A secular representation for the long-term resonant dynamics beyond - - PowerPoint PPT Presentation

Introduction Non-resonant case Resonant case Exploration Application Conclusion

A secular representation for the long-term resonant dynamics beyond Neptune

Melaine Saillenfest Marc Fouchard, Giacomo Tommei, Giovanni Valsecchi

IMCCE, Observatoire de Paris Dipartimento di Matematica, Università di Pisa

20/09/2016

Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

Plan

1

Introduction

2

Secular non-resonant theory

3

Secular theory for a single resonance

4

Exploration of the parameter space

5

Application to known objects

6

Conclusion

Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

Introduction

Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

A well-known but rather unexplored mechanism

Mean-motion resonances with Neptune :

• rigin of large orbital variations beyond the planetary region

strong captures are relatively rare variety of possible behaviours yet to be explored Goal : general analysis of the resonant dynamics beyond Neptune (extensive exploration of what can be done by the known planets) A quasi-integrable dynamics : smooth long-term behaviour typical time-scales > 1 Gyr = ⇒ can be described by a secular theory

Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

What is a secular theory ?

The two-body problem is degenerate :      ˙ M = 2π/T ˙ ω = 0 ˙ Ω = 0 Only one varying angle in a 3D space Effect of a small perturbation : ω and Ω become slow angles

π 2π 1 2 3 ω (rad) time (Gyrs)

     ˙ M = 2π/T + O(ε) ˙ ω = O(ε) ˙ Ω = O(ε) Secular theory : study of the slow (dominant) motion

Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

Secular non-resonant theory

Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

Hamiltonian of the problem

Hamiltonian in heliocentric coordinates : µ = GM⊙ H = − µ2 2L2 +

N

• i=1

ni Λi −

N

• i=1

Gmi

• 1

|r − ri| − r · ri |ri|3

• Canonic coordinates (Delaunay elements) :

         ℓ = M g = ω h = Ω λ1, λ2 . . . λN et            L = √µa G =

• µa (1 − e2)

H =

• µa (1 − e2) cos I

Λ1, Λ2 . . . ΛN Secular Hamiltonian F (1st order of the masses) : average of H with respect to the fast independent angles ℓ and λ1, λ2 . . . λN.

Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

Secular non-resonant Hamiltonian

General form of the secular Hamiltonian : F = F(L, G, H, g) conservation of secular momenta L and H with L and H as parameters, the dynamics is described by the level curves of F in the (g, G) plane Simpler version of the parameters :

• a = L2/µ

CK = (H/L)2 = (1 − e2) cos2 I ...and of the variables :    ω = g q = a

• 1 −
• 1 − (G/L)2
• Melaine Saillenfest

A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

Study of the lowest order terms

CK > 1/5 CK < 1/5 π/2 π 3π/2 2π ω (rad) 35 40 45 50 55 q (AU) π/2 π 3π/2 2π ω (rad)

100 200 300 400 500 600 700 800 900 1000 a (AU) 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 CK 0.1 1 10 102 103 104 105 106 Oscillation period (Gyrs) q < aN Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

Semi-analytical model

Lowest order terms : only accurate for large semi-major axis and small eccentricity = ⇒ Numerical "exact" secular Hamiltonian :

F(L, G, H, g) = −

N

• i=1

1 4π2 2π 2π Gmi

• r(L, G, H, ℓ, g, h) − ri(λi)
• dλi dℓ

0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 30 40 50 60 70 80 90 CK a (AU) no island for q > aN single island at ω = 0 two islands at ω = 0 and π/2 single island at ω = π/2

Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

Numerical exploration of the parameter space

5 10 15 20 25 Maximum width (AU) 16.4 AU 30 50 100 200 1000 10000 a (AU) 0.05 0.1 0.15 0.2 0.25 CK 8 10 12 14 16 18 20 22 24 Width in perihelion (AU)

= ⇒ a non-resonant secular evolution allows a maximum perihelion excursion of about 16.4 AU

Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

Secular theory for a single resonance

Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

Coordinate change

Principal resonant angle : σ = k λ − kp λp − (k − kp) ̟ with k, kp ∈ N and k > kp New canonical coordinates : matrices A and (AT )−1 A       M λp ω Ω {λi=p}       =       σ γ u v {λi=p}       ← − resonant angle ← − fast angle    ← − no change Three time-scales :

