Stability Dynamics Habitability of Planets in Binary Systems Facts and Questions:
Binary and Multi-Star Systems in the Solar neighbourhood: >60% Do the gravitational perturbations allow planetary formation ? (about 60 binary systems hosting planets) Information about the area of formation Long-term stability is of great interest search for habitable planets Outline
Types of planetary motion in binary star systems Stability of planetary motion Examples of planetary motion in tight binaries Habitable planets in binaries? Solar System in a wide binary (Innanen et al. 1997)
S-Typ Planetary motion in binaries: Trojans – type motion Near L4 and L5
L4 and L5 (stable for µ < 1:25) are at the third vertex of an equilateral triangle (Sun-Jupiter-Asteroid)
Studies by : Harrington (1977) Graziani & Black (1981) and Black (1982) planar case: No serious threat to the long-term stability if the star is at a distance > 5 x planetary distance inclined case: at high iBinary the planetary orbits becomes unstable Kozai mechanism Pendleton & Black (1983)
Some General Numerical Studies of planetary motion in binary systems:
Dvorak R.(1984 and 1986) Rabl & Dvorak R. (1988) Dvorak R., Froeschle C. & Froeschle Ch. (1989)
Holman M. & Wiegert P. (1999) Pilat-Lohinger E. & Dvorak R. (2002) Szenkovits F. & Mako Z. (2008)
Stability analysis
Initial Conditions: abinary =1 AU ebinary=[0, ….0.9]
aplanet=[0.1, …0.9] eplanet=[0,… 0.9] i, Ώ,ω, = 0° M= 0°,90°,180°,270° Numerical Methods: Chaos Indicators: Long-term numerical Fast Lyapunov Indicator integration: (FLI) C. Froeschle, R.Gonczi, E. Lega Stability-Criterion: (1996) No close encounters within the Hill‘ sphere MEGNO RLI (i)Escape time Helicity Angle (ii) Study of the eccentricity: maximum eccentricity
Lyapunov Exponent
General Studies of S-type motion in binary systems:
Dvorak R.(1984 and 1986) Rabl & Dvorak R. (1988) Dvorak R., Froeschle C. & Froeschle Ch. (1989) (200-500 periods)
Holman M. & Wiegert P. (1999)
(mass-ratios: 0.1-09, eBinary: 0 – 0.9; 10000 periods)
Pilat-Lohinger & Dvorak (2002):
The Fast Lyapunov Indicator (FLI) (see Froeschle et al., CMDA 1997) 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) 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.
Stability analysis
Initial Conditions: abinary =1 AU ebinary=[0, ….0.9]
aplanet=[0.1, …0.9] eplanet=[0,… 0.9] i, Ώ,ω, = 0° M= 0°,90°,180°,270° mass-ratio = 0.5 S-type motion mass-ratio e_binary 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.0 0.45 0.38 0.37 0.30 0.26 0.23 0.20 0.16 0.13 0.1 0.37 0.32 0.29 0.27 0.24 0.20 0.18 0.15 0.11 0.2 0.32 0.27 0.25 0.22 0.19 0.18 0.16 0.13 0.10 0.3 0.28 0.24 0.21 0.18 0.16 0.15 0.13 0.11 0.09 0.4 0.21 0.20 0.18 0.16 0.15 0.12 0.11 0.10 0.07 0.5 0.17 0.16 0.13 0.12 0.12 0.09 0.09 0.07 0.06 0.6 0.13 0.12 0.11 0.10 0.08 0.08 0.07 0.06 0.045 0.7 0.09 0.08 0.07 0.07 0.05 0.05 0.05 0.045 0.035 0.8 0.05 0.05 0.04 0.04 0.03 0.035 0.03 0.025 0.02
Stable zone (in units of length) of S-type motion for all computed mass-ratios and eccentricities of the binary. The given size for each (mu,e_binary) pair is the lower value of the studies by Holman &Wiegert (AJ,1999) and Pilat-Lohinger &Dvorak (CMDA, 2002)
planets in binaries tight binary systems
binary abinary [AU] ebinary M1/M2
Gliese86 22 ? 0.7/0.5 γ Cephei 20 0.44 1.6/0.4 HD41004 23 ? 0.7/0.4 HD188753 12.3 0.5 1.06/1.63 HD196885 25 ? 1.3/0.6
Examples of Planets in Tight Binary Systems: binary abinary [AU] ebinary m1 m2 [mSun] Gliese86 ~22 ? 0.7 0.5 γ Cephei ~20 0.4 1.6 0.4 HD41004 ~23 ? 0.7 0.4
Gliese86 γ Cephei HD 41004
m sin i = 2.3 MJ m sin i = 1.6 MJ m sin i = 4 MJ a = 1.3 AU ( 1.6 or 1.7) a = 2.15 AU a = 0.11AU e = 0.39 +/- 0.17 ? e = 0.045 e = 0.11 – 0.2 Influence of the eccentricity of the planet?
Influence of the eccentricity mass-ratio = 0.2
γ Cephei : a_binary= 20 AU e_binary= 0.44 e_planet= 0.2
acrit = 0.18
SZ= 3.6 AU
a_planet=2.15 AU
Pilat-Lohinger & Dvorak, (CeMDA 2002) The mass of the planet ?
