Dynamical instabilities and the Onset of Chaos in Tight Planetary Systems
Almog Yalinewich - 26.12.19
Preamble
The Two Body Problem
Kepler's second law
In velocity space, the particle always moves on a circle
Cross with angular momentum and get the conserved LRL vector
always points toward the periapse
Types of Structure Formation
Monolithic
Polylithic
Planetary Collisions
- Composition
- Atmospheric Mass
- Spin
- Orbital Parameters
Collision Course
Isolation mass
How long?
Teetering on the Verge of Stability
Dawson 2018
High Energy Connection
Every Gyr per solar system =>
1 per week in the galaxy
Survival Time of Tight Co-planar Planetary Systems
Numerical Experiments
Rice et al. 2018
Previous Paradigm -
Chaotic Diffusion
Impulse Approximation
Planet - Planet Scattering
Higher order interaction gives
Not steep enough
Diffusion Time
Evolution of Orbital Parameters
Diffusion predicts gradual increase
Evolution of Orbital Parameters
Energy / Angular momentum unchanged, but the position of periapse does change
Periapse Drift
Failure of Chaotic Diffusion
Ellipse anti - alignment
Angles are not random
Theoretical explanation for survival time steepness?
KAM Theory
Hamiltonian Perturbation Theory
KAM result: for a sufficiently small
the deviation is bounded forever
Original Proof
Try to find a canonical change of variable
Such that the new Hamiltonian is
Solution by iterations => convergence for finite
Why doesn't this always work?
Periodic Points
Coordinates are just labels. Nothing special about them
What would prevent you from re - labeling the perturbed trajectory to the unperturbed trajectory?
Mapping preserves periodic points
If the perturbed trajectory has periodic points, it cannot be mapped to the unperturbed trajectory
System with Two Degrees of Freedom
No fixed points
System with Two Degrees of Freedom
Complete circle
Treasure Hunt Model
Creating Fixed Points
Every irrational number can be approximated by a rational number
Dirichlet's Theorem
Multiple Steps
declines polynomially (Dirichlet)
What about the perturbation ?
Fourier Transform
This sum looks complicated, but it is actually dominated by a single term
Asymptotics
Critical value
Bound on Perturbations
Attains maximum for some finite
No fixed points / system is stable if
Nekhoroshev Estimates
Opposite extreme: very unstable conditions
Hence
Substituting back to the exponent
Exponential dependence on the strength of the perturbation
Nekhoroshev Estimates in Hamiltonian Systems
Actions are bounded for exponentially long time
Nekhoroshev Estimates for Tight Planetary Systems
Dependence on Separation
Interaction between planets
Fourier transform - Laplace coefficients
Asymptotic expansion
Dependence on Mass
Quillen 2011 argues for three body resonance overlap
Three body resonant Hamiltonian independent of periapse direction
6 DOF
Two conserved quantities (Energy & Angular momentum)
12 DOF
4 DOF
Small parameter
Laplace Resonance
Ansatz for The Survival Time
Yalinewich & Petrovich 2019
Packing Metric
Previous paradigm:
Hill Parameter
We predict
Actually fits better to simulations
Packing Metric
Chambers et al. 1996
Two Body Resonances
Obertas et al 2017
Conclusion
Theoretical model for the survival time of tightly packed, co - planar, non resonant planetary systems
Future directions: effects of inclination & resonance
Questions?
Paradigm shift:
- Chaotic diffusion
- Packing metric
stability of planetary systems
By almog yalinewich
stability of planetary systems
- 261