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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

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SLIDE 1

Dissipation in resonant systems: Implications of observed orbital configurations

J.-B. Delisle, J. Laskar, A. C. M. Correia

Geneva Observatory - Switzerland

October 8, 2015

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SLIDE 2

Resonant/near resonant systems

Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 2 / 14

1.9 1.95 2 2.05 2.1

  • 100

100 200 300 400 P2/P1 2λ2 − λ1

  • What is a resonance between 2 planets?

– P2/P1 = p/q (p, q integers) – Example: 2/1

  • Resonant or near resonant system?

Resonance width depends on mi, ei

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SLIDE 3

Kepler near-resonant planets

Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 3 / 14

  • Distribution of period ratio in Kepler data

Fabrycky et al. (2014)

  • Peaks at resonances −→ convergent migration (P2/P1 ց)
  • Peaks slightly shifted to the right

(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)

−→ tidal dissipation?

Lissauer et al. (2011), Fabrycky et al. (2014)

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SLIDE 4

Formation scenario

Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 4 / 14

t 2.0 2.1 2.2 2.3 2.4 2.5 P2 P1

End of migration Capture in resonance Convergent migration (in protoplanetary disk) Evolution under tides (slow)

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SLIDE 5

Kepler near-resonant planets

Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 5 / 14

  • 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)

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SLIDE 6

Why tidal dissipation?

Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 6 / 14

  • Distribution of period ratio close to resonances (2:1 + 3:2)

RES −1 −0.5 0.5 1 1.5 2 2.5 3 ·10−2 0.2 0.4 0.6 0.8 1 P2/P1 − (p + 1)/p CDF

P1 < 5 d 5 d ≤ P1 < 15 d P1 ≥ 15 d

Delisle, Laskar (2014)

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SLIDE 7

Why tidal dissipation?

Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 6 / 14

  • Distribution of period ratio close to resonances (2:1 + 3:2)

RES −1 −0.5 0.5 1 1.5 2 2.5 3 ·10−2 0.2 0.4 0.6 0.8 1 P2/P1 − (p + 1)/p CDF

P1 < 5 d 5 d ≤ P1 < 15 d P1 ≥ 15 d

Shift for close-in systems

  • KS-tests

– Close-in vs Farthest: 0.08% – Close-in vs Intermediate: 3.5% – Intermediate vs Farthest: 10%

Delisle, Laskar (2014)

Evidence for tidal dissipation

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SLIDE 8

Analytical model of resonances

Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 7 / 14

  • First order resonances (2/1, 3/2, etc.)

Sessin & Ferraz-Mello (1984), Henrard et al. (1986), Wisdom (1986), Batygin & Morbidelli (2013) Integrable approximation is straightforward

  • Higher order resonances (3/1, 5/2, etc.)

2 degrees of freedom (not integrable) – New simplifying assumption

e1/e2 ≈ (e1/e2)forced

(ecc. ratio at resonance center)

1.9 1.95 2 2.05 2.1

  • 100

100 200 300 400 P2/P1 2λ2 − λ1

Integrable pendulum-like approx.

H = −(I − δ)2 + 2R cos(qθ)

Delisle, Laskar, Correia, Bou´ e (2012) Delisle, Laskar, Correia (2014)

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SLIDE 9

Dissipative evolution in resonance

Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 8 / 14

1.9 1.95 2 2.05 2.1

  • 100

100 200 300 400 P2/P1 2λ2 − λ1

  • Dissipation affects the resonant motion in 2 ways

Width change Spiraling of trajectory

  • Relative amplitude: A = Amplitude

Width

– if A ց Locked in resonance, P2/P1 ≈ p/q – if A ր Escape from resonance, P2/P1 no more locked

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SLIDE 10

Migration in protoplanetary disk

Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 9 / 14

t 1.9 2.0 2.1 2.2 2.3 2.4 2.5 P2 P1 t 1.6 1.8 2.0 2.2 2.4 P2 P1

  • A ր (unstable res.) ⇐⇒ Te,1

Te,2 <

e1

e2

2

forced

  • ecc. damping timescales

(by disk-planet interactions)

Delisle, Correia, Laskar (2015) Escape with P2/P1 ց (convergent migration)

A ց A ր

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SLIDE 11

Migration in protoplanetary disk

Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 9 / 14

  • A ր (unstable res.) ⇐⇒ Te,1

Te,2 <

e1

e2

2

forced

  • ecc. damping timescales

(by disk-planet interactions)

  • Observed resonant systems

constraints on disk properties Delisle, Correia, Laskar (2015) Escape with P2/P1 ց (convergent migration) (ex: aspect ratio, surface density profile...)

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SLIDE 12

Constraints on disk properties

Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 10 / 14

5 10 15 20 25 ai (AU)

τ = +∞, K = 70 a1 a2 τ = 10, K = 34 τ = 8, K = 30 τ = 4, K = 17

2.7 2.8 2.9 3.0 3.1 3.2 3.3 P2/P1 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 ei

e1 e2

1 2 3 4 5 e1/e2 0.0 0.2 0.4 0.6 0.8 1.0 t (yr) ×106 50 100 150 200 250 300 350 (deg)

̟1 − ̟2 3λ2 − λ1 − 2̟1

0.0 0.2 0.4 0.6 0.8 1.0 t (yr) ×106 0.0 0.2 0.4 0.6 0.8 1.0 t (yr) ×106 0.0 0.2 0.4 0.6 0.8 1.0 t (yr) ×106

LOCKED IN RES. ESCAPE Varying disk properties

a

P2 P1

e

e1 e2

angles ex: HD 60532 b, c Observed in 3/1 res.

→ Did not escape → Constraints on disk

(aspect ratio...)

Delisle, Correia, Laskar (2015)

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SLIDE 13

Tidal dissipation

Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 11 / 14

τ = T1

T2

τc ≈ L e1

e2

2 4+|k2|(1+L)

4L−|k1|(1+L)

τα = e1

e2

2

L ≈ m1

m2

  • k1

k2

  • 1/3
  • τ < τc: Amplitude ր −→ separatrix crossing possible

– τ < τα: Diverging

P2/P1 > k2/k1 EXT

– τ > τα: Converging

P2/P1 < k2/k1 INT

  • τ > τc: Amplitude ց −→ evolution close to libration center

– q = 1: Diverging

P2/P1 > k2/k1 EXT

– q > 1: Staying in resonance

P2/P1 ≈ k2/k1 RES

Delisle, Laskar, Correia (2014)

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SLIDE 14

Constraints on planets nature

Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 12 / 14

ex: GJ 163

Parameter [unity] b c d m sin i [M⊕] 10.661 7.263 22.072 P [days] 8.633 25.645 600.895 a [AU] 0.06069 0.12540 1.02689 e 0.0106 0.0094 0.3990

  • Planets b, c close to 3:1 MMR (order 2)

P2 P1 = 2.97 < 3

Internal circulation (converging)

τα < τ < τc

Delisle, Laskar, Correia (2014)

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SLIDE 15

Constraints on planets nature

Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 13 / 14

M1 (◦) ∆t2/κ∆t1 ∆t2/∆t1 20 40 60 80 100 120 140 160 180 200 400 600 800 1000 1200 1400 500 1000 1500 2000 2500 3000 2.8 2.9 3 3.1 3.2 3.3 (P2/P1)f

EXT INT RES

Initial Amplitude GJ 163b, c are here GJ 163b: gaz GJ 163c: rock

Delisle, Laskar, Correia (2014)

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SLIDE 16

Conclusion

Jean-Baptiste DELISLE (Geneva) Dissipation in resonance October 8, 2015 14 / 14

  • 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