Secular ( long-term ) dynamics: Replacing point masses with wires (longitude-averaged evolution)
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1 Planetary Dynamics
2 Secular ( long-term ) dynamics: Replacing point masses with wires (longitude-averaged evolution)
3 Linear secular theory (keeping quadratic terms in Hamiltonian): N planets N eccentricity modes e k z k ω k z kα = e kα exp[i(g α t + β α )] N z k = e k exp(i ω k )= A α z kα α=1
4 Fomalhaut Eccentric planet begets eccentric ring closed non-crossing orbits planetary wire (secular approximation) Equilibrium belt orbits are eccentric and aligned with the planet s orbit
5 beta Pic b 8 AU 5.7 AU line of nodes of the perturbing planet VLT 8-m with L Adaptive Optics L ~ L, t ~ 12 Myr M ~ 9 MJ semimajor axis 8-15 AU; consistent with evolving vertical warp
6 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)
7 J z 1 e 2 cos I conserved
8 3.2:2 conjunction 3:2 3:2 3:2 2.8:2 2.9:2 b Δ Neptune-Pluto Orbit-Orbit Resonance
9 Resonant KBOs (~26%)
10 Plutino (3:2) Snapshot Wave pattern rotates rigidly with Neptune
11 Resonant clustering of Plutinos (3:2) in the Kuiper Belt Epsilon Eridani 80 AU 678
12 GJ 876: Resonant Planetary System
13 1 N/N TOT < 5 Kepler adjacent pairings RV adjacent pairings Period Ratio N/N TOT Kepler adjacent pairings RV adjacent pairings Period Ratio
14 x Ω p 2x M p a p x a = a p + x Deriving the chaotic zone width I. The kick at conjunction e v v x a Kick in eccentricity e 1 v GM p x 2 M p M ( a x t ) 2 Kick in semimajor axis x Use Jacobi constant: C J GM 2(a + x) Ω p M (a + x)(1 e 2 ) x x a 2 ( e) 2 x a ( Mp M ) 2 ( a ) 5 x
15 chaotic zone J S U N Constraining a p and M p using the sharp inner belt edge Lecar et al Inner belt edge = Outer edge of planet s chaotic zone Quillen 06 Chaotic zone width ~ (M planet /M star ) 2/7 a planet Surface brightness from Kalas et al. 05
16
17 Resonance Sweeping
18 Dissipative relaxation of parent bodies onto non-crossing (forced eccentric) orbits Relaxation occurs during: Present-day collisional cascade Prior coagulation e forced (a) = b(2) 3/2 (a planet/a) b (1) 3/2 (a planet/a) e planet closed non-crossing orbits planetary wire (secular approximation) Wyatt et al. 99 Kalas et al. 05
19 70 AU AU AU 678
20 Planetary chaotic zone x m +1:m m : m 1 x s = Region where first-order resonances overlap
21 Regular #3 Chaotic zone Chaotic zone #2 conjunction #1 #1 Chaotic #3 Random walk... in a and e Orbit crossing and ejection #2 Wisdom 1980 Duncan et al Quillen 2006
22 Candidate planet (0.5 Jupiter mass) not confirmed: only 2 epochs not thermal emission from planetary atmosphere 40 R J reflective dust disk? Variable Hα emission?
