Stability of PlanetsStability of Planetsin Binary Star Systemsin Binary Star SystemsÁkos Bazsó

in collaboration with:E. Pilat-Lohinger, D. Bancelin, B. Funk

ADG Group

Outline

Exoplanets in multiple star systems

Secular perturbation theory

Application: tight binary systems

Summary + Outlook

About

NFN sub-project SP8“Binary Star Systems and Habitability”

Stand-alone project“Exoplanets: Architecture, Evolution and Habitability”

Basic dynamical types

S-type motion (“satellite”)around one star

P-type motion (“planetary”)around both stars

Image: R. Schwarz

Exoplanets in multiple star systems

Observations: (Schwarz 2014, Binary Catalogue)

● 55 binary star systems with 81 planets

● 43 S-type + 12 P-type systems

● 10 multiple star systems with 10 planets

Example: γ Cep (Hatzes et al. 2003)

● RV measurements since 1981

● Indication for a “planet” (Campbell et al. 1988)

● Binary period ~57 yrs, planet period ~2.5 yrs

Multiplicity of stars

~45% of solar like stars (F6 – K3) with d < 25 pc in multiple star systems (Raghavan et al. 2010)

Known exoplanet host stars:

single double triple+ source

77% 20% 3% Raghavan et al. (2006)

83% 15% 2% Mugrauer & Neuhäuser (2009)

88% 10% 2% Roell et al. (2012)

Exoplanet catalogues

The Extrasolar Planets Encyclopaediahttp://exoplanet.eu

Exoplanet Orbit Databasehttp://exoplanets.org

Open Exoplanet Cataloguehttp://www.openexoplanetcatalogue.com

The Planetary Habitability Laboratoryhttp://phl.upr.edu/home

NASA Exoplanet Archivehttp://exoplanetarchive.ipac.caltech.edu

Binary Catalogue of Exoplanetshttp://www.univie.ac.at/adg/schwarz/multiple.html

Habitable Zone Galleryhttp://www.hzgallery.org

Binary Catalogue

Binary Catalogue of Exoplanetshttp://www.univie.ac.at/adg/schwarz/multiple.html

Dynamical stability

Stability limit for S-type planetsRabl & Dvorak (1988), Holman & Wiegert (1999), Pilat-Lohinger & Dvorak (2002)

Parameters (abin

, ebin

, μ)

Outer limit at roughly max. ¼ of stellar separation (for μ = 0.5)

Implications for planet formation → truncation of protoplanetary disk

Secular perturbation theory in a nutshell

Secular perturbation theory

“secular” = long time-scales:

min. 3 interacting massive bodies (m0, m

1, m

2)

gravitational perturbations lead to ...

● mean motion resonances (MMR)

● secular resonances (SR)

resonance = integer ratio of 2 frequencies

T sec≫T rev

f 1/ f 2=p /q∈ℚ

Single star – single planet

two-body problem = indefinitely stable

Basic parameters

● Semi-major axis a● Eccentricity e

0 < e < 1● Solar system planets:

e ≤ 0.2

pericenter apocenter

Binary star system

precession of pericenter (and line of nodes) with time

Laplace-Lagrange linear theory

Developed for solar system (low mass-ratio)

Limits = low eccentricity / inclination

Objects = host star + 2 perturbers + massless test planet

Simple analytical formula (Murray & Dermott 1999)

g = free (proper) secular frequency of test planet

gj = fundamental Eigenfrequencies of system

Free/forced eccentricity

h(t)=efree sin(g t+ϕ)+∑ jA (g , g j ,e j)sin(g j t+ϕ j)

Example 1: frequencies for planets

Example 2: asteroid main-belt

Image: Tsiganis (2008)

Application

Binary star systems with separation a < 100 AU

Typical setting:

● host star (“primary”)

● companion star (“secondary”)

● giant planet (Jupiter like) around primary

● stability of additional (terrestrial) planets ?

Explanation for numerical results

Semi-analytical method

● Determine secular frequency of giant planet

● Find intersection with analytical curve of free frequency

Aim of study

Investigated systemsStar 1 Star 2 Planet

name MassM

sun

SpectralType

MassM

sun

DistanceAU

Ecc. MassM

jup

DistanceAU

Ecc.

GJ 3021 0.90 G6V 0.15 (?) 68 0.20 (?) 3.37 0.49 0.51

Gliese 86 0.83 K0V 0.49 19 0.40 4.01 0.11 0.05

94 Cet 1.34 F8V 0.20 (?) ≥ 100 (?) 0.20 (?) 1.68 1.42 0.30

HD 41004 0.70 K2V 0.15 (?) 23 0.20 (?) 2.54 1.64 0.20 (?)

τ Boo 1.30 F6IV 0.40 (?) 45 0.20 (?) 5.90 0.046 0.02

HD 177830 1.47 K0IV 0.23 97 0.20 (?) 1.49 1.22 ≈ 0.00

HD 196885 1.33 F8V 0.45 21 0.42 2.98 2.60 0.48

γ Cep 1.40 K1III 0.41 19 0.41 1.85 2.05 0.05

(?) = estimated values; minimum masses M sin(i)

Selected systemsStar 1 Star 2 Planet

name MassM

sun

SpectralType

MassM

sun

DistanceAU

Ecc. MassM

jup

DistanceAU

Ecc.

GJ 3021 0.90 G6V 0.15 68 0.20 3.37 0.49 0.51

Gliese 86 0.83 K0V 0.49 19 0.40 4.01 0.11 0.05

94 Cet 1.34 F8V 0.20 ≥ 100 0.20 1.68 1.42 0.30

HD 41004 0.70 K2V 0.15 23 0.20 2.54 1.64 0.20

τ Boo 1.30 F6IV 0.40 45 0.20 5.90 0.046 0.02

HD 177830 1.47 K0IV 0.23 97 0.20 1.49 1.22 ≈ 0.00

HD 196885 1.33 F8V 0.45 21 0.42 2.98 2.60 0.48

γ Cep 1.40 K1III 0.41 19 0.41 1.85 2.05 0.05

Numerical results for HD 41004

Image: E. Pilat-Lohinger

Why not the other systems ?

HD 1237

HD 41004

Dependence on secondary star 1

increasing a2

Dependence on secondary star 2

increasing m2

HD 196885

Outlook

Ongoing work

Determine width of perturbation – dependence on parameters (mass, distance, eccentricity, …)

Goals:

● Catalogue for observers

● Binary star systems allowing “unperturbed” habitable zone (small eccentricity)

That's all folks ...That's all folks ...

References

Campbell, Walker & Yang (1988), ApJ 331, 902

Hatzes, Cochran, Endl et al. (2003), ApJ 599, 1383

Holman & Wiegert (1999), AJ 117, 621

Mugrauer & Neuhäuser (2009), A&A 494, 373

Murray & Dermott (1999), Solar System Dynamics, Cambridge Univ. Press

Pilat-Lohinger & Dvorak (2002), CMDA 82, 143

Rabl & Dvorak (1988), A&A 191, 385

Raghavan, Henry, Mason et al. (2006), ApJ 646, 523

Raghavan, McAlister, Henry et al. (2010), ApJS 190, 1

Roell, Neuhäuser, Seifahrt & Mugrauer (2012), A&A 542, 92

Tsiganis (2008), Lecture Notes in Physics 729, 111, Springer