Almog Yalinewich - 26.12.19
Preamble
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
Monolithic
Polylithic
Isolation mass
How long?
Dawson 2018
Every Gyr per solar system =>
1 per week in the galaxy
Rice et al. 2018
Higher order interaction gives
Not steep enough
Diffusion predicts gradual increase
Energy / Angular momentum unchanged, but the position of periapse does change
Ellipse anti - alignment
Angles are not random
KAM result: for a sufficiently small
the deviation is bounded forever
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?
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
No fixed points
Complete circle
Every irrational number can be approximated by a rational number
declines polynomially (Dirichlet)
What about the perturbation ?
This sum looks complicated, but it is actually dominated by a single term
Critical value
Attains maximum for some finite
No fixed points / system is stable if
Opposite extreme: very unstable conditions
Hence
Substituting back to the exponent
Exponential dependence on the strength of the perturbation
Actions are bounded for exponentially long time
Interaction between planets
Fourier transform - Laplace coefficients
Asymptotic expansion
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
Yalinewich & Petrovich 2019
Previous paradigm:
Hill Parameter
We predict
Actually fits better to simulations
Chambers et al. 1996
Obertas et al 2017
Theoretical model for the survival time of tightly packed, co - planar, non resonant planetary systems
Future directions: effects of inclination & resonance
Questions?
Paradigm shift: