Modelling the Formation of Planets

Transcrição

Modelling the Formation of Planets
Wilhelm Kley
Institut für Astronomie & Astrophysik
Abtlg. Computational Physics
20. Mai, 2010
W. Kley
Planet Formation
Organisation
• Context
- Solar and extrasolar Planets
- Planet Formation
• Accretion Disks
• Planet-Disk Modelling
- Numerics
- Results
• Summary/Outlook
(A. Crida)
W. Kley
Stuttgart: 20. Mai, 2010
1
Solar System
The Planets
Sun
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptun
(Pluto)
W. Kley
2
Solar System
Summary
8 Planets: Mercury to Neptune
3 Dwarf Planets: Ceres, Pluto, Eris, Makemake, Haumea
Minor bodies: TNO, asteroids, comets
Tiny bodies: meteorites, dust
•
•
•
•
•
•
•
coplanar, circular, uniform orbits
prograde rotation (with exceptions)
99% of mass in Sun
99% of angular momentum in planets
Age: about 4.5 billion years
Solid and gaseous planets
Titius-Bode rule
W. Kley
3
Exoplanets
Main Characteristics
Planets orbiting other solar-like stars
• Total detected: ≈ 450 (May 2010)
(by RV method, Transits,
direct imaging, microlensing)
• Transiting Planets: ≈ 80
• Systems with 2 or more planets: ≈ 40
• Planetary Systems in Binaries: 50
• Planetary Systems in
Mean Motion Resonance: 8
• Fraction of stars with planets: ≈ 15%
List: http://exoplanet.eu/
W. Kley
Source:
http://www.oklo.org/
(by Greg Laughlin)
4
Exoplanets
First Object: 51 Peg
K = 55 m/s
P = 4.23 days
e = 0.00
M = 0.45 MJup/ sin i
(Mayor & Queloz, 1995)
Kepler III: Amplitude (K), Period (P ) and Stellar mass (M∗)
⇒ planet mass (Mp) & distance (a)
Sinusoidal radial velocity curve ⇒ circular orbit
W. Kley
5
Exoplanets
Direct Imaging (Nov. 2008)
17. brighest star, 25 Light yrs, Age: 200 Mio. yrs, Debris disk
W. Kley
6
Exoplanets
Direct Imaging II
Star:
Fomalhaut: (Mag: 1.16)
2.59MSun
Planet:
Distance 115 AU
Orbit 900 Yrs
Mp ≈ 3MJup
at inner edge of
Debris Ring
W. Kley
7
Exoplanets
Mass vs Distance
Sonnensystem
Hot Jupiters
at small distances
M sin i [MJup]
10
MEarth =
aM ercury
(Data: exoplanet.eu)
1
.1
W. Kley
1
MJup
300
= 0.4AU
.1
1
Abstand [AE]
10
8
Exoplanets
.8
Eccentricity Distribution
Sonnensystem
Extrasolar
OGLE
Exzentrizität
.6
Large eccentricity spread
- similar to binary stars
.4
Gravitational scattering
(Data: exoplanet.eu)
.2
0
W. Kley
.1
1
Abstand (AE)
10
9
Planet Formation
Grand Picture
Planets form in protoplanetary disks ≡ Accretion Disks
W. Kley
10
Accretion disk
Overview
Circumstellar disk around young stars
(angular momentum conservation)
Matter spirals inwards & angular momentum outwards
Require: Turbulent Transport
Interaction with magnetic field of star
Stellar activity → ionisation of disk ?
W. Kley
11
Accretion disk
Magnetized Disks
For a Keplerian Disk:
ΩK (r) =
GM
r3
1/2
Stability Criterion for Purely Hydrodynamic Disks (Rayleigh)
dL
> 0,
dr
(LK = ΩK r2 ∝ r1/2)
−→ stable
Stability Criterion for magnetized Disks (Chrandrasekhar, et al.)
dΩ
> 0,
dr
(ΩK ∝ r−3/2)
−→ unstable
Magneto-Rotational-Instability - MRI
(Balbus & Hawley, 1993)
W. Kley
12
Accretion disk
MRI-Principle
