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Planetary Dynamics
Secular (“long-term”) dynamics:
Replacing point masses with wires
(longitude-averaged evolution)
Linear secular theory (keeping quadratic terms in Hamiltonian):
N planets N eccentricity modes
ek
zk
z̃kα = ekα exp[i(gα t + βα )]
ω̃k
zk = ek exp(iω̃k ) =
N
!
α=1
Aα z̃kα
Fomalhaut
Eccentric planet begets
eccentric ring
Equilibrium belt orbits
are eccentric
and aligned with
the planet’s orbit
closed non-crossing
orbits
planetary wire
(secular approximation)
beta Pic b
8 AU
5.7 AU
2003
2009
line of nodes of the perturbing planet
VLT 8-m with L’ Adaptive Optics
L ~ 10-3.7 L☉ , t ~ 12 Myr → M ~ 9 MJ
semimajor axis 8-15 AU; consistent with evolving vertical warp
Mode amplitudes vary with time in nonlinear secular theory
Mercury’s chaotic orbit
(Laskar 96)
Production of
hot Jupiters in
multi-planet systems
by secular chaos
(Wu & Lithwick 10)
!
Jz ∝ 1 − e2 cos I
conserved
3.2:2
3:2
conjunction
3:2
3:2
2.8:2
2.9:2
b
resonant width Δ a res
Neptune-Pluto Orbit-Orbit Resonance
Resonant KBOs (~26%)
Plutino (3:2) Snapshot
Wave pattern rotates rigidly with Neptune
Resonant clustering
of Plutinos (3:2) in the
Kuiper Belt
Epsilon Eridani
80 AU
678
GJ 876:
Resonant
Planetary System
N/NTOT < 5
1
Kepler adjacent pairings
RV adjacent pairings
0
1
1.5
2
2.5
3
3.5
4
4.5
Period Ratio
1
N/NTOT
0.8
0.6
0.4
Kepler adjacent pairings
RV adjacent pairings
0.2
0
1
10
Period Ratio
100
5
2x
∆x
x
Deriving the chaotic zone width
I. The kick at conjunction
x!a
Mp
Kick in eccentricity ∆e
ap
Ωp
a
= ap + x
1 GMp
∆v
∼
∆t
∆e ∼
2
v
v x
Mp ! a "2
∼
M∗ x
Kick in semimajor axis ∆x
!
GM∗
Use Jacobi constant: CJ ≈ −
− Ωp M∗ (a + x)(1 − e2 )
2(a + x)
x∆x
∆x
2
∼
⇒ 2 ∼ (∆e) ⇒
a
a
!
"2 # $
Mp
a 5
M∗
x
chaotic zone
N
J S
U
Constraining ap and Mp
using the sharp inner
belt edge
Lecar et al. 2001
Inner belt edge =
Outer edge of planet’s
“chaotic zone”
Chaotic zone width ~
(Mplanet/Mstar)2/7 aplanet
Quillen 06
Surface brightness
from Kalas et al. 05
Resonance Sweeping
Dissipative relaxation of
parent bodies
onto non-crossing
(forced eccentric)
orbits
Relaxation occurs during:
• Present-day collisional
cascade
• Prior coagulation
(2)
eforced (a) =
closed non-crossing
orbits
planetary wire
(secular approximation)
b3/2 (aplanet /a)
eplanet
(1)
b3/2 (aplanet /a)
Wyatt et al. 99
Kalas et al. 05
U8
A
0
7
6
80 AU
678
7
130 AU
678
Planetary chaotic zone
= Region where first-order
resonances overlap
m+1:m
x
m:m−1
∆x
∆s
Regular
Chaotic zone
Chaotic zone
#3
#2
conjunction #1
Chaotic
#3
#1
Random
walk
Orbit
crossing
and
ejection
. . .
in a
and e
#2
Wisdom 1980
Duncan et al. 1989
Quillen 2006
Candidate planet (0.5 Jupiter mass)
not confirmed: only 2 epochs
not thermal emission from
planetary atmosphere
40 RJ reflective dust disk?
Variable Hα emission?
