Resonance
Capture
When does it
Happen?
Alice Quillen
University of
Rochester
Department of Physics and
Astronomy
Resonance Capture
Astrophysical 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

 ~ e2
resonant angle
fixed -- Capture
Escape
Resonance Capture
Theoretical Setup
K (  , ;  ,  )  a 2  b  c

k
  k , p ( k  p)/2 cos(  ( k  p)  p p )
p 0
Mean motion resonances can be written
a 2
b
k /2




cos( k )
k
,0
2
k
k
k corresponds to order 2:1 3:2 (first order)
Coefficients depend on time in
drifting/migrating systems
b sets the distance to the resonance
db
gives the drift rate
dt
H ( ,  ) 
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 Theory
for Integrable Drifting Resonances
V  Rate of Volume swept by upper separatrix
V  Rate of Volume swept by lower separatrix
V  V  Rate of Growth of Volume in resonance
V  V
Pc 
 Probability of Capture
V
Theory introduced by Yoder
and Henrard, applied toward
mean motion resonances by
Borderies and Goldreich
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 nonadiabatic regime
• Subterms in resonances can cause chaotic motion.
K (, ; ,  )  a  b  c
2
k
 e
p
   k , p  ( k  p ) / 2 cos(  ( k  p)  p p )
p 0
temporary
capture in a
chaotic system
Rescaling
By rescaling momentum and time

 k ,0 k
2 2 /(4  k )

a
(2  k ) /(4  k )
a
t
2
k
The Hamiltonian can be written as
   k2,0/(4 k )
2
K    b  
k /2
cos 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)
By rescaling momentum and time

 k ,0 k
2 2 /(4  k )

a
(2  k ) /(4  k )
a
t
2
k
The Hamiltonian can be written as
   k2,0/(4 k )
2
K    b  
k /2
cos k
Drift rates
2
have tunits
First order
resonances
have critical
drift rates
that scale
with planet
mass to the
power of -4/3.
Second order
toConfirming
the power
and going
ofbeyond
-2. previous theory
by Friedland and
numerical work by Ida,
Wyatt, Chiang..
Scaling (continued)
We are used to thinking of a resonance width
  e for first order resonances.
However this predicts the wrong scaling.
For a particle initially with zero eccentricity
at the separatrix
it is perturbed to an eccentricity e
2
 k ,0 k
2 2 /(4  k )
a
 (2  k ) /(4  k )
2 /(4  k ) a
oscillating with period P  k ,0
k2
Note same factors which set the adiabatic limit.
t 2
Numerical Integration of rescaled systems
Capture probability as a function of drift rate and initial
eccentricity for first order resonances
Probability of
Capture
First
order
At low
initial
eccentricity
the
transition is
very sharp
Critical
drift rate–
above this
capture is
not possible
drift rate 
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
Probability of
Capture
Second
order
resonance
s
drift rate 
High
initial
particle
eccentrici
ty allows
capture at
faster
drift rates
When there are
two resonant terms
drif
t
2
K (, ; ,  )    b  c
1/ 2

Capture is prevented by
corotation resonance
corotation resonance
strength
depends on planet
eccentricity
separation set by
secular
precession
escape
capture
drift rate 
cos(  )   cos(   p )
depends on
planet
eccentricity
Capture
is
prevente
d by a
high
drift
rate
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.
Lynnette Cook’s graphic for 55Cnc
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
s m 
M p / M* 
5/ 3 
j 

3
10


4 / 3
 L*  M * 
 

L
M
 

1/ 2
 a 


 AU 
Applications (continued)
• Capture into j:j+1 resonance for drifting planets requires
4/3
dn p
M


p
 j 5/ 3 
 in units of planet rotation period
dt
 M* 
• 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