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Runaway growth in protoplanetary disks
Chris W. OrmelAlexander von Humboldt fellow
Max-Planck-Institute for Astronomy, Heidelberg
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Chris Ormel | runaway growth in protoplanetary disks | Tübingen, 03.03.09 | 2
Outline
Characteristics of Runaway Growth (RG)
Modeling RG
Continuous vs discrete methods
High dynamic range Monte Carlo method
Planetesimal accretion in the Kuiper Belt
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Terminology
Usage of “runaway growth”
Positive feedback (e.g., thermal runaway)
Accretion of gas on planet
Within a single population due to coagulatione.g., dust grains, planetesimals, stellar systems
Mathematically, referred to as “gelation”Instantaneous!
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Gravitational focussing
Massive particles (planetesimals) gravitationally attract each other, increasing :σ
Thus, σ ∝(Mbig)4/3 if Δv < vesc
Potential for RG in grav.dominated regime.
bcoll >> R
ij=geom [1 vesc
v 2
]; vesc2=
GMbig
R
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Runaway growth (RG)
Define RG as
Particles separate in mass
Condition independent of absolute rate of growth
dM/dt determined by the collision kernel
It follows [e.g., Lee 2000; Ormel & Spaans (2008)]: If ν>1: RG, instantaneous (!) in the limit N -> ∞
If ν≤1, β>1: RG after a time t0
Gravitational focusing σ ∝M4/3; ν>1 possible
K ij= ijvij ; if K ij ∝ M im j
− M i≫m j then
d M1 /M 2
dt0
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Modelling RG
Difficult for continuous methodsBut see Lee (2000, 2001)
Batch (dynamic)
mass
Wetherill (1990) batching algorithm#objects
Minimum distance among batches
Mass transfer between batches
Enforce discretization [e.g., Wetherill 1990, Kenyon & Luu (1999), Inaba et al. (2001)]
Use a Monte Carlo method [Ormel & Spaans 2008]
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Continuous vs Monte Carlo (MC)
number density/unit mass
Continuous view(distribution method) MCview
Interactions between mass bins
Interactions between particlesParticle mass
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+ =
Blum (2000)
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Monte Carlo coagulation
Main advantage: include “particle” properties
E.g., porosity, composition, eccentricity, etc.
See poster P5.5 by A. Zsom et al.
But how to manage 10X dust particles/planetesimals with MC?
MC at high dynamic range: 2 tricks
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Ingredient #1: grouping
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Ingredient #2: flexible grouping
mass
Mas
s de
nsity
Small particles still dominate(require little resolution)
Particles in tail will start runaway(resolve individually)
Sketch of particle distribution incipient to RG
Fixed #MCparticles per log mass bin(Determines the level of grouping)
groupingHigh Low/None
resolutionLow High
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Strong runaway growth Test behavior of
kernels K ∝ (mi mj)ν
with =2,3ν
Prediction: runaway time tR scales as [Malyskin & Goodman (2001)]
slope -1, -2
t R∝log N 1−
101001010N~300
Note: N (initial #particles) sets simulation size; number density same in all models!
[Ormel & Spaans (2008)]
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Application: runaway growth in Kuiper Belt
Model characteristics/approximations
Include velocity evolution; i.e., encounters can result in:
Collisions Dynamical friction (energy equipartition) Viscous stirring
Use order-of-magnitude expressions [Greenberg et al. (1991); Goldreich et al. (2004)]
Assume eccentricities~inclinations; calculate (random) velocity change.
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Approximations (ctd)
.....
No fragmentation; all collisional collisions result in accretion; no gas drag
No spatial inhomogeneities
Simulate accretion @35 AU in a 6AU annulus (few Earth masses;
Start monodisperse planetesimal population of 8km and ∆v = 3 m/s. (cf. Kenyon & Luu, 1999)
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Results: w/o velocity evol.
Distribution plotted after each factor of 2 increase in radius
-Y-axis indicates relative contribution by mass
1 particle/bin line
3m/s < Hill velocity big bodies: runaway growth ceases
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w/ velocity evolutionLarge gap between m1 and m2
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Conclusions/outlook
Runaway accretion code tested
RG required for Kuiper Belt [cf. Kenyon & Luu, 1999]
Follow collisional evolution in more detail
E.g. model internal state of bodies (e.g., asteroids), binarity, etc.
TBDs
Include fragmentation, gas drag, spatial inhomogeneities (oligarchy!), etc.
Compare/merge with N-body code