HR 8799
A-type star 30-60 Myr old
with 4 Super-Jupiters
Orbital resonances afford stability
d:c = 2:1 resonance
Other possibilities include
d:c:b = 4:2:1
e:d:c = 4:2:1
dynamical masses < 20 Jupiter
masses each
Cloudy spectra unlike
brown dwarfs
log(L/L! )
6-7
MJ
7-10
MJ
log(age/yr)
2:1 = Planetary Speedometer
tmigrate ≡ a/(da/dt) ≥ 107 yr
6
tmigrate = 10 yr
Gemini Planet Imager (GPI)
2012
Pan-STARRS (once a week,
mag 24)
and LSST (once every few
days, mag 24.5)
PS1
LSST site
M ~ 0.1 M⊕♁
3 AU-3
π (100
km)2
1 km s-1
KBOs = test particles
The Kuiper Belt: The Global View
Big KBOs are excited
Deriving the chaotic zone width
Resonance overlap
Resonance spacing
∆s ! x "2
∼
a
a
if ∆x > ∆s
∆x
i.e., if x <
m:m−1
m+1:m
x
!
Mp
M∗
then chaos
∆s
"2/7
a
g., if ωplanet = ωbelt (nested ellipses)
aplanet = 115 AU
then eplanet = 0.12
Mplanet = 0.5MJ
Mbelt > Mparent bodies ∼ 3M⊕
Enough material for
gas giant core
Asymmetric capture:
Migration-shifted potentials
V
π
Direct
Φ
t2librate
∆Φ ∼
torbital tmigrate
Indirect
Results
If Mplanet ↑ then aplanet ↓
Planet position too far
from dust belt
→
∴ Mplanet < 3 MJ
Planet also evacuates
Kirkwood-type gaps
Chiang et al. 2008
Kalas et al. 2008
Number of stable parent bodies (in 0.1 AU bins)
1000
100
0.3 MJ
11:9
6:5
5:4 4:3
9:7
10
4:3
100
1 MJ
7:5
10
3:2
100
3 MJ
10
5:3
100
10 MJ
7:4
10
1
90 100 110 120 130 140 150
Time-averaged semimajor axis (AU)
Results
1.0
If Mplanet ↑ then edust ↑
0.8
→
∴ Mplanet < 3 MJ
Relative oŒ
Surface brightness
profiles broaden too much
K05 model
0.1 MJ, apl=120 AU
0.3 MJ, apl=115.5 AU
1 MJ, apl=109 AU
3 MJ, apl=101.5 AU
10 MJ, apl=94 AU
0.6
0.4
Mbelt > Mparent bodies ∼ 3M⊕
Enough material for gas giant
core
0.2
0.0
100
120
140
160
180
Semimajor axis a (AU)
200
• The Kuiper belt comprises tens
of thousands of icy, rocky
objects having sizes greater
than 100 km
• Many KBOs occupy highly
eccentric and inclined orbits
that imply a violent past
• Pluto and other Resonant
KBOs share special gravitational
relationships with Neptune
e.g., if ωplanet = ωbelt (nested ellipses)
aplanet = 115 AU
then eplanet = 0.12
Mplanet = 0.5MJ
• Extrasolar debris disks are
nascent Kuiper belts
•
Belts are gravitationally
sculpted by planets
Fomalhaut b: Planet-Debris Disk Interaction
chaotic
width
t = 100
0 Myr
Myr
Step 1: Screen parent bodies for gravitational stability
tage ∼ 108 yr
Step 2: Replace parent bodies with dust grains
∆t = tcollision = 0.1 Myr
Step 3: Integrate dust grains with radiative force
for collisional lifetime
tcollision ∼ torb /τ ∼ 0.1 Myr
Parent bodies collide once in system age
Collisional Cascade
Distribution of sizes of
Kuiper belt objects
6
1000
Nominal Diameter at 42 AU (km)
100
10
log N(<mR) (deg-2)
4
Most visible grains are just large
enough to avoid stellar blow-out
Cl
2
a
ic
ss
1
al
Excited
Measured Extrapolated
0
-2
-4
20
25
mR
30
35
Results
If Mplanet ↑ then aplanet ↓
Planet position too far
from dust belt
→
∴ Mplanet < 3 MJ
Planet also evacuates
Kirkwood-type gaps
Mbelt > Mparent bodies ∼ 3M⊕
Enough material for gas giant
core
Chiang et al. 2008
Kalas et al. 2008
Number of stable parent bodies (in 0.1 AU bins)
1000
100
0.3 MJ
11:9
6:5
5:4 4:3
9:7
10
4:3
100
1 MJ
7:5
10
3:2
100
3 MJ
10
5:3
100
10 MJ
7:4
10
1
90 100 110 120 130 140 150
Time-averaged semimajor axis (AU)
