Post on 28-Dec-2015
transcript
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