Things that matter during the first stages of formation

of giant planets

Andrea FortierPhysikalisches Institut – UniBe

02/03/2011

Things that matter during the first stages of formation

of giant planets

Andrea FortierPhysikalisches Institut – UniBe

02/03/2011

Some of the important things

Introduction: context and motivation

Jupiter Saturn Uranus Neptune

M[M]

317.8 94.3 14.6 17.2

a[UA]

5.2 9.5 19.2 30.1

Internal structure:The basics

“solid” core

gaseous envelope

The giant planets of the solar system

Introduction: context and motivation

(Guillot 1999)

Internal structure of the giant planets of the Solar System

Jupiter: 0 < Mc < 11 M

1 < Mz < 39 M

Saturn: 9 < Mc < 22 M 1 < Mz < 8 M

Uranus: 9 < Mc < 14 M

Neptune: 12 < Mc < 16 M

(EOS: SCVH 1995)

Solid content:

The nucleated instability model (Mizuno 1980)

Formation of planetesimals Formation of the embryos

Accretion of gas and solids Cross-over mass (Mc=Menv)

Runaway accretion of gas Gap opening and

termination of the process

(Armitage 2007)

Example

TIME

MA

SS

X cross-over mass

On what depends the cross-over massand the time of cross-over?

FIRST STAGE

But before that …

Keep in mind that:

o The formation of the giant planets must be completed before the protoplanetary disk dissipates, then form < 107 years.

o The final masses of the cores have to be in good agreement with current estimations.

0 < M c [M

]< 18

9 < M c [M

]< 14

9 < M c [M

]< 22

12 < M c [M

]< 16

PROTOPLANETARY DISK

Recipe to make a planet

TO FORM A GIANT PLANET

MODEL FOR THEGAS COMPONENT

MODEL FOR THESOLID COMPONENT

GAS COMPONENT:Internal structure and growth of the envelope

2

4

1(1) mass conservation equation

4

(2) hydrostatic equalibrium equation4

(3) energy conservation equation

(4)

r

r

r

acr

r

M r

GMP

M r

L u VP

M t t

T

4 transport equation

4r

r

GM T

M r P

+ Internal and external boundary conditions

+ Equation Of State (EOS)

+ Opacity () tables

GAS COMPONENT:The growth of the envelope

How do planets grow? By accreting solids (details later): the embryo

increases its gravitational field. By accreting gas:

The embryo is immersed in a gaseous disk so … where does it ends?

2

1/3

min( , )

Accretion Radius

Hill Radius3

P a H

pa

s

PH

R R R

GMR

c

MR a

M

The external boundary condition gives the accretion rate … how???

GAS COMPONENT:The growth of the envelope

How does gas accretion proceed?

Hydrostatic equilibrium should be satisfied:

grows because of solid accretion

44r

r

GMP

M r

The condition RP=min(Ra, RH) must be fulfilled at any time, so the contraction of the envelope implies accretion of gas from the disk

GAS COMPONENT:Opacity matters

(Hubickyj et al. 2005)

4

Transport equation

4r

r

GM TT

M r P

if rad

The lower the opacity,the faster the formation

and

44r

r

GMP

M r

GAS COMPONENT:Solids accretion matters

Sudden cutoff of the solids accretion:

(Hubickyj et al. 2005)

acr

L u VP

M t t

×

The cutoff speeds up the formation

The cutoff delays the formation

2

4

1(1) mass conservation equation

4

(2) hydrostatic equalibrium equation4

(3) energy conservation equation

(4)

r

r

r

acr

r

M r

GMP

M r

L u VP

M t t

T

4 transport equation

4r

r

GM T

M r P

SUMMARY GAS COMPONENT SOLID COMPONENT

THE MASS OF THE CORE CONTRIBUTES TO THE TOTAL MASS

PLANETESIMALS ARE THE MAIN LUMINOSITY SOURCE

And also:o ablation of planetesimals energy deposition, EOS, o the core is not inerto …

SOLID COMPONENT

FORMATION OF PLANETESIMALS

GROWTH OF PLANETESIMALS THROUGH MUTUAL COLLISIONS

GROWTH OF SOLID PLANETARY EMBRYOS

????

N-BODYCALCS.

SOLID COMPONENT:The growth of the core

2

2 eff rel

dMF R v

dt h

Density of solidsEffective cross-section

Relativevelocity of the approachingplanetesimals

STATISTICAL APPROXIMATION:Particle-in-a-box approximation(Safronov 1969)

Accretion rate of solids

v

vt

SOLID COMPONENT:The effective cross-section

2

2 2 1 esceff g

rel

vR R

v

runaway growth

GRAVITATIONAL FOCUSING

ENLARGES THE CROSS-SECTION

Enhancement factor

Gravitational focusing favors the growth of the biggest planetesimals: vesc increases faster than vrel

SOLID COMPONENT:The effective cross-section

The growing embryo “heats” the planetesimal disk.

Vrel increases, gravitational focusing decreases

The growth of the big body becomes self-regulated: the stirring rate of the small planetesimals is determined by the one

that accretes them.

oligarchic growth (e.g. Ida & Makino 1993, Kokubo & Ida 1996, 1998, 2000, 2002)

SOLID COMPONENT:Runaway-oligarchic growth transition

Roughly speaking, a body of the mass of the Moon (~10-2 M ) is already an oligarch.