• short periods < 103 years

{λi=p} and γ

• semi-secular periods ∼ 105 years

σ

• secular periods > 109 years

ω and Ω

Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

Semi-secular Hamiltonian

Semi-secular Hamiltonian K (1st order of the masses) : average of H with respect to the fast angles γ and {λi=p}. general form : K = K(Σ, U, V, σ, u)

⇒ two degrees of freedom ⇒ semi-secular momentum conserved : V =√µa √ 1 − e2 cos I − kp

k

• Methods to describe the secular dynamics :

1) Poincaré sections 2) reduction to a one-degree-of-freedom system : = ⇒ use the adiabatic hypothesis with Tu ≫ Tσ

Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

The adiabatic invariance

Action-angle coordinates of K with (U, u) fixed : θ = mean angle along the trajectory J ∝ enclosed area = adiabatic invariant

Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

Resonant secular Hamiltonian

Secular Hamiltonian : F = F0(J, U, V, u) + O(ξ) where ξ is related to the ratio of frequencies secular/semi-secular. conservation of secular momenta J and V with J and V as parameters, the dynamics is described by the level curves of F in the (u, U) plane Simpler version of the parameters (for a chosen a0) :

• η0 = V/√µa0 + kp/k =
• 1 − ˜

e2 cos ˜ I J (inchanged) ...and of the variables :      ω = u ˜ q = a0

• 1 −
• 1 − (U/√µa0 + kp/k)2
• Melaine Saillenfest

A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

Separatrix crossing

Stretching of the island during the secular evolution :

π 2π σ (rad) Σ = √µa/k (U1, u1) π 2π σ (rad) (U2, u2) π 2π σ (rad) (U3, u3)

Corresponding secular model : trajectory integrable by parts (piecewise model) slow chaotic diffusion

Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

Double resonance island

For resonances of type 1:k, a range of parameters allows two resonance islands = ⇒ three possible oscillating types for (Σ, σ) Disappearance of one island during the secular evolution :

π 2π σ (rad) Σ = √µa/k (U1, u1) π 2π σ (rad) (U2, u2) π 2π σ (rad) (U3, u3)

Corresponding secular model : frequent separatrix crossings long time-scale chaos

Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

Exploration of the parameter space

Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

Resonances other than 1:k (single island)

30 40 50 60 70 80 ˜ q (AU) −0.60 −0.36 −0.35 −0.33 −0.20 π/2 π ω (rad) 30 40 50 60 70 80 ˜ q (AU) 0.30 π/2 π ω (rad) 0.38 π/2 π ω (rad) 0.45 π/2 π ω (rad) 0.50 π/2 π ω (rad) 0.60

Example : resonance N2:37 (a0 = 210.99 AU)

Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

Resonances other than 1:k (single island)

30 40 50 60 70 80 ˜ q (AU) −0.430 −0.262 −0.248 −0.230 −0.100 π/2 π ω (rad) 30 40 50 60 70 80 ˜ q (AU) 0.200 π/2 π ω (rad) 0.275 π/2 π ω (rad) 0.320 π/2 π ω (rad) 0.350 π/2 π ω (rad) 0.430

Example : resonance N2:115 (a0 = 449.36 AU)

Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

Resonances of type 1:k (double island)

30 40 50 60 70 80 90 100 ˜ q (AU) −0.60 −0.41 −0.38 −0.36 −0.30 π/2 π ω (rad) 30 40 50 60 70 80 90 100 ˜ q (AU) 0.00 π/2 π ω (rad) 0.40 π/2 π ω (rad) 0.45 π/2 π ω (rad) 0.50 π/2 π ω (rad) 0.60

Example : resonance N1:19 (a0 = 214.78 AU)

Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

Observations about the resonant secular dynamics

General geometry of the phase space : an equilibrium point at ω = 0 mod π an additional equilibrium at ω = π/2 for prograde orbits large oscillations of q near the equilibrium points Resonances of type 1:k are specific : same features but distorted (asymmetric islands) truncation of the secular trajectories by the "green line" = ⇒ possible segregation of ω and/or wide excursions of q whatever the resonance, provided that η0 is in the required range.

Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

Application to known objects

Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

The "zone of interest"

Zone of interest : interval of parameters allowing stable equilibrium points

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100 200 300 400 500 600 η = √ 1 − e2 cos I semi-major axis (AU) 1:k 2:k 3:k 4:k

2012VP113 2013RF98 2004VN112 2010GB174 2007TG422 Sedna Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

Out of the zone of interest : Sedna

0.0 ◦ 17.1 ◦ 25.0 ◦ 30.2 ◦ 34.1 ◦ ˜ I (deg) Neptune 2:141 a0 = 514.7638 AU ; η0 = 0.5119 π 5π/4 3π/2 7π/4 2π ω (rad) 70 80 90 100 110 ˜ q (AU) Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

2004VN112 and 303775

0.0 ◦

12.8 ◦ 28.4 ◦ 35.9 ◦ 40.9 ◦ 44.5 ◦ 47.2 ◦ 49.5 ◦ ˜ I (deg) Neptune 1:35 (a0 = 322.7465 AU) η0 = 0.4703 ; 2πJ = −0.0006 AU2rad2/yr 0.0 ◦ 20.1 ◦ 30.2 ◦ 35.6 ◦ 39.0 ◦ 41.2 ◦ 42.6 ◦ 43.4 ◦ ˜ I (deg) Neptune 1:7 (a0 = 110.3783 AU) η0 = 0.7234 ; 2πJ = −0.0050 AU2rad2/yr −π/2 −π/4 π/4 π/2 ω (rad) 30 40 50 60 70 80 90 100 ˜ q (AU) π 5π/4 3π/2 7π/4 2π ω (rad) 30 40 50 60 70 80 90 100 ˜ q (AU) Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

181902

8.8 ◦ 28.7 ◦ 36.4 ◦ 41.0 ◦ 44.1 ◦ 46.2 ◦ 47.6 ◦ 48.6 ◦ ˜ I (deg) Neptune 1:8 (a0 = 120.6548 AU) η0 = 0.6521 ; 2πJ = −0.0002 AU2rad2/yr 8.5 ◦ 28.8 ◦ 36.6 ◦ 41.2 ◦ 44.3 ◦ 46.5 ◦ 48.0 ◦ 49.1 ◦ ˜ I (deg) Neptune 2:17 (a0 = 125.6307 AU) η0 = 0.6414 ; 2πJ = −0.0004 AU2rad2/yr π/4 π/2 3π/4 π ω (rad) 30 40 50 60 70 80 90 100 ˜ q (AU) 3π/4 π 5π/4 3π/2 ω (rad) 30 40 50 60 70 80 90 100 ˜ q (AU) Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

A high-perihelion trapping mechanism

a (AU) q (AU) I (deg) ω (rad) Ω (rad) σ (rad) time (Gyrs) ˜ I (deg)

Neptune 1:13 (a0 = 166.7680 AU) η0 = 0.5606 ; 2πJ = −0.00120 AU2rad2/yr

100 200 300 400 500 20 40 60 80 10 20 30 40 50 60 π 2π π 2π π 2π −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 11.6 ◦ 30.4 ◦ 38.3 ◦ 43.1 ◦ 46.5 ◦ 49.0 ◦ −π/2 −π/4 π/4 π/2 ω (rad) 30 40 50 60 70 80 ˜ q (AU)

Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

A high-perihelion trapping mechanism

Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

A high-perihelion reservoir

30 35 40 45 50 55 60 65 70 perihelion distance (AU) 20 40 60 80 100 120 140 50 100 150 200 250 300 350 400 450 500 inclination (deg) semi-major axis (AU) Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

Conclusion

Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune

Introduction Non-resonant case Resonant case Exploration Application Conclusion

Conclusion

Secular resonant dynamics beyond Neptune :

• ne-degree-of-freedom system with two parameters

straightforward way to explore the dynamics Observed geometries : wide variations of q for some ranges of η0 dynamical paths from low to high q and I high-perihelion trapping mechanism for 1:k resonances Implication for the known high-perihelion objects : confinement of ω at 0 or π (but no way to favour 0 against π) possible accumulation of resonant high-perihelion objects (with circulating ω) 4 × 106 trapped objects from the Oort Cloud (1/250000)

Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune