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Chris Ormel: planetesimal accretion || Bern 26.05.2010 || 1/46

Planetesimal Accretion

Chris W. Ormel

Max-Planck-Institute for Astronomy, Heidelberg

and

Kees Dullemond, Marco Spaans

MPIA + U. of Heidelberg || U. of Groningen


Contents

1. Introduction

2. Monte Carlo model for collisions

3. Planetesimal growth simulations

4. Transition between runaway growth & oligarchy


Contents

1.Introduction Planet formation stages

Gravitational focusing

Runaway growth & oligarchic growth

2.Monte Carlo model for collisional evolution

3.Planetesimal growth simulations

4.Application: transition to Oligarchy

Chris Ormel: planetesimal accretion || Bern 26.05.2010 || 4/46 [Michiel Hogerheijde]

Planet formation


Planetary distance ladder

μm mm m km

ChondrulesChondrulesISM-dustISM-dust BouldersBoulders PlanetesimalsPlanetesimals

103 km

(proto)Planets

(proto)Planets1 1? 2?

Growth mechanisms:1.Surface forces2.Gravity3.Particle concentration + collapse (GI)

3

– bind matter– timescales– observations


1. Dust to planetesimals

● Chondrule formation, planetesimal formation

● Sticking by surface forces [Blum & Wurm 2008]

● Relative velocity: gas drag– Meter size barrier

● (Particle) Instabilities [Johansen et al. 2007, 2009; Cuzzi et al. 2010]


Sound speedcg~105 cm/s

Turbulence strengthα~10-4

Pressure parameterη ~ 10-3

1mm 1cm 1m

radial drift

Velocities

[Weidenschilling 1977; Ormel & Cuzzi 2007]

10 m/s

1 cm/s

Stokes number


2. From planetesimals to protoplanets

● Sticking mechanism: gravity● Velocity: mutual grav. stirring

– Systematic (Kepl.) & random

● Runaway growth, oligarchic growth● Isolation mass:

M iso≈10−3M E

1 g cm2 3/2

R1 AU

3


3. Protoplanets → Planets

● Inner solar system: chaotic growth

● Outer solar system: Build-up of a ~10 ME core + gas accretion

– Planet synthesis[Mordasini et al. 2009a,b; Ida & Lin 2004, 2008...]

[e.g., Chambers 2001; Raymond 2006]


Gravitational focusing

Rc o l

Rg e o

dM big

dt=Rbig

2 1 vesc2

va2

Σ: Surface density planetesimals

Ω: orbital frequency

vesc : escape velocity of body

va: approach velocity Gravitational focusing factor(can be >> 1)


Keplerian shear (low v regime)

Ω(a)Ω(a+Rh)

va~vh

● Hill radius Rh

─ Minimum approach velocity va~vh

─ Max. GFF ~(vesc/vh)2 ~103

Rh=a M3Mc 1/3

; vh=Rh

dM big

dt=Rbig

2 1 vesc2

va2

Circu

lar orb

it

va=vran ~eaΩ

Relative velocity


Viscous stirring

● Collisionless gravitational encounters:

─ Convert potential energy to random E

─ Increases the random motion v (inclination+ eccentricities)

─ Decreases GF

Total motion

Low random motions

Total motion

Large random motions


GF – velocity regimes

Random velocity, v (mutual eccentricity)

Dispersion-dominated regimeShear-dominated regime

Superescape regime

Approach ve

locity,

v a

Rc o l

Rv s

Max GF

No GF

Hill velocity, vh Escape velocity, vesc

vran vranStirring


GF – velocity regimes

Random velocity, v (mutual eccentricity)

Dispersion-dominated regimeShear-dominated regime

Superescape regime

vran

Rc o l

Rv s

Growth

Escape velocity, vesc

Growth vran

Hill velocity, vh


Runaway & oligarchy

Log (mass)

Log (mass)

T1 = T2 :neutral growth

T1 = ½ T2:runaway growth

T ac=M

dM /dt

dM big

dt=Rbig

2 1 vesc2

va2 vesc= 2GM big

Rbig

T ac∝M big−1 /3

● Growth timescale

● RG (va = cnst):


Oligarchy

T1

T2

T2

Dynamically hot (large v), cool (low v)

T1 < T2 in same zone: RGT1 > T2 different zones:

no RG

mass

position

─ Heating locally slows down growth (viscous stirring)

─ Bodies in same spatial zone separate

─ ........... neighboring zones converge


Runaway growth/Oligarchy

● Chronologically, we expect: [Ida & Makino 1993; Kokubo Ida 1996,1998, 2000]

─ Runaway growth phase (GF-factor increases), big bodies grow quickly

─ Gradual heating of plts. through viscous-stirring of protoplanets

─ Transition to oligarchy, self-regulated [slow] growth “Oligarch heats its own food” [Goldreich et al. 2004]

2 component distribution of oligarchs & planetesimals


Contents

1.Introduction

2.Monte Carlo model for collisional evolution● Key ingredients

● Statistical codes

● Monte Carlo method




Collisional evolution codes

● As function of time, resolve:

─ Masses

─ Random velocities (inclination, eccentricity)

─ Semi-major axis

─ (other properties of bodies)


Approaches

● N-body

─ Solve e.o.m. for every particle [e.g., Kokubo & Ida 1998 2000; Barnes et al. 2009]

● Monte Carlo (relaxation)

● Statistical/Particle in a box

─ Binning approach [e.g., Goldreich et al. 1978; Wetherill & Stewart 1989, Weidenschilling et

al. 1997; Inaba et al. 2001]

─ Monte Carlo (probability) [Ormel & Spaans 2008]

● Hybrid (statistical + N-body) [Bromley & Kenyon 2006; Glaschke et al. 2006]


Binning method

number density/unit mass

Continuous view(binning method)

Interactions between mass bins

Particle mass

● Group particles by mass (bins)

● Consider interactions between bins

─ Fast & easy to implement

─ Mass as only independent variable


Monte Carlo [OS08] approach I.

● Use representative bodies (RB)

─ Each RB represents Ng planetesimals

─ Fix: mass, eccentricity, inclination, semi-major axis

─ Randomize: phase angles

● Calculate the collision probability between each of these RB.

● Perform (group) collision; update probabilities


Monte Carlo approach II.Semi major axes, a

Comp. body: #total (physical) bodies

Particle type I: 30 bodies Particle type II: 5 bodies……Particle group N

CR=N 1N 2Rcol

2 va

2heff Area


Monte Carlo III. Collisions

+ →

● Bookkeeping:

─ Update collision rates

─ Number of collision partners

─ Add/remove new RB (=comp. bodies)

─ Dynamically change group size Ng: the number of physical particles a single RB represents

1 body of group 1 6 bodies of group 2 1 body of New group


Flexible grouping [Ormel & Spaans 2008]

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

groupingHigh Low/None

resolutionLow High


Contents

1.Introduction


3.Planetesimal growth simulations— Movies @1, 6, & 35 AU

— Low and high Σ



Simulation Results

● Simulation properties

─ Single planetesimal size initially (e.g. r=8km)

─ Local (1 disk radius)

─ Multi-zone

─ Until 2000km

─ Viscous stirring, dynamical friction, gas drag, (fragmentation), etc.

─ NO: spatial scattering (close encounters)


1 AU simulation (w/o fragm.)

● Indicated are:

─ Radius plant. (X)

─ Position plant. (Y)

─ Group total mass: Area dot~m1/3tot; mtot = Ng midv

─ Grav. focusing factor w.r.t. biggest particle (v/vh, color)

Single body

Hill radius vh: Hill velocity of biggest body, vh~R1


1 AU simulation (no fragm.)

OligarchsLeftover plts.

Σ = 17 g cm-2


6 AU simulation (w/o fragm.)


35 AU run (w/o fragm.)


35 AU w/o fragm.


35 AU high density + fragmt.

High Σ Low Σ


35 AU: different Σ


End states


Contents

1.Introduction



4.Application: transition to Oligarchy [Ormel et al. 2009]

— Runaway growth & oligarchy phase

— Runaway growth timescale, Trg

— New criterion for transition size, Rtr


Size distribution

1 AU


Key statistics

Radius of biggest body(evolutionary parameter)

(inverse)Gravitational FF

Runaway growth Oligarchy

RG-timescale, Trg


Transition RG/Oligarchy?

1.Runaway growth ↔ increasing GF

2.Oligarchy ↔ decreasing GF

3.RG proceeds exponentially:

We find initially Tr g < Tv s :RG out-paces the stirring!

T rg=K rg

R0s

T vs=a

9Rh log vvh 5

[Ida & Makino 1993; Ormel et al. 2010]


Transition RG/Oligarchy II.

● Ida & Makino (1993) transition:

─ 2 component model

─ Comparison of stirring power among populations

● (Our) equate timescales:

─ The point where the 2comp approximation becomes first valid

2M =m

T rg=T vs-2c=T ac-2c

M,Σ: mass, surface density in big bodies

m,σ: mass, surface density of small bodies


Transition RG/Oligarchy III.

● Ida & Makino (1993) criterion:

● New criterion: [Ormel et al. 2010]

Rtr≈90 km

10 g cm2 1/5

a1 AU

2 /5

R0

10 km 3 /5

Rtr≈320 km

10 g cm2 2/7

a1 AU

5/7

R0

10 km 3 /7


Transition RG/Oligarchy

● New criterion provides conditions at transition:

─ Conditions for core accretion phase

─ Radii oligarchs, timescales, size distr. Planetesimals

─ Speculate Kuiper Belt' size distribution fossil of RG phase

Fraser & Kavelaars (2009)

Diameter [km]


Summary

● Investigated growth ~km → ~103 km

● Introduced novel MC model

● Identified the runaway growth & oligarchy stages

● Set a new criterion for the transition size

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