Resonance Capture

When does it Happen?

Alice Quillen University of Rochester

Department of Physics and Astronomy

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 depend

kk p

k p pp

kk

K a b c

k p p

a bH k

kkk

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• In complex systems many resonances are available. Weak

resonances are quickly passed by. • 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

By rescaling momentum and time

The Hamiltonian can be written as

cos

k

k

k kk

k

k

k

a

at

k

K b k

This power sets dependence on initial eccentricity

This power sets dependence on drift rate

All k-order resonances now look the same

Rescaling (continued)

2/(4 )2,0

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

2 / 2

By rescaling momentum and time

The Hamiltonian can be written as

cos

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..

Scaling (continued) We are used to thinking of a resonance width

for first order resonances.

However this predicts the wrong scaling.

For a particle initially with zero eccentricity

at the separatrix

it is perturbe

e

2/(4 )2,02

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

d to an eccentricity

oscillating with period

Note same factors which set the adiabatic limit.

k

k

k kk

k

ke

a

aP

k

2t

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)

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

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

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

• Neptune’s eccentricity is ~ high enough to prevent capture of TwoTinos.

• A migrating extrasolar planet can easily be captured into the 2:1 resonance but must be moving very slowly to capture into the 3:1 resonance. -- Application to multiple planet extrasolar systems.

• Drifting dust particles can be sorted by size.

Capture for dust particles into j:j+1 resonance

Lynnette Cook’s graphic for 55Cnc

1/ 24/3

*5/3 * *3

/

10p

m

M M L M as j

L M AU

Applications (continued)

• Capture into j:j+1 resonance for drifting planets requires

• Formalism is very general – if you look up resonance coefficients critical drift rates can be estimated for any resonance.

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

-- more work needs to be done to relate critical eccentricities to those actually present in simulations

4/3

5/3

*

in units of planet rotation periodp pdn Mj

dt M