Dissipation in resonant systems: Implications of observed orbital configurations J.-B. Delisle, J. Laskar, A. C. M. Correia Geneva Observatory - Switzerland October 8, 2015
Resonant/near resonant systems • What is a resonance between 2 planets? – P 2 / P 1 = p / q ( p , q integers) – Example: 2/1 2.1 • Resonant or near resonant system? 2.05 P 2 / P 1 Resonance width 2 depends on m i , e i 1.95 1.9 -100 0 100 200 300 400 2 λ 2 − λ 1 Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 2 / 14
Kepler near-resonant planets • Distribution of period ratio in Kepler data Fabrycky et al. (2014) Lissauer et al. (2011), Fabrycky et al. (2014) • Peaks at resonances −→ convergent migration ( P 2 / P 1 ց ) • Peaks slightly shifted to the right −→ tidal dissipation? (Systems near but outside of resonances) Papaloizou & Terquem (2010), Lithwick & Wu (2012), Delisle et al (2012), Batygin & Morbidelli (2013), Lee et al (2013), Delisle et al (2014), Delisle & Laskar (2014) Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 3 / 14
Formation scenario 2 . 5 Convergent migration 2 . 4 (in protoplanetary disk) 2 . 3 Evolution under P 2 P 1 tides (slow) 2 . 2 2 . 1 2 . 0 t Capture in resonance End of migration Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 4 / 14
Kepler near-resonant planets • Other possible explanations for the shift: – protoplanetary disk - planets interactions Rein (2012), Baruteau & Papaloizou (2013) – planetesimals - planets interactions Chatterjee & Ford (2015) – in-situ formation of planets Petrovitch, Malhotra, Tremaine (2013), Xie (2014) Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 5 / 14
Why tidal dissipation? • Distribution of period ratio close to resonances (2:1 + 3:2) 1 P 1 < 5 d 5 d ≤ P 1 < 15 d 0 . 8 P 1 ≥ 15 d 0 . 6 CDF RES 0 . 4 0 . 2 0 − 1 − 0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 P 2 / P 1 − ( p + 1) / p · 10 − 2 Delisle, Laskar (2014) Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 6 / 14
Why tidal dissipation? • Distribution of period ratio close to resonances (2:1 + 3:2) 1 P 1 < 5 d 5 d ≤ P 1 < 15 d 0 . 8 P 1 ≥ 15 d 0 . 6 CDF RES Evidence for tidal 0 . 4 dissipation Shift for close-in systems 0 . 2 0 − 1 − 0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 P 2 / P 1 − ( p + 1) / p · 10 − 2 • KS-tests – Close-in vs Farthest: 0.08% – Close-in vs Intermediate: 3.5% – Intermediate vs Farthest: 10% Delisle, Laskar (2014) Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 6 / 14
Analytical model of resonances • First order resonances (2/1, 3/2, etc.) Integrable approximation is straightforward Sessin & Ferraz-Mello (1984), Henrard et al. (1986), Wisdom (1986), Batygin & Morbidelli (2013) • Higher order resonances (3/1, 5/2, etc.) 2 degrees of freedom (not integrable) – New simplifying assumption e 1 / e 2 ≈ ( e 1 / e 2 ) forced (ecc. ratio at resonance center) 2.1 Integrable pendulum-like approx. 2.05 P 2 / P 1 H = − ( I − δ ) 2 + 2 R cos( q θ ) 2 1.95 Delisle, Laskar, Correia, Bou´ e (2012) 1.9 -100 0 100 200 300 400 Delisle, Laskar, Correia (2014) 2 λ 2 − λ 1 Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 7 / 14
Dissipative evolution in resonance • Dissipation affects the resonant motion in 2 ways 2.1 2.05 Width change P 2 / P 1 2 Spiraling of trajectory 1.95 1.9 -100 0 100 200 300 400 2 λ 2 − λ 1 • Relative amplitude: A = Amplitude Width – if A ց Locked in resonance, P 2 / P 1 ≈ p / q – if A ր Escape from resonance, P 2 / P 1 no more locked Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 8 / 14
