1. No category

Planet disk interaction
Wilhelm Kley
Institut fur
¨ Astronomie & Astrophysik
& Kepler Center for Astro and Particle Physics Tubingen
¨
March 2015
4. Planet-Disk:
Organisation
Lecture overview:
4.1 Introduction
4.2 Type I
4.3 Type II
4.4 Type III
4.5 Stochastic
4.6 Elements
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
2
4.1 Introduction:
Planet Formation Problems
Not possible to form hot Jupiters in situ
- disk too hot for material to condense
- not enough material
Difficult to form massive planets
- gap formation
Eccentric and inclined orbits
- planets form in flat disks (on circular orbits)
But planets grow and evolve in disks:
⇒ Have a closer look at planet-disk interaction
Consider Late Phase of Planet Formation:
Assume protoplanets (embryos) have already been formed
Investigate subsequent evolution in disk,
consider gravitational interaction with star and disk
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
3
4.1 Introduction:
Two main contributions
1) Spiral arms
(Lindblad Torques)
2) Horseshoe region
(Corotation Torques)
(Kley & Nelson, ARAA, 50, 2012)
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
4
4.1 Introduction:
Evidence for migration
many multi-planet extrasolar planetary systems
high fraction in or close to a low-order mean-motion resonance
(MMR)
In Solar System: 3:2 between Neptune and Pluto (plutinos)
Resonant capture through convergent migration process
dissipative forces due to disk-planet interaction
Existence of resonant systems
- Clear evidence for planetary migration
Hot Jupiters (Neptunes) & Kepler systems
- Clear evidence for planetary migration
The main Problem:
Migration Speed
(for Type-I and Type-II)
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
5
4.1 Introduction:
Torque calculation
2 approaches:
a) Linearization: for low mass planets only
basic state: axisymmetric disk, in Keplerian rotation, and given Σ(r )
add planet potential as a small disturbance of planet
linearize equations
calculate new (perturbed) disk structure
calculate torque acting on planet
b) Non-linear: for all planet masses
full non-linear hydrodynamical evolution of embedded planets in disks
Will not go into details of these methods but just quote the main results
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
6
4.1 Introduction:
Torques and migration
Angular momentum of planet (z-direction)
Jp = (mruϕ )|p = mp rp2 Ωp
(1)
change in time by torques (z-component) acting on the planet
J˙ p = Γp
(2)
Γp is calculated over the whole disk
Z
Z
Z
~ ) df =
Γp = −
Σ(~rp × F
Σ(~rp × ∇ψp ) df =
disk
disk
disk
Σ
∂ψp
df (3)
∂ϕ
⇒ Migration Timescale τM
Γp
1 drP
1
=
= 2
rP dt
τM
JP
(4)
Compare τm to evolution of disk
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
7
4. Planet-Disk:
Organisation
Lecture overview:
4.1 Introduction
4.2 Type I
4.3 Type II
4.4 Type III
4.5 Stochastic
4.6 Elements
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
8
4.2 Type I:
Γtot = 2π
R
Lindblad torque: radial torque density
dΓ
(r ) Σ(r ) rdr
dm
0.08
isothermal
adiabatic: γ = 1.40
adiabatic: γ = 1.01
0.06
0.04
dΓ/dm / (dΓ/dm)0
0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1
-0.12
W. Kley
-4
Planet Formation,
-3
-2
-1
0
(r-ap)/H
1
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
2
3
4
9
4.2 Type I:
The linear Torque
Results for a locally isothermal disk. Fixed T (r )).
Torque on the planet: (for lower mass planets, a few M⊕ )
3D analytical (Tanaka et al. 2002) and numerical (D’Angelo & Lubow, 2010)
Γtot = −(1.36 + 0.62βΣ + 0.43βT ) Γ0 .
(5)
where
Σ(r ) = Σ0 r −βΣ
and
T (r ) = T0 r −βT
(6)
Σp rp2 rp2 Ω2p ,
(7)
and normalisation
Γ0 =
mp
M∗
⇒ Migration
2 H
rp
−2
J˙ p = Γtot
⇒
a˙ p
Γtot
=2
ap
Jp
(8)
Classic Type I migration: Inward and fast
Too fast to be in agreement with observations (see lecture 5)
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
10
4.2 Type I:
Torque Saturation
2D hydro-simulations:
- small mass
- inviscid
Total torque vs. time
- isothermal
- adiabatic
Result:
Torques are positive
initially and then diminish with time and settle
to the Lindblad torques
(from spirals)
That means:
Corotation torques decline = saturate
How to maintain the
full corotation torques
?
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
11
4.2 Type I:
Torque Saturation
How to prevent corotation-torque saturation ?
From linear analysis:
Corotation torque depends on the radial gradients of the entropy, S, and
vortensity ω
˜ where the vortensity is defined as:
ω
˜ =
ωz
(∇ × ~u )|z
=
Σ
Σ
(9)
Now: For inviscid and barotropic (p = p(ρ)) flows:
d ωz =0
dt Σ
and
dS
=0
dt
(10)
These quantities are constant along streamlines.
