Resonance Capturegeneralization to non-

adiabatic regime

Observatoire de Cote d’Azur

, Jan 2007

Alice QuillenUniversity of Rochester

This Talk• Resonance Capture – extension to non-

adiabatic regime – when do resonances capture?

• Astrophysical settings: – Dust spiraling inward when collisions are not

important

– Neptune or exoplanet or satellite migrating outward

– Jupiter or exoplanet migrating inward

– Multiple exoplanets in resonance

Production of Star Grazing and Star-Impacting Planetesimals via Planetary Orbital Migration

Quillen & Holman 2000, but also see Quillen 2002 on the Hyades metallicity scatterFEB

Impact

Eject

Resonance Capture

Simple Hamiltonian systems

2( , ) cos( )

2

pH p K

2 2( , ) ( , )

2 2

is constant

0 is conserved

p qH p q H I I

H dI dtH dI

Idt

Harmonic oscillator

Pendulum

Stable fixed point

Libration

Oscillation

p

Separatrix

pq

I

Resonant angle

In the frame rotating with the planet

( 1) 0

mean motions are integer multiples

( 1) constant like 0 or

remains fixed

p

p

jn j n

j j

Resonant angle

Librating resonant angle in resonance

Oscillating resonant angle outside resonance

Resonance CaptureAstrophysical Settings

Migrating planets moving inward or outward capturing planets or planetesimals –

Dust spiraling inward with drag forces

What we would like to know: capture probability as a function of: --initial conditions--migration or drift rate--resonance properties

resonant angle fixed -- Capture

Escape

2~ e

Resonance CaptureTheoretical Setup

2

( )/ 2,

0

2/ 2

,02

( , ; , )

cos( ( ) )

Mean motion resonances can be written

( , ) cos( )

corresponds to order 2:1 3:2 (first order) Coefficients depen

kk p

k p pp

kk

K a b c

k p p

a bH k

kkk

d on time in drifting/ migrating systems sets the distance to the resonance

gives the drift rate

bdbdt

Resonance Capture in the Adiabatic Limit• Application to tidally drifting satellite

systems by Borderies, Malhotra, Peale, Dermott, based on analytical studies of general Hamiltonian systems by Henrard and Yoder.

• Capture probabilities are predicted as a function of resonance order and coefficients.

• Capture probability depends

on initial particle eccentricity.

-- Below a critical eccentricity capture is ensured.

Adiabatic Capture Theoryfor Integrable Drifting Resonances

Rate of Volume swept by upper separatrix

Rate of Volume swept by lower separatrix

Rate of Growth of Volume in resonance

Probability of Capturec

V

V

VP

V

V

VV

Theory introduced by Yoder and Henrard, applied toward mean motion resonances by Borderies and Goldreich

2

( ) / 2,

0

( , ; , )

cos( ( ) )k

k pk p p

p

K a b c

k p p

Limitations of Adiabatic theory

• At fast drift rates resonances can fail to capture -- the non-adiabatic regime.

• Subterms in resonances can cause chaotic motion.

temporary capture in a chaotic system

pe

Rescaling

2/(4 )2,0

(2 ) /(4 )2 /(4 ),0 2

2/ 2

,02

2 / 2

By rescaling momentum and time

The Hamiltonian ( , ) cos( )

can then be written

( , ) cos

k

k

k kk

k

kk

k

k

a

at

k

a bH k

k k

K b k

This factor sets dependence on initial eccentricity

This factor sets dependence on drift rate

All k-order resonances now look the same opportunity to develop a general theory

Rescaling

2/(4 )2,0

(2 ) /(4 )2 /(4 ),0 2

2 / 2cos

k

k

k kk

k

k

k

a

at

k

K b k

Drift rates have units

First order resonances have critical drift rates that scale with planet mass to the power of -4/3. Second order to the power of -2.

2t

Confirming and going beyond previous theory by Friedland and numerical work by Ida, Wyatt, Chiang..