Lower Orbit:
Larger Ω, smaller L
Spring: Transferral of
Angular Momentum
from lower to higher orbit
=⇒ Instability
Study the MRI in accretion disks numerically (saturation)
Calculate transport coefficients for angular momentum
W. Kley
13
Accretion disk
∂%
+ ∇ · (%v)
∂t
∂(%v)
+ ∇ · (%vv)
∂t
∂e
+ ∇ · (ev)
∂t
∂B
− ∇ × (v × B)
∂t
∇·B
Equation of state
Ideal RMHD-Equations
= 0
= −∇p − ρ∇Ψ
= −p∇ · v
1
+ (∇ × B) × B
4π
+(Q : ∇)v
+∇ · Q
−∇ · F
= 0
= 0
p = (γ − 1)e
Ideal Hydrodynamics & Viscosity (Q) ⇒ Viscous Disks
Magnetic Terms (B)
⇒ Magnetic disks
radiative Diffusion (F)
W. Kley
14
Accretion disk
The local Shearing Box
Effect of Keplerian shear flow
Numerical implementation
Small section of accretion disk in equatorial plane (scale free)
Easy: Periodic boundaries in y ≡ ϕ and z
Tricky: shearperiodic in x ≡ r (conservation properties)
W. Kley
15
Accretion disk
No vertical stratification
Shearing box with dimension: x = 1.0 x y = 4.0 x z = 1.0
Numerical resolution 64x128x64
~
No vertical gravity, vertical B-field,
plasma-β=400 (Pgas/Pmag )
Animation of magnetic field: y −z slice, covers 16 Orbital periods
Initial random perturbation of velocity and temperature
(M. Flaig)
W. Kley
16
Accretion disk
vertically stratified disks
3D Magneto-hydrodynamical Turbulence with Radiation Transport
in Accretion disks, stratified Local Shearing Box, Movie: 6 Orbits
Finite Volume Methode (Grid-Resolution 64×128×512)
Resources on hpc-bw (BW-GRID):
* 128 Cores
* 5 MB Memory per Core
* 100 GB Storage sapce
* 125.000 CPU hours
* Runtime: 40 days
(M. Flaig)
W. Kley
17
Turbulent state
Accretion disk
Here non-zero vertical magnetic field, Bz
Magnetic field strength
(Emag /p0 vs. Time)
α = hTT + TM i/p0
TR = ρδvr δvφ
TM = −δBr δBφ
W. Kley
Angular momentum transport
(α vs. Time)
(∝ Trφ efficiency of angular momentum transport)
(Reynolds Stress)
(Magnetic Stress)
18
Planet Formation
Two main scenarios
Gravitational-Instability (top-down)
Sequential Accretion (bottom-up)
(L. Mayer)
(NASA, U2)
Density-Fluctuations grow
Fast Formation (103 years)
Require: Fast Cooling
large disk mass
From small to large particles
Slow Formation (106 Years)
Need: High Sticking probability
(small timescales, distant planets)
W. Kley
(Comets, asteroids, solid planets,
cores of planets)
19
Planet Formation
I. Gravitational-Instability
Consider local density perturbation in disk
Analytical
Numerical
Stability-Criterion (Toomre)
Evolution of an isothermal Disk
• Finite-Difference Hydrodynamics
• Viscous Disk
csκ0
>1
Q≡
πGΣ0
with
cs = sound velocity
κ0 = Epicyclic-Frequency (ΩK )
Σ0 = surface density
• Pressure & Rotation stabilize
• Density destabilizes
(Tobias Müller)
Disk heats up upon compression, need fast cooling
Require: coolingtime ≈ period
Only in large distances from star
W. Kley
⇒
fast formation
(for ca. 30-50 AU, no cores)
20
Particle dynamics in disk
Planet Formation
( Jürgen Blum (Braunschweig))
Particles have relative velocity with respect to the gas ⇒ Forces
Problem I: Fast radial drift towards star (for 1m Size: 1 AU / 100 Years)
