Dust dynamics (in GANDALF) Richard Booth

Transcription

1 Institute of Astronomy Dust dynamics (in GANDALF) Richard Booth

2 Why Dust Dynamics? SAO-26462, micron wavelength HL Tau, mm~ wavelength

3 Why Dust Dynamics? Turbulent clouds Meshless Finite Volume Hopkins+ (215)

4 Why Dust Dynamics? Turbulent Clustering Clustering important for grain growth Pan & Padoan (211)

5 Dust Dynamics: Key equations Acceleration of Gas & Dust: d D vg ρg = P+ρ D v g g a g +ρ g ρd K ( v d v g ) Dt ρg = %nablaρg K s (v d v g )+ f D Dt D vd md =m d a d ρ g K m d ( v d v g) Dt Epstein Regime: vt K= ρs s 1 t s= K (ρg +ρd ) opping time is very short for small particles!

6 Dust Dynamics: Equations for centre of mass / relative velocity: D v COM = a F (Δ v ) Dt ρg a g +ρd a d ( a= ρ +ρ ) d g D Δ v Δ v P = +( a d a g + ρ )+Δ v v COM G ( Δ v ) g Dt ts Terms F & G can often be neglected since (see Laibe & Price, , Youdin & Goodman 25) - Depend on density / velocity gradients - Second order in Delta v Analytical solution (approx): P Δ v (t+ Δ t )=Δ v (t )exp( Δ t /t s )+(a d a g + ρ ) t s (1 exp( Δ t /ts)) g v COM (t +Δ t )=v COM (t )+a Δ t (Constant accelerations and stopping time)

7 Interpolation: Direct ρ =Σ i mi W (r ij, h j ) mi v =Σi v i ρ W (r ij, h j ) i Drag force not aligned with particle separation: - Angular momentum not conserved - Use projected forces? Laibe & Price (212) 7

8 Interpolation: Projected Time integration Zero drag case + drag correction Loren-Aguilar & Bate (215) 8

9 Interpolation: Projected Time integration S=Δ v ζ (Δ t) Δ a Λ (Δ t) 1 exp( Δ t /t s) ζ (Δ t )= 1+ϵ Δt Λ (Δ t)=(δ t +t s )ζ (Δ t ) 1+ϵ ρd ϵ= ρ g 9

10 Interpolation: Projected Force Error Loren-Aguilar & Bate (215) Force Error for projected forces: - Uniform density

11 Interpolation: Settling Test K/m = vz vz z.9 K/m = Test particle z Loren-Aguilar+ (214) vz vz Full Two-fluid Booth, Sijacki & Clarke (215) 11

12 Current atus in GANDALF: Test Particle limit only: t s=.25, t =.75 t s=.25, t =.75 12

13 Tests: andard tests are easy t =.4 ts =.1, L2 =.76 vz t = vx 1 4. ts =.1, L2 = x z.9 K/m = z z.5 K/m =..8.6 vz vx t = vz vz vx ts =1, L2 =.758 K/m = K/m = z.5 s 13

14 Tests: andard tests are easy (in the test particle limit) vx 1 4 t = 1 4 ts =1, L2 =.758 vx 1 4 t = 1 4 ts =.1, L2 =.76 vx 1 4 t = 1 4. ts =.1, L2 = x Full two-fluid equations are dissipative though Low resolution will over-damp waves Semi-implicit approach helps Damping weaker in when dust-to-gas ratio is low 14

15 Tests: Multi-dimensional ρ.4 S tandard S P H.1.6 G as D ens ity D us t D ens ity 1 v 1 G as Vortic ity..1 x Inte g r al S P H.1..1 x.1 15

16 Tests: Multi-dimensional ρ.4 S tandar d S P H.1.6 G as D ens ity D us t D ens ity 1 v 1 G as Vortic ity..1 x Inte g ral S P H.1..1 x.1 16

17 Shear Test: Shu, Milione, Roberts (1973) Isothermal gas Impose a background potential Flat rotation curve: v (R)=v max F b ϵb exp( ϵb R)+1 exp(ϵd R) Ω Logarithmic spiral perturbation p V s = A R exp ( ϵ s R) cos( χ) m χ= ln( R) m(θ Ω p t ) tani Pattern speed, Ωp Find solutions along stream lines Gittins & Clarke (24) 17

18 Shear Test: ream-line solution ream-lines used to build the full disc structure 18

19 Modelling: Test Problems..4 (Booth+ 215).8 Gas Dust y Richard Booth. x x

20 Modelling: Test Problems ρ/ ρ Gas gas.4 h/ H = N NGB = 4 st r eam 2 η 6 ρ/ ρ N NGB = 5 η η x Richard Booth Dust3.2.8 Vξ y h/ H =.16 h/ H =.5 Vη 6. (Booth+ 215) x η

21 Modelling: Test Problems (Booth+ 215) ρ/ ρ h/ H =.16 h/ H = η N NGB = η 2 h/ H =.35 h/ H = Dust η x Richard Booth 2. Vξ. σv / cs.5 ρ/ ρ 1.6 N NGB = 4 st r eam 2 1. y Vη 6. gas.4.8 Gas h/ H =.35 h/ H =.5 2 ok es.94 N um ber x η

22 Application: Dust in self-gravitating protoplanetary discs Gas Dust Rice+ (24) 3/11/15 22

23 Application: Dust in self-gravitating protoplanetary discs Gas FragmentDust Mass: -3 MEarth Gibbons+ (214) Rice+ (24) 3/11/15 23

24 Key Questions: Growth & Fragmentation Most likely place for this to happen: Class 1 discs Gas mm/cm grains seen(miotello+ 214) Q~1 requires ~.1 to.1 at 3au Is this consistent with self-gravity? Dust How large do grains need to be for trapping to be effective enough? Do collisions lead to fragmentation? Can growth to ~ 1 occur? = 1 corresponds to few cm (high density) Fragmentation velocity ~ 1 m/s for silicates (Guttler+ 2) few m/s for ices (Wada+ 29; Gundlach & Blum, 215) Large velocity dispersion for planetesimals 3/11/15 Of order c_s: ~ few m/s (Walmswell+ 213) How much does coupling reduce this? 24

25 Modelling: Simulations (Dimensionless!) Gas ΣG = 3 Gas D Simulations: Easier to reduce noise Need to resolve scales < H for < 1 1, 4 & 16 million particles per phase 3/11/15 Dust -3 ΣD -2 = -1 1 Beta Cooling: t c =βω 1 Fixed okes number: t s= Ω Test particle limit Disc mass =.1 ar mass ( β= ) 25

26 Density enhancement p( > Σ D / Σ G ) =. =.3 = 1. = 3. =. = 3. =. 1 ΣD / ΣG 2 3 Fraction of particles in high density regions: Density enhancement > for.3 < < 3 Gravitational collapse needs >.3 3/11/15 26

27 Relative velocities: Equal sized particles Measure distribution of relative velocities, Using r.m.s relative velocity Δ v= Δ v Rate of collisions: -1-2 = 1. = 3. =. = 3. = Γ Δ v P (Δ v ) v p( v) P( Δ v ) v/ cs Intermediate regime for < 3 3/11/15 27

28 Relative velocities: Equal Sized Particles λstop = Δv ts λstop < H for < 3 1 λ st op = H Large : Gravitationally driven random walk 1/ 2 cs λ st op = hsph vmed/ cs Δ v /11/ Small : Gravity ineffective What is driving?

29 Relative velocities: Inhomogeneity.4 y σv / cσs G Dust Collision velocity 3/11/ v Gas vorticity 29

30 Bi-disperse case: 1 6 dust particles Velocity distribution Dominated by velocity dispersion if one particle large Radial drift larger at low = 1. = 3. =. = dust particles vmed/ cs vmed/ cs = 1. = 3. =. = 3. =. =.3 2 3/11/15 3

31 Physical units Radial scaling: Collision velocity 3 1 km Particle size: 1cm to 1km v [m /s ] 2 Can constrain grain growth in self-gravitating discs 1 1 cm Planetesimal formation: May be possible beyond 3 au /11/ R [AU ]

32 Summary Dust dynamics can be fundamentally different to gas Dust implementation in GANDALF (SPH) is under way Include feed back on the gas Hope to include dust in Meshless Finite Volume Testing required! andard tests are passed relatively easily Accurate dust and gas simulations are possible 3/11/15 Proper testing in astrophysical context is important Many interesting prospects for ALMA observations 32