Runaway Accretion &KBO Size Distribution

Re’em SariCaltech

Minimum Mass Solar Nebula

y1=Py12=P y250=P

)AU(r

Sur

face

den

sity

(g

cm-2)

d88=P

Geometric Accretion• Collision cross section is geometric

• Scale height h~v/– n /h

• In terms of surface density:– Independent of velocity

• For MMSN:– Earth (6,400km) 108 yr

– Jupiter’s core 109 yr

– Neptune (25,000km) 1012 yr

– Pluto (1,100km) 1011 yr

€

collision rate = nπR2 v

€

dR

dt=

ΣΩ

8ρ≈ 3cm

yr(a /AU)−3

Gravitational Focusing• Larger collision cross section

• Growth rate depends on Safronov’s number (vesc/v)2

• Safronov: velocity, v, is not a free parameter.– Controlled by interactions between bodies.

• Form MMSN: geometric required limits time (yr) time (yr) on

eccentricity

– Earth (6,400km) 108 yr t<108 none

– Jupiter’s core 109 yr t<107 e<0.1

– Neptune (25,000km) 1012 yr 107<t<1010 0.01<e<0.4

– Pluto (1,100km) 1011 yr t<1010 e<0.05

€

collision rate = nπR2 1+ (vesc /v)2[ ]v

Runaway Accretion• Without gravitational focusing

– Bodies tend to become of equal size

• With focusing

– Few large bodies become larger than their peers.

€

1

R

dR

dt∝

1

R

€

1

R

dR

dt∝

1

R

vescv

⎛

⎝ ⎜

⎞

⎠ ⎟2

∝ R+1

•Could eccentricities be excited to required levels?•Is there enough mass in the excited region?•Does oligarchic stage stall accretion?

Most of the Bodies & mass

Exponentialcutoff

Runawaytail

m

m N(m)

Mass Spectrum ?• With focusing

• Mapping of initial size

• Almost generally

€

1

R

dR

dt∝ R+1

Most of the Bodies & mass

Exponentialcutoff

Runawaytail

m

m N(m)

€

R0 → R

€

n(R) = n0(R0)dR0

dR

€

n(R)∝ R−2

•Does not agree with Kenyon’s simulations!•Does not agree with observed KBO size distribution!

€

n(R)∝ R−4

QuickTime™ and aPNG decompressorare needed to see this picture.

Interaction with a cold surrounding

Simplified Runaway Accretion

• Setup:– Many small bodies: 0.3 g cm-2 (from min solar nebula)

– Few large bodies: 10-3 R 100km (We see them)

• Processes:

⎟⎟⎠

⎞⎜⎜⎝

⎛=

2v1

uRdt

dR

Resc

ρ

σ

⎟⎟⎠

⎞⎜⎜⎝

⎛=

4v1

uRdt

du

uesc

ρ

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛422 v

vv

v

uRResc

H

esc

ρ

σ

ρ

, u, v

stirring

friction

accretion

stirring=

Simple Solution

• Consistency:– We assume u>vH & v<vH which requires– We neglected collisions between small bodies

• Generalization yields thecomplete velocity spectrum.

• Mass spectrum more subtle

• Agrees with detailedsimulations (Kenyon)

€

u =Σ

σ

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

Θ−1/ 2vH ≈ 3vH HH v3.0vv 12/3

≈Θ⎟⎠

⎞⎜⎝

⎛= −

3/2Θ<

<Θ

km1.0≈

> Rs

(Kenyon 2002)

Physical Collisions during runaway?

• Collisions are negligible if

• Initial body size (GW)– About 1 km for MMSN– Independent of a

€

u =Σ

σ

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

Θ−1/ 2vH ≈ 3vH HH v3.0vv 12/3

≈Θ⎟⎠

⎞⎜⎝

⎛= −

€

r >Σ

σR ≈1km

€

ρr

<ΣΩ

ρR

Vescu

⎛

⎝ ⎜

⎞

⎠ ⎟4

Collisions in KB after runaway?• Observed velocity dispersion vdis ~ 1 km/s

• Observed surface density ~ 310-4 gr/cm2

• Age of the solar system T ~ 4 Gyr

• Collisions between bodies of similar size r occur if:

• Collision of size r on size R:

€

ρr

> T−1 ⇒ r < 0.3km

€

ρR

R

r

⎛

⎝ ⎜

⎞

⎠ ⎟3

> T−1

r3vdis2 > R3vesc

2

⎫

⎬

⎪ ⎪ ⎪

⎭

⎪ ⎪ ⎪

⇒ R ≈ 70km r ≈12km

Collisions After Runaway

• Bodies above 70 km are safe.

• The 10 km bodies, can be destroyed by smaller bodies.

€

ρR

R

r

⎛

⎝ ⎜

⎞

⎠ ⎟3

> T−1

r3vdis2 > R3vesc

2

⎫

⎬

⎪ ⎪ ⎪

⎭

⎪ ⎪ ⎪

⇒ R ≈ 70km r ≈12km

suffi

cient

rate

sufficient energy

•Bernstein et. al. astro-ph/0308467 suggest 40 km cutoff.•Agrees also with Stern (95) calculations.•Needs self consistent calculation, e.g. Dohnanyi spectrum.