Stability of Extra-solar Planetary Systems C. Beaugé (UNC) S. Ferraz-Mello (USP) T. A. Michtchenko...

Stability of Extra-solar Planetary Systems

C. Beaugé (UNC)S. Ferraz-Mello (USP)T. A. Michtchenko (USP)

USP-UNC team on Exoplanets:

0.01 0.10 1.00 10.00a(AU) x Star Mass (Msun)

0.1

1.0

10.0

Plan

et M

ass (M

jup)

x sin

i

update 01/10/2004 before 06/1999 after 05/2003

(Sequence Number)1

10

100Pe

riod

Ratio

III

II

I

28 planets in 12 multiplanet systems (*)

(*) Eri not included since too uncertain.

3 (4) classes

Ia – Planets in mean-motion resonances

Ib – Low-eccentricity Non-resonant Planet Pairs

II – Non-resonant Planets with a Significant Secular Dynamcis

III – Weakly interacting Planet Pairs

CLASS Ia – Planet Pairs in Mean Motion Resonance

Star Period m.sin i a Period Eccentricity planets ratio (m_Jup) (AU) (days) HD 82943 1.99 1.7 0.75 219.5 0.39 c,b 1.8 1.18 436.2 0.15 GJ 876 2.02 0.56 0.13 30.12 0.27 c,b 1.89 0.21 61.02 0.10 55 Cnc 2.99 0.78 0.115 14.7 0.02 b,c(?) 0.22 0.24 43.9 0.44

Data from Ferraz-Mello et al (2004) [HD 82943], Fisher et al. (2003) [GJ 876], McArthur et al.(2004) [55 Cnc].

GJ 876

Solar System without Saturn50 M

yr

Collis

ion

Ch

aos

Ord

er

025

40

60

80

Grid: 33x251Ref: Michtchenko (unpub.)

8.0 8.2 8.4 8.6 8.8 9.0 9.2 9.4 9.6 9.8 10.0 10.2 10.4

semi-major axis (AU)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

ecce

ntric

ity

2/1 7/3 5/2 8/3 .

EARTH

MARSB

CVENUS

HD 82943

M0=1.15 Msun

m1=1.7 Mjup/sin im2=1.8 Mjup/sin i

-0.2 -0.1 0.0 0.1 0.2-0.2

-0.1

0.0

0.1

0.2

B

A

M

HD 82943i=90 deg

Ref: Ferraz-Mello et al. (2005)

Axes:x = e2.cos y = e2.sin

RED = collision in t<260,000 yrsGRAY=very chaoticWHITE=mild or almost no chaos

0 20000 40000 60000TIME (yr)

0

2

4

6SE

MI-MAJ

OR A

XIS

(AU)


HD 82943i=30 deg

-0.2 -0.1 0.0 0.1 0.2-0.2

-0.1

0.0

0.1

0.2

B

A

M

Axes:x = e2.cos y = e2.sin

RED = collision in t<260,000 yrsGRAY=very chaoticWHITE=mild or almost no chaos


51000 51500 52000 52500 53000DATE (JD-2,400,000)

-0.02

-0.01

0.00

0.01

0.02km

/sSol. B

(O-C) of solution B vs. (O-C) of the Mayor et al. solution (squares)

Ref: Ferraz-Mello et al. 2005

Solar System without Saturn50 M

yr

Collis

ion

Ch

aos

Ord

er

025

40

60

80

Grid: 33x251Ref: Michtchenko (unpub.)

8.0 8.2 8.4 8.6 8.8 9.0 9.2 9.4 9.6 9.8 10.0 10.2 10.4

semi-major axis (AU)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

ecce

ntric

ity

2/1 7/3 5/2 8/3 .

CLASS Ib – Low Eccentricity Near-Resonant Planet Pairs

Extrasolar Planets (MS stars) Star Period m.sin i a Period Eccentricity planets ratio (m_Jup) (AU) (days) 47 UMa 2.64 2.9 2.1 1079.2 0.05 b,c(?) 1.1 4.0 2845.0 0

Ref: Fisher et al. (2003).