Migration in protoplanetary disk � e 1 � 2 • A ր (unstable res.) ⇐⇒ T e , 1 T e , 2 < e 2 forced ecc. damping timescales (by disk-planet interactions) Escape with P 2 / P 1 ց (convergent migration) 2 . 5 A ց A ր 2 . 4 2 . 4 2 . 2 2 . 3 P 2 P 1 2 . 2 P 2 P 1 2 . 0 2 . 1 1 . 8 2 . 0 1 . 6 1 . 9 t t Delisle, Correia, Laskar (2015) Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 9 / 14
Migration in protoplanetary disk � e 1 � 2 • A ր (unstable res.) ⇐⇒ T e , 1 T e , 2 < e 2 forced ecc. damping timescales (by disk-planet interactions) Escape with P 2 / P 1 ց (convergent migration) • Observed resonant systems constraints on disk properties (ex: aspect ratio, surface density profile...) Delisle, Correia, Laskar (2015) Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 9 / 14
Constraints on disk properties Varying disk properties τ = + ∞ , K = 70 τ = 10 , K = 34 τ = 8 , K = 30 τ = 4 , K = 17 25 a 1 20 a 2 a a i (AU) 15 10 5 0 3 . 3 3 . 2 P 2 3 . 1 P 2 / P 1 3 . 0 P 1 ex: HD 60532 b, c 2 . 9 2 . 8 2 . 7 Observed in 3/1 res. 0 . 35 0 . 30 0 . 25 e e 1 → Did not escape 0 . 20 e i 0 . 15 e 2 0 . 10 0 . 05 → Constraints on disk 0 . 00 5 (aspect ratio...) 4 e 1 3 e 1 / e 2 e 2 2 1 0 angles 350 300 ̟ 1 − ̟ 2 250 3 λ 2 − λ 1 − 2 ̟ 1 (deg) 200 150 100 50 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 × 10 6 × 10 6 × 10 6 × 10 6 t (yr) t (yr) t (yr) t (yr) Delisle, Correia, Laskar (2015) ESCAPE LOCKED IN RES. Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 10 / 14
Tidal dissipation � e 1 � 2 4 + | k 2 | (1 + L ) � 2 � e 1 τ = T 1 τ c ≈ L τ α = T 2 e 2 4 L −| k 1 | (1 + L ) e 2 1 / 3 � � L ≈ m 1 k 1 � � � � m 2 k 2 � � • τ < τ c : Amplitude ր −→ separatrix crossing possible – τ < τ α : Diverging P 2 / P 1 > k 2 / k 1 EXT – τ > τ α : Converging P 2 / P 1 < k 2 / k 1 INT • τ > τ c : Amplitude ց −→ evolution close to libration center – q = 1 : Diverging P 2 / P 1 > k 2 / k 1 EXT – q > 1 : Staying in resonance P 2 / P 1 ≈ k 2 / k 1 RES Delisle, Laskar, Correia (2014) Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 11 / 14
Constraints on planets nature ex: GJ 163 Parameter [unity] b c d [ M ⊕ ] m sin i 10 . 661 7 . 263 22 . 072 [days] P 8 . 633 25 . 645 600 . 895 [AU] a 0 . 06069 0 . 12540 1 . 02689 0 . 0106 0 . 0094 0 . 3990 e • Planets b, c close to 3:1 MMR (order 2) P 2 P 1 = 2 . 97 < 3 Internal circulation (converging) τ α < τ < τ c Delisle, Laskar, Correia (2014) Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 12 / 14
Constraints on planets nature ∆ t 2 / ∆ t 1 0 500 1000 1500 2000 2500 3000 180 3.3 160 EXT INT Initial Amplitude 140 3.2 120 ( P 2 / P 1 ) f M 1 ( ◦ ) 3.1 100 80 3 60 RES 2.9 40 20 2.8 0 0 200 400 600 800 1000 1200 1400 GJ 163b, c are here ∆ t 2 /κ ∆ t 1 GJ 163b: gaz Delisle, Laskar, Correia (2014) GJ 163c: rock Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 13 / 14
Conclusion • Classification of outcome of dissipative process in resonance • Constraints on systems properties from period ratio – Disk properties (disk-planet interactions) – Planets nature (tidal dissipation efficiency) • Analytical model – Better understanding of these complex process – First approximation of constraints – Need numerical simulations for precise constraints Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 14 / 14
Recommend
More recommend