Is the elliptic restricted three body problem a good model ?
Circular Planetary Motion: -- Size of Stable Region
RTP 1Jup 3Jup 5Jup e_bin
0 0.38 0.38 0.35 0.35 0.1 0.34 0.31 0.31 0.31 0.2 0. 30 0.24 0.24 0.24 0.3 0.24 0.21 0.21 0.21 0.4 0. 20 0.20 0.20 0.20 0.5 0.16 0.14 0.14 0.14
Elliptic Planetary Motion: e_bin 0.1 0.2 0.3 0.4 0.5 e_plan 0.0 0.31 0.24 0.22 0.21 0.14 0.1 0.31 0.24 0.22 0.19 0.14 0.2 0.28 0.24 0.18 0.16 0.14 0.3 0.26 0.24 0.19 0.15 0.13 0.4 0.25 0.21 0.16 0.13 0.11 0.5 0.19 0.16 0.14 0.12 0.09
Stable Region for a 5Jupiter-mass Planet e_bin=0.4
RTBP 1Mj 3Mj 5Mj 8Mj e_plan
0.0 0.2 0 0.20 0.20 0.20 0.20 0.1 0.2 0 0.20 0.20 0.19 0.19 0.2 0.18 0.18 0.17 0.16 016 0.3 0.17 0.16 0.16 0.15 0.15 0.4 0.17 0.15 0.14 0.13 0.13 0.5 0.15 0.12 0.11 0.09 0.09 gamma Cephei a_bin ~ 20 AU, e_bin ~ 0.4
RTBP 1Mj 3Mj 5Mj 8Mj e_plan
0.0 4.0 4.0 4.0 4.0 4.0 0.1 4.0 4.0 4.0 3.8 3.8 0.2 3.8 3.6 3.4 3.2 3.2 0.3 3.4 3.6 3.2 3.0 3.0 0.4 3.4 3.0 2.8 2.6 2.6 0.5 3.0 2.4 2.2 1.8 1.8 mass-ratio = 0.2 γ Cephei
a_bin = 22 AU e_bin = 0.44 3.6AU a_p = 2 AU e_p=0.21
a_bin=18.5AU e_bin=0.36 3.2 AU a_p=2.13AU e_p=0.12 Stability limits
mass-ratio border of stable motion Gamma Cephei: 0.2 3.2 – 3.8 AU HD41004 AB: 0.36 (e=0.39) 4.6 – 2.8 AU (e=0.5) 4.6 – 2.53AU Gliese 86: 0.42 close-in planet
γ Cephei mass-ratio = 0.2 aBinary = 20 AU eBinary ~ 0.4 increase e γ Cep b : 0.11 0.5.
Planet’s eccentricity border of stable region [AU] 0.0 4.0 0.1 3.8 0.2 3.6 0.3 3.4 04 3.2 0.5 3.0 gamma Cephei eBinary not known ! binary abinary [AU] ebinary m1 m2 Gliese86 ~22 ? 0.7 0.5 γ Cephei ~20 0.4 1.6 0.4 HD41004 ~23 ? 0.7 0.4
HD 41004A:
m sin i = 2.3 MJ a = 1.3 AU ( 1.6 or 1.7) e = 0.39 +/- 0.17 ? HD41004 AB Influence of the secondary on the planetary motion ? with without secondary secondary
gammaCep e_b=0.44 e_pl=0.209
HD41004 e_b=0.2 e_pl=0.22 Differences of the two planetary systems:
semi-major axis of the planet eccentricity of the binary mass-ratio of the binary mass of the giant planet different e_binary different mass-ratio
Gamma Cephei
Planet is close to the host-star: The region is mainly influenced by the mean motion resonances
If the planet is closer to the secondary -> an arc-like structure of chaos appears which depends on: a_planet, e_binary, masses, 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, …)
The Habitable Zone based on the definition given by Kasting et al. (1993). Different HZs
Binary Star Configurations:
Different Binary Configurations Insolation
Next Studies: Inclined planetary systems Results for HD41004A
stable motion in the HZ only for a < 0.7 AU the eccentricity of the binary ebin < 0.7
we have found two zones for habitable planets: for nearly circular planetary motion around 0.5 AU the eccentricity of the detected planet has to be < 0.36 for higher eccentric efpl < 0.06 e < 0.23 planetary motion around fpl 0.6 AU the eccentricity of the maximum eccentricity detected planet has to be < 0.3 gamma Cephei
Primary and Secondary: m1 = 1.6 MS m2 = 0.4 MS a = 20AU e = 0.44 Planet: mP = 1.7 Mjup a = 2.15 AU e = 0.2
Gliese 86
aBinary=22 AU eBinary=0.7 HZ of HD41004 A
red: low ecc. blue: high ecc.
4:1 7:2 3:1 8:3
Region of nearly circular motion exists size and position depends on (i) a_giant planet (ii) e_giant planet (iii) m_giant planet No strong dependence on i (i<25) Stable circular motion for close-in planets Stable region around i=35°- eBinary=0.2 : ePlanet=0.1, 0.3 ePlanet =0.2 : eBinary=0.1, 0.3 Innanen et al. (1997) : Solar System in a Wide Binary Star System