Measuring Debris Disk Masses
N
dN/ds ∝ s−q
s
sblow svisible
∼ 0.2µm
850 µm flux consistent w/
MSED (stop ) ∼ 0.01M⊕
q = 7/2(Dohnanyi)
stop ∼ 10 cm
[from tcollision (stop ) ∼ tage ]
ssubmm
∼ 100 µm
stop
Ṁ∗
Ṁ∗
Ṁ∗
Ṁ∗
= 103 Ṁ"
= 102 Ṁ"
= 10Ṁ"
= 1Ṁ"
Packed planetary systems
Packed
formation
Instability
and ejection
Stability
restored
occurs when
planets outweigh
parent disk
Signature recorded in Kuiper belt
Regularization
by dynamical
friction
Dynamical friction:
Small bodies slow
big bodies
Numerical simulations of packed planetary systems
Solar system-like outcomes emerge from chaos
Observed
Simulated
Stirring
of KBOs
by
Rogue
Ice Giants
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Short-Period Comets
Raising Sedna’s Peri
by Stellar Encounters
Before
Typical open
cluster
Praesepe cluster M44
After
· n∗ ∼ 4 stars/pc3 (R1/2 ∼ 2 pc)
· t ∼ 200 Myr
· !v∗2 "1/2 ∼ 1 km/s
Fernandez & Brunini 00
Discovery of
Kuiper Belt Object
(KBO) #3
1992 QB1: “Smiley”
David Jewitt (U Hawaii)
Jane Luu (UC Berkeley)
Size depends on observed brightness and
intrinsic reflectivity (albedo)
Quaoar
Ixion
Planetary Protection Mechanism:
Orbit-Orbit Resonance
Neptune makes 3 orbits
for every
2 orbits of Pluto
“Dance of the Plutinos”
The Orbit of Sedna
Number
Resonant KBOs
12
6
0
Number
5:1 4:1 3:1 5:2 7:3 2:1 9:5 7:4
80
40
0
Number
5:3 8:5 3:2 10:7 7:5 11:8 4:3
6
3
0
13:10 9:7 14:11 5:4 11:9 6:5 1:1
2003 UB313
“Xena”
Bigger than Pluto
Palomar 48-inch / M. Brown, C. Trujillo, & D. Rabinowitz (Caltech,Yale)
Hubble Space Telescope
2003 UB313 (mapp = 19)
Diameter = 2397±100 km
vs.
Pluto
Diameter = 2274 km
“Dwarf Planets”
Nix
“Easter Bunny”
Hydra
“Santa”
I.A.U. definition
(a) orbits the Sun
(b) hydrostatic (round)
shape
(c) not a satellite
(d) not cleared its
neighborhood
What we know: • The Kuiper belt comprises tens of
thousands of icy, rocky objects
having sizes greater than 100 km
•
The Kuiper belt is the source of
short-period comets
•
Pluto and other Resonant KBOs
share special gravitational
relationships with Neptune
•
Many KBOs, especially large ones,
occupy highly eccentric and inclined
orbits that imply a violent past
•
Other star systems have their own
Kuiper belts
First discovered Neptune Trojan (1:1)
1 Large Neptune Trojan in 60
⇒
°
~10-30 Large Neptune Trojans
vs. ~1 Large Jovian Trojan
(“Large” ≡ 130-230 km diameter
assuming 12-4% visual albedo)
Neptune
based on Mike Brown’s survey limits:
1 Earth mass at less than 200 AU
1 Neptune at less than 500 AU
1 Jupiter (in reflected light) at less than 1000 AU
based on Hipparcos and Tycho-2
cannot be a self-luminous main-sequence star above
the hydrogen burning limit
infrared detection of a Jupiter or brown dwarf
could be interesting
birth ring
daughter dust particle
Theoretical Snapshots of Resonant KBOs
⇐ Plutinos 3:2
•
•
Twotinos 2:1 ⇒
Observational Facts and Theoretical Deductions
1. Pluto is the largest known member of a swarm of
billions of outer solar system bodies that supply
new comets.
2. Pluto and the Plutinos are locked in an orbital
resonance established by Neptune.
3. The orbits of many Kuiper Belt Objects are dynamically
excited.
4. Pluto is not alone in having an
orbital companion.
Pluto
Charon
1999 TC 36
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