based on the definition given by Kasting et al. (1993).

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based on the definition given by Kasting et al. (1993).

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The Habitable Zone

based onthe

definition

given by

Kasting et al.

(1993).

Habitable Zone

- Zone around a star where liquid water can exist on the surface of a terrestial-like planet
- This zone depends on:
- the spectraltype , the mass , the age, …. of the star
- the orbit of the planet
- the mass, the composition, the atmosphere , ……of the planet
- the parameters of other planets in this system (mass, orbit, …)

Types of Habitable Zones:

- hot-Jupiter type
- Solar system type
- +(4) giant planet type: habitable moon or trojan planet

- 164 Extra-solar planetary systems
- 194 Planets near other solar-type stars
- 19 Mulitple planetary systems
- 21 Planets in binaries

- 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

Distribution of the detected Extra-Solar Planets

Mercury Earth Mars

Venus

Jupiter

- Multi-planetary systems

- Binaries

- Single Star and Single Planetary Systems

.

- the orbital line-of-sight inclination i is not known
from radial velocities measurements we get only

a lower limit for the planetary masses;

- the relative inclination irbetween planetary orbital planes is usually unknown.
- Are the orbital parameters reliable -- using two body keplerian fits
(the strong dynamical interactions between planets)

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

Major catastrophe in less than 100000 years

(S. Ferraz-Mello, 2004)

Numerical Methods

Chaos Indicators:

Fast Lyapunov Indicator (FLI)

C. Froeschle,R.Gonczi, E. Lega (1996)

Mean Exponential Growth

factor of Nearby Orbits

(MEGNO)

Cincotta & Simo (2000)

- Long-term numerical integration:
- Stability-Criterion:
- No close encounters within theHill‘ sphere
- (i)Escape time
- (ii) Study of the eccentricity:maximum eccentricity

Multi-planetary

systems

- Class Ia –> Planets in mean motion resonance (HD82943, Gliese876,HD128311,55Cnc,HD202206)
- Class Ib Low-eccentricity near-resonant planet pairs (47Uma)
- Class II Non-resonant planets with significant secular dynamics (55 Cnc, Ups And, HD12661, HD169830,HD37124, HD160691)
- Class IIIHierarchical planet pairs (HD168443, HD74156,HD11964,HD38529,55Cnc)

MMR

3:1

2:1

2:1

2:1

7:3/5:2

Class

II

III

Ia

III

Ia

III

III

II

Ia

II

III

II

II

Ib

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

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 in pairs, depending on the resonance

HD82943

Aligned

Anti-aligned

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

Planet m sin i a e w P

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

360

~

0.9

wc (deg)

320

0.8

280

0.7

240

0.6

ec

200

0.5

160

0.4

120

0.3

80

0.2

40

0.1

0.0

0

Due to high eccentricities

of the orbits and despite

relatively small semi-major

axis, the relative distances

between the two planets

may remain sufficiently

large over the whole

evolutionary time scale of

The system.

It was shown by several authors

(e.g. Rivera & Lissauer 2000, Laughlin & Chambers 2001, Chiang & Murray 2002; Lee & Peale 2002, 2003; Ji et al. 2003, 2004, Zhou & Sun 2003, Bois et al. 2003)

that the orbits in almost all multi-planet systems

(except HD38529, HD168443,HD74156)

are locked in the so-called

Apsidal Synchronous Precession (ASP)

meaning that the two orbital planes precess at the same rate, i.e. the relative apsidal longitude θ3 of two planetary orbits librates about 0 (aligned topology) or π (anti-aligned topology).

, where

- Low order Mean Motion Resonance +
- Favorable relative initial orbital phases of planets +
- High planetary eccentricities, especially of the outer planet +
- Anti-aligned Apsidal Synchronous Precession
=

NO close approaches between planets =>

NO strong dynamical interactions =>

STABILITY over long evolutionary timescale

- 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 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

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)

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

Unstable orbits

2:1 1.3 AU

3:1 1 AU

SR 0.8 – 0.9 AU

4:1 0.82 AU

Stable orbits

Between resonances

Terrestrial planet is possible!