Outer Solar System planets Period Mass a Period Eccentricity ratio (m_Jup) (AU) (years) Jupiter 2.500 1.0 5.204 11.866 0.0489 Saturn 2.831 0.30 9.584 29.668 0.0571 Uranus 1.958 0.046 19.178 83.987 0.0468 Ndeptune 0.054 30.004 164.493 0.0112

Resonant Pulsar Planets

star Period Mass a Period Eccentricity ratio (m_Earth) (AU) (days) PSR 1257+12 2.63 0.02 0.19 25.262 0.0 1.48 4.3 0.36 66.5419 0.0186 3.9 0.47 98.2114 0.0252

Ref: Konacki & Wolszczan (2003)

Neighborhood of Uranus

Ref: Michtchenko & Ferraz-Mello, 2001

CLASS Ib – Low-Eccentricity Near-Resonant Planet Pairs

Extrasolar Planets (MS stars) Star Period m.sin i a Period Eccentricity planets ratio (m_Jup) (AU) (days) 47 UMa 2.64 2.9 2.1 1079.2 0.05 b,c(?) 1.1 4.0 2845.0 0

Ref: Fisher et al. (2003).

Outer Solar System planets Period Mass a Period Eccentricity ratio (m_Jup) (AU) (years) Jupiter 2.500 1.0 5.204 11.866 0.0489 Saturn 2.831 0.30 9.584 29.668 0.0571 Uranus 1.958 0.046 19.178 83.987 0.0468 Ndeptune 0.054 30.004 164.493 0.0112

Resonant Pulsar Planets

star Period Mass a Period Eccentricity ratio (m_Earth) (AU) (days) PSR 1257+12 2.63 0.02 0.19 25.262 0.0 1.48 4.3 0.36 66.5419 0.0186 3.9 0.47 98.2114 0.0252

Ref: Konacki & Wolszczan (2003)

0.44 0.45 0.46 0.47 0.48 0.49

SEMI-MAJOR AXIS (AU)

0.0

0.1

0.2

ECCE

NTRICITY

11/8 7/5 10/7 3/2 11/7

Grid: 21x101

Neighborhood of the 3rd planet of pulsar B1257 +12

0.465 0.466 0.467 0.468 0.469 0.470 0.471 0.472 0.473 0.474 0.4750.00

0.05

0.10

0.15

0.20

Grid 21x51

CHAOS

ORDER

Neighborhood of the 3rd planet of pulsar B1257 +12

X

Class II – NR Pairs with significant Secular Dynamics

Star Period m.sin i a Period Eccentricity planet ratio (m_Jup.) (AU) (days) 55 Cnc 0.045 0.038 2.81 0.2 e,bc 5.2 0.78 0.115 14.7 0.02 15.6 0.22 0.24 43.9 0.44 HD 160691 4.63 1.67 1.50 645 0.20 b,c 3.1 4.2 2986 0.6 And 5.29 1.79 0.805 241.16 0.27 c,d 3.53 2.543 1276.15 0.25 HD 12661 5.48 2.30 0.823 263.6 0.35 b,c 1.57 2.557 1444.5 0.20 HD 169830 9.32 2.88 0.81 225.62 0.31 b,c 4.04 3.6 2102 0.33 HD 37124 12.7 0.86 0.543 153.3 0.1 b,c 1.0 2.952 1942 0.4 Data from Fisher et al. (2003) [all except ...], McArthur et al.(2004) [55 Cnc], McCarthy et al. (2004) [HD 160691b,c] , Santos et al. (2004) [HD 160691d], Schneider (2004) [HD 169830].

Dynamical map of the neighborhood of planet And D cf. Robutel & Laskar (2000) unpublished

White line: Colision line with planet C Black spot: actual position of And D

chaotic regular

1/5 2/11

Consequences

Poisson-Laplace invariance of semi-major axes in theSolar System (at order m )

In Extra-solar systems (with large eccentricities)

< a >=0

as far as a close approximation of the two planets does not occur.

i

2

Consequences

In a system formed by two coplanar planets the eccentricities vary in anti-phase because of

N.B. cos i = 0

Eccentricity variation of upsilon Andromeda planets B(green), C(red) , D(blue)in a 100,000-yr simulations.

0 20000 40000 60000 80000 100000TIME (yrs)

0.0

0.1

0.2

0.3

ECCE

NTRI

CITY

Class III – Weakly Interacting Planet Pairs

??

Star Period m.sin i a Period Eccentricity ratio (Jupiter mass) (AU) (days) HD 168443 30.5 7.64 0.295 58.1 0.53 b,c 16.96 2.873 1770 0.20 HD 74156 >51.3 1.61 0.745 51.6 0.65 b,c >8.2 > 3.82 > 2650 0.35 And 0.64 0.058 4.617 0.01 b,cd 52.2 1.79 0.805 241.16 0.27 276.4 3.53 2.543 1276.15 0.25 HD 160691 0.044 0.09 9.55 0 d,bc 67.5 1.67 1.50 645 0.20 312 3.1 4.2 2986 0.6 HD 38529 151.9 0.78 0.129 14.309 0.29 b,c 12.7 3.68 2174.3 0.36 55 Cnc 1610 0.045 0.038 2.81 0.2 ebc,d 307 0.78 0.115 14.7 0.02 103 0.22 0.24 43.9 0.44

3.91 5.26 4517 0.3 Data from Fisher et al. (2003) [all except ...], McArthur et al.(2004) [55 Cnc], McCarthy et al. (2004) [HD 160691b,c] , Santos et al. (2004) [HD 160691d].

