N-body Simulationen
Gas aus dem System entwichen
Gasplaneten und eine Scheibe von Planetesimalen
Störende Wechselwirkung zwischen Planeten und den Planetesimalen Migration The Kuiper Belt
- 3 Populations • Classical (stable) Belt • Resonant Objects, 3/4, 2/3, 1/2 with Neptune • Scattered Disk Objects Orbital distribution cannot be explained by present planetary perturbations planetary migration Standard migration model: Fernandez & Ip (1984) - Semi-major axes of the planets - ~ Kuiper-belt structure - constrains the size of the initial
disc (<30-35 AU , m~35-50 MΕ) Problem #1: The final orbits of the planets are circular
Problem #2: If everything ended <108 yr, what caused the … Ein neues Migration Modell
Sind die Planeten weiter auseinander (Neptun bei ~20 AU) - leichte Migration
Ein kompaktes System kann instabil werden aufgrund von Resonanzen zwischen den Planeten (und nicht close encounters) ! Alessandro Morbidelli (OCA, France) Kleomenios Tsiganis (Thessaloniki, Greece) Hal Levison (SwRI, USA) Rodney Gomes (ON-Brasil)
Tsiganis et al. (2005), Nature 435, p. 459 Morbidelli et al. (2005), Nature 435, p. 462 Gomes et al. (2005), Nature 435, p. 466
Nice Model N-body simulations:
Sun + 4 giant planets + Disc of planetesimals
43 simulations t~100 My: ( e , sinΙ ) ~ 0.001 2/3 aJ=5.45 AU , aS=aJ2 - Δa , Δa < 0.5 AU
U and N initially with a < 17 AU ( Δa > 2 AU )
Disc: 30-50 ME , edge at 30-35 AU (1,000 – 5,000 bodies)
8 simulations for t ~ 1 Gy with aS= 8.1-8.3 AU
Evolution of the planetary system
• A slow migration phase with (e,sinI) < 0.01, followed by • Jupiter and Saturn crossing the 1:2 resonance eccentricities are increased chaotic scattering of U,N and S (~2 My) inclinations are increased • Rapid migration phase: 5-30 My for 90% Δa Crossing the 1:2 resonance The final planetary orbits Statistics: • 14/43 simulations (~33%) failed (one of the planets left the system)
• 29/43 67% successful simulations: all 4 planets end up on stable orbits, very close to the observed ones • Red (15/29) U – N scatter • Blue (14/29) S-U-N scatter Better match to real solar system data Jupiter Trojans
Trojans = asteroids that share Jupiter’s orbit but librate around the Lagrangian points, δλ ~ ± 60o
We assume a population of Trojans with the same age as the planet
A simulation of 1.3 x 106 Trojans all escape from the system when J and S cross the resonance !!!
Is this a problem for our new migration model? … No! Chaotic capture in the 1:1 resonance
• The total mass of captured Trojans depends on migration speed
• For 10 My < Tmig < 30 My we trap 0.3 - 2 MTro
This is the first model that explains the distribution of Trojans in the space of proper elements ( D , e , I ) The timing of the instability
• What was the initial distribution of planetesimals like ?
