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:

- Semi-major axes of the planets

- ~ Kuiper-belt structure

- constrains the size of the initial

disc (<30-35 AU , m~35-50 MΕ)

Fernandez & Ip (1984)

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

aJ=5.45 AU , aS=aJ22/3 - 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 – 5,000bodies)

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

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

• What was the initial

distribution of

planetesimals like ?

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

Late Heavy Bombardment

Petrological data (Apollo, etc.) show:

• Same age for 12 different impact sites

• Total projectile mass ~ 6x1021 g

• Duration of ~ 50 My

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

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 !

ConclusionsThe 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

• 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 + ...

Solar system architecture

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

m_Jup

semi-major axis: a =

0.052 AU

period: p = 4.23 days

eccentricity: e = 0

a of Mercury: 0.387 AU

Discovered by:Michel Mayor Didier Queloz

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

VenusJupiter

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

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

Facts about Extra-Solar

Planetary Systems:

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

Binaries

Single Star and Single Planetary Systems

Multi-planetary systems

Major catastrophe in less than 100000 years

0 20000 40000 60000TIME (yr)

0.00

4.00

8.00

SEMI-MAJO

R A

XIS

(S. Ferraz-Mello,

2004)

Long-term numerical

integration:

Stability-Criterion:No close encounters within

the Hill‘ sphere

(i)Escape time(ii) Study of the eccentricity:

maximum eccentricity

Chaos Indicators:

Fast Lyapunov Indicator

(FLI)

C. Froeschle, R.Gonczi, E. Lega

(1996)

MEGNO

RLI

Helicity Angle

LCE

Numerical Methods

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.

Binaries

Single Star and Single Planetary Systems

Multi-planetary systems

www.univie.ac.at/adg/exostab/

ExoStabappropriate 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)

Computationsdistance star-planet: 1 AUvariation 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]

(0.93 [Msun]) a_p= 2.08 AU

e_p=0.31

m=0.001

HZ: 0.7 – 1.3 AU

m = 0.005

HD10697: m_p= 6.12 [Mjup]

(1.15 Msun) a_p = 2.13 AU

e_p = 0.11

HZ: 0.85 – 1.65 AU

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

(D Pi / Pi) gives the periodic scaling for the planets-1

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?

Gliese 876 HD82943 HD160691

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]

Major catastrophe in less than 100000 years

0 20000 40000 60000TIME (yr)

0.00

4.00

8.00

SEMI-MAJO

R A

XIS

(S. Ferraz-Mello,

2004)

HD 82943 c,b : A case study (S.Ferraz-Mello)

.

HD82943

Aligned

Anti-aligned

S PPA A 1 212

Periastra in the same direction

S PPA A1212

Periastra in opposite directions

Periastra in the same direction

S - P1 - P2

S - A1 - A2

A1 - S - P2

P1 - S - A2

Periastra in opposite directions

S - P1 - A2

S - A1- P2

P1 - S – P2

A1 - S – A2

Equivalent pairs, depending on the

resonance