CLASS Ia – Planet Pairs in Mean Motion Resonance

Star Period m.sin i a Period Eccentricity planets ratio (m_Jup) (AU) (days) HD 82943 1.99 1.7 0.75 219.5 0.39 c,b 1.8 1.18 436.2 0.15 GJ 876 2.02 0.56 0.13 30.12 0.27 c,b 1.89 0.21 61.02 0.10 55 Cnc 2.99 0.78 0.115 14.7 0.02 b,c(?) 0.22 0.24 43.9 0.44

Data from Ferraz-Mello et al (2004) [HD 82943], Fisher et al. (2003) [GJ 876], McArthur et al.(2004) [55 Cnc].

Ref: Papaloizou,Cel. Mech.Dyn. Astron87 (2003).

Model:2 planets & 1 gapa=0.8 and a=1.4

Fig. scale 2x2

Evolution of a 2-planet system under [2/1 resonance] non-conservative forces (mass ratio 0.54)

2

1

2 1

2 1

[arbitrary units]

SYMMETRIC APSIDAL COROTATIONS

(0,0)

Stable stationary solutions with aligned periastraAt conjunction both planets are at the perihelion.

N.B. m < m (“internal case”)1 2

2 1m /m = const.

Ex: HD82943m2/m1=1.06

+

++

[2/1 resonance]

Stable asymmetric stationary solutions. 2/1 resonance

0.11

0.22

0.33

0.44

0.56

0.67

0.78

0.89

1.00

m < m2 1(external cases)

Ref: Ferraz-Mello et al. Cel. Mech. Dyn. Astron. (2003)

ASYMMETRIC APSIDAL COROTATIONS


Stable Stationary Solutions. 3/1 resonance.

Asymmetric

Anti-alligned

0.11

0.22

0.33

0.44

0.56

0.67

0.78

0.89

1.1

1.2

1.0

External Internal

PROBLEM: 55 Cnc b,c ..... e1=0.02 e2=0.44 m2/m1=0.28 (?????)

http://www.astro.iag.usp.br/~dinamica/usp-unc.htm

@ArXiv: Astro-ph/0404166 /0402335 /0301252 /0210577

References downloadable from:

THE PLANETS OF THE STAR GJ 876 (Gliese 876)

MASSES[unit: Jupiter mass]

(B) 1.89 /sin i(C) 0.56

TERRA

MERCÚRIO

BC

VENUS

1/sin i ~ 1.25-2.0

THE PLANETS OF UPSILON ANDROMEDA

MASSES[unit: Jupiter mass]

(B) 0.69 /sin i (C) 1.89(D) 3.75

JÚPITER

MARTE

D

CB

0 10 20

mutualinclination

HD 82943

Mayor et al. (2004) planar elements and masses and initial mutual inclination 10 .o

o

planar case

Radial Velocity of a star with 2 planets – HD 82943

Roughly 50 percent of the stars for which one planet was detected show evidence of additional companions.Ref. Marcy et al Astroph.J. 556 (2001)

51500 52000 52500

Mayor et al.A&A, 2004

And

F8 VL=3L_sund=13.47 pcM=1.3 M_sunagee 2-3 Gyrs

Absence of astrometric variation means sin i > 0,4 (i > 25deg)Absence of transits means sin i < 0.9925 (i < 83 deg)

planet B C D units m.sin i 0.69 1.89 3.75 M_jup a 0.059 0.83 2.53 AU period 4.617 241.5 1284 days eccentricity 0.012 0.28 0.27 - perihelion 73 250 260 deg

Corotations as functions of the eccentricities

104 deg

conjunction line

Intersecting orbits of an asymmetric stationary solution

EccentricitiesInner 0.17Outer 0.38

Mass ratio0.44 (ext.)

Asymmetric solutions bifurcating from solutions with aligned periastra

Asymmetric solutions bifurcating from solutions with anti-aligned periastra

Long-term evolution of the eccentricity (same solution)

Evolution of the eccentricities in the same solution

Stability of Extra-solar Planetary Systems C. Beaugé (UNC) S. Ferraz-Mello (USP) T. A. Michtchenko...

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