6. growth of plnets: an overview 6.1. observational constraints a. the planets’ masses and radii...

6. GROWTH OF PLNETS: AN OVERVIEW

6.1. Observational Constraints

a. The planets’ masses and radii and the age of the Solar System

17.147 ME

3.883 RE

NeptuneUranus

Earth

4.007 RE

14.536 ME

Formed within the age of the Solar System : 4.567 Gyr ± a few Myr (CAI age)

6. GROWTH OF PLNETS: AN OVERVIEWb. a few Earth masses of hydrogen and helium (Guillot 1999)

• Mass M• Equatorial radius a• Gravitational moments J2, J4, and J6 (Laplace series expansion of gravitational moment)

Pioneer and Voyager

about 25% rock, 60 to 70% ice5 to 15% H and He

Uranus and Neptune were formed prior to gas-disk dissipation.

• Solar-type stars suggest that circumstellar disks dissipate over a timescale of several million years [Brice˜no, Vivas & Calvet 2001; Haisch, Lada & Lada 2001; Hillenbrand 2004; Lagrange, Backman & rtymowicz 2000; Strom, Edwards & Skrutskie 1993]


Uranus and Neptune were formed within ~ 0.1Gyrs


Why Uranus and Neptune?

• Easy to make within 0.1 Gyrs : core accretion (0.1Gyr) or streaming instability (0.1 Myr) Hard to stay there (no migrations of giants)

• Pure gravitational instability : less than ~Myrs (severe depends on disk model)• Streaming instability triggers GI: less than ~0.1Myrs• Core accretion : need longer than solar system age

• Not much H and He : Probably formed at the end of gas-disk dissipation Enough time to form planetesimals


6.2. Minimum Mass Solar Nebula

• The Minimum Mass Solar Nebula (MMSN) [Hayashi 1981]: A protoplanetary disk that contains the minimum amount of solids necessary to build the planets of the solar system

• Size of Uranus and Neptune, core size for Jupiter and Saturn:

• Default model in this study : MMSN• But, possibly higher (ejected mass: Oort cloud 1 – 30 ME)


6.3. Need for Gravitational Focusing

• From equation 41, mass accretion time is

esc

col

v

uF

Rdt

dM

M

~1

dtdMM

t form ~

2

1~

escform v

uRt

escH vuv ,

2

2/3

10~

gcmAU

aMMSN

Gyrv

u

AU

at

escform

23

1015~

, kmR 000,25~


Gyrv

u

AU

at

escform

23

1015~

escvuAUaUranus 20AUaNeptune 30

Gyrt form 120~

Gyrt form 400~

Uranus :

Neptune :

About hundred times longer than age of the solar system

There must be a mechanism to shorten formation time scale.

Gravitational focusing

Alternative: Uranus and Neptune are formed between Jupiter and Saturn

6. GROWTH OF PLNETS: AN OVERVIEW6.4. Cooling is Necessary, so Accreted Bodies were Small

• Initial random velocity : Collision cross section is smaller than viscous stirring cross section.

escvu

• No gravitational focusing

• Viscous stirring dominates.• Planetesimals heats upto escvu ~

)/()/( escvs

esccol vuFvuF

• Embryo eat up most of planetesimals in entire region

• Takes too long time to grow

• Only some of planetesimals within Hill radius collide with embryo.

• Need to keep random velocity slow to shorten formation time Confirmed by N-body simulation : Levison & Stewart 2001

6. GROWTH OF PLNETS: AN OVERVIEW• How small should be the planetesimals?

4

~1

u

v

Rdt

du

uesc

vs sdt

du

u colin

~1

.

assuming ~

4

~

escv

u

R

s

• To form Neptune within age of the solar system, escvu 1.0

• Size of planetesimals : a few kilometers

6. GROWTH OF PLNETS: AN OVERVIEW• Formation of Earth without gravitational focusing

GyrGyrv

u

AU

at

escform 1.0~

1015~

23

6. GROWTH OF PLNETS: AN OVERVIEW6.5. Completion: When Small Bodies Have been Consumed

6.5.1. Complete with au ~

GyrAU

aGyr

v

u

AU

at

escform

223

102~

1015~

• 20 Gyr for Neptune

• At this speed, small bodies will frequently collide each others

• escua fragmentation

• ua Kinetic energy loss :


6.5.2. Completion with au

• Planet can be easily formed within 10 Myr.

• But, accrete only small body within annulus of 5 RHill

• Not enough mass : about factor 5 small

334)5(2 RRa H )/(62.0 SUNH RaRR ,

kmAU

aRiso

4/1

10000,12

• Possible modifications

Migration of embryo

Migration of small bodies

More massive disk

6. growth of plnets: an overview 6.1. observational constraints a. the planets’ masses and radii...

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