Vertical Oscillation of Protoplanetary Disk (PP disk): 1D multi - - PowerPoint PPT Presentation

Vertical Oscillation of Protoplanetary Disk (PP disk): 1D multi color Radiation Hydrodynammical Simulations

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Tomoyuki Hanawa Tetsuya Harada (Chiba U.) Hot upper atmospheres and cold main disk oscillate in the opposite directions.

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PP disk@ 1.6μm(H)+345 GHz

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Lin+06 & Ohashi+07 taken with SMA

• verlaid on Fukagawa+02 (Subaru)

Much better images will be taken with ALMA

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Williams & Cieza ’11

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Star Disk direct (optical) Scattered

Emission (mid IR)

absorption

Structure of Irradiated PP Disk

Chiang & Goldreich ’97

Two Layer Model

Hot Surface Layer + Cool Main Disk

Ts >> Td

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1D Grazing Recipe

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Eν = E′

ν + E′′ ν ,

F ν = F ′

ν + F ′′ ν ,

stellar scattered+emission

r z

τ 0

ν

τ 00

ν

∂E00

ν

∂t + ∂F 00

ν

∂z = ρc  κν,a ✓ −E00

ν + 4πBν

c ◆ + κν,sE0

ν

• ,

∂Fν ∂t + ∂ ∂z

• c2χ00

νE00 ν

• = −ρc (κν,a + κν,s) F 00

ν ,

M1 model Eq.

E0

ν(z) = πR2 ⇤Bν (Teff)

r2 + z2 exp − 2 7 √ r2 + z2 z κν Z 1

z

ρ(z0) dz0 !

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Radiation Hydrodynamics

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∂E00

ν

∂t + ∂F 00

ν

∂z = ρc  κν,a ✓ −E00

ν + 4πBν

c ◆ + κν,sE0

ν

• ,

∂Fν ∂t + ∂ ∂z

• c2χ00

νE00 ν

• = −ρc (κν,a + κν,s) F 00

ν ,

α α speed reduction : α = 10−4 → c = 30 km s−1

∂ρ ∂t + ∂ ∂z (ρvz) = 0 , ∂vz ∂t + vz ∂vz ∂t + 1 ρ P ∂z + GMz (r2 + z2)3/2 = 0 , T ds dt = Z ∞ κν,a [cEν − 4πBν(T)] dν .

We solve the above partial differential equations explicitly.

assumption : Tgas = Tdust

Our finite difference scheme is designed so that all the physical variables approach to the equilibrium ones in the limit of Δt = ∞.

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Upwind Reconstruction

• f the Radiation Field

Iν(n) = 3Eν 8π (1 − β2)3 3 + β2 (1 − β · n)−4 β = 3f 2 +

• 4 − 3f 2,

β = β F |F |

Kinetic Reconstruction

Kanno, Harada, & Hanawa 2013, PASJ in press

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Absorption & Emission within Cell

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F (+)

ν,x,i+1/2,j,k

= e−∆τi/2F (+)

ν,x,i+1/2,j,k +

• 1 − e−∆τi/2 Sν

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• Flux at boundary absorption Flux at center Emission

e−∆τi/2F

e−∆τi

ν,xx,i+1/2,j,k ν,xx,i+1/2,j,k

P (+)

ν,x,i+1/2,j,k

= e−∆τi/2P (+)

ν,x,i+1/2,j,k +

• 1 − e−∆τi/2 Sν

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• S

e−∆τi : optical depth

approaching to diffusion limit when is large

e−∆τi

MUSCL for 2nd order in space

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41 colors

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0.1 µm ≤ λ ≤ 1 mm ∆ log λ = ∆ log ν = 0.1

λ (µm)

Opacity: Draine (2003)

M* = 2.2 Mo Teff = 6250 K R* = 3.8 Ro

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Initial model (Equilibrium)

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Model 1: Σ = 7 g cm-2

r = 100 AU

zmax = 70AU, Δz = 0.5 AU

Σ = 7, 20, 70 g cm−2

ρ T

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Model 1: overview

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Early density oscillation at z = 0.25 AU

D

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density fluctuation at various heights

Period = 420 yr, e-folding growth timescale = 2,000 yr.

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velocity perturbation

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node node Upper layers expands to receive more stellar light, when the disk main body is compressed.

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thermal engine

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I PdV > 0

rotates clockwise Volume z = 0.25 AU Pressure

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Variation in Radiative Flux @ z = 19.75 AU

density max density min

Z ∆Fνdν

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Excitation Mechanism

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Compressed Surface + Compressed Main Disk Expanded Surface + Compressed Main Disk Excess Heating Heating Deficiency

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Limit Cycle Oscillation

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Mass Ejection

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ρ = 10-14 g cm-3 10-15 g cm-3 10-13 g cm-3

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Light variation and mass ejection

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Doppler shift in CO lines The period is 2/3 of the Keplerian.

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High Surface Density (Σ = 70 g cm-2)

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PV diagram

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Σ =7 g cm-2 Σ =70 g cm-2 almost adiabatic

τth τdyn

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Summary and Implications

• PP disks are overstable against vertical
• scillation with a node, since they

have hot cold inner disk and hot surface layers.

• The vertical oscillation affects

appearance and evolution of PP disks.

• 2D RHD simulations are desired.

Flaring of an annulus may result in a

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