Post on 20-Jan-2016
Alice Quillen University of Rochester
Department of Physics and Astronomy
Oct, 2005
Submillimeter imaging by Greaves and collaborators
Morphology of Circumstellar Disks
•Structure observed in circumstellar disks: arcs, spiral arms, edges, warps, clumps, holes, lopsidedness
•Planets can affect the morphology a planet detection technique,
complimentary to radial velocity and transit searches.
•Jupiter is 10-3 times the Mass of the Sun resonant effects or long timescales (secular) required for a planet to affect the morphology
•Models are sensitive to dust production sites, the gas and planetesimals as well as planetary properties.
•Models are sensitive to the past history –evolution of the planetary systems.
Production of Star Grazing
and Star-Impacting
Planetesimals via Planetary
Orbital Migration
(Quillen & Holman 2000)
FEB
Impact
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
Resonance Capture vs Stability of Particles in Resonance
• Poynting Robertson drag and that from solar wind cause in-spiral of dust particles. These particles will cross orbital resonances with a planet. If captured, resonances can delay the in-spiral.
• Alternatively dust particles released from planetesimals trapped in resonance are likely to be trapped.
• The particle distribution is dominated by long lived particles. Consequently the morphology can be affected by resonances.
Possible explanations proposed for Epsilon Eridani (me) and Vega disk morphologies (Kuchner and Wyatt and collaborators). For all of these (Deller). Also see Ozernoy et al. 2001. On the dust in the Kuiper Belt (see Liou & Zook and Moro-Martin et al).
Resonance Capture -Astrophysical Settings
Migrating planets or satellites moving inward or outward capturing planets or planetesimals into resonance–
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
,02
Mean motion resonances can be written
( , ) cos( )
corresponds to order of resonance. : Coefficients depend on time in drifting/ migrating systems sets the distance
t
kk
a bH k
kkk j j k
b
o the resonance
gives the drift rate
Particle eccentricity depends on
dbdt sh
ap
e o
f re
son
an
ce d
ep
en
ds
on
b
Non zero resonant angle. Large radius here corresponds to large eccentricity
2
( ) / 2,
0
cos( ( ) )k
k pk p p
p
K a b c
k p p
p
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 but not on drift rate
d
rift
ing
syste
m
Capture here
Limitations of Adiabatic theory• In complex drifting 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
,
* planet mass/ planet eccentricity
pk p e
Me
2 ( ) / 2,
0
cos( ( ) )k
k pk p p
p
K a b c k p p
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
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 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..
Numerical Integration of rescaled systems
Capture probability as a function of drift rate and initial eccentricity for first order resonances
Pro
bab
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
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
curves to the right correspond to higher initial eccentricities
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
• 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.
General formulism We have derived a general recipe to predict capture
probabilities for drifting Hamiltonian orbital systems. Coefficients given in appendix of Murray and Dermott can now be used to predict the capture probability for ANY migrating or drifting system into ANY resonance.
First Applications• Neptune’s eccentricity is ~ high enough to
limit 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 are sorted by size. We can predict which size particle will be captured into which resonance.