Rescaling and dimensional analysis

2/ 2

,02( , ) cos( )

Combine , , to yeild units of time or momentum

one finds the same scaling factors

The timescale does not seem to be

the libration period in any particular part of the r

kk

a bH k

k ka b

esonance

Numerical Integration of rescaled systemsCapture probability as a function of drift rate and initial

eccentricity for first order resonancesP

rob

ab

ilit

y o

f C

ap

ture

drift rate

First order

Critical drift rate– above this capture is not possible

At low initial eccentricity the transition is very sharp

Transition between “low” initial eccentricity and “high” initial eccentric is set by the critical eccentricity below which capture is ensured in the adiabatic limit (depends on rescaling of momentum)

40( ) 2.2,1,0.1,10

1.5crit

t

Capture probability as a function of initial eccentricity and drift rate

drift rate

Pro

bab

ilit

y o

f C

ap

ture

Second order resonances

High initial particle eccentricity allows capture at faster drift rates

2 4 60( ) 1,0.1,10 ,10 ,10

0.125crit

t

When there are two resonant terms

drift rate

coro

tati

on

reson

an

ce

str

en

gth

dep

en

ds o

n p

lan

et

eccen

tric

ity

Capture is prevented by corotation resonance

Capture is prevented by a high drift rate

capture

escape

drift separation set by secular precession

depends on planet eccentricity

2

1/ 2

( , ; , )

cos( ) cos( )p

K b c

Limits

• Critical drift rates defining the adiabatic limit for each resonance can be computed from resonance strengths.

• Critical planet eccentricities for prevention of capture by corotation resonance can also be computed

4/3 2/31, 1,0

2 22,0

5/ 41,0 31

2

,

232

2( 1) for 1

0.5( 2) for 2

where (+ possibly an indirect term)

p c it

c

r

p rit

n j a k

j k

f

a j

n

Trends• Higher order (and weaker) resonances require slower drift rates for

capture to take place. Dependence on planet mass is stronger for second order resonances.

• If the resonance is second order, then a higher initial particle eccentricity will allow capture at faster drift rates – but not at high probability.

• If the planet is eccentric the corotation resonance can prevent capture

• For second order resonances e-e’ resonant term behave likes like a first order resonance and so might be favored.

• Resonance subterm separation (set by the secular precession frequencies) does affect the capture probabilities – mostly at low drift rates – the probability contours are skewed.

Applications to Neptune’s migration

• Neptune’s eccentricity is ~ high enough to prevent capture of TwoTinos. 2:1 resonance is weak because of indirect terms, but corotation term not necessarily weak.

• Critical drift rate predicted is more or less consistent with previous numerical work.

Few objects captured in 2:1 and many objects interior to 2:1 suggesting that capture was inefficient

image from wiki

Application to multiple planet systems

This work has not yet been extended to do 2 massive bodies, but could be. Resonances look the same but coefficients must be recomputed.

4/32:1

23:1

drift timescale

For capture:

0.4

15

p

p

a

a

P

P

A migrating extrasolar planet can easily be captured into the 2:1 resonance but must be moving fairly slowly to capture into the 3:1 resonance. ~Consistent with timescales chosen by Kley for hydro simulations.

Lynnette Cook’s graphic for 55Cnc

Future Applications

• Formalism is very general – if you look up resonance coefficients, critical drift rates for non adiabatic limit can be estimated for any resonance : (Quillen 2006 MNRAS 365, 1367) See my website for errata!

• Particle distribution could be predicted following migration given an initial eccentricity distribution and a drift rate.

• Drifting dust particles can be sorted by size.

Future work• Extension to consider stochastic migration? • While we have progress in predicting the

probability of resonance capture we still lack good theory for predicting lifetimes.

• This theory has not yet been checked numerically --- more work needs to be done to relate critical eccentricities to those actually present in simulations