Problem II: Destructive Collisions
W. Kley
21
Planet Formation
Planetesimals I
Growth of planetesimals by collisions/accretion
SPH (Smoothed-Particle-Hydrodynamics)
Here: elastic-plastic strength model, formation and evolution of cracks
2 Basalt Spheres: ρ = 3g/cm3, R1=9m, R2=7.5m, Vrel=25m/s
∆ t = 30ms, tmax = 4.5 sec
(Christoph Schäfer, PHD-Thesis, Univ. of Tübingen 2005)
W. Kley
22
Planet Formation
Planetesimals II
Solution: study porous particles
Particle method Smoothed Particle Hydrodynamics (250,000 Particles)
Calibration with real experiments (TU Braunschweig)
Fragmentation (Test-Case)
Vrel = 10m/s
(R. Geretshauser)
Vrel = 1m/s
Computer time on hpc-bw: 1-3 days on 24-32 cores (240k-480k Particles)
=⇒ DFG-Forschergruppe: The Formation of Planets (2007-2012)
W. Kley
23
Migration
Planet Formation
Evolution of planet in Disk
(here viscous disk
parametrization by α )
Require:
Torques
(disk pulls on planet)
=⇒ Migration
=⇒ motion in disk
Accretion
(planet attracts material)
=⇒ Mass growth rate
On the right:
Saturn mass planet in disk
- Spiral arms
- Gap formation
(A. Crida)
W. Kley
24
Modelling
A test project
Actors
Star (1 M), Disk (0.01 M), Planet (Mp = MJup)
Disk: Hydrodynamical Evolution
20
- In potential of Star & Planet
- 2D (r, ϕ or x − y)
- rmin = 2.08, rmax = 10.4 AU
- locally isothermal: T ∝ r−1
- const. viscosity: ν = const.
Planet: At 5.2 AU
- here on fixed orbit
- point mass (smoothed potential)
Compare 17 Codes in
De Val-Borro & 22 Co-authors, MN
(2006)
(Grid-Sample: 128x128)
10
0
-10
-20
-20
-10
0
10
20
EU comparison project: http://www.astro.su.se/groups/planets/comparison/
W. Kley
25
Modelling
Equations-2D: Conservative
Mass Conservation
∂Σ
+ ∇ · (Σu) = 0
∂t
u = (ur , uϕ) = (v, rω)
Radial Momentum
∂p
∂Φ
∂(Σv)
2
+ ∇ · (Σvu) = Σ r(ω + Ω) −
−Σ
+ fr
∂t
∂r
∂r
Angular Momentum
∂Φ
∂[Σr2(ω + Ω)]
∂p
2
+ ∇ · [Σr (ω + Ω)u] = −
−Σ
+ fϕ
∂t
∂ϕ
∂ϕ
Energy
H/r = const. =⇒ T ∝ 1/r
W. Kley
p = Σ T /µ
26
Modelling
Typical Result
Viscous disk: Mp = 1 MJup, ap = 5.2 AU (WK, 2002)
W. Kley
27
Modelling
EU comparison: Results
Types: solid - Upwind, HighOrder, short-dashed - Riemann,
long-dashed SPH;
(Cyl. vs. Cartesian)
W. Kley
28
Modelling
Coriolis Terms I
Angular Momentum Equation
Conserved (as before)
∂[Σr2(ω + Ω)]
+ ∇ · [Σr2(ω + Ω)u] =
∂t
∂p
∂Φ
−
−Σ
+ fϕ
∂ϕ
∂ϕ
Not-Conserved (Explicit)
∂[Σr2ω]
+ ∇ · [Σr2ωu] = −2 Σ vΩ r
∂t
∂p
∂Φ
−
−Σ
+ fϕ
∂ϕ
∂ϕ
W. Kley
29
Modelling
Coriolis Terms II
Conservative
Non-Conservative
Conservative results: identical to inertial frame
=⇒ If rotating frame: Use conservative formulation
W. Kley
30
Modelling
FARGO - Basics
A fast eulerian transport algorithm for differentially rotating disks
(Masset, Astronomy & Astrophysics 2000) Code-page: http://fargo.in2p3.fr/
The problem:
- Cylindrical coord. system
- explicit code
→ Courant condition:
∆ϕ
∆t <
ω
with ω ∝ r−3/2 → ∆t ∝ r3/2
The solution:
Calculate average velocity ω̄i
of ring i, and perform Shift with
ni = Nint(ω̄i∆t/∆ϕ)
Transport on with remaining velocity
=⇒ much larger ∆t
Take care: no ring separation
W. Kley
31
Modelling
FARGO - Results
Azimuthally averaged surface density
1.4
Fargo vs. Standard
inertial vs. rotating
1.2
Jupiter mass planet
after 25 Orbits
Used Timesteps
– 51000
– 39000
– 5800
– 5800
Σ
1
t=25 orbits
r−phi (128x384)
inertial, std.
rotating, std.
inertial, Fargo
rotating, Fargo
.8
.6
.5
W. Kley
1
1.5
r
2
=⇒ Identical results !
=⇒ Huge Savings !
larger savings for
log-grid
2.5
http://fargo.in2p3.fr/
32
Modelling
The Grid
Azimuthally averaged surface density
1.4
Cartesian vs. Cylindrical
(inertial frame)
Jupiter mass planet
after 25 Orbits
1.2
Models
– Cyl.: 128×384
– Cart.: 320×320
– Cart.: 640×640
Σ
1
.8
t=25 orbits
q01, cyl, std
q04, cart, 320
q05, cart, double
.6
.5
W. Kley
1
1.5
r
2
=⇒ Cartesian problematic !
– no operator-splitting
– 2nd order RK-integrator
[only PENCIL code reasonable
(HiOrder)]
2.5
33
Modelling
Nested Grid I
Nested-Grid System: Centered on Planet - corotating frame
(D’Angelo et al. 2002/03)
l=1
l=3
W. Kley
l=2
34
Modelling
Nested Grid II 1 MJ
(here 2D, ϕ − r plots)
(D’Angelo et al., 2002, PhD Thesis Tübingen, 2004)
W. Kley
35
Torque calculation
Modelling
Sum over all grid-cells,
P
Tz =
cells (xp Fy
Which parts belong to planet ?
− ypFx)
Migration of Saturn mass planet
Mp = 3.10-4M* ; Nr = 460 ; Ns = 1572
SEMI - MAJOR - AXIS
1
Use Truncation Radius
Rtrunc ≈ 0.5 − 1.0 Rroche
Smooth tapering-function fb
fb(s)
1
b=0.2
0.5
b=0.4
b=0.6
0.95
0.9
0.85
Method :
(A)
(B) b = 1.
(B) b = 0.9
(B) b = 0.8
(B) b = 0.7
(B) b = 0.6
(B) b = 0.5
(B) b = 0.4
(B) b = 0.3
(B) b=0. <=> none
0.8
300
b=0.8
b = 1.
305
310
315
TIME [orbit]
320
325
(A. Crida)
0
0
W. Kley
0.2
0.4
0.6
0.8
s / rHill
1
1.2
1.4
– (A) fully self-gravitating
=⇒ need b ≈ 0.5 − 0.6
36
Disk physics
Planet Formation
∂Σcv T
+ ∇ · (Σcv T u) = −p∇ · u +D − Q −2H∇ · F~
∂t
4
Adiabatic: Pressure Work, Viscous heating (D), radiative cooling (Q) (∝ Tef
f)
radiative Diffusion: in disk plane (realistic opacities)
4e-05
local cool/heat
Specific Torque [(a2 Ω2)Jup]
3e-05
2e-05
fully radiative
Mp = 20Mearth
(Kley&Crida ’08)
1e-05
0
adiabatic
-1e-05
Disk Physics determines
Direction of motion
-2e-05
isothermal
-3e-05
-4e-05
0
100
200
300
Time [Orbits]
W. Kley
37
Planet Formation
Migration (Type I)
Isothermal and radiative Models: Migration inward or outward possible
Vary Mp
(Kley&Crida ’08)
W. Kley
38
Planet Formation
Summary/Outlook
New properties of extrasolar Planets
• High Masses
• Large eccentricities
• Close distances
Planet-disk simulations
• Coord. System
• Angular momentum conservation
• FARGO-Accelaration
• Grid-Refinement
• Thermodynamics
Accurate accretion & migration history of planets requires
• Full 3D simulations
• High resolution (locally)
• Radiation transport
• Self-Gravity
• Magnetic Fields
W. Kley
39
Planet Formation
The End
Thank you for your attention !
(A. Crida)
W. Kley
40

Modelling the Formation of Planets

Transcrição

Documentos relacionados

Exoplanets at the E

Vorgehensweise bei Berechnung eines - software