Initial gradients (in S and ω
˜ ) are wiped out in the corotation region
=⇒ Need mechanism to maintain gradients
These are viscosity and radiative diffusion (or cooling)
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
12
4.2 Type I:
Saturation - Origin
Red and Blue
Orbits have
different periods
⇒ Phase mixing
(F. Masset)
(Balmforth & Korycanski)
W. Kley
Planet Formation,
3D radiative disk (B. Bitsch)
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
13
4.2 Type I:
Disk physics
dΣcv T
~
+ ∇ · (Σcv T u) = −p∇ · u +D − Q −2H∇ · F
dt
Adiabatic: Pressure Work, Viscous heating (D), radiative cooling (Q)
radiative Diffusion: in disk plane (realistic opacities)
4e-05
local cool/heat
Mp = 20Mearth
(Kley&Crida ’08)
Specific Torque [(a2 Ω2)Jup]
3e-05
2e-05
fully radiative
Disk Physics determines
Direction of motion
1e-05
0
adiabatic
-1e-05
see also:
( Paardekooper&Mellema ’06
Baruteau&Masset ’08
Paardekooper&Papaloizou ’08)
-2e-05
isothermal
-3e-05
-4e-05
0
100
200
300
Time [Orbits]
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
14
4.2 Type I:
Role of viscosity
Total torque
vs. viscosity:
in viscous α-disk
2D radiative model
- in equilibrium
Efficiency depends
of ratio of timescales
τvisc /τlibrat
τrad /τlibrat
Local disk properties
matter
(Kley & Nelson 2012)
⇒ Need viscosity to prevent saturation !
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
15
4.2 Type I:
Mass dependence
Isothermal and radiative models. Outward migration for Mp ≤ 40 MEarth
Vary Mp
(Kley & Crida 2008)
In full 3D
(Kley ea 2009)
With irradition:
(Bitsch ea. 2012)
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
16
4. Planet-Disk:
Organisation
Lecture overview:
4.1 Introduction
4.2 Type I
4.3 Type II
4.4 Type III
4.5 Stochastic
4.6 Elements
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
17
4.3 Type II:
Gap formation: Type I ⇒ Type II
Mp
Mp
Mp
Mp
Mp
= 0.01 MJup
= 0.03 MJup
= 0.1 MJup
= 0.3 MJup
= 1.0 MJup
Depth depends on:
- Mp
- Viscosity
- Temperature
Torques reduced :
Migration slows
Type I ⇒ Type II
linear ⇒ non-linear
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
18
4.3 Type II:
Gap opening criteria
Thermal condition
Hill sphere RH = rp (q/3)1/3 comparable/larger than disk scale height H
with q = mp /M∗
q≥3
H
r
3
= 3hp3
(11)
p
with a disk aspect ratio h = 0.05: q ≥ 1.25 · 10−4 , or mp ≥ 0.13mJup
Viscous condition
Viscosity tends to close gap against gap opening torques. From
impulse approximation (Lecture 3) we find
q ≥ 30παh2
(12)
with h = 0.05 and α = 10−2 : q ≥ 2.4 · 10−3 , or mp ≥ 2.5mJup
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
19
4.3 Type II:
A combined gap opening criterion
Using 2D hydrodynamical viscous disk simulations, (Crida ea. 2006) find gap
opening (minimum density within gap smaller 1/10 ambient density) for
P=
3 H
50
+
≤1
4 RH
qRe
(13)
with the Reynoldsnumber Re = Ωp rp2 /ν.
In type II regime the planet remains in the middle of the gap which moves
with the disk. The disk evolves on a viscous timescale and the planet will
move with the same timescale
τmig,II = τvisc =
rp2
rp2
1
=
=
ν
αcs H
αh2 Ωp
(14)
Question:
How well is this assumption really satisfied?
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
20
4.3 Type II:
New simulations
2D hydro simulations, with net mass flow through the disk. Planet migrates
˙ disk . Black dot: planet at r = 0.7r0 .
in disk. Different colours ≡ different M
Compare migration rate to the viscous speed. Can be smaller/larger than
˙ = 10−7 M⊕ /yr.
viscous speed. ΣS : reference density for M
20.0
10.0
aP ‘uvisc
r
5.0
0.02SS
0.05SS
0.10SS
0.20SS
0.50SS
1.00SS
2.00SS
2.0
1.0
0.5
0.2
0.1
(Durmann
& Kley, 2015)
¨
W. Kley
Planet Formation,
0.2
0.5
SP a2P ‘MJ
1.0
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
2.0
5.0
21
4.3 Type II:
The migration process
Result implies that during the migration process material can cross
the gap and is transferred from one side to the other
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
22
4.3 Type II:
Evidence gap formation
Moon (S/2005 S1) in Keeler Gap, Cassini (1. May 2005)
Gap width ≈ 40 km, Moon-diameter ≈ 7 km)
Here: very clean gap, no pressure, no viscosity
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
23
4.4 Type III:
Type-III migration principle
Stationary Planet
Migrating Planet
(F. Masset, 2002)
Inner material crosses horseshoe region: gains ang.mom.
