Zooming In on Star Formation - Napflio - June 10-14, 2019

Streaming Instability for Particle-size Distributions

Niels Bohr Institute - Copenhagen - Denmark

Martin Pessah

Leonardo Krapp, Pablo Benitez-Llambay, Oliver Gressel, & MEP (arXiv:1905.13139)

• Drift velocity depends on Stokes number and dust-to-gas mass ratio

Streaming Instability - One Dust Species

•Homogeneous shearing box

•Radial pressure gradient

• Dust particles drift wrt gas

Youdin & Goodman 2005

Youdin & Johansen 2007

Johansen & Youdin 2007

Jacquet, Balbus, & Latter 2011

Squire & Hopkins 2018

…

✏Ts

Particle-size DistributionNX

i=1

✏i = ✏

•Discrete representation of a continuous distribution with N dust species {✏i, Tsi}

d✏

dTs

Ts

✏i✏

Stokes number (particle size)

min

max

Multi-Fluid Shearing Box & Steady State

•Multi-fluid shearing box {✏i, Tsi} for i = 1, . . . , N

•Steady state solution AN =NX

i=1

✏iTsi

1 + T 2si

BN = 1 +NX

i=1

✏i1 + T 2

si

v0g = �q⌦0x y +�v0

g(A,B)

v0i = �q⌦0x y +�v0

i (A,B)

Benitez-Llambay et al. 2019ApJS, 241, 25

•Small amplitude perturbations lead to a 4(N+1) linear equations

�f(x, z, t) ⇠ Re[�f(kx, kz) ei(kxx+kzz)�!t]•Assume and find the fastest growing modes

Krapp et al. 2019

Exercise•@ fixed dust-to-gas mass ratio and Stokes number range; increase the number of species

�T Is =

⇥10�4 , 10�1

⇤

�T IIs =

⇥10�4 , 1

⇤

•Two examples (equal gas and dust mass)

•Maximum growth rate behaves quite differently!

10�5 10�4

✏i

10�3

10�2

�/⌦

0

✏ = 0.001

DistributionTs = 1

10�4 10�3

✏i

✏ = 0.01

10�3 10�2

✏i

✏ = 0.1

10�2 10�1

✏i

✏ = 1

What Is Going On?•Fix dust-to-gas-mass ratio and fix range of Stokes numbers

•Increase the number of species so that the mass-per-species decreases•Consider a distribution with equal mass-per-bin

ϵ1 ϵ1 ϵ1 ϵ1

Some Insights

At very low dust-to-gas mass ratios:• species with largest Stokes number dominates• problem reduces to one-dust species with decreasing mass• growth-rate scales with (RDI, Squire & Hopkins 2018)ϵ1

At higher dust-to-gas mass ratios:• background flow is determined by the ensemble• no longer describable as a one-dust problem• unclear current RDI framework can describe this• growth-rate scales with ϵ1

Krapp et al. 2019

particle-size distribution

10�4 10�3 10�2 10�1 100

Ts

10�2

10�1

100

✏Two fluids

10�3 10�2 10�1 100

Ts,max

Particle-size distribution

10�410�310�210�1

Ts,min

Particle-size distribution

�6 �5 �4 �3 �2 �1 0

log10 (�/⌦0)

10�

410

�3

10�

210

�1

100

Ts

10�

2

10�

1

100

✏

Tw

ofluid

s

10�

310

�2

10�

110

0

Ts,

max

Par

ticl

e-si

zedis

trib

ution

10�

410

�3

10�

210

�1

Ts,

min

Par

ticl

e-si

zedis

trib

ution

�6

�5

�4

�3

�2

�1

0

log10(�

/⌦

0)

Reduced Growth for Particle-size Distributionsone-dust species

Krapp et al. 2019

Unless we can identify mechanisms to produce mono disperse particle populations efficiently,the scope of the streaming instability may be narrowed down profoundly.