withwith significantsignificant secularsecular dynamicsdynamics (55 Cnc, Ups And, HD12661, HD169830,HD37124, HD160691) ClassClass IIIIIIÆÆ HierarchicalHierarchical planetplanet pairspairs (HD168443, HD74156,HD11964,HD38529,55Cnc) Class II -- Non-resonant planets with significant secular dynamics
Planet pairs of this class can have strong gravitational interactions, where long-term variations are ascribed to secular perturbations, large variations of the eccentricities and dynamical effects like the alignment and anti-alignment for the apsidal lines.
For the long-term stability of such a system, it is not necessary hat the planets are in MMR. Star Planet mass_P a_P e_P Period [M_Sun] [M_Jup] [AU] [days]
55 Cnc e 0.045 0.038 0.17 2.808 (1.03) b 0.784 0.115 0.02 14.67
HD169830 b 2.88 0.81 0.31 225.62 (1.4) c 4.04 3.6 0.33 2102
HD37124 b 0.72 0.54 0.1 153 (0.91) c ? 1.3 2.5 0.7 1595
(Ups And, HD12661, HD160691) (in collaboration with Erdi and Sandor)
HD 169830 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 HD160691 b HD160691 c A [AU]: 1.5 2.3 e: 0.31 0.8 M .sin i: 1.7 1.0 [M_jup]
MEGNO – Stability map
Stability condition: 2:1 mean motion resonance (exact location: a_c=2.381 AU)
Bois, E., Kiseleva-Eggleton, L., Rambaux, N., Pilat-Lohinger, E., 2003, ApJ 598, 1312 0.9 0.8 Due to high eccentricities 0.7 of the orbits and despite 0.6 relatively small semi-major ec 0.5 axis, the relative distances 0.4 between the two planets 0.3 may remain sufficiently 0.2 large over the whole 0.1 evolutionary time scale of 0.0 The system. thethe orbitsorbits areare lockedlocked inin thethe soso--calledcalled Apsidal Synchronous Precession (ASP) meaning that the two orbital planes precess at the same rate
Æ θθ3 of two planetary orbits librateslibrates aboutabout 00(aligned)(aligned) oror ππ (anti(anti--alignedaligned ).).
where
AA suitablesuitable mechanismmechanism forfor compactcompact multimulti-- planetaryplanetary systemssystems
¾ Low order Mean Motion Resonance + ¾ Favorable relative initial orbital phases of planets + ¾ High planetary eccentricities, especially of the outer planet + ¾ Anti-aligned Apsidal Synchronous Precession = NO close approaches between planets => NO strong dynamical interactionsinteractions =>=> STABILITY over long evolutionary timescale PlanetPlanet mm sinsin ii aa ee ωω PP
HD160691b 1.67 +/- 0.11 1.50 +/- 0.02 0.2 +/- 0.03 294 +/- 9 645.5 +/- 3
c 3.1+/- 0.71 4.17+/- 0.07 0.57+/- 0.1 161 +/- 8 2986+/-30
d 0.04405 0.09 0 (+0.02) 4+/- 2 9.55+/0.03 5:1 9:2 0.9
11 0 e e 0.4 4 3.5 newnew systemsystem 0.3 3 2.5 0.2 2 1.5 0.1 1 0.5 HD160691HD160691 0 0 1.35 1.4 1.45 1.5 1.55 1.6 1.65 a1 (UA)
HD160691 e2=0.21 5:1 9:2 4:1 5:1 9:2 0.9 0.9
0 11 11 0 e e e e 0.4 0.4 2.4 2.4 0.3 0.3 2.2 2.2 2 0.2 2 0.2 1.8 1.8 0.1 0.1 1.6 1.6 0 0 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.35 1.4 1.45 1.5 1.55 1.6 1.65 a1 (UA) a1 (UA)
ClassificationClassification ofof thethe knownknown multimulti-- planetaryplanetary systemssystems (S.Ferraz(S.Ferraz--Mello,Mello, 2005)2005)
ClassClass IaIa ––>> PlanetsPlanets inin meanmean motionmotion resonanceresonance (HD82943, Gliese876,HD128311,55Cnc,HD202206) ClassClass IbIb ÆÆ LowLow--eccentricityeccentricity nearnear--resonantresonant planetplanet pairspairs (47Uma) ClassClass IIIIÆÆ NonNon--resonantresonant planetsplanets withwith significantsignificant secularsecular dynamicsdynamics (55 Cnc, Ups And, HD12661, HD169830,HD37124, HD160691) ClassClass IIIIIIÆÆ HierarchicalHierarchical planetplanet pairspairs (HD168443, HD74156,HD11964,HD38529,55Cnc) Class III -- Hierarchical planet pairs
Roughly speaking this class is for all planet pairs with a large ratio of their orbital periods -- P1/P2 > 10.
Due to the large ratio of periods the gravitational interaction are not so strong like in class II and the probability of a capture in a MMR is negligible.
The weaker interactions lead to stable motion in the numerical simulations, even if the orbits of the planet are not so good determined. HD 74156 • The orbital parameters were taken from the Mstar = 1.05 MSun Geneva group of observers • Masses are Minimum Masses
HD 74156 b m sini = 1.6 Mjup a = 0.28 AU e = 0.647
HD 74156 c m sin i= 8.2 Mjup a = 3.82 AU e = 0.354 The resonant region and the habitable zone e= 0.30
e=0.35
e=0.40
e=0.45 • Initial Conditions
• 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 Inner Resonances with the outer planet: i = 0° Outer Resonances with the inner planet: i = 0° 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 (in collaboration with Erdi and Sandor) M = 1.39 M HD 38529 star Sun HD 168443 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 = 1.01 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