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Assembling Kepler's tightlypacked planetary systems

Tom Hands Richard Alexander and Walter Dehnen
tomhands.com & tomhands.com/exovis
Kepler-11, ar tist's impression. Image Credit: NASA/Tim Pyle


Known exoplanets (8/1/2014)
100.0000
























































Planetary mass MSin(i) PlanetaryorMm ss //Jupiter Masses a Mjup












1.0000


































































































































































































































































































































































0.0100






































































































































































































































j




s













































n



u















Transit ( 288 ) Transit Timing Variations ( 10 ) Radial Velocity ( 432 ) Microlensing ( 18 ) Directly Imaged ( 8 ) Kepler Candidates (M ~ r^2.06) ( 3469 )











v



e






















0.0001

m

M

0.01

0.10

Data source: exoplanets.org

SemiOrbital semi-major axis //AU U -majora axis A

1.00

10.00 p

100.00


Kepler-11: A prototype tightly-packed system

· · · ·

6 low mass planets Low mutual inclination Very low eccentricities This is potentially the dominant mode of planet formation (Batalha et al. 2013)

Image Credit: Lissauer et al. 2011


Sun

Kepler-32
system

Kepler-11 M Kepler-80
ju p

10Meart Meart 0.01 0.1 1 semi-major axis / AU
h

h

10



Why is this interesting?

· · · · ·

Dynamically cold - why? Low e and i suggests weak interactions or damping Planets probably too close to have formed in situ (Raymond & Cossou 2014) Not all pairs are resonant So how were these things formed?


Disc-driven migration

·

Key physics:

· Low mass - type I migration · Planet-planet interactions · Eccentricity damped by disc · Turbulence in disc
Image Credit: Phil Armitage


A parameterised approach

· ·

Bespoke N-body code Additional forces

· - migration timescale · - disc turbulence parameter · K - ratio of e to a damping

·

Modified 2nd order Leapfrog integrator


Our simulations

· · ·

Explore 3D parameter space 10,000 runs for each system Initial conditions

· Outside of snow-line · Random initial spacing based on
oligarchic growth argument

· Random initial phases


Example systems: Kepler-11

0.35 0.30 0.25
Fraction of runs

0.20 0.15 0.10 0.05 0.00 1.2 1.0 0.8 0.6 0.4 log10(a) 0.2 0.0 0.2 0.4


Parameter space: Kepler-11
6.0 6.0 5.0 5.5 5.5 5.5 6.0 5.0 5.0
log10(b ) log10(t ) log10(t )

6.5 7.0

4.5

4.5 7.5

4.0

8.0 7.5 7.0 6.5 6.0 5.5 5.0 log10(b ) 0.1 0.2

4.0 0.5

1.0

1.5 2.0 log10(K )

2.5

8.0 0.5

1.0

1.5 2.0 log10(K ) 0.9

2.5

0.0

0.3 0.4 0.5 0.6 0.7 Fraction of systems surviving in correct order

0.8

1.0


Summary

· · · ·

It is possible to build such tightly-packed systems via simultaneous, disc-driven migration with sensible parameters Preferred migration rate is slow Level of turbulence is likely very important Moderate to high e damping required


Caveats & Future work

· · ·

This model is a proof of concept Initial conditions

· One can envisage many improvements · Planetary order · Initial spacing
Modifications to the scheme

· Asynchronous formation/migration · Stopping criteria


Summary

· · · · ·

It is possible to build tightly-packed systems via simultaneous, disc-driven migration with sensible parameters Preferred migration rate is slow Level of turbulence is likely very important Moderate to high e damping required Future work to concentrate on more realistic models