1 My < Τinst < 1 Gyr
Depending on the density (or inner edge) of the disc
LHB timing suggests an external disc of planetesimals in agreement with the short dynamical lifetimes of particles in the proto-solar nebula Late Heavy Bombardment
A brief but intense bombardment of the inner solar system, presumably by asteroids and comets ~ (3.9±0.1) Gyrs ago, i.e. ~ 600 My after the formation of the planets
Petrological data (Apollo, etc.) show: • Same age for 12 different impact sites • Total projectile mass ~ 6x1021 g • Duration of ~ 50 My
We need a huge source of small bodies, which stayed intact for ~600 My and some sort of instability, leading to the bombardment of the inner solar system 1 Gyr simulation of the young solar system The Lunar Bombardment
Two types of projectiles: asteroids / comets
~ 9x1021 g comets ~ 8x1021 g asteroids
(crater records 6x1021 g)
The Earth is bombarded by ~1.8x1022 g comets (water) 6% of the oceans
Compatible with D/H measurements ! Conclusions The NICE model assumes:
An initially compact and cold planetary system with PS / PJ < 2 and an external disc of planetesimals
3 distinct periods of evolution for the young solar system:
1. Slow migration on circular orbits 2. Violent destabilization 3. Calming (damping) phase
Main observables reproduced: 1. The orbits of the four outer planets (a,e,i) 2. Time delay, duration and intensity of the LHB 3. The orbits and the total mass of Jupiter Trojans Solar system architecture
• Inner (terrestrial) planets: Mercury – Venus – Earth - Mars (1.5 AU) • Main Asteroid Belt (2 – 4 AU) • Gas giants: Jupiter (5 AU), Saturn (9.5 AU) • Ice giants: Uranus (19 AU), Neptune (30 AU) • Kuiper Belt (36 – 50 AU) + Pluto + ... Are there Planets outside the Solar System ? First answer : 1992 Discovery of the first Extra- Solar Planet around the pulsar PSR1257+12 (Wolszczan &Frail)
Are there Planets moving around other Sun-like stars ? The EXO Planet: 51 Peg b
Mass: M sin i = 0.468 Discovered by: m_Jup Michel Mayor semi-major axis: a = Didier Queloz 0.052 AU period: p = 4.23 days eccentricity: e = 0 a of Mercury: 0.387 AU Status of Observations
452 Extra-solar planets
43 Mulitple planetary systems
43 Planets in binaries Questions
How frequent are other planetary systems ? Are they like our Solar System ? (no. of planets, masses, radii, albedos, orbital paramenters , …. ) What type of environments do they have? (atmospheres, magnetosphere, rings, … ) How do they form and evolve ? How do these features depend on the type of the central star (mass, chemical composition, age, binarity, … ) ? Extra-solare Planeten
ca. 130 Planeten entdeckt
massereich (~Mjup) enge Umlaufbahnen
Radialgeschwindigkeits- messungen Distribution of the detected Extra-Solar Planets
Mercury Earth Mars Jupiter Venus Mass distribution 55 Cancri
5 Planeten bei 55 Cnc: 55Cnc d -- the only known Jupiter-like planet in Jupiter- distance
Binary: a_binary= 1000 AU Facts about Extra-Solar Planetary Systems:
Only 28% of the detected planets have masses < 1 Jupitermass About 33% of the planets are closer to the host-star than Mercury to the Sun Nearly 60% have eccentricities > 0.2 And even 40% have eccentricities > 0.3 Sources of uncertainty in parameter fits:
the unknown value of the orbital line-of-sight inclination i allows us to determine from radial velocities measurements only the lower limit of planetary masses;
the relative inclination ir between planetary orbital planes is usually unknown.
In most of the mulitple-planet systems, the strong dynamical interactions between planets makes planetary orbital parameters found – using standard two-body keplerian fits – unreliable (cf. Eric Bois)
All these leave us a substantial available parameter space to be explored in order to exclude the initial conditions which lead to dynamically unstable configurations Multi-planetary systems
Binaries
Single Star and Single Planetary Systems Major catastrophe in less than 100000 years
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(S. Ferraz-Mello, 2004) Numerical Methods Chaos Indicators: Fast Lyapunov Indicator Long-term numerical (FLI) integration: C. Froeschle, R.Gonczi, E. Lega (1996) Stability-Criterion: No close encounters within the Hill‘ sphere MEGNO RLI (i)Escape time Helicity Angle (ii) Study of the eccentricity: maximum eccentricity LCE The Fast Lyapunov Indicator (FLI) (see Froeschle et al., CMDA 1997) a fast tool to distinguish between regular and chaotic motion length of the largest tangent vector:
FLI(t) = sup_i |v_i(t)| i=1,.....,n
(n denotes the dimension of the phase space) it is obvious that chaotic orbits can be found very quickly because of the exponential growth of this vector in the chaotic region. For most chaotic orbits only a few number of primary revolutions is needed to determine the orbital behavior. Multi-planetary systems
Binaries
Single Star and Single Planetary Systems www.univie.ac.at/adg/exostab/
ExoStab appropriate for single-star single-planet system
- Stability of an additional planet - Stability of the habitable zone (HZ) - Stability of an additional planet with repect to the HZ Stability maps
Inner region (Solar system type)
Outer region (Hot-Jupiter-type) Computations
distance star-planet: 1 AU variation of - a_tp:[0.1,0.9] [1.1,4] AU - e_gp: 0 – 0.5 - M_gp: 0 and 180 deg - M_tp: [0, 315] deg
Dynamical model: restricted 3 body problem
Methods: (i) Chaos Indicator: - FLI (Fast Lyapunov) - RLI (Relative Lyapunov) (ii) Long-term computations - e-max ANIMATION How to use the catalogue
HD114729: m_p=0.82 [Mjup] m=0.001 (0.93 [Msun]) a_p= 2.08 AU e_p=0.31 HZ: 0.7 – 1.3 AU m = 0.005
HD10697: m_p= 6.12 [Mjup] (1.15 Msun) a_p = 2.13 AU HZ: 0.85 – 1.65 AU e_p = 0.11 Multi-planetary systems: Multi-Planeten Systeme
Stabiliät des Planetensystems muss überprüft werden
Gliese 876
d c b a [AU] 0.0208 0.13 0.2078 P[days] 1.9377 30.1 60.94 e 0. 0.27 0.0249 m sin i 0.018 0.56 1.935 CoRoT Exo 7 b Spacing of Planets -- Hill criterion
Convenient rough proxy for the stability of planetary systems
In its simpliest form for planets of equal mass on circular orbits around a sun-like star Two adjacent orbits with separation
Dai=ai+1 – ai have to fullfill: mass-dependency
closer spacing for smaller mass planets Determination of Planetary Periods:
Using Kepler‘s third law in its log-differential Form (d ln(P) = 3/2 d ln(a)) obtain the periods Pi
-1 (D Pi / Pi) gives the periodic scaling for the planets Numerical Study
Fictitious compact planetary systems: Sun-mass star up to 10 massive planets (4/17/30 Earth- masses) aP= 0.01AU …… 0.26 AU e, incl, omega, Omega, M: 0 Compact planetary system using the Hill- Criterion: mp=1Earth-mass Class Ia -- Planets in mean-motion resonance
This class contains planet pairs with large masses and eccentric orbits that are relatively close to each other, where strong gravitational interactions occur.
Such systems remain stable if the two planets are in mean motion resonance (MMR). Star Planet mass_P a_P e_P Period [M_Sun] [M_Jup] [AU] [days]
GJ 876 b 0.597 0.13 0.218 30.38 (0.32) c 1.90 0.21 0.029 60.93
55 Cnc b 0.784 0.115 0.02 14.67 (1.03) c 0.217 0.24 0.44 43.93
HD82942 b 1.7 0.75 0.39 219.5 (1.15) c 1.8 1.18 0.15 436.2
HD202206 b 17.5 0.83 0.433 256.2 (1.15) c 2.41 2.44 0.284 1296.8 How important are the resonances for the long term stability of multi-planet systems?
Systems in 2:1 resonance
GJ876 b GJ876c HD82 b HD82 c HD160 b HD160 c A [AU]: 0.21 0.13 1.16 0.73 1.5 2.3 e: 0.1 0.27 0.41 0.54 0.31 0.8 M .sin i: 1.89 0.56 1.63 0.88 1.7 1.0 [M_jup]
Gliese 876 HD82943 HD160691 Major catastrophe in less than 100000 years
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(S. Ferraz-Mello, 2004) HD 82943 c,b : A case study (S.Ferraz-Mello)
. HD82943
Aligned
Anti-aligned Periastra in the same direction
S - P1 - P2 S P A2 A1 1 P2 S - A1 - A2 A1 - S - P2 P1 - S - A2 Periastra in opposite Periastra in the same direction directions
S - P1 - A2 S - A1- P2 A P S P A 2 1 2 1 P1 - S – P2 A1 - S – A2
Equivalent pairs, depending on the Periastra in opposite directions resonance