23 HR 8799 A-type star 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
24 log(age/yr) Cloudy spectra unlike brown dwarfs 6-7 MJ log(l/l ) 7-10 MJ
25 2:1 = Planetary Speedometer t migrate a/(da/dt) t migrate
26
27 Gemini Planet Imager (GPI) 2012 Pan-STARRS (once a week, mag 24) and LSST (once every few days, mag 24.5) PS1 LSST site
28
29 M ~ 0.1 M 3 AU -3 π (100 km) 2 1 km s -1 KBOs = test particles
30 The Kuiper Belt: The Global View Big KBOs are excited
31 Deriving the chaotic zone width Resonance overlap Resonance spacing ) 2 s a ( x a if x s x m +1:m m : m 1 x s i.e., if x< ( Mp M then chaos ) 2/7 a
32 ., if ω planet = ω belt (nested ellipses) then a planet =115AU e planet =0.12 M planet =0.5M J M belt M parent bodies 3M Enough material for gas giant core
33 Asymmetric capture: Migration-shifted potentials Direct π Φ t 2 librate Φ t orbital t migrate Indirect
34 Results If M planet then a planet Planet position too far from dust belt M planet < 3 M J Planet also evacuates Kirkwood-type gaps Chiang et al Kalas et al Number of stable parent bodies (in 0.1 AU bins) :9 0.3 M J 6:5 5:4 9:7 4:3 1 M J 4:3 7: M J 3: M J 5:3 7: Time-averaged semimajor axis (AU)
35 Results If M planet then e dust Surface brightness profiles broaden too much M planet < 3 M J M belt M parent bodies 3M Enough material for gas giant core Relative K05 model 0.1 M J, a pl =120 AU 0.3 M J, a pl =115.5 AU 1 M J, a pl =109 AU 3 M J, a pl =101.5 AU 10 M J, a pl =94 AU Semimajor axis a (AU)
36 .g., if ω planet = ω belt (nested ellipses) a planet =115AU then e planet =0.12 M planet =0.5M J 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 Extrasolar debris disks are nascent Kuiper belts Belts are gravitationally sculpted by planets
37 Fomalhaut b: Planet-Debris Disk Interaction chaotic width Step 1: Screen parent bodies for gravitational stability t age 10 8 yr t = Myr Myr
38 Step 2: Replace parent bodies with dust grains
39 Step 3: Integrate dust grains with radiative force for collisional lifetime t collision t orb /τ 0.1 Myr t = t collision =0.1 Myr
40 Parent bodies collide once in system age Collisional Cascade Distribution of sizes of Kuiper belt objects 6 Nominal Diameter at 42 AU (km) log N(<mR) (deg -2 ) Classical Excited Measured Extrapolated Most visible grains are just large enough to avoid stellar blow-out m R
41 Results If M planet then a planet Planet position too far from dust belt M planet < 3 M J Planet also evacuates Kirkwood-type gaps Chiang et al Kalas et al M belt M parent bodies 3M Enough material for gas giant core Number of stable parent bodies (in 0.1 AU bins) :9 0.3 M J 6:5 5:4 9:7 4:3 1 M J 4:3 7: M J 3: M J 5:3 7: Time-averaged semimajor axis (AU)
42 Measuring Debris Disk Masses N dn/ds s q s blow 0.2µm s visible s submm s top 100 µm s 850 µm flux consistent w/ M SED (s top ) 0.01M q =7/2(Dohnanyi) s top 10 cm [from t collision (s top ) t age ] Ṁ = 10 3 Ṁ Ṁ = 10 2 Ṁ Ṁ = 10Ṁ Ṁ = 1Ṁ
43 Packed planetary systems Packed formation Instability and ejection Stability restored Regularization by dynamical friction occurs when planets outweigh parent disk Signature recorded in Kuiper belt
44 Dynamical friction: Small bodies slow big bodies Numerical simulations of packed planetary systems Solar system-like outcomes emerge from chaos
45 Observed Simulated Stirring of KBOs by Rogue Ice Giants
46 Short-Period Comets l
47 Raising Sedna s Peri by Stellar Encounters Praesepe cluster M44 Before After Typical open cluster n 4 stars/pc 3 (R 1/2 2 pc) t 200 Myr v 2 1/2 1km/s Fernandez & Brunini 00
48 Discovery of Kuiper Belt Object (KBO) # QB1: Smiley David Jewitt (U Hawaii) Jane Luu (UC Berkeley)
49 Size depends on observed brightness and intrinsic reflectivity (albedo) Quaoar Ixion
50 Planetary Protection Mechanism: Orbit-Orbit Resonance Neptune makes 3 orbits for every 2 orbits of Pluto Dance of the Plutinos
51 The Orbit of Sedna
52 Resonant KBOs Number Number Number :1 4:1 3:1 5:2 7:3 2:1 9:5 7:4 5:3 8:5 3:2 10:7 7:5 11:8 4:3 13:10 9:7 14:11 5:4 11:9 6:5 1:1
53 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
54 Dwarf Planets Nix Easter Bunny Hydra I.A.U. definition (a) orbits the Sun (b) hydrostatic (round) shape (c) not a satellite (d) not cleared its neighborhood Santa
55 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
56 ~10-30 Large Neptune Trojans vs. ~1 Large Jovian Trojan ( Large km diameter assuming 12-4% visual albedo)
57 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
58 birth ring daughter dust particle
59 Theoretical Snapshots of Resonant KBOs Plutinos 3:2 Twotinos 2:1
60 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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