Planet looses ang.mom. −→ Runaway
Need massive disk and large radii
(Masset & Papaloizou, 2003; P. Artymowicz)
Here inward, but outward same argument.
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
24
4.4 Type III:
Regime
(Masset & Papaloizou, 2003) Need massive disk and large radii.
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
25
4. Planet-Disk:
Organisation
Lecture overview:
4.1 Introduction
4.2 Type I
4.3 Type II
4.4 Type III
4.5 Stochastic
4.6 Elements
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
26
Turbulent disks
4.5 Stochastic:
self-gravitating disk
MRI turbulent disk
(M. Flock)
(G. Lodato)
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
27
4.5 Stochastic:
Planets in turbulent disks
Disks are MHD turbulent and embedded planets will feel random
(stochastic) perturbations due to the density fluctuations.
Analyze impact on migration
planet with mp = 10Mearth
in unstratified turbulent disk
with H/r = 0.05
~
and toroidal B-field
plasma-β = pgas /pmag = 50
with effective α = 0.01
(R. Nelson)
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
28
4.5 Stochastic:
Planets in turbulent disks
evolution of semi-major axis
smooth lines show migration in
equivalent laminar disks
On very long time scales similar
to laminar cases
(R. Nelson)
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
29
Organisation
4. Planet-Disk:
Lecture overview:
4.1 Introduction
4.2 Type I
4.3 Type II
4.4 Type III
4.5 Stochastic
4.6 Elements
(ESO April 2010: Turning Planetary Theory Upside Down)
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
30
Planets on eccentric Orbits
4.6 Elements:
Torque on planet due to disk
Z
Tdisk = −
disk
Power: Energy loss of planet
Z
~ df
~r˙p · F
Pdisk =
~ ) df
(~r × F
1
z
disk
e0 = 0.00
e0 = 0.02
e0 = 0.05
e0 = 0.10
e0 = 0.15
e0 = 0.20
e0 = 0.30
0.8
0.6
0.4
Torque
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
10
10.5
Lp = mp
11
Time [Orbits]
p
GM∗ a
11.5
p
1 − e2
12
Ep = −
L˙ p
1 a˙
e2 e˙
Tdisk
=
−
=
Lp
2a
1 − e2 e
Lp
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
1 GM∗ mp
2
a
E˙ p
a˙
Pdisk
=
=
Ep
a
Ep
31
4.6 Elements:
Eccentric planets in 3D radiative disks
Vary Planet Mass 10 − 200MEarth
- Fixed e0 = 0.40
Planet mass Mp = 20MEarth
- Vary Eccentricity
10
0.45
0.45
e0=0.0
e0=0.05
e0=0.10
e0=0.15
e0=0.20
e0=0.25
e0=0.30
e0=0.35
e0=0.40
eccentricity
0.35
0.3
0.25
0.2
20 MEarth
30 MEarth
50 MEarth
80 MEarth
100 MEarth
200 MEarth
0.4
0.35
eccentricity
0.4
0.15
0.3
0.25
0.2
0.15
0.1
0.1
0.05
0.05
0
0
0
50
100
150
200
250
0
Time [Orbits]
50
100
150
200
time [Orbits]
(Bitsch & Kley 2010)
• e-damping for all planet masses.
Small e: exponential damping, large e: e˙ ∝ e−2
• Damping timescale: ≈ (H/r )2 shorter than migration time
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
32
4.6 Elements:
Inlined planets in 3D radiative disks
Vary Planet Mass 20 − 100MEarth
- Same i0 = 4deg
Fix planet mass Mp = 20MEarth
- Vary initial Inclination
4.5
i(t) for a 20 Earth-mass planet
10
3.5
Inclination
i (degrees)
8
20 MEarth
25 MEarth
30 MEarth
35 MEarth
40 MEarth
50 MEarth
80 MEarth
100 MEarth
4
i0 = 2o
i0 = 5o
i0 = 10o
i2 di/dt = C
6
4
3
2.5
2
1.5
1
2
0.5
0
0
0
100
200
300
t (orbits)
400
500
0
50
100
150
200
Time [Orbits]
(Cresswell ea 2007; Bitsch&Kley 2011)
• i-damping for all planet masses.
Small i: exponential damping, large i: i˙ ∝ i −2
• Damping timescale: ≈ (H/r )2 shorter than migration time
Planets are confined to the disk midplane and are on circular orbits
W. Kley
Planet Formation,
Planet-disk interaction, 45th Saas-Fee Lectures, 2015
33