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 !!!
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