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Resonant dynamics of trojan exoplanets I: overview of resonant - - PowerPoint PPT Presentation

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Resonant dynamics of trojan exoplanets I: overview of resonant structure and diffusion Roco Isabel Paez Universita' degli Studi di Roma Tor Vergata paez@mat.uniroma2.it in collaboration with Christos Efthymiopoulos RCAAM, Academy of

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Resonant dynamics of trojan exoplanets I:

  • verview of resonant structure and diffusion

Rocío Isabel Paez

Universita' degli Studi di Roma “Tor Vergata”

paez@mat.uniroma2.it

in collaboration with Christos Efthymiopoulos RCAAM, Academy of Athens

cefthim@academyofathens.gr

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Contents

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Formulation hierarchical construction of the hamiltonian computation of resonant proper elements Numerical experiments production of phase portraits and stability maps parametric study of μ and e' identification of the corresponding web of resonances numerical computation and characterization of chaotic diffusion Introduction motivation Conclusions

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Introduction

3

Motivation

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SLIDE 4

Introduction

4

Motivation

t r

  • j

a n p l a n e t

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SLIDE 5

Construction of the hamiltonian

5 5

Starting point: pERTBP and further generalizations

Modified Delaunay variables Complete Hamiltonian

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SLIDE 6

Construction of the hamiltonian

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Forced equilibrium

Forced equilibrium position

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SLIDE 7

Construction of the hamiltonian

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Expansion around the forced equilibrium

where

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SLIDE 8

Construction of the hamiltonian

8 8

Considerations about

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SLIDE 9

Construction of the hamiltonian

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Classification of resonances

secondary resonances 1:n transverse secular

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Construction of the hamiltonian

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Planar ERTBP

then

shift with respect to L4

with

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SLIDE 11

Numerical Experiments

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Parametric Study

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Phase portraits for e' = 0 (CRTBP)

μ = 0.0041

a) ep = 0.0001 b) ep = 0.06 c) ep = 0.1

μ = 0.0031

d) ep = 0.0001 e) ep = 0.05 f) ep = 0.1

pericenter crossing condition 0.001 ≤ μ ≤ 0.01 Δμ = 0.001 35 initial conditions along the 0.0 ≤ Δu ≤ 1.0 Δ(Δu) ~ 0.03 line x = B(u-u0)

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Parametric Study

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FLI stability maps

ep Δu

400x400

0 ≤ Δu ≤ 1 0 ≤ ep ≤ 0.1 Δt = 1/300 T ~ 1000 periods e' = 0 e' = 0.02 e' = 0.6

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Parametric Study

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Survey of resonances

μ = 0.0041 (1:6)

A) e' = 0, B) e' = 0.02, C) e' = 0.04, D) e' = 0.06, E) e' = 0.08, F) e' = 0.1

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Parametric Study

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Survey of resonances - Another example

μ = 0.0012 (1:12), 0.0014 (1:11), 0.0016 (1:10), 0.0021 (1:9), 0.0024 (1:8), 0.0031 (1:7), 0.0041 (1:6) and 0.0056 (1:5)

A) e' = 0, B) e' = 0.02, C) e' = 0.04, D) e' = 0.06, E) e' = 0.08, F) e' = 0.1

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Parametric Study

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Dependence on μ

400x400 ep = e' = 0.02

Resonances

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Chaotic Diffusion

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Kinds of diffusion

Parameters e' = 0.02 μ = 0.0041 ep = 0.01625 Initial Conditions x = 0 φ = π/3 Yf = 0 Δu = 0.299 Δu = 0.376 Integration 10 periods

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Chaotic Diffusion

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Main paradigm of diffusion: modulational

μ = 0.0041 ep = 0.01675 Δu = 0.376

e' = 0 e' = 0.02 e' = 0

ep,0 ep ep,0 ep

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SLIDE 19

Statistics of Orbits

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Classification of orbits

regular orbits escaping orbits transition orbits

> Snapshots at T = 10,10,10,10,10 periods

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Statistics of Orbits

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Stastistical results

(a) T = 10, (b) T = 10, (c) T = 10, (d) T = 10, (e) T = 10

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Statistics of Orbits

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Stastistical results

Power law α ~ 0.8

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Statistics of Orbits

Comparision between FLI and escaping times

Tesc < 10

Transition orbits Tesc > 10 Escaping orbits regular orbits

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Conclusions

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Hamiltonian Formalism in modified Delaunay variables secular effects, due to one or more planets hierarchy of Hamiltonian models corresponding to different levels of perturbation resonant proper elements characterisation of resonances Visualization of the resonant web - dependence on physical parameters phase portraits FLI maps Chaotic diffusion & statistics of escapes different paradigms and rates of chaotic diffusion statistical study of an ensemble of orbits in the resonant domain two characteristic peaks in the escaping times distribution correlation between the escaping times and the structure of the resonant web

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Thanks for your attention! Questions?

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Secular dynamics adding more planets

frequencies of the leading terms in the quasi-periodic representation of the oscillations of the planets' eccentricity vectors with and the amplitudes of oscillation of the ecc. vector and trigonometric on

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Correlation between the proper libration and Δu

Action labels libration motion around the forced eq. point We define Δu in the following way: for given ep, we compute the position of the fixed point. We then consider all the invariant curves around the eq. point (x = 0,u = u0) of the 1 d.o.f. We also take the line x = B(u-u0). We call up the point where the invariant curve intersects the

  • line. We finally define Δu = (up-u0). Up to quadratic terms in Δu, one has

In general, for B ≠ 0, Δu ≠ Dp (half-width of the oscillation of the variable u along the invariant curve of corresponding to the action variable Js), which is the common definition of the proper libration. Instead, one has