Timescales:Runaway growth: Tgrow M-1/3

(order of magnitude ~104 - 105 yrs)

Oligarchic growth: Tgrow M1/3

(order of magnitude ~106 - 107 yrs)

IN PRACTICE, THE FORMATION OF A 10 M EMBRYO IS GOVERNED BY THE OLIGARCHIC GROWTH. THIS INTRODUCES A SERIOUS PROBLEM IN PLANETARY FORMATION: SOLID EMBRYOS FORM TOO SLOW.

Example: After 10 Myrs, at 5 AU only a 1 M embryo is formed (Thommes et al. 2003)

But protoplanets have a gaseous envelope that enlarge the cross-section more than the gravitational focusing alone:

GAS COMPONENT SOLID COMPONENT The effective cross-section

Gas drag of the envelopematters!!Moreover, there is a strong dependence on the planetesimal size.

The protoplanetary disk

3/ 20

-20

g

solid surface density

(1 ) 10 g cm

1 if

4 if

gas/dust (gas/dust) gas surface density,

s

s

iceice

ice

a

AU

a a

a a

The Minimum Mass Solar Nebula (MMSN) (Hayashi 1981)

… but in general the MMSN does not work (i.e. can not form the giant planets of the Solar System in reasonable timescales).Then, usually people consider disks more massive than the MMSN(some factor×MMSN), other indexes for the power law (a-p) or more complex models for the protoplanetary disk and its evolution.

SOLID COMPONENT: Dependence on the solids disk density

2

2 eff rel

dMF R v

dt hThe surface solids density at the is very

important in determining the accretion rate:

But evolves with time.

Simplest case: in situ formation (a fixed), decreasing due to the accretion

Where?

In the feeding zone of the planet: (a-a, a+a)with a=3-5 RHill

SOLID COMPONENT: The feeding zone

a ~ 4 RH aMP1/3

SOLID COMPONENT: The feeding zone

a ~ 4 RH aMP

1/3

ExamplesAt a=5.2 AU we have:

1 M 0.4 AU10 M 0.9 AU100 M 1.9 AUJupiter 2.8 AU

What’s the limiting mass?

a a MP1/3

MP 4a a Miso (a2)3/2

Isolation mass

SOLID COMPONENT: Dependence on the solids’ surface density

2

2 eff rel

dMF R v

dt h

(A.F. PhD Thesis)

SOLID COMPONENT: Dependence on the solids’ surface density

2

2 eff rel

dMF R v

dt h

(A.F. PhD Thesis)

SOLID COMPONENT: Dependence on the solids’ surface density

2

2 eff rel

dMF R v

dt h

(A.F. PhD Thesis)

The importance of the oligarchic growth in giant planet calculations

Parameters:

a=6 AU0= 16 g cm-2 Rpsimal= 100 km

Oligarchic growthfor the core

Runaway growthfor the core

What else matters?

Planetesimal size:

0(5.2AU) = 15 g cm-2 (~ 5 MMSN)

21 M25 M

29 M

Time [106 yrs.]

Mas

s [M

]

(Fortier et al. 2007, 2009)

What else matters?

Giant planet formation adopting a size distribution for the accreted planetesimals

the mass of solids isin small planetesimals

all planetesimal sizesare equally abundant

the mass of solids isin big planetesimals rmin=30 m

rmax=100 km

(Benvenuto et al, submitted)

What else matters?

Planetesimal size: How big were planetesimals born?This problem is under debate.Recent models claim that planetesimals were born big (> or >> 100 km, e.g. Johansen et al. 2007)

What was the original size distribution?We don’t know.

How did this distribution evolve?By mutual collisions that lead to both accretion andfragmentation.

What else matters?Planetesimal migration

(Thommes et al. 2003)

0 = 10 MMSN

What else matters?Planet migration

What else matters?Planet migration

Interaction between planets and the gaseous protoplanetary disk.Orbital migration is a consequence of angular momentum exchangebetween the planet and the gas disk. The type of migration dependson the planet’s mass.

Type I: (low mass planets)In the “classical version” migration rates ~ MP

~0.1 Myr for planet core

Must be slower in reality

Local thermal effects reduce the migration rate

Type II: (the planet is massive enough to open a gap)• Mp << local Mdisk : the planet is coupled to the viscous evolution of the disk and migrates with the gas viscous timescale.• MP ~ local Mdisk : the disk is not capable to give the planet the angular momentum it needs to migrate with the gas. Migration eventually stops.

What else matters?Planet migration

In situ formation

Formation with migration

Parameters:

a=6 AU0= 16 g cm-2 Rpsimal= 100 km

What else matters?Simultaneous formation

In situ, simultaneous formation considering planetesimal migration

• The different cases correspond to different density profiles and planetesimal size distribution.• Planets do not migrate in any of these cases.• (a,t) is affected by planetesimal migration due to the gas drag of the disk, planet accretion and the presence of another growing embryo.

Planets don’t see each other

Steep profile:Formation of P1 is delayed by P2

P2 forms firstFormation of P1 is accelerated

P1 forms firstFormation of P2 is delayed

density wave

P1 P2

(Guilera et al. 2010, Guilera et al. sumbitted)

What else matters?Simultaneous formation with planet migration

(Preliminary results)

PROTOPLANETARY DISK

Recipe to make a planet

TO FORM A GIANT PLANET

MODEL FOR THEGAS COMPONENT

MODEL FOR THESOLID COMPONENT

Recipe to make a planet

GIANT PLANET

MORE THAN ONE PLANET?INTERACTIONS!!

MODEL FOR THESOLID COMPONENT

MODEL FOR THEGAS COMPONENT

PROTOPLANETARY DISK

MODEL FOR THEGAS COMPONENT

MODEL FOR THESOLID COMPONENT

Thank you !!!